Bourgain–Morrey spaces meet structure of Triebel–Lizorkin spaces.

Let 0 < q ≤ p ≤ ∞ , r ∈ (0 , ∞ ] , and M q , r p (R n) denote the Bourgain–Morrey space which was introduced by J. Bourgain and has proved important in the study of some linear and nonlinear partial differential equations. In this article, via cleverly combining both the structure of Bourgain–Morrey spaces and the structure of Triebel–Lizorkin spaces and adding an extra exponent τ ∈ (0 , ∞ ] , the authors introduce a new class of function spaces, called Triebel–Lizorkin–Bourgain–Morrey spaces M F ˙ q , r p , τ (R n) . The authors show that M F ˙ q , r p , τ (R n) is just a bridge connecting Bourgain–Morrey spaces and global Morrey spaces. In addition, by fully using exquisite geometrical properties of cubes of Euclidean spaces, the authors also explore various fundamental real-variable properties of M F ˙ q , r p , τ (R n) as well as its relations with other Morrey type spaces, such as Besov–Bourgain–Morrey spaces and local Morrey spaces. Finally, via finding an equivalent quasi-norm of Herz spaces and making full use of both the Calderón product and the sparse family of dyadic grids, the authors obtain the sharp boundedness on M F ˙ q , r p , τ (R n) of classical operators including the Hardy–Littlewood maximal operator, the Calderón–Zygmund operator, and the fractional integral.

This project is supported by the National Natural Science Foundation of China (Grant Nos. 11971058 and 12071197) and the National Key Research and Development Program of China (Grant No. 2020YFA0712900).

Introduction

Let $p\in (0,\infty ]$ and E be a measurable set of ${\mathbb{R}}^n$ . Recall that the Lebesgue space $L^p(E)$ is defined to be the set of all the measurable functions f on E such that, when $p\in (0,\infty )$ ,

$$\begin{array}{c}\hfill {\Vert f\Vert }_{L^p(E)}:={\left[{\int }_E,{|f(x)|}^p,,d,x\right]}^{\frac{1}{p}}<\infty \end{array}$$

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and

$$\begin{array}{c}\hfill {\Vert f\Vert }_{L^{\infty }(E)}:=\underset{x\in E}{\mathrm{ess}\sup }|f(x)|:=\underset{\genfrac{}{}{0.0pt}{}{F\subset E}{|F|=0}}{inf}\underset{x\in E\backslash F}{sup}|f(x)|<\infty .\end{array}$$

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In addition, let $x\in {\mathbb{R}}^n$ and $r\in (0,\infty )$ . Then a ballB(x, r) in ${\mathbb{R}}^n$ centered at $x\in {\mathbb{R}}^n$ with the radius r is defined by setting

$$\begin{array}{c}\hfill B(x,r):=\{y\in {\mathbb{R}}^n:|y-x|<r\}.\end{array}$$

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Throughout this article, we denote by ${\mathbf{1}}_E$ the characteristic function of any set $E\subset {\mathbb{R}}^n$ and, for any $p\in (0,\infty ]$ and any open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , the symbol $L_{\mathrm{loc}}^p(\mathrm{\Omega })$ is defined to be the set of all the measurable functions f on $\mathrm{\Omega }$ such that, for any $x\in \mathrm{\Omega }$ , there exists a ball $B(x,r)\subset \mathrm{\Omega }$ such that $f{\mathbf{1}}_{B(x,r)}\in L^p(\mathrm{\Omega })$ . In 1938, to study both the local behavior of solutions of second order elliptic partial differential equations and the calculus of variations, Morrey [[29]] introduced the classical Morrey space. Indeed, let $0<q\le p\le \infty$ and, for any $\nu \in \mathbb{Z}$ and $m:=(m_1,...,m_n)\in {\mathbb{Z}}^n$ , let

$$\begin{array}{c}\hfill Q_{\nu ,m}:=\prod _{j=1}^n\left[\frac{m_j}{2^{\nu }},,,\frac{m_j+1}{2^{\nu }}\right)\end{array}$$

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be the dyadic cube of ${\mathbb{R}}^n$ ,

$$\begin{array}{c}\hfill {\mathcal{D}}_{\nu }:=\left\{Q_{\nu ,m},:,,m,\in ,{\mathbb{Z}}^n\right\},\text{and}\mathcal{D}:=\left\{Q_{\nu ,m},:,,\nu ,\in ,\mathbb{Z},,,,m,\in ,{\mathbb{Z}}^n\right\}.\end{array}$$

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Then the Morrey space ${\mathcal{M}}_q^p({\mathbb{R}}^n)$ is defined to be the set of all the $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ such that

$$\begin{array}{c}\hfill {\Vert f\Vert }_{{\mathcal{M}}_q^p({\mathbb{R}}^n)}:=\underset{Q\in \mathcal{D}}{sup}{|Q|}^{\frac{1}{p}-\frac{1}{q}}{∥f,{\mathbf{1}}_Q∥}_{L^q({\mathbb{R}}^n)}<\infty .\end{array}$$

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After that, function spaces based on Morrey spaces have proved very important in the study of both harmonic analysis and partial differential equations; see, for instance, [[11], [26], [28], [32]–[35], [38]–[40]] for their applications in harmonic analysis and [[6], [9], [27], [44], [49]] for their applications in partial differential equations as well as the monographs Adams [[1]], Lemarié-Rieusset [[19]], Sawano et al. [[41]], and Yuan et al. [[48]].

On the other hand, in order to study both the restriction and the Bochner–Riesz multiplier problems in ${\mathbb{R}}^3$ , Bourgain [[3]] refined the Stein–Tomas (Strichartz) estimate via a special Bourgain–Morrey space. Later on, to explore two minimization problems on non-scattering solutions of nonlinear Schrödinger equations, Masaki [[23]] introduced Bourgain–Morrey spaces for the full range of exponents. Nowadays, Bourgain–Morrey spaces have proved very important in the study related to the Strichartz estimate and the nonlinear Schrödinger equation; see, for instance, [[2], [4], [30]]. In addition, Bourgain–Morrey spaces have also been widely used in other linear and nonlinear partial differential equations, we refer the reader to [[25], [43]] for their applications in Airy equations as well as [[18], [24]] for their applications in KdV equations. We now recall the exact definition of Bourgain–Morrey spaces as follows, which is just [[23], Definition 1.8]; see also [[24], Definition 2.1] or [[16], Definition 1.1].

Definition 1.1

Let $0<q\le p\le \infty$ and $r\in (0,\infty ]$ . The Bourgain–Morrey space ${\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)$ is defined to be the set of all the $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ such that

$$\begin{array}{c}\hfill {\Vert f\Vert }_{{\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)}:={\left\{\sum _{\nu \in \mathbb{Z},m\in {\mathbb{Z}}^n},{\left[|,Q_{\nu ,m},{|}^{\frac{1}{p}-\frac{1}{q}},{∥f,{\mathbf{1}}_{Q_{\nu ,m}}∥}_{L^q({\mathbb{R}}^n)}\right]}^r\right\}}^{\frac{1}{r}},\end{array}$$

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with the usual modification made when $r=\infty$ , is finite.

Observe that, when $r=\infty$ , the space ${\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)$ is just the classical Morrey space ${\mathcal{M}}_q^p({\mathbb{R}}^n)$ . Recently, Hatano et al. [[16]] gave various important real-variable properties of Bourgain–Morrey spaces, including the approximation property, the interpolation, and the boundedness of several classical operators on them. The article [[16]] of Hatano et al. opens the door to study the real-variable theory of function spaces based on Bourgain–Morrey spaces. Very recently, motivated by the structure of Bourgain–Morrey spaces, amalgam-type spaces, and Besov spaces, Zhao et al. [[50]] introduced the Besov–Bourgain–Morrey space (see also Definition 3.3 below for the definition) which unifies and generalizes both Bourgain–Morrey spaces and amalgam-type spaces. Also, some important basic real-variable properties of Besov–Bourgain–Morrey spaces corresponding to Bourgain–Morrey spaces were investigated in [[50]]. Then it is a natural and meaningful question to ask whether or not it is possible to introduce the Bourgain–Morrey space with the structure of Triebel–Lizorkin spaces and to develop its corresponding real-variable theory.

The main target of this article is to give an affirmative answer to this interesting question. To be precise, via cleverly combining both the structure of Bourgain–Morrey spaces and the structure of Triebel–Lizorkin spaces and adding an extra exponent $\tau \in (0,\infty ]$ in the quasi-norm of Bourgain–Morrey spaces ${\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)$ , we introduce a new class of function spaces, called Triebel–Lizorkin–Bourgain–Morrey spaces $\mathcal{M}{\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . We show that the space $\mathcal{M}{\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is just a bridge connecting Bourgain–Morrey spaces and global Morrey spaces. In addition, by fully using exquisite geometrical properties of cubes of Euclidean spaces, we also explore various fundamental real-variable properties of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , including the nontriviality, the approximation property, and the diversity as well as the relations with other Morrey type spaces, such as Besov–Bourgain–Morrey spaces and local Morrey spaces. Finally, via finding an equivalent quasi-norm of Herz spaces and making full use of both the Calderón product and the sparse family of dyadic grids, we obtain the sharp boundedness on $\mathcal{M}{\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ of classical operators including the Hardy–Littlewood maximal operator, the Calderón–Zygmund operator, and the fractional integral. All these have laid a solid foundation for the further development of the real-variable theory of function spaces based on Bourgain–Morrey spaces.

The remainder of this article is organized as follows.

In Sect. 2, we first introduce the concept of the Triebel–Lizorkin–Bourgain–Morrey space ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ (see Definition 2.1 below). Then we give some fundamental properties of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . To be exact, we find two equivalent quasi-norms of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ (see Theorem 2.3 below) and obtain various inclusion relations on different indices and the completeness as well as the dilation invariance, the translation invariance, and the orthogonality invariance of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Via making full use of both the dilation invariance and the translation invariance of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , we characterize the nontriviality of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ (see Theorem 2.9 below). Finally, when all the exponents are finite, we establish both the approximation and the density properties of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ using the aforementioned equivalent quasi-norms.

Section 3 is devoted to investigating the relations between Triebel–Lizorkin–Bourgain–Morrey spaces and other various Morrey type spaces and then studying the diversity of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ themselves. Indeed, we first prove that ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ with some special indices goes back to Lebesgue spaces. Next, we explore the embedding between ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and (Besov–)Bourgain–Morrey spaces. Via constructing a concrete function which essentially characterizes the exponents r and $\tau$ (see Lemmas 3.9 and 3.11 below), we prove that the aforementioned embedding is proper and hence the Triebel–Lizorkin–Bourgain–Morrey space is indeed a new space different from the Besov–Bourgain–Morrey space. In particular, we show that

1.1 $$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,r}^{p,r}({\mathbb{R}}^n)={\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)\end{array}$$

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with equivalent quasi-norms. On the other hand, we give an equivalent quasi-norm of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ based on local Morrey spaces and then obtain

1.2 $$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)=G{M}_{n(\frac{1}{p}-\frac{1}{q})}^{q,\tau }({\mathbb{R}}^n),\end{array}$$

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where the later space is the global Morrey space from [[15]] (see also Definition 3.12 below). Combining (1.1) and (1.2), we conclude that the Triebel–Lizorkin–Bourgain–Morrey space is just a bridge connecting Bourgain–Morrey spaces and global Morrey spaces. Finally, as an application of all the aforementioned relations, we study the diversity of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ about the parameters p, q, and r and also obtain the diversity of ${\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)$ about the parameter $\tau$ . However, when $r\in (0,\infty )$ , the diversity of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ on $\tau$ is still unclear.

The main target of Sect. 4 is to study the sharp boundedness on Triebel–Lizorkin–Bourgain–Morrey spaces of several classical operators. First, via proving the equivalence between Herz spaces and local Morrey spaces as well as using the equivalent quasi-norms of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ in terms of local Morrey spaces obtained in Sect. 3, we establish a boundedness criterion of operators on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ via Herz spaces (see Theorem 4.2 below) and then obtain the sharp boundedness of the Hardy–Littlewood maximal operator, the Fefferman–Stein vector-valued inequality, and the Calderón–Zygmund operator on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Then we also establish the sharp boundedness of the fractional integral and the fractional maximal operator on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ via making full use of both the Calderón product of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and the sparse family of dyadic grids on Euclidean spaces.

Section 5 is devoted to presenting several further remarks. To be precise, we introduce two new distribution spaces associated with Triebel–Lizorkin–Bourgain–Morrey spaces and Besov–Bourgain–Morrey spaces, which are the generalizations of Triebel–Lizorkin spaces and Besov spaces. We will develop a real-variable theory of these new spaces in forthcoming articles.

Finally, we make some conventions on symbols. Let $\mathbb{N}:=\{1,2,...\}$ , ${\mathbb{Z}}_{+}:=\mathbb{N}\cup \{0\}$ , $\mathbb{Z}$ denote the set of all integers, and ${\mathbb{Z}}^n:=\mathbb{Z}\times \cdots \times \mathbb{Z}$ (repeated n times). For any $s\in \mathbb{R}$ , $\lfloor s\rfloor$ denotes the largest integer not greater than s. We denote by C a positive constant which is independent of the main parameters, but may vary from line to line. We also use $C_{(\alpha ,\beta ,...)}$ to denote a positive constant depending on the indicated parameters $\alpha$ , $\beta ,$ $...$ . The symbol $f\lesssim g$ means $f\le Cg$ . If $f\lesssim g$ and $g\lesssim f$ , we then write $f\sim g$ . If $f\le Cg$ and $g=h$ or $g\le h$ , we then write $f\lesssim g=h$ or $f\lesssim g\le h$ . Moreover, we use $\mathbf{0}$ to denote the origin of ${\mathbb{R}}^n$ . In this article, any cubeQ has finite edge length and all its edges parallel to the coordinate axes. For any cube Q, we denote by l(Q) the edge length of Q and by $\lambda Q$ for any given $\lambda \in (0,\infty )$ the cube with edge length $\lambda l(Q)$ and the same center as Q. For any $y\in {\mathbb{R}}^n$ and $t\in (0,\infty )$ , Q(y, t) denotes a cube of ${\mathbb{R}}^n$ centered at y with $l(Q)=t$ . In particular, for any $y\in {\mathbb{R}}^n$ and $\nu \in \mathbb{Z}$ , we simply write $Q_{\nu }(y):=Q(y,2^{-\nu }).$ For a given function f on ${\mathbb{R}}^n$ , let $\mathrm{supp}(f):=\{x\in {\mathbb{R}}^n:f(x)\ne 0\}$ be the support of f. Furthermore, let X and Y be two quasi-normed vector spaces equipped, respectively, with the quasi-norms ${\Vert ·\Vert }_X$ and ${\Vert ·\Vert }_Y$ . Then we use $X\hookrightarrow Y$ to denote $X\subset Y$ and there exists a positive constant C such that, for any $f\in X$ , ${\Vert f\Vert }_Y\le C{\Vert f\Vert }_X.$ In the end, when we prove a theorem or the like, we always use the same symbols in the wanted proved theorem or the like.

Triebel–Lizorkin–Bourgain–Morrey spaces

In this section, we first introduce the concept of Triebel–Lizorkin–Bourgain–Morrey spaces. Then, in Sect. 2.1, we give two equivalent quasi-norms of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and obtain their inclusion relations on different indices, the dilation invariance, the translation invariance, and the orthogonality invariance of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , and also the completeness of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Next, in Sect. 2.2, we characterize the nontriviality of Triebel–Lizorkin–Bourgain–Morrey spaces. Finally, in Sect. 2.3, when all the exponents are finite, we establish both the approximation and the density properties of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

First, we introduce Triebel–Lizorkin–Bourgain–Morrey spaces as follows.

Definition 2.1

Let $p,q,r,\tau \in (0,\infty ].$ The Triebel–Lizorkin–Bourgain–Morrey space (for short, TLBM space) ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is defined by setting

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n):=\left\{f,\in ,L_{\mathrm{loc}}^q,({\mathbb{R}}^n),:,,{\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},<,\infty \right\},\end{array}$$

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where

$$\begin{array}{c}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}:={∥{\left\{{\int }_0^{\infty },{\left[{|B(·,t)|}^{\frac{1}{p}-\frac{1}{q}-\frac{1}{r}},{∥f,{\mathbf{1}}_{B(·,t)}∥}_{L^q({\mathbb{R}}^n)}\right]}^{\tau },,\frac{\mathit{dt}}{t}\right\}}^{\frac{1}{\tau }}∥}_{L^r({\mathbb{R}}^n)}\end{array}$$

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when $\tau \in (0,\infty )$ and where

$$\begin{array}{c}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}:={∥\underset{t\in (0,\infty )}{sup},\left\{{|B(·,t)|}^{\frac{1}{p}-\frac{1}{q}-\frac{1}{r}},{∥f,{\mathbf{1}}_{B(·,t)}∥}_{L^q({\mathbb{R}}^n)}\right\}∥}_{L^r({\mathbb{R}}^n)}\end{array}$$

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when $\tau =\infty$ .

Remark 2.2

• We should point out that, in Definition 2.1, if $\tau =r$ , then, in this case, the quasi-norm ${\Vert ·\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}$ is just an equivalent quasi-norm of the Bourgain–Morrey space ${\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)$ , which was established in [[50], Theorem 2.9] (see also Lemma 3.5 below). This further implies that ${\mathcal{M}\dot{F}}_{q,r}^{p,r}({\mathbb{R}}^n)={\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)$ and hence ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is just a generalization of ${\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)$ [see Proposition 3.6(iii) below for the details].
• The quasi-norm of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ essentially contains the structure of Triebel–Lizorkin spaces. Indeed, in the definition of ${\Vert ·\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}$ , if we replace f therein by the Littlewood–Paley functions ${\{{\mathrm{\Delta }}_t^{\phi }(f)\}}_{t\in (0,\infty )}$ [see (5.3) below for the definition], then we obtain a new function space which is just a generalization of Triebel–Lizorkin spaces (see Definition 5.1 and Remark 5.2 below for more details).

Basic properties

The main target of this subsection is to investigate some fundamental properties of TLBM spaces. All these results play important roles throughout this article.

First, we give the following two equivalent quasi-norms of TLBM spaces.

Theorem 2.3

Let $p,q,r,\tau \in (0,\infty ].$

• Then $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ if and only if $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ and
• ‖|f|‖MF˙q,rp,τ(Rn):=∑ν∈Z2νn(1p-1q-1r)f1B(·,2ν)Lq(Rn)τ1τLr(Rn),

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• with the usual modification made when $\tau =\infty$ , is finite. Moreover, there exist two positive constants $C_1$ and $C_2$ such that, for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ,
• C1‖f‖MF˙q,rp,τ(Rn)≤‖|f|‖MF˙q,rp,τ(Rn)≤C2‖f‖MF˙q,rp,τ(Rn).

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• For any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ , let
• [f]MF˙q,rp,τ(Rn):=∑ν∈Z2-νn(1p-1q-1r)f1Qν(·)Lq(Rn)τ1τLr(Rn)

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• with the usual modification made when $\tau =\infty$ . Then the same conclusions of (i) hold true with $\Vert |·{|\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}$ therein replaced by ${[·]}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}$ .

Proof

We only show the case that both r and $\tau$ are finite because the proofs of the other cases that $r=\infty$ or $\tau =\infty$ are quite similar and hence we omit the details.

First, we prove (i). For the sufficiency, let $f\in L_{\text{loc}}^q({\mathbb{R}}^n)$ be such that ${\Vert |f|\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}<\infty$ . Observe that, for any $\nu \in \mathbb{Z}$ and $t\in [2^{\nu -1},2^{\nu }]$ , we have

2.1 $$\begin{array}{c}\hfill t^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})}\sim 2^{\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})}.\end{array}$$

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By this, we find that

2.2 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \sim {\left\{{\int }_{{\mathbb{R}}^n},{\left[{\int }_0^{\infty },{\left\{t^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})},{∥f,{\mathbf{1}}_{B(y,t)}∥}_{L^q({\mathbb{R}}^n)}\right\}}^{\tau },,\frac{\mathit{dt}}{t}\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & ={\left\{{\int }_{{\mathbb{R}}^n},{\left[\sum _{\nu \in \mathbb{Z}},{\int }_{2^{\nu -1}}^{2^{\nu }},{\left\{t^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})},{∥f,{\mathbf{1}}_{B(y,t)}∥}_{L^q({\mathbb{R}}^n)}\right\}}^{\tau },,\frac{\mathit{dt}}{t}\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & \lesssim {\left\{{\int }_{{\mathbb{R}}^n},{\left[\sum _{\nu \in \mathbb{Z}},{\left\{2^{\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})},{∥f,{\mathbf{1}}_{B(y,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}\right\}}^{\tau }\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & ={\Vert |f|\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}<\infty .\hfill \end{array}$$

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This implies that $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and hence the sufficiency of (i) holds true.

Conversely, we next prove the necessity of (i). Indeed, from (2.1) again, it follows that, for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ,

2.3 $$\begin{array}{cc}\hfill {\Vert |f|\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \sim {\left\{{\int }_{{\mathbb{R}}^n},{\left[\sum _{\nu \in \mathbb{Z}},\left({\int }_{2^{\nu }}^{2^{\nu +1}},\frac{\mathit{dt}}{t}\right),{\left\{2^{(\nu +1)n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})},{∥f,{\mathbf{1}}_{B(y,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}\right\}}^{\tau }\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & \lesssim {\left\{{\int }_{{\mathbb{R}}^n},{\left[\sum _{\nu \in \mathbb{Z}},{\int }_{2^{\nu }}^{2^{\nu +1}},{\left\{t^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})},{∥f,{\mathbf{1}}_{B(y,t)}∥}_{L^q({\mathbb{R}}^n)}\right\}}^{\tau },\frac{\mathit{dt}}{t}\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & ={\left\{{\int }_{{\mathbb{R}}^n},{\left[{\int }_0^{\infty },{\left\{t^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})},{∥f,{\mathbf{1}}_{B(y,t)}∥}_{L^q({\mathbb{R}}^n)}\right\}}^{\tau },,\frac{\mathit{dt}}{t}\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & ={\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}<\infty ,\hfill \end{array}$$

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which completes the proof of the necessity of (i). Moreover, combining this estimate and (2.2), we further obtain, for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\sim {\Vert |f|\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\end{array}$$

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This then finishes the proof of (i).

