Endpoint L1 estimates for Hodge systems.

Introduction

Let d ≥ 2 . In this paper we give a simple proof of the endpoint Besov-Lorentz estimate ‖ I α F ‖ B ˙ d / (d - α) , 1 0 , 1 (R d ; R k) ≤ C ‖ F ‖ L 1 (R d ; R k) for all F ∈ L 1 (R d ; R k) which satisfy a first order cocancelling differential constraint, where α ∈ (0 , d) and I α is a Riesz potential. We show how this implies endpoint Besov–Lorentz estimates for Hodge systems with L 1 data via fractional integration for exterior derivatives.

In the $L^1$ theory for linear elliptic systems it is quite difficult to obtain better than weak-type bounds. A program in this direction was pioneered in the seminal work of J. Bourgain and H. Brezis [[5]] (see also [[6], [29]]) and received remarkable contributions from Lanzani and Stein [[14]] and Van Schaftingen [[30]–[32]], while endpoint fine parameter improvements on the Lorentz [[10], [25]] and Besov-Lorentz [[28]] scales have only recently been obtained.

The purpose of this paper is to give a simple proof of the Besov-Lorentz estimates obtained in [[28]] for a restricted class of operators and to show how this estimate can be used to resolve several open questions in the theory, in particular estimates for Hodge systems [[31], Open Problems 1 & 2] and the endpoint extension of [[32], Propositions 8.8 & 8.10] in the case of first order operators. Our starting place is an estimate the first and third named authors proved in [[10]], that for $d\ge 2$ and $\alpha \in (0,d)$ there exists a constant $C>0$ for which one has the inequality

1.1 $$\begin{array}{c}\hfill \Vert I_{\alpha }{F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}\le C{\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}\end{array}$$

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for all $F\in L^1({\mathbb{R}}^d;{\mathbb{R}}^d)$ such that $\text{*}\mathit{div}F=0$ . Here $L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)$ is a Lorentz space (see Sect. 2 for a precise definition) and $I_{\alpha }$ denotes the Riesz potential of order $\alpha \in (0,d)$ , defined for $F\in L^1({\mathbb{R}}^d;{\mathbb{R}}^k)$ by

1.2 $$\begin{array}{c}\hfill I_{\alpha }F(x):=\frac{1}{\mathrm{\Gamma }\left(\frac{\alpha }{2}\right)}{\int }_0^{\infty }{t}^{\alpha /2-1}{p}_t\ast F(x)dt\equiv \frac{1}{\gamma (\alpha )}{\int }_{{\mathbb{R}}^d}\frac{F(y)}{{|x-y|}^{d-\alpha }}dy,\end{array}$$

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where $p_t(x):={(4\pi t)}^{-d/2}{exp(-|x|}^2/4t)$ is the heat kernel in ${\mathbb{R}}^d$ and

$$\begin{array}{c}\hfill \gamma (\alpha )=\frac{{\pi }^{d/2}{2}^{\alpha }\mathrm{\Gamma }(\alpha /2)}{\mathrm{\Gamma }\left(\frac{d}{2},-,\frac{\alpha }{2}\right)}\end{array}$$

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is a normalization constant (see, e.g. [[26], p. 117]).

The estimate (1.1) is a partial replacement for the failure of the Hardy–Littlewood–Sobolev embedding in the $L^1$ endpoint, cf. [[26], p. 119], while a comprehensive resolution of the question of a replacement has been given by D. Stolyarov, who in [[28]] (see also [[3], Conjecture 2] where such an inequality was conjectured to hold) establishes the sharper inequality

1.3 $$\begin{array}{c}\hfill \Vert I_{\alpha }{F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\le C{\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^k)}\end{array}$$

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for a very general class of subspaces of $L^1({\mathbb{R}}^d;{\mathbb{R}}^k)$ that includes the kernels of J. Van Schaftingen's class of cocancelling operators [[32]] (see Definition 2.5 below where we recall this class). The argument in [[28]] is quite involved, and it is there commented by Stolyarov that whether the inequality (1.3) admits a simpler proof if one only seeks its validity for the more restrictive class of divergence free measures is unknown. We will shortly give such a proof, which benefited from several insights from his paper and the series of lectures[1] he gave on the topic.