Now, we turn to show (ii). Indeed, from (i), we deduce that, to prove (ii), it suffices to show that, for any $f\in L_{\text{loc}}^q({\mathbb{R}}^n)$ , ${[f]}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\sim {\Vert |f|\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}$ . Notice that, for any $y\in {\mathbb{R}}^n$ and $\nu \in \mathbb{Z}$ ,

$$\begin{array}{cc}\hfill B\left(y,,,2^{-\nu -1}\right)& \subset Q_{\nu }(y)=Q\left(y,,,2^{-\nu }\right)\hfill \\ \hfill & \subset B\left(y,,,\sqrt{n},2^{-\nu -1}\right)\subset B\left(y,,,2^{-\nu +{\nu }_0}\right),\hfill \end{array}$$

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where ${\nu }_0:=\lfloor {log}_2\sqrt{n}\rfloor$ . Applying this and repeating the arguments similar to those used in both (2.2) and (2.3), we conclude that, for any $f\in L_{\text{loc}}^q({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {[f]}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\sim {\Vert |f|\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\end{array}$$

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This finishes the proof of (ii) and hence Theorem 2.3. $\square$

Now, we consider the inclusion properties of TLBM spaces as follows. We point out that Proposition 2.4(i) follows from both Theorem 2.3 and the monotonicity of ${\ell }^{\tau }$ and that Proposition 2.4(ii) is a direct corollary of the Hölder inequality; we omit the details.

Proposition 2.4

Let $p,q,r,\tau \in (0,\infty ].$

• If $0<{\tau }_1\le {\tau }_2\le \infty$ , then ${\mathcal{M}\dot{F}}_{q,r}^{p,{\tau }_1}({\mathbb{R}}^n)\hookrightarrow {\mathcal{M}\dot{F}}_{q,r}^{p,{\tau }_2}({\mathbb{R}}^n)$ .
• If $0<q_1\le q_2\le \infty$ , then ${\mathcal{M}\dot{F}}_{q_2,r}^{p,\tau }({\mathbb{R}}^n)\hookrightarrow {\mathcal{M}\dot{F}}_{q_1,r}^{p,\tau }({\mathbb{R}}^n)$ . Moreover, for any $f\in {\mathcal{M}\dot{F}}_{q_2,r}^{p,\tau }({\mathbb{R}}^n)$ ,
• ‖f‖MF˙q1,rp,τ(Rn)≤‖f‖MF˙q2,rp,τ(Rn).

Graph

Recall that a real $n\times n$ matrix $\mathcal{O}$ is said to be orthogonal if ${\mathcal{O}}^{\text{t}}\mathcal{O}=I_n$ , where ${\mathcal{O}}^{\text{t}}$ denotes the transpose of $\mathcal{O}$ and where $I_n$ denotes the $n\times n$ identity matrix. We next give the dilation invariance, the translation invariance, and the orthogonality invariance of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ as follows.

Proposition 2.5

Let $p,q,r,\tau \in (0,\infty ].$ Then

• for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and $\beta \in (0,\infty )$ ,
• ‖f(β·)‖MF˙q,rp,τ(Rn)=β-np‖f‖MF˙q,rp,τ(Rn);

Graph

• for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and $\xi \in {\mathbb{R}}^n$ ,
• ‖f(·-ξ)‖MF˙q,rp,τ(Rn)=‖f‖MF˙q,rp,τ(Rn);

Graph

• for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and any $n\times n$ orthogonal matrix $\mathcal{O}$ ,
• fO·MF˙q,rp,τ(Rn)=‖f‖MF˙q,rp,τ(Rn).

Graph

Proof

Let $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Then, using the definition of ${\Vert ·\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}$ and the change of variables, we easily find that, for any $\beta \in (0,\infty )$ and $\xi \in {\mathbb{R}}^n$ and for any $n\times n$ orthogonal matrix $\mathcal{O}$ ,

$$\begin{array}{cc}\hfill & {\Vert f(\beta ·)\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}={\beta }^{-\frac{n}{p}}{\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},\hfill \\ \hfill & {\Vert f(·-\xi )\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}={\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},\hfill \end{array}$$

Graph

and

$$\begin{array}{c}\hfill {∥f,\left(\mathcal{O},·\right)∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}={\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\end{array}$$

Graph

These then finish the proof of Proposition 2.5. $\square$

At the end of this subsection, we consider the completeness of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . In what follows,we use the symbol $\mathcal{M}({\mathbb{R}}^n)$ to denote the set of all measurable functions on ${\mathbb{R}}^n$ . Then we have the following lemma about the completeness of general quasi-normed linear spaces, which is just [[20], Proposition 1.2.36] (see also [[8], Theorem 2]).

Lemma 2.6

Let $X\subset \mathcal{M}({\mathbb{R}}^n)$ be a quasi-normed linear space, equipped with a quasi-norm ${\Vert ·\Vert }_X$ which makes sense for all functions in $\mathcal{M}({\mathbb{R}}^n)$ . Further assume that X satisfies

• for any $f,g\in \mathcal{M}({\mathbb{R}}^n)$ , $|g|\le |f|$ almost everywhere implies that ${\Vert g\Vert }_X\le {\Vert f\Vert }_X$ ;
• for any ${\{f\}}_{m\in \mathbb{N}}\subset \mathcal{M}({\mathbb{R}}^n)$ and $f\in \mathcal{M}({\mathbb{R}}^n)$ , $0\le f_m\uparrow f$ as $m\to \infty$ almost everywhere implies that $\Vert f_m\Vert \uparrow \Vert f\Vert$ as $m\to \infty$ .

Then X is complete.

The following completeness of TLBM spaces is just a direct corollary of both Lemma 2.6 and the monotone convergence theorem; we omit the details.

Theorem 2.7

If $p,q,r,\tau \in (0,\infty ]$ , then ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is complete.

Nontriviality

In this subsection, via making use of the characteristic function of $B(\mathbf{0},1)$ and both the dilation invariance and the translation invariance of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , we give a sufficient and necessary condition for the nontriviality of TLBM spaces.

We first consider the sufficient and necessary condition for the characteristic function ${\mathbf{1}}_{B(\mathbf{0},1)}$ belonging to TLBM spaces as follows.

Proposition 2.8

Let $p,q,r,\tau \in (0,\infty ]$ . Then ${\mathbf{1}}_{B(\mathbf{0},1)}\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ if and only if one of the following three statements holds true:

• 0 $\tau \in (0,\infty )$ ;
• 0

• 0

Proof

For any $y\in {\mathbb{R}}^n$ , let

$$\begin{array}{c}\hfill {\mathcal{A}}_{p,q,r,\tau }(y):={\left\{{\int }_0^{\infty },{\left[{|B(y,t)|}^{\frac{1}{p}-\frac{1}{q}-\frac{1}{r}},{∥{\mathbf{1}}_{B(\mathbf{0},1)},{\mathbf{1}}_{B(y,t)}∥}_{L^q({\mathbb{R}}^n)}\right]}^{\tau },,\frac{\mathit{dt}}{t}\right\}}^{\frac{1}{\tau }}\end{array}$$

Graph

when $\tau \in (0,\infty )$ and let

$$\begin{array}{c}\hfill {\mathcal{A}}_{p,q,r,\tau }(y):=\underset{t\in (0,\infty )}{sup}\left\{{|B(y,t)|}^{\frac{1}{p}-\frac{1}{q}-\frac{1}{r}},{∥{\mathbf{1}}_{B(\mathbf{0},1)},,{\mathbf{1}}_{B(y,t)}∥}_{L^q({\mathbb{R}}^n)}\right\}\end{array}$$

Graph

when $\tau =\infty$ . Then $\Vert {\mathbf{1}}_{B(\mathbf{0},1)}{\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}={\Vert {\mathcal{A}}_{p,q,r,\tau }\Vert }_{L^r({\mathbb{R}}^n)}$ . Therefore, to prove the present proposition, we only need to estimate $\Vert {\mathcal{A}}_{p,q,r,\tau }{\Vert }_{L^r({\mathbb{R}}^n)}$ . For this purpose, we consider the following three cases on y.

Case (1) $y\in B(\mathbf{0},1)$ . In this case, via some simple computations, we have

2.4 $$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|=|B(y,t)|\sim t^n\text{when}t\in [0,1-|y|),\end{array}$$

Graph

2.5 $$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|\le |B(y,t)|\sim t^n\text{when}t\in [1-|y|,1+|y|],\end{array}$$

Graph

and

2.6 $$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|=|B(\mathbf{0},1)|\sim 1\text{when}t\in (1+|y|,\infty ).\end{array}$$

Graph

By these, we conclude that, if $\tau \in (0,\infty )$ and $\frac{1}{p}-\frac{1}{r}\in \mathbb{R}\backslash (0,\frac{1}{q})$ , then

$$\begin{array}{c}\hfill {\mathcal{A}}_{p,q,r,\tau }(y)\gtrsim {\left[{\int }_0^{1-|y|},t^{n(\frac{1}{p}-\frac{1}{r})\tau },\frac{\mathit{dt}}{t}\right]}^{\frac{1}{\tau }}+{\left[{\int }_{1+|y|}^{\infty },t^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },\frac{\mathit{dt}}{t}\right]}^{\frac{1}{\tau }}=\infty ;\end{array}$$

Graph

if $\tau \in (0,\infty )$ and $\frac{1}{p}-\frac{1}{r}\in (0,\frac{1}{q})$ , then

$$\begin{array}{cc}\hfill {\mathcal{A}}_{p,q,r,\tau }(y)& \lesssim {\left[{\int }_0^{1+|y|},t^{n(\frac{1}{p}-\frac{1}{r})\tau },,\frac{\mathit{dt}}{t}\right]}^{\frac{1}{\tau }}+{\left[{\int }_{1+|y|}^{\infty },t^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },,\frac{\mathit{dt}}{t}\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \sim {(1+|y|)}^{n(\frac{1}{p}-\frac{1}{r})}+{(1+|y|)}^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})}\sim 1.\hfill \end{array}$$

Graph

From these, we deduce that, if $\tau \in (0,\infty )$ , then ${\mathcal{A}}_{p,q,r,\tau }{\mathbf{1}}_{B(\mathbf{0},1)}\in L^r({\mathbb{R}}^n)$ if and only if

2.7 $$\begin{array}{c}\hfill \frac{1}{p}-\frac{1}{r}\in \left(0,,,\frac{1}{q}\right).\end{array}$$

Graph

On the other hand, by (2.4), (2.5), and (2.6) again, we find that, if $\tau =\infty$ and $\frac{1}{p}-\frac{1}{r}\in \mathbb{R}\backslash [0,\frac{1}{q}]$ , then

$$\begin{array}{c}\hfill {\mathcal{A}}_{p,q,r,\tau }(y)\gtrsim \underset{t\in (0,1-|y|)}{sup}{t}^{n(\frac{1}{p}-\frac{1}{r})}+\underset{t\in [1-|y|,\infty )}{sup}{t}^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})}=\infty ;\end{array}$$

Graph

if $\tau =\infty$ and $\frac{1}{p}-\frac{1}{r}\in [0,\frac{1}{q}]$ , then

$$\begin{array}{cc}\hfill {\mathcal{A}}_{p,q,r,\tau }(y)& \lesssim \underset{(0,1+|y|]}{sup}{t}^{n(\frac{1}{p}-\frac{1}{r})}+\underset{(1+|y|,\infty )}{sup}{t}^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})}\hfill \\ \hfill & \sim {(1+|y|)}^{n(\frac{1}{p}-\frac{1}{r})}+{(1+|y|)}^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})}\sim 1.\hfill \end{array}$$

Graph

Therefore, if $\tau =\infty$ , then ${\mathcal{A}}_{p,q,r,\tau }{\mathbf{1}}_{B(\mathbf{0},1)}\in L^r({\mathbb{R}}^n)$ if and only if

2.8 $$\begin{array}{c}\hfill \frac{1}{p}-\frac{1}{r}\in \left[0,,,\frac{1}{q}\right].\end{array}$$

Graph

Case (2) $y\in B(\mathbf{0},2)\backslash B(\mathbf{0},1)$ . In this case, by some simple computations, we have

$$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|=0\text{when}t\in [0,|y|-1],\end{array}$$

Graph

$$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|\le |B(y,t)|\sim t^n\text{when}t\in (|y|-1,|y|+1),\end{array}$$

Graph

and

$$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|=|B(\mathbf{0},1)|\sim 1\text{when}t\in [|y|+1,\infty ).\end{array}$$

Graph

Similarly to the proof of Case (1), we easily find that ${\mathcal{A}}_{p,q,r,\tau }{\mathbf{1}}_{B(\mathbf{0},2)\backslash B(\mathbf{0},1)}\in L^r({\mathbb{R}}^n)$ if and only if

2.9 $$\begin{array}{c}\hfill \frac{1}{p}-\frac{1}{r}\in \left(-,\infty ,,,\frac{1}{q}\right)\text{and}\tau \in (0,\infty )\end{array}$$

Graph

or

2.10 $$\begin{array}{c}\hfill \frac{1}{p}-\frac{1}{r}\in \left(-,\infty ,,,\frac{1}{q}\right]\text{and}\tau =\infty .\end{array}$$

Graph

Case (3) $y\in {\mathbb{R}}^n\backslash B(\mathbf{0},2)$ . In this case, observe that

$$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|=0\text{when}t\in [0,|y|-1],\end{array}$$

Graph

$$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|\le |B(\mathbf{0},1)|\sim 1\text{when}t\in (|y|-1,|y|+1),\end{array}$$

Graph

and

$$\begin{array}{c}\hfill |B(y,t)\cap B(\mathbf{0},1)|=|B(\mathbf{0},1)|\sim 1\text{when}t\in [|y|+1,\infty ).\end{array}$$

Graph

Similarly to the proof of Case (1) again, we conclude that ${\mathcal{A}}_{p,q,r,\tau }{\mathbf{1}}_{{\mathbb{R}}^n\backslash B(\mathbf{0},2)}\in L^r({\mathbb{R}}^n)$ if and only if

2.11 $$\begin{array}{c}\hfill \frac{1}{p}-\frac{1}{q}\in (-\infty ,0)\text{and}min\{r,\tau \}<\infty \end{array}$$

Graph

or

2.12 $$\begin{array}{c}\hfill \frac{1}{p}-\frac{1}{q}\in (-\infty ,0]\text{and}\tau =r=\infty .\end{array}$$

Graph

Therefore, combining (2.7), (2.8), (2.9), (2.10), (2.11), and (2.12), we conclude that ${\mathbf{1}}_{B(\mathbf{0},1)}\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ if and only if (i), (ii), or (iii) of the present proposition holds true. This then finishes the proof of Proposition 2.8. $\square$

Now, we have the following characterization of the nontriviality of TLBM spaces.

Theorem 2.9

If $p,q,r,\tau \in (0,\infty ]$ , then ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is nontrivial if and only if (i), (ii), or (iii) of Proposition 2.8 holds true.

To show Theorem 2.9, we need the following Young inequality.

Lemma 2.10

If $p,q,r,\tau \in [1,\infty ]$ , then, for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and $g\in L^1({\mathbb{R}}^n)$ , $f\ast g\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and

$$\begin{array}{c}\hfill {\Vert f\ast g\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\le {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}{\Vert g\Vert }_{L^1({\mathbb{R}}^n)}.\end{array}$$

Graph

Proof

From the Minkowski integral inequality and Proposition 2.5(ii), we infer that, for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and $g\in L^1({\mathbb{R}}^n)$ ,

$$\begin{array}{cc}\hfill {\Vert f\ast g\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \le \left\{{\int }_{{\mathbb{R}}^n},\left[{\int }_0^{\infty },\{,{\int }_{{\mathbb{R}}^n},{|B(y,t)|}^{\frac{1}{p}-\frac{1}{q}-\frac{1}{r}}\right)\right)\hfill \\ \hfill & {\left({\left(\times \{\Vert f(·-\xi ),{\mathbf{1}}_{B(y,t)},{\Vert }_{L^q({\mathbb{R}}^n)},|g(\xi )|,,d,\xi ,{\}}^{\tau },,\frac{\mathit{dt}}{t}\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^n}{\Vert f(·-\xi )\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}|g(\xi )|d\xi ={\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}{\Vert g\Vert }_{L^1({\mathbb{R}}^n)}.\hfill \end{array}$$

Graph

This finishes the proof of Lemma 2.10. $\square$

Based on this Young inequality, the estimate of ${\mathbf{1}}_{B(\mathbf{0},1)}$ , and both the dilation invariance and the translation invariance of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , we show Theorem 2.9 as follows.

Proof of Theorem 2.9

We first show the sufficiency. To do so, assume that (i), (ii), or (iii) of the present theorem holds true. Then, by Proposition 2.8, we find that ${\mathbf{1}}_{B(\mathbf{0},1)}\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . This further implies that the TLBM space under consideration is nontrivial and hence finishes the proof of the sufficiency.

To prove the necessity, let $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)\backslash \{0\}$ . Then notice that, for any $u\in (0,min\{p,\tau ,q,r\})$ ,

$$\begin{array}{c}\hfill {\left[{∥{|f|}^u∥}_{{\mathcal{M}\dot{F}}_{q/u,r/u}^{p/u,\tau /u}({\mathbb{R}}^n)}\right]}^{\frac{1}{u}}={∥f∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}<\infty .\end{array}$$

Graph

Thus, we may assume that $q,p,r,\tau \in (1,\infty ]$ . Without loss of generality, we may also assume that $f\ge 0$ and $f\in L^{\infty }({\mathbb{R}}^n)$ via replacing f by $min\{1,|f|\}$ . Then, using Lemma 2.10, we conclude that ${\mathbf{1}}_{B(\mathbf{0},1)}\ast f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)\backslash \{0\}$ . Since ${\mathbf{1}}_{B(\mathbf{0},1)}\ast f$ is a non-negative and non-zero continuous function, it follows that there exist $x_0\in {\mathbb{R}}^n$ , $ϵ\in (0,\infty )$ , and $s\in (0,\infty )$ such that

$$\begin{array}{c}\hfill ϵ{\mathbf{1}}_{B(x_0,s)}\le {\mathbf{1}}_{B(\mathbf{0},1)}\ast f.\end{array}$$

Graph

Thus, ${\mathbf{1}}_{B(x_0,s)}\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . From this and both (i) and (ii) of Proposition 2.5, it follows that

$$\begin{array}{cc}\hfill \Vert {\mathbf{1}}_{B(\mathbf{0},1)}{\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& ={∥{\mathbf{1}}_{B(\mathbf{0},s)},(s·)∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}=s^{-\frac{n}{p}}{∥{\mathbf{1}}_{B(\mathbf{0},s)}∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\hfill \\ \hfill & =s^{-\frac{n}{p}}{∥{\mathbf{1}}_{B(x_0,s)}∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}<\infty ,\hfill \end{array}$$

Graph

which, together with Proposition 2.8, further implies that one of the above four statements in the present theorem holds true. Therefore, the necessity holds true. This finishes the proof of Theorem 2.9. $\square$

Remark 2.11

By Theorem 2.9, we find that, only when $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ , the TLBM space ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ may be nontrivial. Thus, in what follows, we always assume that $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ when we consider the TLBM space ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

Approximation and density properties

In this subsection, we investigate both the approximation and the density properties of TLBM spaces when all the exponents are finite. First, combining the equivalent quasi-norms established in Theorem 2.3(ii) and the exquisite geometrical properties of cubes of Euclidean spaces, we obtain the approximation property of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Then, due to the Lebesgue dominated convergence theorem, we also give the density property of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

We first give some preliminary concepts. Let Q be a cube of ${\mathbb{R}}^n$ . The average operator $m_Q$ is defined by setting, for any $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ ,

2.13 $$\begin{array}{c}\hfill m_Q(f):=\frac{1}{|Q|}{\int }_{Q}f(x)dx.\end{array}$$

Graph

Moreover, for any $q\in (0,\infty )$ , any cube Q, any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ , and any $k\in \mathbb{Z}$ , let

$$\begin{array}{c}\hfill m_Q^{(q)}(f):={\left[m_Q,\left({|f|}^q\right)\right]}^{\frac{1}{q}}\text{and}{E}_k^{(q)}(f):=\sum _{Q\in {\mathcal{D}}_k}{m}_Q^{(q)}(f){\mathbf{1}}_Q.\end{array}$$

Graph

Then, via the operators ${\{E_k^{(q)}\}}_{k=1}^{\infty }$ , we have the following approximation property of TLBM spaces.

Theorem 2.12

If $0<q<p<r<\infty$ and $\tau \in (0,\infty )$ , then, for any non-negative function $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , ${\{E_k^{(q)}(f)\}}_{k=1}^{\infty }$ converges to f in ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

To show this theorem, we need two auxiliary lemmas. First, repeating the proof of [[50], (2.23)] with $Q_{j,m}$ therein replaced by Q, we obtain the following locally approximation property of $L^q({\mathbb{R}}^n)$ via the operators ${\{E_k^{(q)}\}}_{k=1}^{\infty }$ , which plays a vital role in the proof of Theorem 2.12; we omit the details.

Lemma 2.13

If $q\in (0,\infty )$ , then, for any cube Q and any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{∥\left[f,-,E_k^{(q)},(f)\right],{\mathbf{1}}_Q∥}_{L^q({\mathbb{R}}^n)}=0.\end{array}$$

Graph

In addition, we also require the following estimate of operators ${\{E_k^{(q)}\}}_{k=1}^{\infty }$ .

Lemma 2.14

Let $0<q<p<r<\infty$ and $\tau \in (0,\infty )$ . Then there exists a positive constant C, depending only on p, q, r, $\tau$ , and n, such that, for any $k\in \mathbb{N}$ , $y\in {\mathbb{R}}^n$ , and $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ,

2.14 $$\begin{array}{cc}& {\left[\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥E_k^{(q)},(f),{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \le C{\left[\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥f,{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}.\hfill \end{array}$$

Graph

Proof

Let $k\in \mathbb{N}$ , $y\in {\mathbb{R}}^n$ , and $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Then we have

2.15 $$\begin{array}{cc}& {\left[\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥E_k^{(q)},(f),{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \lesssim {\left[\sum _{\nu =-\infty }^k,2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥E_k^{(q)},(f),{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}+{\left[\sum _{\nu =k+1}^{\infty },\cdots \right]}^{\frac{1}{\tau }}\hfill \\ \hfill & =:G_k^{(1)}(y)+G_k^{(2)}(y).\hfill \end{array}$$

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We first estimate $G_k^{(1)}(y)$ . Indeed, for any $\nu \in \mathbb{Z}$ , we have

2.16 $$\begin{array}{cc}\hfill {∥E_k^{(q)},(f),{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}& ={\left\{{\int }_{Q_v(y)},\sum _{Q\in {\mathcal{D}}_k},{\left[m_Q^{(q)},(f)\right]}^q,{\mathbf{1}}_Q,(x),,d,x\right\}}^{\frac{1}{q}}\hfill \\ \hfill & ={\left[\sum _{Q\in {\mathcal{D}}_k,Q\cap Q_{\nu }(y)\ne \mathrm{\varnothing }},\frac{|Q\cap Q_{\nu }(y)|}{|Q|},{\int }_Q,{|f(z)|}^q,,d,z\right]}^{\frac{1}{q}}.\hfill \end{array}$$

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For any $\nu \in \mathbb{Z}\cap (-\infty ,k]$ and $Q\in {\mathcal{D}}_k$ such that $Q\cap Q_{\nu }(y)\ne \mathrm{\varnothing }$ , we find that $Q\subset Q_{\nu -2}(y).$ From this and (2.16), we deduce that, for any $\nu \in \mathbb{Z}\cap (-\infty ,k]$ ,

$$\begin{array}{cc}\hfill {∥E_k^{(q)},(f),{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}& \le {\left[\sum _{Q\in {\mathcal{D}}_k,Q\cap Q_{\nu }(y)\ne \mathrm{\varnothing }},{\int }_Q,{|f(z)|}^q,,d,z\right]}^{\frac{1}{q}}\hfill \\ \hfill & \le {\left[\sum _{Q\in {\mathcal{D}}_k,Q\subset Q_{\nu -2}(y)\ne \mathrm{\varnothing }},{\int }_Q,{|f(z)|}^q,,d,z\right]}^{\frac{1}{q}}\hfill \\ \hfill & \le {\left[{\int }_{Q_{\nu -2}(y)},{|f(z)|}^q,,d,z\right]}^{\frac{1}{q}}={∥f,{\mathbf{1}}_{Q_{\nu -2}(y)}∥}_{L^q({\mathbb{R}}^n)}.\hfill \end{array}$$

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This, combined with the change of variables, further implies that

2.17 $$\begin{array}{cc}\hfill G_k^{(1)}(y)& \le {\left[\sum _{\nu =-\infty }^k,2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥f,{\mathbf{1}}_{Q_{\nu -2}(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \sim {\left[\sum _{\nu =-\infty }^{k-2},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥f,{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \le {\left[\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥f,{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }},\hfill \end{array}$$

Graph

which is just the desired estimate of $G_k^{(1)}(y)$ .

Then we estimate $G_k^{(2)}(y)$ . Indeed, for any $\nu \in \mathbb{Z}\cap [k+1,\infty )$ and $Q\in {\mathcal{D}}_k$ such that $Q\cap Q_{\nu }(y)\ne \mathrm{\varnothing }$ , it holds true that $Q\subset Q_{k-1}(y)$ , which further implies that

$$\begin{array}{c}\hfill \bigcup _{Q\in {\mathcal{D}}_k,Q\cap Q_{\nu }(y)\ne \mathrm{\varnothing }}Q\subset Q_{k-1}(y).\end{array}$$

Graph

From this and (2.16) again, it follows that, for any $\nu \in \mathbb{Z}\cap [k+1,\infty )$ ,

$$\begin{array}{cc}\hfill {∥E_k^{(q)},(f),{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}& \le {\left[\sum _{Q\in {\mathcal{D}}_k,Q\cap Q_{\nu }(y)\ne \mathrm{\varnothing }},2^{-(\nu -k)n},\left({\int }_Q,{|f(z)|}^q,,d,z\right)\right]}^{\frac{1}{q}}\hfill \\ \hfill & \le 2^{-\frac{(\nu -k)n}{q}}{∥f,{\mathbf{1}}_{Q_{k-1}(y)}∥}_{L^q({\mathbb{R}}^n)},\hfill \end{array}$$

Graph

which, together with $0<p<r$ , further implies that

$$\begin{array}{cc}\hfill G_k^{(2)}(y)& \le {\left[\sum _{\nu =k+1}^{\infty },2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },2^{-\frac{(\nu -k)n\tau }{q}},{∥f,{\mathbf{1}}_{Q_{k-1}(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \le 2^{\frac{kn\tau }{q}}{∥f,{\mathbf{1}}_{Q_{k-1}(y)}∥}_{L^q({\mathbb{R}}^n)}{\left[\sum _{\nu =k+1}^{\infty },2^{-\nu n(\frac{1}{p}-\frac{1}{r})\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \sim 2^{-(k-1)n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})}{∥f,{\mathbf{1}}_{Q_{k-1}(y)}∥}_{L^q({\mathbb{R}}^n)}\hfill \\ \hfill & \le {\left[\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥f,{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }},\hfill \end{array}$$

Graph

which completes the estimation of $G_k^{(2)}(y)$ . Using this estimate, (2.15), and (2.17), we then conclude that (2.14) holds true. This finishes the proof of Lemma 2.14. $\square$

Via both Lemmas 2.13 and 2.14 and the equivalent quasi-norms of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ established in Theorem 2.3(ii), we now establish the approximation property of TLBM spaces as follows.

Proof of Theorem 2.12

Let $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ be a non-negative function. Then, by Theorem 2.3, we find that

2.18 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \sim {[f]}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\hfill \\ \hfill & ={\left\{{\int }_{{\mathbb{R}}^n},{\left[\sum _{\nu \in \mathbb{Z}},{\left\{2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})},{∥f,{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}\right\}}^{\tau }\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & =:{\left\{{\int }_{{\mathbb{R}}^n},{[F(y)]}^r,,d,y\right\}}^{\frac{1}{r}}.\hfill \end{array}$$

Graph

Now, for any $k\in \mathbb{N}$ and $y\in {\mathbb{R}}^n$ , let

$$\begin{array}{c}\hfill H_k(y):={\left\{\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥\left[f,-,E_k^{(q)},(f)\right],{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}.\end{array}$$

Graph

Then, using Lemma 2.14, we conclude that, for any $k\in \mathbb{N}$ and $y\in {\mathbb{R}}^n$ ,

2.19 $$\begin{array}{cc}\hfill H_k(y)& \lesssim {\left[\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥f,{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & +{\left[\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥E_k^{(q)},(f),{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \lesssim F(y).\hfill \end{array}$$

Graph

Let $D:=\{y\in {\mathbb{R}}^n:F(y)=\infty \}$ . From (2.18), we deduce that $F\in L^r({\mathbb{R}}^n)$ and hence $|D|=0$ . We next show that, for any $y\in {\mathbb{R}}^n\backslash D$ ,

2.20 $$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{H}_k(y)=0.\end{array}$$

Graph

To this end, fix an $\epsilon \in (0,1)$ and a $y\in {\mathbb{R}}^n\backslash D$ . Then, by (2.19), we conclude that, for any $k\in \mathbb{N}$ ,

$$\begin{array}{c}\hfill H_k(y)\lesssim F(y)<\infty ,\end{array}$$

Graph

which, combined with the assumption $\tau \in (0,\infty )$ , further implies that there exists a $J\in \mathbb{N}$ such that

2.21 $$\begin{array}{cc}\hfill H_k^{(1)}(y):=& {\left\{\sum _{\nu \in \mathbb{Z}\backslash [-J,J]},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥\left[f,-,E_k^{(q)},(f)\right],{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill \lesssim & {\left\{\sum _{\nu \in \mathbb{Z}\backslash [-J,J]},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥f,{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}<\epsilon .\hfill \end{array}$$

Graph

On the other hand, for any $k\in \mathbb{N}$ and $y\in {\mathbb{R}}^n$ , let

$$\begin{array}{c}\hfill H_k^{(2)}(y):={\left\{\sum _{\nu =-J}^J,2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{∥\left[f,-,E_k^{(q)},(f)\right],{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}.\end{array}$$

Graph

Then, for any $k\in \mathbb{N}$ , we have

2.22 $$\begin{array}{c}\hfill H_k(y)\lesssim H_k^{(1)}(y)+H_k^{(2)}(y).\end{array}$$

Graph

By Lemma 2.13, we find that, for any $\nu \in \mathbb{Z}\cap [-J,J]$ ,

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{∥\left[f,-,E_k^{(q)},(f)\right],{\mathbf{1}}_{Q_{\nu }(y)}∥}_{L^q({\mathbb{R}}^n)}=0.\end{array}$$

Graph

This further implies that ${lim}_{k\to \infty }{H}_k^{(2)}(y)=0$ , and hence there exists a $K_1\in \mathbb{N}$ such that, for any $k\in \mathbb{N}\cap (K_1,\infty )$ ,

$$\begin{array}{c}\hfill H_k^{(2)}(y)<\epsilon .\end{array}$$

Graph

By this, (2.21), and (2.22), we conclude that (2.20) holds true.

From Theorem 2.3, (2.19), the Lebesgue dominated convergence theorem, and (2.20), it follows that

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}{∥f,-,E_k^{(q)},(f)∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \sim \underset{k\to \infty }{lim}{\left[f,-,E_k^{(q)},(f)\right]}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}=\underset{k\to \infty }{lim}{\left\{{\int }_{{\mathbb{R}}^n},{[H_k(y)]}^r,,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & ={\left\{{\int }_{{\mathbb{R}}^n},\underset{k\to \infty }{lim},{[H_k(y)]}^r,,d,y\right\}}^{\frac{1}{r}}=0.\hfill \end{array}$$

Graph

This finishes the proof of Theorem 2.12. $\square$

In what follows, for any real function $f\in \mathcal{M}({\mathbb{R}}^n)$ , let $f_{+}:=max\{f,0\}$ and $f_{-}:=max\{-f,0\}$ . For any $z\in \mathbb{C}$ , we denote by $\Re z$ the real part of z and by $\Im z$ the imaginary part of z. As a simple application of Theorem 2.12 and the quasi-triangle inequalities of both $l^p$ and $L^p({\mathbb{R}}^n)$ , we have the following approximation properties; we omit the details.

Corollary 2.15

If $0<q<p<r<\infty$ and $\tau \in (0,\infty )$ , then, for any $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ,

• {Ek(q)((ℜf)+)+iEk(q)((ℑf)+)-[Ek(q)((ℜf)-)+iEk(q)((ℑf)-)]}k=1∞ converges to f in ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , where $i:=\sqrt{-1}$ ;
• {Ek(q)(f)}k=1∞ converges to |f| in ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

Let $L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ denote the set of all the bounded functions on ${\mathbb{R}}^n$ with compact support and ${\mathcal{C}}_{\text{c}}^{\infty }({\mathbb{R}}^n)$ the set of all the infinitely differentiable functions on ${\mathbb{R}}^n$ with compact support. Now, with the help of the Lebesgue dominated convergence theorem, we show the following density property, that is, $L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ and ${\mathcal{C}}_{\text{c}}^{\infty }({\mathbb{R}}^n)$ are both dense in ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ when all the exponents are finite.

Proposition 2.16

If $0<q<p<r<\infty$ and $\tau \in (0,\infty )$ , then

• Lc∞(Rn) is dense in ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ;
• Cc∞(Rn) is dense in ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

Proof

We first show (i). To do this, let $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and, for any $m\in \mathbb{N}$ , let

2.23 $$\begin{array}{c}\hfill f_m:=f{\mathbf{1}}_{[0,m]}(|f|){\mathbf{1}}_{B(\mathbf{0},m)}.\end{array}$$

Graph

Obviously, for any $m\in \mathbb{N}$ , $f_m\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ . Moreover, it is easy to show that $f_m$ converges to f almost everywhere in ${\mathbb{R}}^n$ as $m\to \infty$ and that, for any $m\in \mathbb{N}$ , $|f-f_m|\le 2|f|\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . From these and the Lebesgue dominated convergence theorem, we deduce that

2.24 $$\begin{array}{c}\hfill \underset{m\to \infty }{lim}{\Vert f-f_m\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}=0,\end{array}$$

Graph

which completes the proof of (i).

Next, we show (ii). For this purpose, let $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and fix any $ϵ\in (0,\infty )$ . Then, by (i), we find that there exists a $g_{ϵ}\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ , depending on both f and $ϵ$ , such that

2.25 $$\begin{array}{c}\hfill \Vert f-g_{ϵ}{\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}<ϵ.\end{array}$$

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Then there exists an $M\in (0,\infty )$ , depending on $g_{ϵ}$ , such that $|g_{ϵ}|\lesssim {\mathbf{1}}_{B(\mathbf{0},M)}$ almost everywhere in ${\mathbb{R}}^n$ with the implicit positive constant depending on both $ϵ$ and M. Moreover, let $\eta \in {\mathcal{C}}_{\text{c}}^{\infty }({\mathbb{R}}^n)$ be such that $0\le \eta \le {\mathbf{1}}_{B(\mathbf{0},1)}$ and ${\int }_{{\mathbb{R}}^n}\eta (x)dx=1$ . For any $k\in \mathbb{N}$ , let ${\eta }_k(·):=k^n\eta (k·)$ . Then, by this, $\mathrm{supp}(g_{ϵ})\subset B(\mathbf{0},M)$ , and [[12], Proposition 8.6 and Theorem 8.15], we find that $g_{ϵ}\ast {\eta }_k\in {\mathcal{C}}_{\text{c}}^{\infty }({\mathbb{R}}^n)$ for any $k\in \mathbb{N}$ and

2.26 $$\begin{array}{c}\hfill g_{ϵ}\ast {\eta }_k\to g_{ϵ}\end{array}$$

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as $k\to \infty$ almost everywhere in ${\mathbb{R}}^n$ . In addition, from the definition of ${\eta }_k$ , $0\le \eta \le {\mathbf{1}}_{B(\mathbf{0},1)}$ , and $|g_{ϵ}|\lesssim {\mathbf{1}}_{B(\mathbf{0},M)}$ almost everywhere in ${\mathbb{R}}^n$ with the implicit positive constant depending on both $ϵ$ and M, we deduce that, for any $k\in \mathbb{N}$ and $x\in {\mathbb{R}}^n$ ,

$$\begin{array}{cc}\hfill |g_{ϵ}\ast {\eta }_k(x)-g_{ϵ}(x)|& \le \left|{\int }_{B(x,k^{-1})},{\eta }_k,(x-z),g_{ϵ},(z),,d,z\right|+|g_{ϵ}(x)|\hfill \\ \hfill & \le k^n{\int }_{B(x,k^{-1})}\eta (k(x-z))|g_{ϵ}(z)|dz+|g_{ϵ}(x)|\hfill \\ \hfill & \lesssim \frac{1}{|B(x,k^{-1})|}{\int }_{B(x,k^{-1})}|g_{ϵ}(z)|dz+|g_{ϵ}(x)|\hfill \\ \hfill & \lesssim {\mathbf{1}}_{B(\mathbf{0},M+1)}(x)\hfill \end{array}$$

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with the implicit positive constants depending on both $ϵ$ and M, but independent of both k and x. This, combined with the Lebesgue dominated convergence theorem, (2.26), Propositions 2.8 and 2.5(i), and Theorem 2.9, further implies that

$$\begin{array}{c}\hfill {∥g_{ϵ},,\ast ,{\eta }_k,-,g_{ϵ}∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\to 0\end{array}$$

Graph

as $k\to \infty$ . By this and (2.25), we conclude that there exists a $k_{ϵ}\in \mathbb{N}$ such that

$$\begin{array}{cc}& \Vert g_{ϵ}\ast {\eta }_{k_{ϵ}}{-f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\hfill \\ \hfill & \lesssim \Vert g_{ϵ}\ast {\eta }_{k_{ϵ}}-g_{ϵ}{\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}+{\Vert g_{ϵ}-f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\lesssim ϵ\hfill \end{array}$$

Graph

with the implicit positive constants independent of $ϵ$ . Therefore, from this and $g_{ϵ}\ast {\eta }_{k_{ϵ}}\in {\mathcal{C}}_{\text{c}}^{\infty }({\mathbb{R}}^n)$ , it further follows that ${\mathcal{C}}_{\text{c}}^{\infty }({\mathbb{R}}^n)$ is dense in ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . This then finishes the proof of (ii) and hence Proposition 2.16. $\square$

Relations with other Morrey type spaces

The targets of this section are fourfold. The first one is to establish the coincidence between special TLBM spaces and Lebesgue spaces. The second one is to study the relation between TLBM spaces and (Besov–)Bourgain–Morrey spaces. In particular, in Sect. 3.2, we show that the Bourgain–Morrey space is a special case of the TLBM space as well as the TLBM space is indeed a new space different from the (Besov–)Bourgain–Morrey space. The third target is to establish an equivalent quasi-norm of TLBM spaces based on local Morrey spaces and then prove that the global Morrey space is also a special TLBM space, which is presented in Sect. 3.3. Thus, we find that the TLBM space is just a bridge connecting Bourgain–Morrey spaces and global Morrey spaces. Finally, the last target is to study the diversity of TLBM spaces via making full use of the above relations, which is the main content of Sect. 3.4.

Coincidence with Lebesgue spaces

In this subsection, we show that a special case of the TLBM space ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is just the Lebesgue space. To do so, recall that the Hardy–Littlewood maximal operator M is defined by setting, for any $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ and $x\in {\mathbb{R}}^n$ ,

$$\begin{array}{c}\hfill M(f)(x):=\underset{B\ni x}{sup}\frac{1}{|B|}{\int }_B|f(y)|dy,\end{array}$$

Graph

where the supremum is taken over all the balls B containing x. Then, as an application of the boundedness of M on Lebesgue spaces, we establish the following coincidence between special TLBM spaces and Lebesgue spaces.

Proposition 3.1

Let $0<q<p\le \infty$ . Then ${\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)=L^p({\mathbb{R}}^n)$ with equivalent quasi-norms.

Proof

We first show ${\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)\subset L^p({\mathbb{R}}^n)$ . To do this, let $f\in {\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)$ . Then, from Definition 2.1, we infer that

3.1 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)}& ={∥\underset{t\in (0,\infty )}{sup},\left\{{|B(·,t)|}^{-\frac{1}{q}},{\Vert f{\mathbf{1}}_{B(·,t)}\Vert }_{L^q({\mathbb{R}}^n)}\right\}∥}_{L^p({\mathbb{R}}^n)}\hfill \\ \hfill & ={∥{\left[\underset{t\in (0,\infty )}{sup},\frac{1}{|B(·,t)|},{\int }_{B(·,t)},{|f(z)|}^q,,d,z\right]}^{\frac{1}{q}}∥}_{L^p({\mathbb{R}}^n)}\hfill \\ \hfill & \sim {∥{\left[M,(,|,f,{|}^q,)\right]}^{\frac{1}{q}}∥}_{L^p({\mathbb{R}}^n)}.\hfill \end{array}$$

Graph

Notice that ${|f(x)|}^q\le {M(|f|}^q)(x)$ for almost every $x\in {\mathbb{R}}^n$ . Thus, using (3.1), we have

3.2 $$\begin{array}{c}\hfill {\Vert f\Vert }_{L^p({\mathbb{R}}^n)}\le {∥{\left[M,(,|,f,{|}^q,)\right]}^{\frac{1}{q}}∥}_{L^p({\mathbb{R}}^n)}\sim {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)},\end{array}$$

Graph

which implies that $f\in L^p({\mathbb{R}}^n)$ and hence ${\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)\subset L^p({\mathbb{R}}^n)$ .

Next, we prove $L^p({\mathbb{R}}^n)\subset {\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)$ . To this end, let $f\in L^p({\mathbb{R}}^n)$ . Then, by (3.1) again, the assumption $q<p$ , and the boundedness of the Hardy–Littlewood maximal operator M on $L^{p/q}({\mathbb{R}}^n)$ , we find that

3.3 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)}& \sim {∥M,(,|,f,{|}^q,)∥}_{L^{p/q}({\mathbb{R}}^n)}^{\frac{1}{p}}\lesssim {∥{|f|}^q∥}_{L^{p/q}({\mathbb{R}}^n)}^{\frac{1}{p}}\hfill \\ \hfill & ={\Vert f\Vert }_{L^p({\mathbb{R}}^n)}<\infty ,\hfill \end{array}$$

Graph

which further implies that $f\in {\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)$ and hence $L^p({\mathbb{R}}^n)\subset {\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)$ . Therefore, from this, (3.2), and (3.3), we deduce that $L^p({\mathbb{R}}^n)={\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)$ with equivalent quasi-norms, which completes the proof of Proposition 3.1. $\square$

Remark 3.2

Let $q\in (0,\infty )$ . Then, applying Proposition 3.1 with $p=\infty$ , we obtain

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,\infty }^{\infty ,\infty }({\mathbb{R}}^n)=L^{\infty }({\mathbb{R}}^n).\end{array}$$

Graph

On the other hand, by Remark 2.2(i) [see also Proposition 3.6(iii) below], we find that

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,\infty }^{\infty ,\infty }({\mathbb{R}}^n)={\mathcal{M}}_q^{\infty }({\mathbb{R}}^n).\end{array}$$

Graph

Therefore, when $p=\infty$ , Proposition 3.1 goes back to the well-known coincidence between the Morrey space ${\mathcal{M}}_q^{\infty }({\mathbb{R}}^n)$ and the Lebesgue space $L^{\infty }({\mathbb{R}}^n)$ (see, for instance, [[42], p. 13]).

Relations with (Besov–)Bourgain–Morrey spaces

In this subsection, we explore the (proper) embedding relations between TLBM spaces and Besov–Bourgain–Morrey spaces. To do this, we first recall the following concept of Besov–Bourgain–Morrey spaces introduced in [[50], Definition 1.2].

Definition 3.3

Let $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ . The Besov–Bourgain–Morrey space $\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is defined to be the set of all the $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ such that

$$\begin{array}{c}\hfill {\Vert f\Vert }_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}:={\left\{\sum _{\nu \in \mathbb{Z}},{\left[\sum _{m\in {\mathbb{Z}}^n},{\left\{|,Q_{\nu ,m},{|}^{\frac{1}{p}-\frac{1}{q}},{∥f,{\mathbf{1}}_{Q_{\nu ,m}}∥}_{L^q({\mathbb{R}}^n)}\right\}}^r\right]}^{\frac{\tau }{r}}\right\}}^{\frac{1}{\tau }},\end{array}$$

Graph

with the usual modifications made when $q=\infty$ , $r=\infty$ , or $\tau =\infty$ , is finite.