To this end, let us recall the approach to (1.1) in [[10]]: For the space of divergence free measures one finds appropriate atoms, one demonstrates the sufficiency of an estimate on an atom, and one establishes the estimate for a single atom. The atoms in this case are oriented piecewise- $C^1$ loops which satisfy the uniform ball-growth condition: To any oriented piecewise- $C^1$ loop $\mathrm{\Gamma }\subset {\mathbb{R}}^d$ one associates the measure which is given by integration along the curve

1.4 $$\begin{array}{c}\hfill \int \mathrm{\Phi }·d{\mu }_{\mathrm{\Gamma }}:={\int }_0^{|\mathrm{\Gamma }|}\mathrm{\Phi }(\gamma (t))·\dot{\gamma }(t)dt,\end{array}$$

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for $\mathrm{\Phi }\in C_0({\mathbb{R}}^d;{\mathbb{R}}^d)$ , where $\gamma :[0,|\mathrm{\Gamma }|]\to {\mathbb{R}}^d$ is the parametrization of $\mathrm{\Gamma }$ by arclength. The atoms are then such piecewise- $C^1$ closed curves for which

1.5 $$\begin{array}{c}\hfill \Vert {\mu }_{\mathrm{\Gamma }}{\Vert }_{{\mathcal{M}}^1({\mathbb{R}}^d)}:=\underset{x\in {\mathbb{R}}^d,r>0}{sup}\frac{||{\mu }_{\mathrm{\Gamma }}||(B(x,r))}{r}\le \overline{C}\end{array}$$

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for some universal constant $\overline{C}>0$ , where $||{\mu }_{\mathrm{\Gamma }}||$ is the total variation measure of ${\mu }_{\mathrm{\Gamma }}$ . The sufficiency of an estimate on these atoms follows in two steps. First, by Smirnov's integral decomposition of divergence free measures [[21]] one has an approximation of such objects in the strict topology by convex combinations of oriented $C^1$ closed loops. Second, a surgery on such loops shows how any oriented $C^1$ closed loop $\mathrm{\Gamma }$ admits a further decomposition into oriented piecewise- $C^1$ closed loops ${\{{\mathrm{\Gamma }}_i\}}_{i=1}^{N(\mathrm{\Gamma })}$ which satisfy (1.5) with some universal constant and whose total length is bounded by a constant times the length of this loop. This approximation/decomposition and the triangle inequality then yields that it suffices to prove the estimate for a single loop which satisfies the ball growth condition (1.5). Finally, the estimate (1.1) for a single loop was argued in [[10]] by a hands on interpolation that utilizes several pointwise estimates for Riesz potentials and bounds for various maximal functions.

While the argument of (1.1) in [[10]] for a single loop with a ball growth condition involves only estimates for various maximal functions, in this paper we observe that it can be further simplified by the consideration of a very natural stronger quantity that arises in Stolyarov's estimates:

$$\begin{array}{c}\hfill {\int }_0^{\infty }{t}^{\alpha /2-1}{\Vert p_t\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}dt.\end{array}$$

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In particular, in [[28]], Stolyarov shows how if one controls a discrete analogue of (1.6) this implies (1.3). As we will see below in Sect. 2, the continuous version (1.6) also controls the Besov-Lorentz norm and therefore, taking into account the reduction to atoms established in [[10]], for the demonstration of the Besov–Lorentz inequality for divergence free functions it suffices to prove the inequality

1.7 $$\begin{array}{c}\hfill {\int }_0^{\infty }{t}^{\alpha /2-1}\Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}dt\le C||{\mu }_{\mathrm{\Gamma }}||({\mathbb{R}}^d)\end{array}$$

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for all oriented piecewise- $C^1$ closed loops $\mathrm{\Gamma }$ which satisfy (1.5). Let us remark that it is not difficult to see that (1.6) controls the Lorentz norm of the Riesz potential of a function, since this follows directly from the representation (1.2) and Minkowski's inequality for integrals. The argument for the Besov-Lorentz case is only slightly more complicated because of the more technical definition of the space.