Remark 3.4

• As was pointed out in [[50], Remark 1.3], the assumptions that $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ in Definition 3.3 are reasonable;
• Observe that, in Definition 3.3, if $\tau =r$ , then
• MB˙q,rp,r(Rn)=Mq,rp(Rn).

Graph

In order to investigate the relation between ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and $\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , we require the following equivalent quasi-norm of Besov–Bourgain–Morrey spaces, which is just [[50], Theorem 2.9].

Lemma 3.5

Let $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ and, for any $h\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ , let

$$\begin{array}{c}\hfill {[h]}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}:={\left[{\int }_0^{\infty },{\left\{{\int }_{{\mathbb{R}}^n},{\left[{|B(y,t)|}^{\frac{1}{p}-\frac{1}{q}-\frac{1}{r}},{∥h,{\mathbf{1}}_{B(y,t)}∥}_{L^q({\mathbb{R}}^n)}\right]}^r,,d,y\right\}}^{\frac{\tau }{r}},,\frac{\mathit{dt}}{t}\right]}^{\frac{1}{\tau }}\end{array}$$

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with the usual modifications made when $q=\infty$ , $r=\infty$ , or $\tau =\infty$ . Then $f\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ if and only if $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ and ${[f]}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}<\infty$ . Moreover, there exist two positive constants $C_1$ and $C_2$ such that, for any $f\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill C_1{\Vert f\Vert }_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\le {[f]}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\le C_2{\Vert f\Vert }_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\end{array}$$

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Applying this lemma and the Minkowski integral inequality, we immediately conclude the following embedding relations between TLBM spaces and Besov–Bourgain–Morrey spaces; we omit the details.

Proposition 3.6

Let $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ . Then,

• when $\tau \in (0,r]$ , $\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)\hookrightarrow {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ;
• when $\tau \in [r,\infty ]$ , ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)\hookrightarrow \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ;
• MF˙q,rp,r(Rn)=MB˙q,rp,r(Rn)=Mq,rp(Rn) with equivalent quasi-norms.

When all the exponents are finite, the above inclusions (i) and (ii) are proper, that is, the following conclusions hold true.

Proposition 3.7

Let $0<q<p<r<\infty$ and $\tau \in (0,\infty )$ .

• If $\tau \in (0,r)$ , then $\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)⫋{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .
• If $\tau \in (r,\infty )$ , then ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)⫋\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

To prove this proposition, we require some preliminary lemmas. First, we give both the dilation invariance and the orthogonality invariance of Besov–Bourgain–Morrey spaces as follows.

Lemma 3.8

Let $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ . Then

• there exist two positive constants $C_1$ and $C_2$ such that, for any $f\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and $\gamma \in (0,\infty )$ ,
• C1γ-np‖f‖MB˙q,rp,τ(Rn)≤‖f(γ·)‖MB˙q,rp,τ(Rn)≤C2γ-np‖f‖MB˙q,rp,τ(Rn);

Graph

• there exist two positive constants $C_1$ and $C_2$ such that, for any $f\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and any $n\times n$ orthogonal matrix $\mathcal{O}$ ,
• C1‖f‖MB˙q,rp,τ(Rn)≤‖f(O·)‖MB˙q,rp,τ(Rn)≤C2‖f‖MB˙q,rp,τ(Rn).

Graph

Proof

(i) is just [[50], Proposition 2.11]. Thus, we only need to show (ii) here. Indeed, by the definition of ${[·]}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}$ and the change of variables, we have, for any $f\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and any $n\times n$ orthogonal matrix $\mathcal{O}$ ,

$$\begin{array}{c}\hfill {[f(\mathcal{O}·)]}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}={[f]}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},\end{array}$$

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which, together with Lemma 3.5, further implies that

$$\begin{array}{cc}\hfill {\Vert f(\mathcal{O}·)\Vert }_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \sim {[f(\mathcal{O}·)]}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}={[f]}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\hfill \\ \hfill & \sim {\Vert f\Vert }_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\hfill \end{array}$$

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This finishes the proof of (ii) and hence Lemma 3.8. $\square$

In what follows, for any $k\in \mathbb{N}$ , let ${\mathrm{\Gamma }}_k:=B(0,2^{-k+1})\backslash B(0,2^{-k}).$ Then the following specific function essentially characterizes the exponent $\tau$ of $\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , which plays an important role in the proof of Proposition 3.7.

Lemma 3.9

Let $0<q<p<r<\infty$ and $a,\tau \in (0,\infty )$ and let

$$\begin{array}{c}\hfill f_a^{(p)}:=\sum _{k\in \mathbb{N}}{2}^{\frac{\mathit{kn}}{p}}{k}^{-a}{\mathbf{1}}_{{\mathrm{\Gamma }}_k}.\end{array}$$

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Then $f_a^{(p)}\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ if and only if $a\tau \in (1,\infty )$ .

In what follows, let $l:=\{x=(x_1,\cdots ,x_n)\in {\mathbb{R}}^n:x_1=\cdots =x_n\}$ and, for any $x=(x_1,...,x_n)\in {\mathbb{R}}^n$ , let

3.4 $$\begin{array}{c}\hfill \tilde{x}:=\frac{1}{n}\sum _{i=1}^n{x}_i\end{array}$$

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and ${|x|}_{\infty }:={max}_{i\in \{1,\cdots ,n\}}|x_i|$ . To prove the last lemma, we need the following geometrical property about cubes.

Lemma 3.10

Let $\alpha :=\frac{15}{16}{n}^{\frac{1}{2}}$ , $\beta :=\frac{1}{30}{n}^{-\frac{1}{2}}$ , and

$$\begin{array}{cc}\hfill E:& =\{x=(x_1,...,x_n)\in {\mathbb{R}}^n:x_1\ge 0,|x|<\alpha ,\hfill \\ \hfill & \text{and}|x-(\tilde{x},...,\tilde{x})|\le \beta |x|\},\hfill \end{array}$$

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where $\tilde{x}$ is the same as in (3.4). For any $k\in \mathbb{N}$ , let

3.5 $$\begin{array}{c}\hfill R_{k,(1,\cdots ,1)}:=Q\left(\left(\frac{9}{2^{k+3}},,,...,,,\frac{9}{2^{k+3}}\right),,,\frac{3}{2^{k+2}}\right).\end{array}$$

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Then one has

$$\begin{array}{c}\hfill E\subset \left[\bigcup _{k\in \mathbb{N}},Q_{k,(1,...,1)}\right]\cup \left[\bigcup _{k\in \mathbb{N}},R_{k,(1,...,1)}\right]\end{array}$$

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(see Fig. 1 above for the case that $n=2)$ .

Graph: Fig. 1This figure reflects the result of Lemma 3.10 when n=2 , where A1:=(α,0) , A2:=(α/2,0) , B1:=(0,α) , and B2:=(0,α/2)

Proof

For any $k\in \mathbb{N}$ , let

$$\begin{array}{cc}\hfill E_k:& =\{x=(x_1,...,x_n)\in {\mathbb{R}}^n:x_1\ge 0,2^{-k}\alpha \le |x|<2^{-k+1}\alpha ,\hfill \\ \hfill & \left(,\text{and},,|,x,-,(,\tilde{x},,,...,,,\tilde{x},),|,\le ,\beta ,|,x,|\right\}.\hfill \end{array}$$

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Then $E={\bigcup }_{k\in \mathbb{N}}{E}_k$ . This implies that, to show the present lemma, it suffices to prove that, for any $k\in \mathbb{N}$ ,

3.6 $$\begin{array}{c}\hfill E_k\subset Q_{k,(1,\cdots ,1)}\cup R_{k,(1,\cdots ,1)}.\end{array}$$

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Observe that, for any $k\in \mathbb{N}$ , $E_k=2^{-k+1}{E}_1:=\{2^{-k+1}x:x\in E_1\}$ . Combining this and the homogeneity of $|·|$ , we find that, in order to show (3.6), we only need to prove (3.6) with $k=1$ . To do this, let $x\in E_1$ . Then, via an obvious geometrical observation, we have $\tilde{x}\in [\frac{7}{16},\frac{15}{16})$ , where $\tilde{x}$ is defined in (3.4). Next, we show (3.6) with $k=1$ by considering the following two cases on $\tilde{x}$ .

Case (i) $\tilde{x}\in [\frac{7}{16},\frac{21}{32})$ . In this case, from ${|y|}_{\infty }\le |y|$ for any $y\in {\mathbb{R}}^n$ , $|x|<\alpha$ , and both $\alpha =\frac{15}{16}{n}^{\frac{1}{2}}$ and $\beta =\frac{1}{30}{n}^{-\frac{1}{2}}$ (hence $\alpha \beta =\frac{1}{32}<\frac{1}{16}$ ), we deduce that

3.7 $$\begin{array}{cc}\hfill {\left|x,-,\left(\frac{9}{16},,,...,,,\frac{9}{16}\right)\right|}_{\infty }& \le {\left|x,-,\left(\tilde{x},,,...,,,\tilde{x}\right)\right|}_{\infty }+{\left|\left(\tilde{x},,,...,,,\tilde{x}\right),-,\left(\frac{9}{16},,,...,,,\frac{9}{16}\right)\right|}_{\infty }\hfill \\ \hfill & \le \left|x,-,\left(\tilde{x},,,...,,,\tilde{x}\right)\right|+{\left|\left(\tilde{x},,,...,,,\tilde{x}\right),-,\left(\frac{9}{16},,,...,,,\frac{9}{16}\right)\right|}_{\infty }\hfill \\ \hfill & <\beta |x|+\frac{1}{8}<\alpha \beta +\frac{1}{8}<\frac{3}{16},\hfill \end{array}$$

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which further implies that $x\in R_{1,(1,\cdots ,1)}$ and hence completes the proof of (3.6) with $k=1$ in this case.

Case (ii) $\tilde{x}\in [\frac{21}{32},\frac{15}{16})$ . In this case, similarly to (3.7), we have

$$\begin{array}{cc}\hfill {\left|x,-,\left(\frac{3}{4},,,...,,,\frac{3}{4}\right)\right|}_{\infty }& \le {\left|x,-,\left(\tilde{x},,,...,,,\tilde{x}\right)\right|}_{\infty }+{\left|\left(\tilde{x},,,...,,,\tilde{x}\right),-,\left(\frac{3}{4},,,...,,,\frac{3}{4}\right)\right|}_{\infty }\hfill \\ \hfill & \le \left|x,-,\left(\tilde{x},,,...,,,\tilde{x}\right)\right|+{\left|\left(\tilde{x},,,...,,,\tilde{x}\right),-,\left(\frac{3}{4},,,...,,,\frac{3}{4}\right)\right|}_{\infty }\hfill \\ \hfill & <\beta |x|+\frac{3}{16}<\alpha \beta +\frac{3}{16}<\frac{1}{4}.\hfill \end{array}$$

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This then implies that $x\in Q_{1,(1,\cdots ,1)}$ and hence finishes the proof of (3.6) with $k=1$ . Thus, (3.6) holds true for any $k\in \mathbb{N}$ , which further implies that

$$\begin{array}{c}\hfill E\subset \bigcup _{k\in \mathbb{N}}{E}_k\subset \left[\bigcup _{k\in \mathbb{N}},Q_{k,(1,...,1)}\right]\cup \left[\bigcup _{k\in \mathbb{N}},R_{k,(1,...,1)}\right].\end{array}$$

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This finishes the proof of Lemma 3.10. $\square$

Applying this lemma and both the dilation invariance and the orthogonality invariance of Besov–Bourgain–Morrey spaces, we now show Lemma 3.9.

Proof of Lemma 3.9

Let

$$\begin{array}{c}\hfill g_a^{(p)}:=\sum _{k\in \mathbb{N}}{2}^{\frac{\mathit{kn}}{p}}{k}^{-a}{\mathbf{1}}_{Q_{k,(1,\cdots ,1)}}.\end{array}$$

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We first claim that

3.8 $$\begin{array}{c}\hfill {∥f_a^{(p)}∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\sim {∥g_a^{(p)}∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\end{array}$$

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Indeed, observe that, for any $x\in {\mathbb{R}}^n$ ,

$$\begin{array}{c}\hfill g_a^{(p)}(x)\le f_a^{(p)}\left(n^{-\frac{1}{2}},x\right).\end{array}$$

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This, combined with Lemma 3.8(i), further implies that

3.9 $$\begin{array}{c}\hfill {∥g_a^{(p)}∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\le {∥f_a^{(p)},\left(n^{-\frac{1}{2}},·\right)∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\sim {∥f_a^{(p)}∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\end{array}$$

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Conversely, notice that, for any $x\in {\mathbb{R}}^n$ ,

$$\begin{array}{cc}\hfill g_a^{(p)}\left(\frac{4}{3},x\right)& =\sum _{k\in \mathbb{N}}{2}^{\frac{\mathit{kn}}{p}}{k}^{-a}{\mathbf{1}}_{Q_{k,(1,\cdots ,1)}}\left(\frac{4}{3},x\right)\hfill \\ \hfill & =\sum _{k\in \mathbb{N}}{2}^{\frac{\mathit{kn}}{p}}{k}^{-a}{\mathbf{1}}_{R_{k,(1,\cdots ,1)}}\left(x\right),\hfill \end{array}$$

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where $R_{k,(1,...,1)}$ is the same as in (3.5). Therefore, from Lemma 3.10, we deduce that there exist two positive constants $\alpha$ and $\beta$ and a cone

$$\begin{array}{cc}\hfill E:& =\{x=(x_1,...,x_n)\in {\mathbb{R}}^n:x_1\ge 0,|x|<\alpha ,\hfill \\ \hfill & \text{and}|x-(\tilde{x},...,\tilde{x})|\le \beta |x|\}\hfill \end{array}$$

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such that

3.10 $$\begin{array}{cc}\hfill E& \subset \left[\bigcup _{k\in \mathbb{N}},Q_{k,(1,...,1)}\right]\cup \left[\bigcup _{k\in \mathbb{N}},R_{k,(1,...,1)}\right]\hfill \\ \hfill & =\mathrm{supp}\left(g_a^{(p)},\left(\frac{4}{3},·\right)\right)\cup \mathrm{supp}\left(g_a^{(p)}\right).\hfill \end{array}$$

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On the other hand, via the compactness of the unit sphere of ${\mathbb{R}}^n$ , we find that there exists an $N\in \mathbb{N}$ and a finite sequence ${\{{\mathcal{O}}_j\}}_{j=0}^N$ of orthogonal matrices such that ${\mathcal{O}}_0=I_n$ and

$$\begin{array}{c}\hfill B(\mathbf{0},\alpha )=\bigcup _{j=0}^N\left({\mathcal{O}}_j,E\right),\end{array}$$

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where $I_n$ denotes the $n\times n$ identity matrix. This, together with (3.10), further implies that, for any $x\in {\mathbb{R}}^n$ ,

$$\begin{array}{c}\hfill f_a^{(p)}\left({\alpha }^{-1},x\right)\le \sum _{j=0}^N\left[g_a^{(p)},\left({\mathcal{O}}_j,\frac{4}{3},x\right),+,g_a^{(p)},\left({\mathcal{O}}_j,x\right)\right].\end{array}$$

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From this and both (i) and (ii) of Lemma 3.8, it follows that

$$\begin{array}{cc}\hfill {∥f_a^{(p)}∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \sim {∥f_a^{(p)},\left({\alpha }^{-1},·\right)∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\hfill \\ \hfill & \le {∥\sum _{j=0}^N,g_a^{(p)},\left({\mathcal{O}}_j,\frac{4}{3},·\right),+,g_a^{(p)},\left({\mathcal{O}}_j,·\right)∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\hfill \\ \hfill & \lesssim \sum _{j=0}^N\left[{∥g_a^{(p)},\left({\mathcal{O}}_j,\frac{4}{3},·\right)∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},+,{∥g_a^{(p)},\left({\mathcal{O}}_j,·\right)∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\right]\hfill \\ \hfill & \sim (N+1)\left[{∥g_a^{(p)},\left(\frac{4}{3},·\right)∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},+,{∥g_a^{(p)}∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\right]\hfill \\ \hfill & \sim {∥g_a^{(p)}∥}_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},\hfill \end{array}$$

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which, combined with (3.9), further implies that $\Vert f_a^{(p)}{\Vert }_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\sim {\Vert g_a^{(p)}\Vert }_{\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}$ and hence the above claim (3.8) holds true.

Next, by [[50], Lemma 2.17(ii)], we find that $g_a^{(p)}\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ if and only of $a\tau \in (1,\infty )$ . Combining this and the above claim (3.8), we further conclude that $f_a^{(p)}\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ if and only if $a\tau \in (1,\infty )$ , which completes the proof of Lemma 3.9. $\square$

In addition, the function $f_a^{(p)}$ defined as in Lemma 3.9 characterizes the exponent r of TLBM spaces ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ as follows, which also plays an essential tool in the proof of Proposition 3.7.

Lemma 3.11

Let $0<q<p<r<\infty$ and $a,\tau \in (0,\infty )$ . Assume that $f_a^{(p)}$ is the same as in Lemma 3.9. Then $f_a^{(p)}\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ if and only if $ar\in (1,\infty )$ .

Proof

For any $y\in {\mathbb{R}}^n\backslash \{\mathbf{0}\}$ , let $s:=-{log}_2|y|$ . Then, from Theorem 2.3(i) and the change of variables, we deduce that

3.11 $$\begin{array}{cc}& {∥f_a^{(p)}∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}^r\hfill \\ \hfill & \sim {∥\left|f_a^{(p)}\right|∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}^r\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^n}{\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu (\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau },{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(y,2^{\nu })\right|\right]}^{\frac{\tau }{q}}\right\}}^{\frac{r}{\tau }}dy\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^n}\left\{\sum _{\nu \in \mathbb{Z}},2^{-\nu (\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }\right)\hfill \\ \hfill & \times {\left({\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,\left(|y|,{\mathbf{e}}_1,,,2^{-\nu }\right)\right|\right]}^{\frac{\tau }{q}}\right\}}^{\frac{r}{\tau }}dy\hfill \\ \hfill & \sim {\int }_{\mathbb{R}}{2}^{-sn}\left\{\sum _{\nu \in \mathbb{Z}},2^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }\right)\hfill \\ \hfill & \times {\left({\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,\left(2^{-s},{\mathbf{e}}_1,,,2^{-\nu }\right)\right|\right]}^{\frac{\tau }{q}}\right\}}^{\frac{r}{\tau }}ds\hfill \\ \hfill & =:{\int }_{\mathbb{R}}{2}^{-sn}{\left[G,(,s,)\right]}^{\frac{r}{\tau }}ds,\hfill \end{array}$$

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here and thereafter, ${\mathbf{e}}_1:=(1,0,...,0)$ .

Now, we prove the sufficiency. For this purpose, assume $ar\in (1,\infty )$ . Observe that, for any $k\in \mathbb{N}$ , $s\in \mathbb{R}$ , and $\nu \in \mathbb{Z}$ ,

3.12 $$\begin{array}{c}\hfill \left|{\mathrm{\Gamma }}_k,\cap ,B,(2^{-s}{\mathbf{e}}_1,2^{-\nu })\right|=0\end{array}$$

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if and only if $B(2^{-s}{\mathbf{e}}_1,2^{-\nu })\subset B(\mathbf{0},2^{-k})$ or $B(2^{-s}{\mathbf{e}}_1,2^{-\nu })\subset {\mathbb{R}}^n\backslash B(\mathbf{0},2^{-k+1})$ , which then further implies that (3.12) holds true if and only if $2^{-s}+2^{-\nu }\le 2^{-k}$ or $2^{-s}-2^{-\nu }\ge 2^{-k+1}$ . Therefore, for any $k\in \mathbb{N}$ , $s\in \mathbb{R}$ , and $\nu \in \mathbb{Z}$ ,

$$\begin{array}{c}\hfill \left|{\mathrm{\Gamma }}_k,\cap ,B,(2^{-s}{\mathbf{e}}_1,2^{-\nu })\right|\ne 0\end{array}$$

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if and only if

3.13 $$\begin{array}{c}\hfill 2^{-k}<2^{-s}+2^{-\nu }\end{array}$$

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and

3.14 $$\begin{array}{c}\hfill 2^{-k+1}>2^{-s}-2^{-\nu }.\end{array}$$

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We next estimate (3.11) by considering the following three cases on s.

Case (i) $s\in (-\infty ,-1)$ . In this case, $2^{-s}>2$ . Using this and (3.14), we find that

$$\begin{array}{c}\hfill 2^{-\nu }>2^{-s}-1>1\end{array}$$

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and hence $\nu \le -1$ . If $\nu >s$ , then $2^{-s}\ge 2^{-\nu }+1$ . On the other hand, from (3.14) again, we infer that

$$\begin{array}{c}\hfill 2^{-\nu }>2^{-s}-2^{-k+1}\ge 2^{-s}-1,\end{array}$$

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which contradicts the fact $2^{-s}\ge 2^{-\nu }+1$ . Therefore, $\nu \le s$ and hence $\nu \le \lfloor s\rfloor$ . In this case, applying the assumption $0<q<p<r$ , we conclude that

3.15 $$\begin{array}{cc}\hfill G(s)& =\sum _{\nu =-\infty }^{\lfloor s\rfloor }{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(2^{-s}{\mathbf{e}}_1,2^{-\nu })\right|\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \lesssim \sum _{\nu =-\infty }^{\lfloor s\rfloor }{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},2^{-kn}\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \sim \sum _{\nu =-\infty }^{\lfloor s\rfloor }{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }\sim 2^{-\lfloor s\rfloor n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }.\hfill \end{array}$$

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This, together with $\lfloor s\rfloor \le s$ and $0<q<p$ , further implies that

3.16 $$\begin{array}{cc}\hfill {\int }_{-\infty }^{-1}{2}^{-sn}{\left[G,(,s,)\right]}^{\frac{r}{\tau }}ds& \lesssim {\int }_{-\infty }^{-1}{\left[2^{-\lfloor s\rfloor n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }\right]}^{\frac{r}{\tau }}{2}^{-sn}ds\hfill \\ \hfill & \le {\int }_{-\infty }^{-1}{2}^{-\lfloor s\rfloor n(\frac{1}{p}-\frac{1}{q})r}ds<\infty ,\hfill \end{array}$$

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which is the desired estimate in this case.

Case (ii) $s\in [-1,1]$ . In this case, $2^{-s}\in [2^{-1},2]$ . If $\nu \ge 2$ , then $2^{-s}-2^{-\nu }\ge 2^{-2}$ . From this and (3.14), we deduce that $2^{-k+1}>2^{-2}$ and hence $k\le 2$ . In this case, by this and the assumption $0<q<p<r<\infty$ , we have

3.17 $$\begin{array}{cc}\hfill G(s)& =\sum _{\nu =-\infty }^1{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(2^{-s}{\mathbf{e}}_1,2^{-\nu })\right|\right]}^{\frac{\tau }{q}}+\sum _{\nu =2}^{\infty }\cdots \hfill \\ \hfill & \lesssim \sum _{\nu =-\infty }^1{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},2^{-nk}\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & +\sum _{\nu =2}^{\infty }{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{2}^{-\nu n\frac{\tau }{q}}{\left[\sum _{k=1}^2,2^{\frac{\mathit{knq}}{p}},k^{-aq}\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \lesssim \sum _{\nu =-\infty }^1{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }+\sum _{\nu =2}^{\infty }{2}^{-\nu n(\frac{1}{p}-\frac{1}{r})\tau }\lesssim 1,\hfill \end{array}$$

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which further implies that

3.18 $$\begin{array}{c}\hfill {\int }_{-1}^1{2}^{-sn}{\left[G,(,s,)\right]}^{\frac{r}{\tau }}ds\lesssim {\int }_{-1}^1{2}^{-sn}ds<\infty .\end{array}$$

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This is the desired estimate in this case.

Case (iii) $s\in (1,\infty )$ . In this case, $2^{-s}\in (0,2^{-1})$ . Notice that

3.19 $$\begin{array}{cc}\hfill G(s)& =\sum _{\nu =-\infty }^0{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(2^{-s}{\mathbf{e}}_1,2^{-\nu })\right|\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & +\sum _{\nu =1}^{\lfloor s\rfloor +1}\cdots +\sum _{\nu =\lfloor s\rfloor +2}^{\infty }\cdots \hfill \\ \hfill & =:{\text{I}}_1+{\text{I}}_2+{\text{I}}_3.\hfill \end{array}$$

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For ${\text{I}}_1$ , applying the assumption $0<q<p<r<\infty$ , we find that

3.20 $$\begin{array}{cc}\hfill {\text{I}}_1& \lesssim \sum _{\nu =-\infty }^0{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},2^{-kn}\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \lesssim \sum _{\nu =-\infty }^0{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }\lesssim 1.\hfill \end{array}$$

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Now, we deal with ${\text{I}}_2$ . Indeed, for any $\nu \in \mathbb{N}\cap [1,\lfloor s\rfloor +1]$ , $2^{-s}+2^{-\nu }\le 3·2^{-\nu }<2^{-\nu +2}$ . From this and (3.13), we deduce that $k\ge \nu -1$ . Thus, by the assumption $q<p$ , we obtain

3.21 $$\begin{array}{cc}\hfill {\text{I}}_2& =\sum _{\nu =1}^{\lfloor s\rfloor +1}{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k=max\{1,\nu -1\}}^{\infty },2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(2^{-s}{\mathbf{e}}_1,2^{-\nu })\right|\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \lesssim \sum _{\nu =1}^{\lfloor s\rfloor +1}{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k=max\{1,\nu -1\}}^{\infty },2^{\frac{\mathit{knq}}{p}},k^{-aq},2^{-kn}\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \sim \sum _{\nu =1}^{\lfloor s\rfloor +1}{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{2}^{\nu n(\frac{1}{p}-\frac{1}{q})\tau }{\nu }^{-a\tau }\sim 2^{\frac{n\lfloor s\rfloor \tau }{r}}{(\lfloor s\rfloor )}^{-a\tau },\hfill \end{array}$$

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which completes the estimation of ${\mathrm{I}}_2$ . Finally, for ${\text{I}}_3$ , observe that, for any $\nu \in \mathbb{Z}\cap [\lfloor s\rfloor +2,\infty )$ , $\nu \ge \lfloor s\rfloor +2>s+1$ and hence $2^{-s}-2^{-\nu }>2^{-s-1}$ . This, combined with (3.14), further implies that $2^{-k+1}>2^{-s-1}$ and hence $k\le \lfloor s\rfloor +2$ . In addition, by $\nu >s+1$ , we have $2^{-s}+2^{-\nu }<2^{-s+1}$ . Using this and (3.13), we obtain $k\ge \lfloor s\rfloor$ . From this, $k\le \lfloor s\rfloor +2$ , and the assumption $0<q<p<r$ again, it follows that

$$\begin{array}{cc}\hfill {\text{I}}_3& \lesssim \sum _{\nu =\lfloor s\rfloor +2}^{\infty }{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k=\lfloor s\rfloor }^{\lfloor s\rfloor +2},2^{\frac{\mathit{knq}}{p}},k^{-aq},2^{-n\nu }\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \lesssim {\left\{2^{\frac{\lfloor s\rfloor nq}{p}},{(\lfloor s\rfloor )}^{-aq}\right\}}^{\frac{\tau }{q}}\sum _{\nu =\lfloor s\rfloor +2}^{\infty }{2}^{-\nu n(\frac{1}{p}-\frac{1}{r})\tau }\sim 2^{\frac{n\lfloor s\rfloor \tau }{r}}{(\lfloor s\rfloor )}^{-a\tau },\hfill \end{array}$$

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which is the desired estimate of ${\text{I}}_3$ . Combining this estimate, (3.19), (3.20), and (3.21), we further obtain

3.22 $$\begin{array}{c}\hfill G(s)\lesssim 1+2^{\frac{n\lfloor s\rfloor \tau }{r}}{(\lfloor s\rfloor )}^{-a\tau }.\end{array}$$

Graph

Using this and the assumption $ar\in (1,\infty )$ , we find that

3.23 $$\begin{array}{cc}\hfill {\int }_1^{\infty }{2}^{-sn}{\left[G,(,s,)\right]}^{\frac{r}{\tau }}ds& \lesssim {\int }_1^{\infty }{2}^{-sn}{\left\{1,+,2^{\frac{n\lfloor s\rfloor \tau }{r}},{(\lfloor s\rfloor )}^{-a\tau }\right\}}^{\frac{r}{\tau }}ds\hfill \\ \hfill & \sim {\int }_1^{\infty }\left\{2^{-sn},+,2^{(\lfloor s\rfloor -s)n},{(\lfloor s\rfloor )}^{-ar}\right\}ds\hfill \\ \hfill & \sim {\int }_1^{\infty }\left(2^{-sn},+,s^{-ar}\right)ds<\infty .\hfill \end{array}$$

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This is the desired estimate in this case. Therefore, combining (3.11), (3.16), (3.18), and (3.23), we have $\Vert f_a^{(p)}{\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}^r<\infty$ and hence $f_a^{(p)}\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . This then finishes the proof of the sufficiency.

Next, we show the necessity. To do this, let $ar\in (-\infty ,1]$ . It suffices to prove $f_a^{(p)}\notin {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Indeed, for any $s\in {\bigcup }_{j=1}^{\infty }(j+1/3,j+2/3)$ , we have $2^{-s}{\mathbf{e}}_1\in {\mathrm{\Gamma }}_{\lfloor s\rfloor +1}$ . In addition, for any $\nu \in \mathbb{Z}\cap [\lfloor s\rfloor +3,\infty )$ , it holds true that

$$\begin{array}{c}\hfill 2^{-s}-2^{-\nu }\ge 2^{-\lfloor s\rfloor -1}\text{and}{2}^{-s}+2^{-\nu }\le 2^{-\lfloor s\rfloor },\end{array}$$

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which imply that $B(2^{-s}{\mathbf{e}}_1,2^{-\nu })\subset {\mathrm{\Gamma }}_{\lfloor s\rfloor +1}$ . From this and the assumption $0<q<p<r<\infty$ , we infer that

$$\begin{array}{cc}\hfill G(s)& \ge \sum _{\nu =\lfloor s\rfloor +3}^{\infty }{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(2^{-s}{\mathbf{e}}_1,2^{-\nu })\right|\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \gtrsim \sum _{\nu =\lfloor s\rfloor +3}^{\infty }{2}^{-\nu n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})\tau }{\left\{2^{\frac{\lfloor s\rfloor nq}{p}},{(\lfloor s\rfloor )}^{-aq},2^{-\nu n}\right\}}^{\frac{\tau }{q}}\sim 2^{\frac{\lfloor s\rfloor n\tau }{r}}{(\lfloor s\rfloor )}^{-a\tau }.\hfill \end{array}$$

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By this, (3.11), and the assumption $ar\in (-\infty ,1]$ , we conclude that

$$\begin{array}{cc}\hfill {∥f_a^{(p)}∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}^r& \sim {\int }_{\mathbb{R}}{2}^{-sn}{\left[G,(,s,)\right]}^{\frac{r}{\tau }}ds\hfill \\ \hfill & \ge {\int }_1^{\infty }{2}^{-sn}{[G(s)]}^{\frac{r}{\tau }}ds\ge \sum _{j\in \mathbb{N}}{\int }_{j+\frac{1}{3}}^{j+\frac{2}{3}}{2}^{-sn}{[G(s)]}^{\frac{r}{\tau }}ds\hfill \\ \hfill & \ge \sum _{j\in \mathbb{N}}{\int }_{j+\frac{1}{3}}^{j+\frac{2}{3}}{2}^{-sn}{\left\{2^{\frac{\lfloor s\rfloor n\tau }{r}},{(\lfloor s\rfloor )}^{-a\tau }\right\}}^{\frac{r}{\tau }}ds\hfill \\ \hfill & =\sum _{j\in \mathbb{N}}{\int }_{j+\frac{1}{3}}^{j+\frac{2}{3}}{(\lfloor s\rfloor )}^{-ar}{2}^{(\lfloor s\rfloor -s)n}ds\gtrsim \sum _{j\in \mathbb{N}}{j}^{-ar}=\infty ,\hfill \end{array}$$

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which further implies that $f_a^{(p)}\notin {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . This finishes the proof of the necessity and hence Lemma 3.11. $\square$

Based on above preparations, we now show Proposition 3.7.

Proof of Proposition 3.7

We first show (i). To this end, assume $\tau \in (0,r)$ and let $a\in (\frac{1}{r},\frac{1}{\tau })$ . Then $a\tau <1<ar,$ which, together with both Lemmas 3.9 and 3.11, further implies that

$$\begin{array}{c}\hfill f_a^{(p)}\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)\backslash \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n).\end{array}$$

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This then finishes the proof of (i).

For (ii), assume $\tau \in (r,\infty )$ and let $a\in (\frac{1}{\tau },\frac{1}{r})$ . Then, from Lemmas 3.9 and 3.11 again, it follows that $ar<1<a\tau$ and hence

$$\begin{array}{c}\hfill f_a^{(p)}\in \mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)\backslash {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n).\end{array}$$

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This implies that (ii) holds true and then finishes the proof of Proposition 3.7. $\square$

Relations with local and global Morrey spaces

This subsection is devoted to investigating the relation between TLBM spaces and local or global Morrey spaces. First, we give an equivalent quasi-norm of TLBM spaces via local Morrey spaces. Then, as an application, we show that the global Morrey space is just a special case of TLBM spaces. We now recall the following concepts of local and global Morrey spaces which are originally introduced in [[15]] (see also [[41], p. 214, Definition 36]).

Definition 3.12

Let $q,\tau \in (0,\infty ]$ , $\alpha \in \mathbb{R}$ , and $\xi \in {\mathbb{R}}^n$ .

• The local Morrey space $L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ is defined to be the set of all the $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ such that
• ‖f‖LMαq,τ,ξ(Rn):=∫0∞tατf1B(ξ,t)Lq(Rn)τdtt1τ<∞.

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• The global Morrey space $G{M}_{\alpha }^{q,\tau }({\mathbb{R}}^n)$ is defined to be the set of all the $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ such that
• ‖f‖GMαq,τ(Rn):=supξ∈Rn∫0∞tατf1B(ξ,t)Lq(Rn)τdtt1τ<∞.

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Remark 3.13

Let q, $\tau$ , $\alpha$ , and $\xi$ be the same as in Definition 3.12. We point out that, in some sense, $\xi$ is the center of the local Morrey space $L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ . Indeed, for any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ , by the change of variables and the fact that, for any $x\in {\mathbb{R}}^n$ , $|(2\xi -x)-\xi |=|x-\xi |$ , we obtain

$$\begin{array}{c}\hfill {∥f,(,2,\xi ,-,·,)∥}_{L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}={∥f∥}_{L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}.\end{array}$$

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Observe that x and $2\xi -x$ are symmetrical about $\xi$ . Therefore, in this sense, $\xi$ can be regarded as the center of $L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ .

Via local Morrey spaces, we have an equivalent quasi-norm of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ as follows.

Proposition 3.14

Let $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ . Then there exist two positive constants $C_1$ and $C_2$ such that, for any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill C_1{\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\le {\left[{\int }_{{\mathbb{R}}^n},{∥f∥}_{L{M}_{{\alpha }_{(p,q,r)}}^{q,\tau ,y}({\mathbb{R}}^n)}^r,,d,y\right]}^{\frac{1}{r}}\le C_2{\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},\end{array}$$

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here and thereafter,

3.24 $$\begin{array}{c}\hfill {\alpha }_{(p,q,r)}:=n\left(\frac{1}{p},-,\frac{1}{q},-,\frac{1}{r}\right).\end{array}$$

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Moreover, if $r=\infty$ , then

3.25 $$\begin{array}{c}\hfill G{M}_{{\alpha }_{(p,q)}}^{q,\tau }({\mathbb{R}}^n)\hookrightarrow {\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n),\end{array}$$

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here and thereafter,

3.26 $$\begin{array}{c}\hfill {\alpha }_{(p,q)}:=n\left(\frac{1}{p},-,\frac{1}{q}\right).\end{array}$$

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Proof

Notice that, for any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ ,

$$\begin{array}{cc}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \sim {\left\{{\int }_{{\mathbb{R}}^n},{\left[{\int }_0^{\infty },{\left\{t^{n(\frac{1}{p}-\frac{1}{q}-\frac{1}{r})},{∥f,{\mathbf{1}}_{B(y,t)}∥}_{L^q({\mathbb{R}}^n)}\right\}}^{\tau },,\frac{\mathit{dt}}{t}\right]}^{\frac{r}{\tau }},,d,y\right\}}^{\frac{1}{r}}\hfill \\ \hfill & ={\left[{\int }_{{\mathbb{R}}^n},{∥f∥}_{L{M}_{{\alpha }_{(p,q,r)}}^{q,\tau ,y}({\mathbb{R}}^n)}^r,,d,y\right]}^{\frac{1}{r}}.\hfill \end{array}$$

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Using this and letting $r=\infty$ , we further conclude that, for any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ ,

3.27 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)}& \sim \underset{y\in {\mathbb{R}}^n}{\mathrm{ess}\sup }{∥f∥}_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,y}({\mathbb{R}}^n)}\hfill \\ \hfill & \le \underset{y\in {\mathbb{R}}^n}{sup}{\Vert f\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,y}({\mathbb{R}}^n)}={\Vert f\Vert }_{G{M}_{{\alpha }_{(p,q)}}^{p,q}({\mathbb{R}}^n)},\hfill \end{array}$$

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which implies that (3.25) holds true and hence completes the proof of Proposition 3.14. $\square$

In particular, the following conclusion shows that the TLBM space ${\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)$ is just the global Morrey space.

Theorem 3.15

Let $0<q<p<\infty$ and $\tau \in (0,\infty )$ , or let $0<q\le p\le \infty$ and $\tau =\infty$ . Then

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)=G{M}_{{\alpha }_{(p,q)}}^{q,\tau }({\mathbb{R}}^n)\end{array}$$

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with equivalent quasi-norms, where ${\alpha }_{(p,q)}$ is the same as in (3.26).

Proof

We first consider the case $\tau =\infty$ . Indeed, in this case, by Proposition 3.6(iii), we have

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,\infty }^{p,\infty }({\mathbb{R}}^n)={\mathcal{M}}_{q,\infty }^p({\mathbb{R}}^n)={\mathcal{M}}_q^p({\mathbb{R}}^n).\end{array}$$

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On the other hand, from [[41], p. 214, Example 68], we infer that

$$\begin{array}{c}\hfill G{M}_{{\alpha }_{(p,q)}}^{q,\infty }({\mathbb{R}}^n)={\mathcal{M}}_q^p({\mathbb{R}}^n).\end{array}$$

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Thus, we have

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,\infty }^{p,\infty }({\mathbb{R}}^n)={\mathcal{M}}_q^p({\mathbb{R}}^n)=G{M}_{{\alpha }_{(p,q)}}^{q,\infty }({\mathbb{R}}^n).\end{array}$$

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This finishes the proof of the present theorem in this case.

In what follows, we consider the case $\tau \in (0,\infty )$ . In this case, due to (3.27), it suffices to show that, for any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {\Vert f\Vert }_{G{M}_{{\alpha }_{(p,q)}}^{q,\tau }({\mathbb{R}}^n)}\le \underset{y\in {\mathbb{R}}^n}{\mathrm{ess}\sup }{∥f∥}_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,y}({\mathbb{R}}^n)}.\end{array}$$

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We first claim that, if $|f|\lesssim {\mathbf{1}}_B$ for some ball B, then, for any $\xi \in {\mathbb{R}}^n$ ,

3.28 $$\begin{array}{c}\hfill {\Vert f\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,\eta }({\mathbb{R}}^n)}\to {\Vert f\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}\end{array}$$

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as $\eta \to \xi$ . To do this, fix an f and a ball B such that $|f|\lesssim {\mathbf{1}}_B$ and let $\xi \in {\mathbb{R}}^n$ . Then, for any $\eta \in {\mathbb{R}}^n$ ,

3.29 $$\begin{array}{cc}& \left|{\Vert f\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}^{\tau },-,{\Vert f\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,\eta }({\mathbb{R}}^n)}^{\tau }\right|\hfill \\ \hfill & \le {\int }_0^{\infty }{\left[t^{{\alpha }_{(p,q)}},\left|{∥f,{\mathbf{1}}_{B(\xi ,t)}∥}_{L^q({\mathbb{R}}^n)},-,{∥f,{\mathbf{1}}_{B(\eta ,t)}∥}_{L^q({\mathbb{R}}^n)}\right|\right]}^{\tau }\frac{\mathit{dt}}{t}\hfill \\ \hfill & ={\int }_0^1{\left[t^{{\alpha }_{(p,q)}},\left|{∥f,{\mathbf{1}}_{B(\xi ,t)}∥}_{L^q({\mathbb{R}}^n)},-,{∥f,{\mathbf{1}}_{B(\eta ,t)}∥}_{L^q({\mathbb{R}}^n)}\right|\right]}^{\tau }\frac{\mathit{dt}}{t}+{\int }_1^{\infty }\cdots \hfill \\ \hfill & =:I(\xi ,\eta )+II(\xi ,\eta ).\hfill \end{array}$$

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By $|f|\lesssim {\mathbf{1}}_B$ , we find that, for any $\eta \in {\mathbb{R}}^n$ and $t\in (0,\infty )$ ,

$$\begin{array}{c}\hfill \left|,{\left|f,{\mathbf{1}}_{B(\eta ,t)}\right|}^q,-,{\left|f,{\mathbf{1}}_{B(\xi ,t)}\right|}^q\right|\lesssim {\mathbf{1}}_B\in L^1({\mathbb{R}}^n)\end{array}$$

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with the implicit positive constant independent of both $\xi$ and $\eta$ . This, combined with the Lebesgue dominated convergence theorem and the fact that $|f{\mathbf{1}}_{B(\eta ,t)}|$ converges to $|f{\mathbf{1}}_{B(\xi ,t)}|$ almost everywhere in ${\mathbb{R}}^n$ as $\eta \to \xi$ , further implies that

3.30 $$\begin{array}{c}\hfill \underset{\eta \to \xi }{lim}{∥f,{\mathbf{1}}_{B(\eta ,t)}∥}_{L^q({\mathbb{R}}^n)}={∥f,{\mathbf{1}}_{B(\xi ,t)}∥}_{L^q({\mathbb{R}}^n)}.\end{array}$$

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Moreover, using the Hölder inequality and $|f|\lesssim {\mathbf{1}}_B$ , we have, for any $t\in (0,\infty )$ and $\eta \in {\mathbb{R}}^n$ ,

$$\begin{array}{cc}& \left|{∥f,{\mathbf{1}}_{B(\xi ,t)}∥}_{L^q({\mathbb{R}}^n)},-,{∥f,{\mathbf{1}}_{B(\eta ,t)}∥}_{L^q({\mathbb{R}}^n)}\right|\hfill \\ \hfill & \le {\Vert f\Vert }_{L^{\infty }({\mathbb{R}}^n)}\left[{|B(\xi ,t)|}^{\frac{1}{q}},+,{|B(\eta ,t)|}^{\frac{1}{q}}\right]\lesssim t^{\frac{n}{q}},\hfill \end{array}$$

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which further implies that

3.31 $$\begin{array}{c}\hfill I(\xi ,\eta )\lesssim {\left\{{\int }_0^1,{\left[t^{{\alpha }_{(p,q)}},t^{\frac{n}{q}}\right]}^{\tau },,\frac{\mathit{dt}}{t}\right\}}^{\frac{1}{\tau }}={\left\{{\int }_0^1,t^{\frac{n\tau }{p}},,\frac{\mathit{dt}}{t}\right\}}^{\frac{1}{\tau }}<\infty .\end{array}$$

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On the other hand, from $|f|\lesssim {\mathbf{1}}_B$ again, we deduce that, for any $t\in (0,\infty )$ and $\eta \in {\mathbb{R}}^n$ ,

$$\begin{array}{c}\hfill \left|{∥f,{\mathbf{1}}_{B(\xi ,t)}∥}_{L^q({\mathbb{R}}^n)},-,{∥f,{\mathbf{1}}_{B(\eta ,t)}∥}_{L^q({\mathbb{R}}^n)}\right|\lesssim {\Vert f\Vert }_{L^q({\mathbb{R}}^n)}\lesssim 1,\end{array}$$

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which further implies that

$$\begin{array}{c}\hfill II(\xi ,\eta )\lesssim {\left\{{\int }_1^{\infty },t^{{\alpha }_{(p,q)}\tau },,\frac{\mathit{dt}}{t}\right\}}^{\frac{1}{\tau }}<\infty .\end{array}$$

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Using this, (3.31), (3.30), and the Lebesgue dominated convergence theorem, we conclude that $I(\xi ,\eta )\to 0$ and $II(\xi ,\eta )\to 0$ as $\eta \to \xi$ . This, together with (3.29), further implies that the above claim (3.28) holds true.

Next, for any $k\in \mathbb{N}$ and $x\in {\mathbb{R}}^n$ , let

$$\begin{array}{c}\hfill f_k(x):=\left\{\begin{array}{cc}f(x){\mathbf{1}}_{B(\mathbf{0},k)}(x)\hfill & \text{when}|f(x)|\le k;\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}\right)\end{array}$$

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Then, obviously, for any $k\in \mathbb{N}$ , $|f_k|\le k{\mathbf{1}}_{B(\mathbf{0},k)}$ and $|f_k|\uparrow |f|$ as $k\to \infty$ . Thus, from the monotone convergence theorem, we deduce that, for any $\xi \in {\mathbb{R}}^n$ ,

3.32 $$\begin{array}{c}\hfill \Vert f_k{\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}{\uparrow \Vert f\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}\end{array}$$

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as $k\to \infty$ . In addition, for any $ϵ\in (0,\Vert f{\Vert }_{G{M}_{{\alpha }_{(p,q)}}^{p,\tau }({\mathbb{R}}^n)})$ , by the definition of ${\Vert ·\Vert }_{G{M}_{{\alpha }_{(p,q)}}^{p,\tau }({\mathbb{R}}^n)}$ , we conclude that there exists a ${\xi }_0$ such that $ϵ\in (0,\Vert f{\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{p,\tau ,{\xi }_0}({\mathbb{R}}^n)})$ . This, combined with (3.32), further implies that there exists a $k_0\in \mathbb{N}$ such that

3.33 $$\begin{array}{c}\hfill ϵ\in \left(0,\Vert ,f_{k_0},{\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{p,\tau ,{\xi }_0}({\mathbb{R}}^n)}\right).\end{array}$$

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From the definition of the essential supremum, we deduce that there exists a set E with $|E|=0$ such that

3.34 $$\begin{array}{c}\hfill \underset{y\in {\mathbb{R}}^n}{\mathrm{ess}\sup }{∥f∥}_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,y}({\mathbb{R}}^n)}=\underset{{\mathbb{R}}^n\backslash E}{sup}{\Vert f\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}.\end{array}$$

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Moreover, notice that $|f_{k_0}(·+{\xi }_0)|\le k_0{\mathbf{1}}_{B(-{\xi }_0,k_0)}$ . Therefore, applying the above claim (3.28) with $f:=f_{k_0}(·+{\xi }_0)$ , (3.33), and the density of ${\mathbb{R}}^n\backslash E$ in ${\mathbb{R}}^n$ , we find that there exists an ${\eta }_0\in {\mathbb{R}}^n\backslash E$ such that $ϵ\in (0,\Vert f_{k_0}{\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,{\eta }_0}({\mathbb{R}}^n)})$ . This, together with $|f_{k_0}|\le |f|$ and (3.34), further implies that

$$\begin{array}{c}\hfill ϵ<\Vert f_{k_0}{\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,{\eta }_0}({\mathbb{R}}^n)}\le {\Vert f\Vert }_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,{\eta }_0}({\mathbb{R}}^n)}\le \underset{y\in {\mathbb{R}}^n}{\mathrm{ess}\sup }{∥f∥}_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,y}({\mathbb{R}}^n)}.\end{array}$$

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Letting $ϵ\to {\Vert f\Vert }_{G{M}_{{\alpha }_{(p,q)}}^{p,\tau }({\mathbb{R}}^n)}$ , we obtain

$$\begin{array}{c}\hfill {\Vert f\Vert }_{G{M}_{{\alpha }_{(p,q)}}^{p,\tau }({\mathbb{R}}^n)}\le \underset{y\in {\mathbb{R}}^n}{\mathrm{ess}\sup }{∥f∥}_{L{M}_{{\alpha }_{(p,q)}}^{q,\tau ,y}({\mathbb{R}}^n)},\end{array}$$

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which completes the proof of Theorem 3.15. $\square$

Remark 3.16

• Combining Theorem 3.15 and Propositions 3.1 and 3.6(iii), we conclude that ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is a new space different from $\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , when $0<q<p<r<\infty$ and $\tau \in (0,\infty ]$ , and
• MF˙q,rp,τ(Rn)=GMn(1p-1q)q,τ(Rn)if0

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• Applying these and Proposition 3.6(iii) again, we conclude that the TLBM space ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is indeed a new space and a bridge connecting Bourgain–Morrey spaces and global Morrey spaces.
• Let $0<q<p=r<\infty$ and $\tau \in (0,\infty )$ . Then, by Theorem 2.9 and Proposition 3.1, we find that
• MF˙q,pp,τ(Rn)={0}⫋Lp(Rn)=MF˙q,pp,∞(Rn).

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• Thus, the index $\tau$ indeed plays a role in ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

Diversity of MF˙q,rp,τ(Rn)

The main target of this subsection is to show the following diversity of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ based on its relations with both (Besov–)Bourgain–Morrey spaces and global Morrey spaces.

Theorem 3.17

Let $0<q_i<p_i<r_i\le \infty$ for any $i\in \{1,2\}$ and let ${\tau }_1,{\tau }_2\in (0,\infty )$ .

• If ${\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)={\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)$ , then $p_1=p_2$ , $q_1=q_2$ , and $r_1=r_2$ .
• MF˙q1,∞p1,τ1(Rn)=MF˙q2,∞p2,τ2(Rn) if and only if $p_1=p_2$ , $q_1=q_2$ , and ${\tau }_1={\tau }_2$ .

Remark 3.18

Theorem 3.17 gives the diversity of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ on the exponents p, q, and r and, when $r=\infty$ , also on $\tau$ . However, we should point out that, when $r\in (0,\infty )$ , the diversity of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ on $\tau$ is still unclear.

To prove Theorem 3.17, we need the following auxiliary lemma of Besov–Bourgain–Morrey spaces, which is just [[50], Theorem 2.22(i)].

Lemma 3.19

Let $0<q_1<p_1<r_1\le \infty$ , $0<q_2<p_2<r_2\le \infty$ , and ${\tau }_1,{\tau }_2\in (0,\infty )$ . Then $\mathcal{M}{\dot{B}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)\subset \mathcal{M}{\dot{B}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)$ if and only if $q_1\ge q_2$ , $p_1=p_2$ , $r_1\le r_2$ , and ${\tau }_1\le {\tau }_2$ .

We also require the following technical conclusion.

Proposition 3.20

Let $0<q<p<\infty$ and $a,\tau \in (0,\infty )$ . Assume that $f_a^{(p)}$ is the same as in Lemma 3.9. Then $f_a^{(p)}\in {\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)$ if and only if $a\tau \in (1,\infty )$ .

Proof

Repeating the arguments similar to those used in the estimations of both (2.2) and (2.3) with the norm ${\Vert ·\Vert }_{L^r({\mathbb{R}}^n)}$ therein replaced by ${sup}_{\xi \in {\mathbb{R}}^n}$ , we have, for any $\xi \in {\mathbb{R}}^n$ and $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ ,

3.35 $$\begin{array}{cc}& {\left[{\int }_0^{\infty },t^{{\alpha }_{(p,q)}\tau },{∥f,{\mathbf{1}}_{B(\xi ,t)}∥}_{L^q({\mathbb{R}}^n)}^{\tau },,\frac{\mathit{dt}}{t}\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \sim {\left[\sum _{\nu \in \mathbb{Z}},2^{-\nu {\alpha }_{(p,q)}\tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{-\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}.\hfill \end{array}$$

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From this, it follows that

3.36 $$\begin{array}{cc}\hfill {∥f_a^{(p)}∥}_{G{M}_{{\alpha }_{(p,q)}}^{q,\tau }({\mathbb{R}}^n)}^{\tau }& =\underset{\xi \in {\mathbb{R}}^n}{sup}{\int }_0^{\infty }{t}^{{\alpha }_{(p,q)}\tau }{∥f_a^{(p)},{\mathbf{1}}_{B(\xi ,t)}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\frac{\mathit{dt}}{t}\hfill \\ \hfill & \sim \underset{\xi \in {\mathbb{R}}^n}{sup}\sum _{\nu \in \mathbb{Z}}{2}^{-\nu {\alpha }_{(p,q)}\tau }{∥f_a^{(p)},,{\mathbf{1}}_{B(\xi ,2^{-\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\hfill \\ \hfill & =\underset{\xi \in {\mathbb{R}}^n}{sup}\sum _{\nu \in \mathbb{Z}}{2}^{-\nu {\alpha }_{(p,q)}\tau }\hfill \\ \hfill & \times {\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(\xi ,2^{-\nu })\right|\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & =\underset{\xi \in {\mathbb{R}}^n}{sup}\sum _{\nu \in \mathbb{Z}}{2}^{-\nu {\alpha }_{(p,q)}\tau }\hfill \\ \hfill & \times {\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B(|\xi |,{\mathbf{e}}_1,,,2^{-\nu },)\right|\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & =:\underset{\xi \in {\mathbb{R}}^n}{sup}F(\xi ).\hfill \end{array}$$

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Next, we claim that

3.37 $$\begin{array}{c}\hfill F(\mathbf{0})<\infty \text{if and only if}a\tau \in (1,\infty ).\end{array}$$

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Indeed, it is easy to show that

$$\begin{array}{c}\hfill \left|{\mathrm{\Gamma }}_k,\cap ,B,(\mathbf{0},2^{-\nu })\right|\ne 0\text{if and only if}k\ge \nu +1.\end{array}$$

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From this, it follows that

$$\begin{array}{cc}& \sum _{\nu \in \mathbb{Z}}{2}^{-\nu {\alpha }_{(p,q)}\tau }{\left[\sum _{k\in \mathbb{N}},2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(\mathbf{0},2^{-\nu })\right|\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \sim \sum _{\nu \in \mathbb{Z}}{2}^{-\nu {\alpha }_{(p,q)}\tau }{\left[\sum _{k=\nu +1}^{\infty },2^{kn(\frac{q}{p}-1)},k^{-aq}\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & =\sum _{\nu =0}^{\infty }{2}^{-\nu {\alpha }_{(p,q)}\tau }{\left[\sum _{k=\nu +1}^{\infty },2^{kn(\frac{q}{p}-1)},k^{-aq}\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & +\sum _{\nu =-\infty }^{-1}{2}^{-\nu {\alpha }_{(p,q)}\tau }{\left[\sum _{k=1}^{\infty },2^{kn(\frac{q}{p}-1)},k^{-aq}\right]}^{\frac{\tau }{q}}\hfill \\ \hfill & \sim \sum _{\nu \in {\mathbb{Z}}_{+}}{\nu }^{-a\tau }+1,\hfill \end{array}$$

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which implies that the above claim (3.37) holds true.

Now, we show the necessity of the present proposition. For this purpose, assume that $f_a^{(p)}\in {\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)$ . Then, from Theorem 3.15, (3.36), and (3.37), we deduce that

$$\begin{array}{c}\hfill F(\mathbf{0})\le {\Vert f\Vert }_{G{M}_{{\alpha }_{(p,q)}}^{q,\tau }({\mathbb{R}}^n)}\sim {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)}<\infty \end{array}$$

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and hence $a\tau \in (1,\infty )$ . This finishes the proof of the necessity of the present proposition.

Next, we prove the sufficiency of the present proposition. To do this, assume $a\tau \in (1,\infty )$ . For any $\xi \in {\mathbb{R}}^n\backslash \{\mathbf{0}\}$ , let $s:=-{log}_2|\xi |$ and

$$\begin{array}{c}\hfill G(s):=\sum _{\nu \in \mathbb{Z}}{2}^{-\nu {\alpha }_{(p,q)}\tau }{\left[\sum _{k=1}^{\infty },2^{\frac{\mathit{knq}}{p}},k^{-aq},\left|{\mathrm{\Gamma }}_k,\cap ,B,(2^{-s}{\mathbf{e}}_1,2^{-\nu })\right|\right]}^{\frac{\tau }{q}}.\end{array}$$

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Then, repeating the arguments similar to those used in the estimations of (3.15), (3.17), and (3.22) with r replaced by $\infty$ , we have

$$\begin{array}{c}\hfill G(s)\lesssim \left\{\begin{array}{cc}{2}^{-\lfloor s\rfloor {\alpha }_{(p,q)}\tau }\hfill & \text{when}s\in (-\infty ,-1),\hfill \\ 1\hfill & \text{when}s\in [-1,1],\hfill \\ 1+{(\lfloor s\rfloor )}^{-a\tau }\hfill & \text{when}s\in (1,\infty ).\hfill \end{array}\right)\end{array}$$

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By these and the assumption $a\tau \in (1,\infty )$ , we find that ${sup}_{s\in \mathbb{R}}G(s)<\infty$ . This, combined with Theorem 3.15, (3.36), (3.37), and the assumption $a\tau \in (1,\infty )$ again, further implies that

$$\begin{array}{cc}\hfill {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)}& \sim {\Vert f\Vert }_{G{M}_{{\alpha }_{(p,q)}}^{q,\tau }({\mathbb{R}}^n)}=max\left\{F,(\mathbf{0}),,,\underset{\xi \in {\mathbb{R}}^n\backslash \{\mathbf{0}\}}{sup},F,(\xi )\right\}\hfill \\ \hfill & =max\left\{F,(\mathbf{0}),,,\underset{s\in \mathbb{R}}{sup},G,(s)\right\}<\infty \hfill \end{array}$$

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and hence $f\in {\mathcal{M}\dot{F}}_{q,\infty }^{p,\tau }({\mathbb{R}}^n)$ . This finishes the proof of the sufficiency and hence Proposition 3.20. $\square$

Now, we show Theorem 3.17 as follows.

Proof of Theorem 3.17

We first show (i). Indeed, from Proposition 3.6 and Proposition 2.4(i), it follows that

$$\begin{array}{cc}\hfill \mathcal{M}{\dot{B}}_{q_1,r_1}^{p_1,min\{{\tau }_1,r_1\}}({\mathbb{R}}^n)& \subset {\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)={\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)\hfill \\ \hfill & \subset \mathcal{M}{\dot{B}}_{q_2,r_2}^{p_2,max\{{\tau }_2,r_2\}}({\mathbb{R}}^n)\hfill \end{array}$$

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and

$$\begin{array}{cc}\hfill \mathcal{M}{\dot{B}}_{q_2,r_2}^{p_2,min\{{\tau }_2,r_2\}}({\mathbb{R}}^n)& \subset {\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)={\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)\hfill \\ \hfill & \subset \mathcal{M}{\dot{B}}_{q_1,r_1}^{p_1,max\{{\tau }_1,r_1\}}({\mathbb{R}}^n).\hfill \end{array}$$

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Using these and Lemma 3.19, we conclude that $q_1=q_2$ , $p_1=p_2$ , and $r_1=r_2$ . This finishes the proof of (i).

Next, we prove (ii). Indeed, by Proposition 2.4(i), we find that, it is enough to show that, if ${\tau }_1<{\tau }_2$ , then

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,\infty }^{p,{\tau }_1}({\mathbb{R}}^n)⫋{\mathcal{M}\dot{F}}_{q,\infty }^{p,{\tau }_2}({\mathbb{R}}^n).\end{array}$$

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To this end, let $f_a^{(p)}$ be the same as in Proposition 3.20, ${\tau }_1<{\tau }_2$ , and $a\in (\frac{1}{{\tau }_2},\frac{1}{{\tau }_1})$ . Then, from Proposition 3.20, it follows that

$$\begin{array}{c}\hfill f_a^{(p)}\in {\mathcal{M}\dot{F}}_{q,\infty }^{p,{\tau }_2}({\mathbb{R}}^n)\backslash {\mathcal{M}\dot{F}}_{q,\infty }^{p,{\tau }_1}({\mathbb{R}}^n),\end{array}$$

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which, together with Proposition 2.4(i) again, further implies that ${\mathcal{M}\dot{F}}_{q,\infty }^{p,{\tau }_1}({\mathbb{R}}^n)⫋{\mathcal{M}\dot{F}}_{q,\infty }^{p,{\tau }_2}({\mathbb{R}}^n)$ . This then finishes the proof of (ii) and hence Theorem 3.17. $\square$

Boundedness of operators

In this section, we consider the sharp boundedness of several classical operators on TLBM spaces. To be precise, in Sect. 4.1, we first establish a boundedness criterion of operators on TLBM spaces via Herz spaces (see Theorem 4.2 below) and then give the sharp boundedness of the Hardy–Littlewood maximal operator, the Fefferman–Stein vector-valued inequality, and the Calderón–Zygmund operator. Next, in Sect. 4.2, we obtain the sharp boundedness of both the fractional integral and the fractional maximal operator on TLBM spaces.

Boundedness of operators via Herz spaces

In this subsection, via establishing the equivalence between Herz spaces and local Morrey spaces, we obtain a boundedness criterion of operators on TLBM spaces. As applications, we establish the boundedness of several classical operators on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , including the Hardy–Littlewood maximal operator and the Calderón–Zygmund operator as well as giving the Fefferman–Stein vector-valued inequality on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

First, we recall the following definition of Herz spaces which were originally introduced in [[21], Definition 1.1(a)] (see also [[22], Chapter 1]).

Definition 4.1

Let $q,\tau \in (0,\infty ]$ , $\alpha \in \mathbb{R}$ , and $\xi \in {\mathbb{R}}^n$ . The homogeneous Herz space ${\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ is defined to be the set of all the $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n\backslash \{\xi \})$ such that

$$\begin{array}{c}\hfill {\Vert f\Vert }_{{\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}:={\left[\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })\backslash B(\xi ,2^{\nu -1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}<\infty .\end{array}$$

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The main result of this subsection is the following boundedness criterion of operators on TLBM spaces via Herz spaces.

Theorem 4.2

Let $0<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ . Let T be an operator mapping $L_{\text{loc}}^q({\mathbb{R}}^n)$ into $\mathcal{M}({\mathbb{R}}^n)$ . Assume that there exists a positive constant C such that, for almost every $\xi \in {\mathbb{R}}^n$ and for any $f\in L_{\text{loc}}^q({\mathbb{R}}^n)$ ,

4.1 $$\begin{array}{c}\hfill {\Vert T(f)\Vert }_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}\le C{∥f∥}_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)},\end{array}$$

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where ${\alpha }_{(p,q,r)}$ is the same as in (3.24). Then T is bounded on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

To prove this theorem, we first establish the following technique lemma about the equivalence between Herz spaces and local Morrey spaces.

Lemma 4.3

Let $q,\tau \in (0,\infty ]$ , $\alpha \in (-\infty ,0)$ , and $\xi \in {\mathbb{R}}^n$ . Then

$$\begin{array}{c}\hfill L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)={\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)\end{array}$$

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with equivalent quasi-norms and the positive equivalence constants independent of $\xi$ .

Proof

We first show $L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)\subset {\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ . Indeed, let $f\in L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ . Then, from (3.35), we deduce that

4.2 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{{\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}& \le {\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & \sim {\Vert f\Vert }_{L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}<\infty .\hfill \end{array}$$

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This implies that $f\in {\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ and hence $L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)\subset {\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ .

Conversely, we now prove that ${\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)\subset L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ . To do this, assume that $f\in {\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ and consider the following four cases on both q and $\tau$ .

Case (i) $q\in [1,\infty ]$ and $\tau \in (0,1)$ . In this case, by $q\in [1,\infty ]$ and the Minkowski inequality of $L^q({\mathbb{R}}^n)$ , we conclude that, for any $\nu \in \mathbb{Z}$ ,

$$\begin{array}{c}\hfill {∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}\le \sum _{i=-\infty }^{\nu }{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)},\end{array}$$

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which, combined with (3.35), the well-known inequality that, for any $p\in (0,1]$ and ${\{a_j\}}_{j\in \mathbb{N}}\subset \mathbb{C}$ ,

4.3 $$\begin{array}{c}\hfill {\left(\sum _{j\in \mathbb{N}},|a_j|\right)}^p\le \sum _{j\in \mathbb{N}}{|a_j|}^p,\end{array}$$

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the Tonelli theorem, and $\alpha \in (-\infty ,0)$ , further implies that

4.4 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}& \sim {\left[\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right]}^{\frac{1}{\tau }}\hfill \\ \hfill & \le {\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{\left[\sum _{i=-\infty }^{\nu },{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}\right]}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & \le {\left\{\sum _{\nu \in \mathbb{Z}},\sum _{i=-\infty }^{\nu },2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & ={\left\{\sum _{i\in \mathbb{Z}},2^{i\alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau },\sum _{\nu =i}^{\infty },2^{(\nu -i)\alpha \tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & \sim {\left\{\sum _{i\in \mathbb{Z}},2^{i\alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & ={\Vert f\Vert }_{{\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}<\infty .\hfill \end{array}$$

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Thus, $f\in L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ . This then finishes the proof of ${\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)\subset L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ in this case.

Case (ii) $q\in [1,\infty ]$ and $\tau \in [1,\infty ]$ . In this case, observe that, for any $\nu \in \mathbb{Z}$ ,

4.5 $$\begin{array}{c}\hfill f{\mathbf{1}}_{B(\xi ,2^{\nu })}=\sum _{k=0}^{\infty }f{\mathbf{1}}_{B(\xi ,2^{\nu -k})\backslash B(\xi ,2^{\nu -k-1})}\end{array}$$

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almost everywhere in ${\mathbb{R}}^n$ . Using this, (3.35), the Minkowski inequality, the change of variables, and $\alpha \in (-\infty ,0)$ , we have

4.6 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}& \sim {\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥\sum _{k=0}^{\infty },f,{\mathbf{1}}_{B(\xi ,2^{\nu -k})\backslash B(\xi ,2^{\nu -k-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & \le \sum _{k=0}^{\infty }{\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu -k})\backslash B(\xi ,2^{\nu -k-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & =\sum _{k=0}^{\infty }{2}^{k\alpha }{\left\{\sum _{\nu \in \mathbb{Z}},2^{(\nu -k)\alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu -k})\backslash B(\xi ,2^{\nu -k-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & \sim {\Vert f\Vert }_{{\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}<\infty ,\hfill \end{array}$$

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which further implies that $f\in {\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ . Therefore, ${\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)\subset L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ in this case.

Case (iii) $q\in (0,1)$ and $\tau \in (0,q)$ . In this case, by the assumption $q\in (0,1)$ and (4.3), we find that, for any $\nu \in \mathbb{Z}$ ,

4.7 $$\begin{array}{cc}\hfill {∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }& \le {\left\{{\int }_{{\mathbb{R}}^n},\sum _{i=-\infty }^{\nu },{\left[f,(x),{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})},(x)\right]}^q,,d,x\right\}}^{\frac{\tau }{q}}\hfill \\ \hfill & ={\left\{\sum _{i=-\infty }^{\nu },{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}^q\right\}}^{\frac{\tau }{q}},\hfill \end{array}$$

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which, together with the assumption $\frac{\tau }{q}\in (0,1)$ and (4.3) again, further implies that

$$\begin{array}{c}\hfill {∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\le \sum _{i=-\infty }^{\nu }{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }.\end{array}$$

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From this, (3.35), the Tonelli theorem, and $\alpha \in (-\infty ,0)$ , it follows that

4.8 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}& \sim {\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & \le {\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },\sum _{i=-\infty }^{\nu },{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & ={\left\{\sum _{i\in \mathbb{Z}},2^{i\alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau },\sum _{\nu =i}^{\infty },2^{(\nu -i)\alpha \tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & \sim {\left\{\sum _{i\in \mathbb{Z}},2^{i\alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^i)\backslash B(\xi ,2^{i-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{1}{\tau }}\hfill \\ \hfill & ={\Vert f\Vert }_{{\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}<\infty .\hfill \end{array}$$

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Thus, $f\in L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ and hence ${\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)\subset L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ in this case.

Case (iv) $q\in (0,1)$ and $\tau \in [q,\infty )$ . In this case, applying (4.5) and repeating an argument similar to that used in the estimation of (4.7), we conclude that

$$\begin{array}{c}\hfill {∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\le {\left\{\sum _{k=0}^{\infty },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu -k})\backslash B(\xi ,2^{\nu -k-1})}∥}_{L^q({\mathbb{R}}^n)}^q\right\}}^{\frac{\tau }{q}}.\end{array}$$

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This, combined with (3.35), $\frac{\tau }{q}\in [1,\infty )$ , the Minkowski inequality, the change of variables, and $\alpha \in (-\infty ,0)$ , further implies that

4.9 $$\begin{array}{cc}\hfill {\Vert f\Vert }_{L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}^q& \sim {\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{q}{\tau }}\hfill \\ \hfill & \le {\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{\left[\sum _{k=0}^{\infty },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu -k})\backslash B(\xi ,2^{\nu -k-1})}∥}_{L^q({\mathbb{R}}^n)}^q\right]}^{\frac{\tau }{q}}\right\}}^{\frac{q}{\tau }}\hfill \\ \hfill & \le \sum _{k=0}^{\infty }{\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu -k})\backslash B(\xi ,2^{\nu -k-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{q}{\tau }}\hfill \\ \hfill & =\sum _{k=0}^{\infty }{2}^{k\alpha q}{\left\{\sum _{\nu \in \mathbb{Z}},2^{(\nu -k)\alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu -k})\backslash B(\xi ,2^{\nu -k-1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{q}{\tau }}\hfill \\ \hfill & \sim {\left\{\sum _{\nu \in \mathbb{Z}},2^{\nu \alpha \tau },{∥f,{\mathbf{1}}_{B(\xi ,2^{\nu })\backslash B(\xi ,2^{\nu -1})}∥}_{L^q({\mathbb{R}}^n)}^{\tau }\right\}}^{\frac{q}{\tau }}\hfill \\ \hfill & ={\Vert f\Vert }_{{\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}^q<\infty .\hfill \end{array}$$

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Therefore, $f\in {\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ . This then finishes the proof of ${\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)\subset L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ in this case.

Combining (4.2), (4.4), (4.6), (4.8), and (4.9), we further conclude that $L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)={\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)$ and, for any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ , ${\Vert f\Vert }_{{\dot{K}}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}\sim {\Vert f\Vert }_{L{M}_{\alpha }^{q,\tau ,\xi }({\mathbb{R}}^n)}$ with the positive equivalence constants independent of $\xi$ . This then finishes the proof of Lemma 4.3. $\square$

Remark 4.4

We should point out that, in Lemma 4.3, if $\tau =\infty$ , then the conclusion of Lemma 4.3 was established in [[42], Theorem 26].

We now show Theorem 4.2.

Proof of Theorem 4.2

Observe that ${\alpha }_{(p,q,r)}\in (-\infty ,0]$ . We now show the present theorem by considering the following two cases on ${\alpha }_{(p,q,r)}$ .

Case (i) ${\alpha }_{(p,q,r)}=0$ . In this case, from the assumption $q\le p$ , we infer that $p=q$ and $r=\infty$ . If $\tau \in (0,\infty )$ , then, by Theorem 2.9, we find that ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is trivial and hence the present theorem obviously holds true. If $\tau =\infty$ , then, applying Proposition 3.6(iii), we have

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{p,\infty }^{p,\infty }({\mathbb{R}}^n)={\mathcal{M}}_{p,\infty }^p({\mathbb{R}}^n)={\mathcal{M}}_p^p({\mathbb{R}}^n)=L^p({\mathbb{R}}^n)\end{array}$$

Graph

with equivalent quasi-norms. This, together with (4.1), further implies that, for any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ ,

$$\begin{array}{cc}\hfill {∥T,(,f,)∥}_{{\mathcal{M}\dot{F}}_{p,\infty }^{p,\infty }({\mathbb{R}}^n)}& \sim {∥T,(,f,)∥}_{L^p({\mathbb{R}}^n)}=\underset{\xi \in {\mathbb{R}}^n,k\in \mathbb{Z}}{sup}{∥T,(f),{\mathbf{1}}_{B(\xi ,2^k)\backslash B(\xi ,2^{k-1})}∥}_{L^p({\mathbb{R}}^n)}\hfill \\ \hfill & =\underset{\xi \in {\mathbb{R}}^n}{sup}{∥T,(,f,)∥}_{{\dot{K}}_0^{p,\infty ,\xi }({\mathbb{R}}^n)}\lesssim \underset{\xi \in {\mathbb{R}}^n}{sup}{∥f∥}_{{\dot{K}}_0^{p,\infty ,\xi }({\mathbb{R}}^n)}\hfill \\ \hfill & ={\Vert f\Vert }_{L^p({\mathbb{R}}^n)}\sim {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{p,\infty }^{p,\infty }({\mathbb{R}}^n)},\hfill \end{array}$$

Graph

which completes the proof of the present theorem in this case.

Case (ii) ${\alpha }_{(p,q,r)}\in (-\infty ,0)$ . In this case, by Proposition 3.14, Lemma 4.3, and (4.1), we conclude that, for any $f\in L_{\mathrm{loc}}^q({\mathbb{R}}^n)$ ,

$$\begin{array}{cc}\hfill {\Vert T(f)\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}& \sim {\left[{\int }_{{\mathbb{R}}^n},{∥T,(,f,)∥}_{L{M}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}^r,,d,\xi \right]}^{\frac{1}{r}}\hfill \\ \hfill & \sim {\left[{\int }_{{\mathbb{R}}^n},{∥T,(,f,)∥}_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}^r,,d,\xi \right]}^{\frac{1}{r}}\lesssim {\left[{\int }_{{\mathbb{R}}^n},{∥f∥}_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}^r,,d,\xi \right]}^{\frac{1}{r}}\hfill \\ \hfill & \sim {\left[{\int }_{{\mathbb{R}}^n},{∥f∥}_{L{M}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}^r,,d,\xi \right]}^{\frac{1}{r}}\sim {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\hfill \end{array}$$

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This further implies that, in this case, T is bounded on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and hence finishes the proof Theorem 4.2. $\square$

As applications of Theorem 4.2, we now establish the boundedness on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ of several important operators from harmonic analysis. First, we have the following boundedness of the Hardy–Littlewood maximal operator on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

Theorem 4.5

If $1<q\le p\le r\le \infty$ and $\tau \in (0,\infty ]$ , then the Hardy–Littlewood maximal operator M is bounded on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

Proof

We consider the following two cases on both p and r.

Case (i) $p<r$ . In this case, applying the assumptions on p, q, r, and $\tau$ of the present theorem, we have ${\alpha }_{(p,q,r)}\in (-\frac{n}{q},0]$ . From this and the boundedness of M on Herz spaces (see, for instance, [[22], Theorem 5.1.1 and Remark 5.1.3] or [[20], Corollary 1.5.4]), we deduce that, for any $\xi \in {\mathbb{R}}^n$ and $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {\Vert M(f)\Vert }_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}\lesssim {\Vert f\Vert }_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}.\end{array}$$

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Applying this and Theorem 4.2, we further find that, for any $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {∥M,(,f,)∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\lesssim {∥f∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)},\end{array}$$

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that is, M is bounded on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ in this case.

Case (ii) $p=r$ . In this case, by Theorem 2.9 and Remark 3.16, we obtain

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q,p}^{p,\tau }({\mathbb{R}}^n)=\left\{\begin{array}{cc}\{0\}\hfill & \text{when}\tau \in (0,\infty ),\hfill \\ L^p({\mathbb{R}}^n)\hfill & \text{when}\tau =\infty \text{and}p\in (1,\infty ),\hfill \\ {\mathcal{M}}_q^{\infty }({\mathbb{R}}^n)=L^{\infty }({\mathbb{R}}^n)\hfill & \text{when}\tau =\infty \text{and}p=\infty .\hfill \end{array}\right)\end{array}$$

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It is well known that M is bounded on $L^p({\mathbb{R}}^n)$ when $p\in (1,\infty ]$ . This finishes the proof of Theorem 4.5. $\square$

Using [[20], Theorem 1.6.1] and repeating an argument similar to that used in the proof of Theorem 4.5, we obtain the following Fefferman–Stein vector-valued inequality on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ; we omit the details.

Theorem 4.6

Let $u\in (1,\infty ]$ . If $1<q\le p<r\le \infty$ and $\tau \in (0,\infty ]$ , or if $1<q<p=r<\infty$ and $\tau =\infty$ , then there exists a positive constant C such that, for any ${\{f_j\}}_{j\in \mathbb{N}}\subset L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {∥{\left\{\sum _{j\in \mathbb{N}},{\left[M,(,f_j,)\right]}^u\right\}}^{\frac{1}{u}}∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}\le C{∥{\left(\sum _{j\in \mathbb{N}},{|f_j|}^u\right)}^{\frac{1}{u}}∥}_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}.\end{array}$$

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Remark 4.7

• In Theorems 4.5 and 4.6, if $\tau =r$ , then, in this case, by Proposition 3.6(iii), we find that ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)={\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)$ and hence Theorems 4.5 and 4.6 go back, respectively, to [[16], Lemma 4.1 and Theorem 4.3].
• We should point out that the assumptions on p, q, r, and $\tau$ of Theorems 4.5 and 4.6 are sharp. Indeed, let $p=q=1$ and $r=\tau =\infty$ . Then, in this case, the TLBM space ${\mathcal{M}\dot{F}}_{1,\infty }^{1,\infty }({\mathbb{R}}^n)$ does not satisfy the assumptions of Theorem 4.5; by Remark 3.16(i), we have ${\mathcal{M}\dot{F}}_{1,\infty }^{1,\infty }({\mathbb{R}}^n)=L^1({\mathbb{R}}^n)$ and hence M is not bounded on ${\mathcal{M}\dot{F}}_{1,\infty }^{1,\infty }({\mathbb{R}}^n)$ (see, for instance, [[13], p. 87]). Thus, the assumption $q>1$ of Theorem 4.5 is sharp. On the other hand, let $u\in (1,\infty )$ and $p=q=r=\tau =\infty$ . Then, in this case, the TLBM space ${\mathcal{M}\dot{F}}_{\infty ,\infty }^{\infty ,\infty }({\mathbb{R}}^n)$ does not satisfy the assumptions of Theorem 4.6; from Remark 3.16(i) again, we infer that ${\mathcal{M}\dot{F}}_{\infty ,\infty }^{\infty ,\infty }({\mathbb{R}}^n)=L^{\infty }({\mathbb{R}}^n)$ and hence the Fefferman–Stein vector-valued inequality does not hold true on ${\mathcal{M}\dot{F}}_{\infty ,\infty }^{\infty ,\infty }({\mathbb{R}}^n)$ (see, for instance, [[13], Exercise 5.6.4]). This further implies that the assumptions of Theorems 4.5 and 4.6 are sharp.

At the end of this subsection, we establish the boundedness of Calderón–Zygmund operators on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . To this end, we first give some symbols. Let

$$\begin{array}{c}\hfill \mathrm{\Delta }:=\left\{(x,y),\in ,{\mathbb{R}}^n,\times ,{\mathbb{R}}^n,:,,x,=,y\right\}.\end{array}$$

Graph

We now present the following definition of standard kernels; see, for instance, [[13], Definition 7.4.1].

Definition 4.8

Let $\delta \in (0,1]$ . Then a measurable function K on $({\mathbb{R}}^n\times {\mathbb{R}}^n)\backslash \mathrm{\Delta }$ is called a $\delta$ -standard kernel if there exists a positive constant C such that

• for any $x,y\in {\mathbb{R}}^n$ with $x\ne y$ ,
• |K(x,y)|≤C|x-y|n;

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• for any $x,y,z\in {\mathbb{R}}^n$ with $|x-y|>2|y-z|$ ,
• |K(x,y)-K(x,z)|≤C|y-z|δ|x-y|n+δ

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• and

• |K(y,x)-K(z,x)|≤C|y-z|δ|x-y|n+δ.

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The following concept of Calderón–Zygmund operators is just [[13], Definition 7.4.2].

Definition 4.9

Let $\delta \in (0,1]$ and K be a $\delta$ -standard kernel the same as in Definition 4.8. A linear operator T is called a $\delta$ -Calderón–Zygmund operator with a $\delta$ -standard kernel K if T is bounded on $L^2({\mathbb{R}}^n)$ and, for any $f\in L^2({\mathbb{R}}^n)$ with compact support and for almost every $x\notin \overline{\mathrm{supp}(f)}$ , T has an integral representation

$$\begin{array}{c}\hfill T(f)(x)={\int }_{{\mathbb{R}}^n}K(x,y)f(y)dy.\end{array}$$

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Next, we show that Calderón–Zygmund operators are bounded on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ as follows.

Theorem 4.10

Let $\delta \in (0,1]$ . If $1<q\le p<r\le \infty$ and $\tau \in (0,\infty ]$ , or if $1<q<p=r<\infty$ and $\tau =\infty$ , then any $\delta$ -Calderón–Zygmund operator T can be extended into a bounded linear operator on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

Proof

We first show that T is well defined on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Indeed, let $f\in {\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Then, combining Lemma 4.3 and Proposition 3.14, we find that

$$\begin{array}{c}\hfill {\left[{\int }_{{\mathbb{R}}^n},{∥f∥}_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}^r,,d,\xi \right]}^{\frac{1}{r}}\sim {\left[{\int }_{{\mathbb{R}}^n},{∥f∥}_{L{M}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}^r,,d,\xi \right]}^{\frac{1}{r}}\sim {\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)}<\infty ,\end{array}$$

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which further implies that there exists a ${\xi }_0\in {\mathbb{R}}^n$ such that $f\in {\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,{\xi }_0}({\mathbb{R}}^n)$ . From this and the boundedness of T on the homogeneous Herz space (see, for instance [[22], Theorem 5.1.1 and Remark 5.1.3]), we infer that $T(f)\in {\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,{\xi }_0}({\mathbb{R}}^n)$ . This then further implies that T is well defined on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ .

Now, we prove that T is bounded on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ . Indeed, if $1<q\le p<r\le \infty$ and $\tau \in (0,\infty ]$ , then, by ${\alpha }_{(p,q,r)}\in (-\frac{n}{q},0]$ and [[22], Theorem 5.1.1 and Remark 5.1.3] again, we conclude that, for any $\xi \in {\mathbb{R}}^n$ and $f\in {\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {\Vert T(f)\Vert }_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}\lesssim {∥f∥}_{{\dot{K}}_{{\alpha }_{(p,q,r)}}^{q,\tau ,\xi }({\mathbb{R}}^n)}.\end{array}$$

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This, combined with Theorem 4.2, further implies that T is bounded on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ in this case. If $1<q<p=r<\tau =\infty$ , then, from Proposition 3.1 and [[14], Theorem 4.2.2], we infer that T is bounded on ${\mathcal{M}\dot{F}}_{q,p}^{p,\infty }({\mathbb{R}}^n)$ . This finishes the proof Theorem 4.10. $\square$

Remark 4.11

• In Theorem 4.10, if $\tau =r$ , then, in this case, applying Proposition 3.6(iii), we have ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)={\mathcal{M}}_{q,r}^p({\mathbb{R}}^n)$ and hence Theorem 4.10 coincides with [[50], Theorem 5.9]. In particular, if further assume that $\delta =1$ and $K(x,y)\equiv K_1(x-y)$ for any $x,y\in {\mathbb{R}}^n$ and some $K_1\in L_{\mathrm{loc}}^1({\mathbb{R}}^n\backslash \{\mathbf{0}\})$ , then Theorem 4.10 just goes back to [[16], Theorem 4.7].
• We should point out that the assumptions on p, q, r, and $\tau$ of Theorem 4.10 are sharp. Indeed, the Calderón–Zygmund operator is not bounded on both $L^1({\mathbb{R}}^n)$ and $L^{\infty }({\mathbb{R}}^n)$ (see, for instance, [[13], Example 5.1.3]). On the other hand, similarly to Remark 4.7(ii), we have ${\mathcal{M}\dot{F}}_{1,\infty }^{1,\infty }({\mathbb{R}}^n)=L^1({\mathbb{R}}^n)$ and ${\mathcal{M}\dot{F}}_{\infty ,\infty }^{\infty ,\infty }({\mathbb{R}}^n)=L^{\infty }({\mathbb{R}}^n)$ do not satisfy the assumptions of Theorem 4.10. Thus, in this sense, the assumptions of Theorem 4.10 are sharp.

Fractional integrals

The main target of this subsection is to explore the boundedness of both the fractional integral and the fractional maximal operator on TLBM spaces.

Let $\alpha \in (0,n)$ . Recall that the fractional integral $I_{\alpha }$ of order $\alpha$ is defined by setting, for any $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ and $x\in {\mathbb{R}}^n$ ,

$$\begin{array}{c}\hfill I_{\alpha }(f)(x):={\int }_{{\mathbb{R}}^n}\frac{f(y)}{{|x-y|}^{n-\alpha }}dy\end{array}$$

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if the right-hand side makes sense; the fractional maximal operator $M_{\alpha }$ is defined by setting, for any $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ and $x\in {\mathbb{R}}^n$ ,

4.10 $$\begin{array}{c}\hfill M_{\alpha }f(x):=\underset{Q\ni x}{sup}\frac{1}{{|Q|}^{1-\frac{\alpha }{n}}}{\int }_Q|f(y)|dy,\end{array}$$

Graph

where the supremum is taken over all the cubes Q of ${\mathbb{R}}^n$ containing x.

Applying Proposition 3.6 and repeating an argument similar to that used in the proof of [[16], Theorem 4.4], we conclude the following boundedness of both $I_{\alpha }$ and $M_{\alpha }$ on ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ , which simultaneously elevates all the four exponents of ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ ; we omit the details.

Theorem 4.12

Let $\alpha \in (0,n)$ , $1<q_1<p_1<r_1<\infty$ , and ${\tau }_1\in (0,\infty )$ . Assume that either $1<q_2<p_2<r_2\le \infty$ and ${\tau }_2\in (0,\infty )$ , or $1<q_2\le p_2\le r_2\le \infty$ and ${\tau }_2=\infty$ satisfy

$$\begin{array}{c}\hfill \frac{1}{p_1}-\frac{1}{p_2}=\frac{\alpha }{n}\text{and}\frac{{\tau }_1}{{\tau }_2}=\frac{r_1}{r_2}=\frac{q_1}{q_2}=\frac{p_1}{p_2}.\end{array}$$

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Then there exists a positive constant C such that, for any $f\in {\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill \Vert I_{\alpha }{(f)\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\le C{\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)}\end{array}$$

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and

$$\begin{array}{c}\hfill \Vert M_{\alpha }{(f)\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\le C{\Vert f\Vert }_{{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)}.\end{array}$$

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Remark 4.13

In Theorem 4.12, if ${\tau }_i=r_i$ for any $i\in \{1,2\}$ , then, in this case, by Proposition 3.6(iii), we have

$$\begin{array}{c}\hfill {\mathcal{M}\dot{F}}_{q_i,r_i}^{p_i,{\tau }_i}({\mathbb{R}}^n)={\mathcal{M}}_{q_i,r_i}^{p_i}({\mathbb{R}}^n)\end{array}$$

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for any $i\in \{1,2\}$ , and hence Theorem 4.12 goes back to [[16], Theorem 4.4 and Corollary 4.5]. Moreover, the other cases of Theorem 4.12 are new.

Next, we are devoted to improving Theorem 4.12 via Calderón products. To do this, we first recall several basic concepts. Recall that a quasi-Banach space X of complex-valued measurable functions is called a quasi-Banach lattice if, for any $f,g\in X$ , $|g|\le |f|$ implies that ${\Vert g\Vert }_X\le {\Vert f\Vert }_X$ . Assume that $X_0$ and $X_1$ are two quasi-Banach lattices and $\theta \in [0,1]$ . Then their Calderón product, denoted by $X_0^{1-\theta }{X}_1^{\theta }$ , is defined to be the set of all the measurable functions f satisfying

4.11 $$\begin{array}{c}\hfill |f|\le {\left|f^{(0)}\right|}^{1-\theta }{\left|f^{(1)}\right|}^{\theta },\end{array}$$

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for some $f^{(0)}\in X_0$ and $f^{(1)}\in X_1$ , and equipped with the quasi-norm

4.12 $$\begin{array}{c}\hfill {\Vert f\Vert }_{X_0^{1-\theta }{X}_1^{\theta }}:=inf\left\{{∥f^{(0)}∥}_{X_0}^{1-\theta },{∥f^{(1)}∥}_{X_1}^{\theta }\right\},\end{array}$$

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where the infimum is taken over all the $f^{(0)}\in X_0$ and $f^{(1)}\in X_1$ satisfying (4.11); see, for instance, [[5], [17]]. Obviously, the TLBM space ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ is a quasi-Banach lattice.

Via borrowing some ideas from the proofs of [[50], Theorems 5.3 and 5.4], we now establish the following improved boundedness of both $I_{\alpha }$ and $M_{\alpha }$ on TLBM spaces via Calderón products.

Theorem 4.14

Let $\alpha \in (0,n)$ and, for any $i\in \{1,2\}$ , $p_i,q_i,r_i,{\tau }_i\in (1,\infty ]$ be such that

$$\begin{array}{c}\hfill \frac{1}{p_1}-\frac{1}{p_2}=\frac{\alpha }{n}\text{and}\frac{{\tau }_1}{{\tau }_2}=\frac{r_1}{r_2}=\frac{q_1}{q_2}=\frac{p_1}{p_2}.\end{array}$$

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• Assume that either
• τi∈(1,∞)and1

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• or

• τi=∞and1

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• Then there exists a positive constant C such that, for any
• f∈MF˙q1,r1p1,τ1(Rn)p1p2M1p1(Rn)1-p1p2,‖Mα(f)‖MF˙q2,r2p2,τ2(Rn)≤C‖f‖[MF˙q1,r1p1,τ1(Rn)]p1/p2[M1p1(Rn)]1-p1/p2.

Graph

• Assume that ${\tau }_i\in (1,\infty )$ and $1<q_i<p_i<r_i<\infty$ for any $i\in \{1,2\}$ . Then there exists a positive constant C such that, for any $f\in {[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{\frac{p_1}{p_2}}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-\frac{p_1}{p_2}}$ ,
• 4.13 $$\begin{array}{c}\hfill \Vert I_{\alpha }{(f)\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\le C{\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}}.\end{array}$$

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Remark 4.15

• In Theorem 4.14(i), let ${\tau }_i=r_i=\infty$ and $p_i\in [q_i,\infty )$ for any $i\in \{1,2\}$ . Then, in this case, ${\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)={\mathcal{M}}_{q_2}^{p_2}({\mathbb{R}}^n)$ and
• [MF˙q1,r1p1,τ1(Rn)]p1p2[M1p1(Rn)]1-p1p2=[Mq1p1(Rn)]p1p2[M1p1(Rn)]1-p1p2

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• and hence Theorem 4.14(i) just coincides with [[37], Theorem 1.1]. Furthermore, in this case, Sawano and Sugano [[37]] pointed out that the interpolation index $\theta :=1-p_1/p_2$ is optimal. Therefore, in this sense, the results obtained in Theorem 4.14 are sharp.
• In Theorem 4.14, when ${\tau }_i=r_i\in (1,\infty )$ for any $i\in \{1,2\}$ , then ${\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)={\mathcal{M}}_{q_2,r_2}^{p_2}({\mathbb{R}}^n)$ and
• [MF˙q1,r1p1,τ1(Rn)]p1p2[M1p1(Rn)]1-p1p2=[Mq1,r1p1(Rn)]p1p2[M1p1(Rn)]1-p1p2.

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• In this case, Theorem 4.14 goes back to [[50], Theorems 5.3 and 5.4 and Remark 5.5]. Moreover, by [[50], Proposition 4.11], we have
• Mq1,r1p1(Rn)⫋[Mq1,r1p1(Rn)]p1p2[M1p1(Rn)]1-p1p2.

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• In this sense, Theorem 4.14 indeed improves Theorem 4.12.

Via the boundedness of the Hardy–Littlewood maximal operator on TLBM spaces, we next show Theorem 4.14(i).

Proof of Theorem 4.14(i)

We only show the present theorem under the assumption that ${\tau }_i\in (1,\infty )$ and, for any $i\in \{1,2\}$ , $1<q_i<p_i<r_i<\infty$ because the cases that ${\tau }_i=\infty$ or $r_i=\infty$ are quite similar and hence we omit the details. Let $f\in {[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{\frac{p_1}{p_2}}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-\frac{p_1}{p_2}}$ , $f_0\in {\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)$ , and $f_1\in {\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)$ satisfy $|f|\le |f_0{|}^{\frac{p_1}{p_2}}{|f_1|}^{1-\frac{p_1}{p_2}}$ almost everywhere on ${\mathbb{R}}^n$ . Then, from the Hölder inequality and the assumption that $1/p_1-1/p_2=\alpha /n$ , we deduce that, for any $R\in (0,\infty )$ and $x\in {\mathbb{R}}^n$ ,

$$\begin{array}{cc}\hfill R^{\alpha }{m}_{Q(x,R)}(|f|)& \le R^{\alpha }{\left[m_{Q(x,R)},(|,f_0,|)\right]}^{\frac{p_1}{p_2}}{\left[m_{Q(x,R)},(|,f_1,|)\right]}^{1-\frac{p_1}{p_2}}\hfill \\ \hfill & \le {\left[M,(f_0),(x)\right]}^{\frac{p_1}{p_2}}{\Vert f_1\Vert }_{{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)}^{1-\frac{p_1}{p_2}}.\hfill \end{array}$$

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This, together with (4.10), further implies that, for any $x\in {\mathbb{R}}^n$ ,

$$\begin{array}{c}\hfill M_{\alpha }(f)(x)\lesssim {\left[M,(f_0),(x)\right]}^{\frac{p_1}{p_2}}{\Vert f_1\Vert }_{{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)}^{1-\frac{p_1}{p_2}},\end{array}$$

Graph

where the implicit positive constant depends only on both n and $\alpha$ . By this, $r_1\in (1,\infty )$ , and Theorem 4.5, we conclude that

$$\begin{array}{cc}\hfill \Vert M_{\alpha }{(f)\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}& \lesssim {∥{\left[M,(,f_0,)\right]}^{\frac{p_1}{p_2}}∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}{\Vert f_1\Vert }_{{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)}^{1-\frac{p_1}{p_2}}\hfill \\ \hfill & \sim \Vert M(f_0){\Vert }_{{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)}^{\frac{p_1}{p_2}}{\Vert f_1\Vert }_{{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)}^{1-\frac{p_1}{p_2}}\hfill \\ \hfill & \lesssim \Vert f_0{\Vert }_{{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)}^{\frac{p_1}{p_2}}{\Vert f_1\Vert }_{{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)}^{1-\frac{p_1}{p_2}},\hfill \end{array}$$

Graph

which, combined with (4.12) and the choice of both $f_0$ and $f_1$ , further implies that

$$\begin{array}{c}\hfill \Vert M_{\alpha }{(f)\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\lesssim {\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}}.\end{array}$$

Graph

This finishes the proof of Theorem 4.14(i). $\square$

Next, we turn to show Theorem 4.14(ii). To achieve this, we require some concepts about sparse families on Euclidean spaces. Recall that a dyadic grid $\mathcal{D}$ is defined to be set of countable cubes, which has the following three properties:

• if $Q\in \mathcal{D}$ , then $l(Q)=2^k$ for some $k\in \mathbb{Z}$ ;
• if $P,Q\in \mathcal{D}$ , then $P\cap Q\in \{\mathrm{\varnothing },P,Q\}$ ;
• for any $k\in \mathbb{Z}$ , one has
• Rn=⋃Q∈D,l(Q)=2kQ.

Graph

For any dyadic grid $\mathcal{D}$ , a set $\mathcal{S}\subset \mathcal{D}$ is said to be sparse if, for any $Q\in \mathcal{S}$ ,

$$\begin{array}{c}\hfill \left|\bigcup _{Q^{\prime }\in \mathcal{S},Q^{\prime }⫋Q},Q^{\prime }\right|\le \frac{1}{2}|Q|.\end{array}$$

Graph

Let $\mathcal{S}$ be a sparse family. Then, as was proved in [[7], (13)], there exist ${\{E_Q\}}_{Q\in \mathcal{S}}$ of disjoint measurable sets such that, for any $Q\in \mathcal{S}$ ,

4.14 $$\begin{array}{c}\hfill E_Q\subset Q\text{and}|E_Q|\ge \frac{1}{2}|Q|.\end{array}$$

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The following pointwise estimate about the fractional integral and the sparse family is a simple corollary of [[7], Propositions 2.2 and 2.3], which plays an essential role in the proof of Theorem 4.14(ii); we omit the details.

Lemma 4.16

Let $\alpha \in (0,n)$ and $f\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ be a non-negative function. Then there exist ${\{{\mathcal{D}}_i\}}_{i=1}^{2^n}$ of dyadic grids and ${\{{\mathcal{S}}_i\}}_{i=1}^{2^n}$ of sparse families satisfying that, for any $i\in \{1,...,2^n\}$ , ${\mathcal{S}}_i\subset {\mathcal{D}}_i$ and there exists a positive constant C, independent of f, such that

$$\begin{array}{c}\hfill I_{\alpha }(f)\le C\underset{i\in \{1,...,2^n\}}{max}\sum _{Q\in {\mathcal{S}}_i}{[l(Q)]}^{\alpha }{m}_Q(f){\mathbf{1}}_Q,\end{array}$$

Graph

where $m_Q(·)$ is the same as in (2.13).

Based on this estimate, the boundedness of the fractional maximal operator established in Theorem 4.14(i), and the Fefferman–Stein vector-valued inequality on TLBM spaces, we now show Theorem 4.14(ii).

Proof of Theorem 4.14(ii)

We first show the present theorem for any non-negative function $f\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ . Indeed, by Lemma 4.16, we conclude that it suffices to prove that, for any sparse family $\mathcal{S}$ and any non-negative function $f\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ ,

4.15 $$\begin{array}{c}\hfill {∥\sum _{Q\in \mathcal{S}},{[l(Q)]}^{\alpha },m_Q,(f),{\mathbf{1}}_Q∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\lesssim {\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}}.\end{array}$$

Graph

To do this, let ${\{E_Q\}}_{Q\in \mathcal{S}}$ be the same as in (4.14). Then, using the fact that ${\mathbf{1}}_Q\le 2M({\mathbf{1}}_{E_Q})$ for any $Q\in \mathcal{S}$ , $r\in (1,\infty ),$ and Theorem 4.6, we find that

$$\begin{array}{cc}\hfill {∥\sum _{Q\in \mathcal{S}},{[l(Q)]}^{\alpha },m_Q,(f),{\mathbf{1}}_Q∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}& \lesssim {∥\sum _{Q\in \mathcal{S}},{\left\{M,\left({[l(Q)]}^{\frac{\alpha }{2}},{[m_Q(f)]}^{\frac{1}{2}},{\mathbf{1}}_{E_Q}\right)\right\}}^2∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\hfill \\ \hfill & ={∥{\left\{\sum _{Q\in \mathcal{S}},{\left[M,\left({[l(Q)]}^{\frac{\alpha }{2}},{[m_Q(f)]}^{\frac{1}{2}},{\mathbf{1}}_{E_Q}\right)\right]}^2\right\}}^{\frac{1}{2}}∥}_{{\mathcal{M}\dot{F}}_{2q_2,2r_2}^{2p_2,2{\tau }_2}({\mathbb{R}}^n)}^2\hfill \\ \hfill & \lesssim {∥{\left[\sum _{Q\in \mathcal{S}},{[l(Q)]}^{\alpha },m_Q,(f),{\mathbf{1}}_{E_Q}\right]}^{\frac{1}{2}}∥}_{{\mathcal{M}\dot{F}}_{2q_2,2r_2}^{2p_2,2{\tau }_2}({\mathbb{R}}^n)}^2\hfill \\ \hfill & ={∥\sum _{Q\in \mathcal{S}},{[l(Q)]}^{\alpha },m_Q,(f),{\mathbf{1}}_{E_Q}∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}.\hfill \end{array}$$

Graph

Therefore, to show (4.15), we only need to prove that, for any non-negative function $f\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {∥\sum _{Q\in \mathcal{S}},{[l(Q)]}^{\alpha },m_Q,(f),{\mathbf{1}}_{E_Q}∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\lesssim {\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}}.\end{array}$$

Graph

Indeed, from the disjointness of ${\{E_Q\}}_{Q\in \mathcal{S}}$ , we infer that, for any non-negative function $f\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill \sum _{Q\in \mathcal{S}}{[l(Q)]}^{\alpha }{m}_Q(f){\mathbf{1}}_{E_Q}\lesssim M_{\alpha }(f).\end{array}$$

Graph

This, together with Theorem 4.14(i), further implies that (4.15) holds true and hence (4.13) also holds true for any non-negative function $f\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ .

Next, we show that (4.13) holds true for any $f\in {[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}$ . To do this, let $f\in {[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}$ . Then $f={(\Re f)}_{+}-{(\Re f)}_{-}+i[{(\Im f)}_{+}-{(\Im f)}_{-}]$ and hence

4.16 $$\begin{array}{cc}\hfill \Vert I_{\alpha }{(f)\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}& \le {∥I_{\alpha },({(\Re f)}_{+})∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}+{∥I_{\alpha },({(\Re f)}_{-})∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\hfill \\ \hfill & +{∥I_{\alpha },({(\Im f)}_{+})∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}+{∥I_{\alpha },({(\Im f)}_{-})∥}_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}.\hfill \end{array}$$

Graph

For any $m\in \mathbb{N}$ , let ${(\Re f)}_{+,m}$ be the same as in (2.23) with f replaced by ${(\Re f)}_{+}$ . Then, for any $m\in \mathbb{N}$ , ${(\Re f)}_{+,m}\in L_{\text{c}}^{\infty }({\mathbb{R}}^n)$ is non-negative. This, combined with the Fatou lemma, $r_2,{\tau }_2\in (1,\infty )$ , and (2.24), further implies that

4.17 $$\begin{array}{cc}\hfill \Vert I_{\alpha }({(\Re f)}_{+}){\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}& \le \underset{m\to \infty }{\underline{lim}}{\Vert I_{\alpha }({(\Re f)}_{+,m})\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\hfill \\ \hfill & \lesssim \underset{m\to \infty }{lim}{\Vert {(\Re f)}_{+,m}\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}}\hfill \\ \hfill & {=\Vert (\Re f)}_{+}{\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}}\hfill \\ \hfill & \le {\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}}.\hfill \end{array}$$

Graph

Similarly, we have

$$\begin{array}{cc}\hfill & \Vert I_{\alpha }({(\Re f)}_{-}){\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\lesssim {\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}},\hfill \\ \hfill & \Vert I_{\alpha }({(\Im f)}_{+}){\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\lesssim {\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}},\hfill \end{array}$$

Graph

and

$$\begin{array}{c}\hfill \Vert I_{\alpha }({(\Im f)}_{-}){\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\lesssim {\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}}.\end{array}$$

Graph

These, combined with both (4.16) and (4.17), further imply that

$$\begin{array}{c}\hfill \Vert I_{\alpha }{(f)\Vert }_{{\mathcal{M}\dot{F}}_{q_2,r_2}^{p_2,{\tau }_2}({\mathbb{R}}^n)}\lesssim {\Vert f\Vert }_{{[{\mathcal{M}\dot{F}}_{q_1,r_1}^{p_1,{\tau }_1}({\mathbb{R}}^n)]}^{p_1/p_2}{[{\mathcal{M}}_1^{p_1}({\mathbb{R}}^n)]}^{1-p_1/p_2}},\end{array}$$

Graph

which completes the proof of Theorem 4.14(ii) and hence Theorem 4.14. $\square$

Further remarks

In this section, we give some remarks.

Let $\mathcal{S}({\mathbb{R}}^n)$ be the space of all Schwartz functions on ${\mathbb{R}}^n$ , whose topology is determined by a family of norms, ${\{\Vert ·\Vert }_{{\mathcal{S}}_M({\mathbb{R}}^n)}{\}}_{M\in {\mathbb{Z}}_{+}}$ , where, for any $M\in {\mathbb{Z}}_{+}$ and $\phi \in \mathcal{S}({\mathbb{R}}^n)$ ,

$$\begin{array}{c}\hfill {∥\phi ∥}_{{\mathcal{S}}_M({\mathbb{R}}^n)}:=\underset{\{\alpha \in {\mathbb{Z}}_{+}^n,|\alpha |\le M\}}{sup}\underset{x\in {\mathbb{R}}^n}{sup}\left\{{\left(1,+,|,x,|\right)}^{n+M+|\alpha |},\left|{\partial }^{\alpha },\phi ,(x)\right|\right\}\end{array}$$

Graph

with $|\alpha |:={\alpha }_1+\cdots +{\alpha }_n$ for any $\alpha :=({\alpha }_1,...,{\alpha }_n)\in {\mathbb{Z}}_{+}^n$ . As in [[45], [47]], let

$$\begin{array}{c}\hfill {\mathcal{S}}_{\infty }({\mathbb{R}}^n):=\left\{\phi ,\in ,\mathcal{S},({\mathbb{R}}^n),:,,{\int }_{{\mathbb{R}}^n},\phi ,(x),x^{\gamma },,d,x,=,0,,\text{for any},,\gamma ,\in ,{\mathbb{Z}}_{+}^n\right\},\end{array}$$

Graph

regarded as a subspace of $\mathcal{S}({\mathbb{R}}^n)$ with the same topology. We denote by ${\mathcal{S}}_{\infty }^{\prime }({\mathbb{R}}^n)$ the space of all continuous linear functionals on ${\mathcal{S}}_{\infty }({\mathbb{R}}^n)$ , equipped with the weak- $\ast$ topology. Let $\phi \in \mathcal{S}({\mathbb{R}}^n)$ be such that

5.1 $$\begin{array}{c}\hfill \mathrm{supp}\left(\widehat{\phi }\right)\subset \left\{x,\in ,{\mathbb{R}}^n,:,,\frac{1}{2},\le ,|x|,\le ,2\right\}\end{array}$$

Graph

and there exists a positive constant C satisfying that

5.2 $$\begin{array}{c}\hfill \left|\widehat{\phi },(x)\right|\ge C>0\text{for any}x\in {\mathbb{R}}^n\text{with}\frac{3}{5}\le |x|\le \frac{5}{3},\end{array}$$

Graph

where, for any $\xi \in {\mathbb{R}}^n$ ,

$$\begin{array}{c}\hfill \widehat{\phi }(\xi ):={\int }_{{\mathbb{R}}^n}f(x)e^{-2\pi ix·\xi }dx.\end{array}$$

Graph

For any $t\in (0,\infty )$ , let ${\phi }_t(·):=t^{-n}\phi (·/t)$ . The Littlewood–Paley operators ${\{{\mathrm{\Delta }}_j^{\phi }\}}_{j\in \mathbb{Z}}$ and ${\{{\mathrm{\Delta }}_t^{\phi }\}}_{t\in (0,\infty )}$ of $\phi$ are defined, respectively, by setting, for any $j\in \mathbb{Z}$ , $t\in (0,\infty )$ , and $f\in {\mathcal{S}}_{\infty }^{\prime }({\mathbb{R}}^n)$ ,

5.3 $$\begin{array}{c}\hfill {\mathrm{\Delta }}_j^{\phi }(f):={\phi }_{2^{-j}}\ast f\text{and}{\mathrm{\Delta }}_t^{\phi }(f):={\phi }_t\ast f.\end{array}$$

Graph

Combining the structure of both ${\mathcal{M}\dot{F}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and $\mathcal{M}{\dot{B}}_{q,r}^{p,\tau }({\mathbb{R}}^n)$ and the Littlewood–Paley operators, we next introduce the following two new function spaces.

Definition 5.1

Let $s\in \mathbb{R}$ , $q,r,\tau \in (0,\infty ]$ , and $\phi \in \mathcal{S}({\mathbb{R}}^n)$ satisfy both (5.1) and (5.2).

• The Bourgain–Morrey–Triebel–Lizorkin space $\dot{F}{\mathcal{M}}_{q,r}^{s,\tau }({\mathbb{R}}^n;\phi )$ is defined to be the set of all the $f\in {\mathcal{S}}_{\infty }^{\prime }({\mathbb{R}}^n)$ such that ${\Vert f\Vert }_{\dot{F}{\mathcal{M}}_{q,r}^{s,\tau }({\mathbb{R}}^n;\phi )}<\infty$ , where
• ‖f‖F˙Mq,rs,τ(Rn;φ):=∫0∞ts-nqΔtφ(f)1B(·,t)Lq(Rn)τdtt1τLr(Rn)

Graph

• when $\tau \in (0,\infty )$ and where
• ‖f‖F˙Mq,rs,τ(Rn;φ):=supt∈(0,∞)ts-nqΔtφ(f)1B(·,t)Lq(Rn)Lr(Rn)

Graph

• when $\tau =\infty$ .
• The Bourgain–Morrey–Besov space $\dot{B}{\mathcal{M}}_{q,r}^{s,\tau }({\mathbb{R}}^n;\phi )$ is defined to be the set of all the $f\in {\mathcal{S}}_{\infty }^{\prime }({\mathbb{R}}^n)$ such that ${\Vert f\Vert }_{\dot{B}{\mathcal{M}}_{q,r}^{s,\tau }({\mathbb{R}}^n;\phi )}<\infty$ , where
• ‖f‖B˙Mq,rs,τ(Rn;φ):=∑j∈Z2jsτ∑m∈ZnΔjφ(f)1Qj,mLq(Rn)rτr1τ

Graph

• with the usual modifications made when $q=\infty$ , $r=\infty$ , or $\tau =\infty$ .

Remark 5.2

We should point out that, in Definition 5.1(i), if $\tau =q$ and $r\in (0,\infty )$ , then, in this case, by [[46], Theorem 2.8], we find that $\dot{F}{\mathcal{M}}_{q,r}^{s,q}({\mathbb{R}}^n;\phi )$ goes back to the Triebel–Lizorkin space ${\dot{F}}_{q,r}^s({\mathbb{R}}^n)$ (see also [[45]]). In addition, if $r=q$ in Definition 5.1(ii), then, in this case, $\dot{B}{\mathcal{M}}_{q,q}^{s,\tau }({\mathbb{R}}^n;\phi )$ is just the Besov space ${\dot{B}}_{q,\tau }^s({\mathbb{R}}^n)$ ; see, for instance, [[36], [45]].

We will develop a real-variable theory of the above two new function spaces in a forthcoming article.

Acknowledgements

Pingxu Hu and Yinqin Li would like to thank Jin Tao and Yirui Zhao for many helpful discussions on the subject of this article. The authors would also like to thank the referee for his/her carefully reading and several useful and helpful comments which definitely improve the presentation of this article.

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Data sharing is not applicable to this article as no data sets were generated or analysed.

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