We therefore proceed to argue the validity of the inequality (1.7). We claim this follows easily from the estimates

1.8 $$\begin{array}{cc}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le \Vert p_t{\Vert }_{L^1({\mathbb{R}}^d)}||{\mu }_{\mathrm{\Gamma }}||({\mathbb{R}}^d)=||{\mu }_{\mathrm{\Gamma }}||({\mathbb{R}}^d),\hfill \end{array}$$

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1.9 $$\begin{array}{cc}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^{\infty }({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le \Vert p_t{\Vert }_{L^{\infty }({\mathbb{R}}^d)}||{\mu }_{\mathrm{\Gamma }}||({\mathbb{R}}^d)=\frac{c}{t^{d/2}}||{\mu }_{\mathrm{\Gamma }}||({\mathbb{R}}^d),\hfill \end{array}$$

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1.10 $$\begin{array}{cc}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le C_1\frac{||{\mu }_{\mathrm{\Gamma }}||{({\mathbb{R}}^d)}^2}{t^{1/2}}\hfill \end{array}$$

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1.11 $$\begin{array}{cc}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^{\infty }({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le \frac{C_2}{t^{(d-1)/2}}{\Vert {\mu }_{\mathrm{\Gamma }}\Vert }_{{\mathcal{M}}^1({\mathbb{R}}^d)}.\hfill \end{array}$$

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The former two are standard (linear) convolution inequalities for $L^1$ functions, while the latter two are nonlinear and only hold because we consider closed loops oriented by their tangent. Indeed, (1.10) follows from the fact such objects admit a generalized minimal surface spanning $\mathrm{\Gamma }$ , while (1.11) utilizes the fact that we work with curves (and we later make use of the fact that they satisfy (1.5)). Note that if we only utilized (1.8) and (1.9) it would not be sufficient for our purposes, since for any $1\le p\le +\infty$ interpolation would yield the estimate

$$\begin{array}{c}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^{p,1}({\mathbb{R}}^d;{\mathbb{R}}^d)}\le \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}^{\theta }{\Vert p_t\ast {\mu }_{\mathrm{\Gamma }}\Vert }_{L^{\infty }({\mathbb{R}}^d;{\mathbb{R}}^d)}^{1-\theta },\end{array}$$

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where $\theta =1/p$ . In particular, when $p=\frac{d}{d-\alpha }$ , using (1.8) and (1.9) we find

$$\begin{array}{c}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}\le C\frac{||{\mu }_{\mathrm{\Gamma }}||({\mathbb{R}}^d)}{t^{\alpha /2}},\end{array}$$

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which is not good enough to get a finite upper bound, as if utilized to estimate the quantity (1.6) gives a logarithmic divergence at both zero and infinity and therefore cannot yield the inequality (1.7).

The subtlety is to notice that the combination of (1.8) and (1.11) gives an estimate with slightly better behavior at zero, while the combination of (1.9) and (1.10) gives an estimate with slightly better behavior at infinity, the inequalities

1.12 $$\begin{array}{cc}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le C_1^{\prime }\frac{||{\mu }_{\mathrm{\Gamma }}||{({\mathbb{R}}^d)}^{(d-\alpha )/d}}{t^{\alpha (d-1)/2d}},\hfill \end{array}$$

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1.13 $$\begin{array}{cc}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le C_2^{\prime }\frac{||{\mu }_{\mathrm{\Gamma }}||{({\mathbb{R}}^d)}^{1+(d-\alpha )/d}}{t^{\alpha /2+(d-\alpha )/2d}}.\hfill \end{array}$$

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Indeed,

$$\begin{array}{cc}\hfill \alpha (d-1)/2d& <\alpha /2,\hfill \\ \hfill \alpha /2+(d-\alpha )/2d& >\alpha /2,\hfill \end{array}$$

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and therefore it remains to divide the integral so as to linearize the estimate, which follows from dividing at ${|\mathrm{\Gamma }|}^2$ (alternatively, one may first reduce to the case $|\mathrm{\Gamma }|=1$ by dilation, though we here avoided this argument because the nonlinearity of the estimates (1.10) and (1.11) becomes less clear).

We postpone further details until Sect. 2, including the proof of the slightly more technical Besov-Lorentz inequality, so that we can continue to a second purpose of this paper, which is to catalog some implications of the inequality (1.3) in the divergence free case. Indeed, a fundamental contribution of J. Van Schaftingen's paper [[32]] is that divergence free vector fields are generic in the class of vector fields which admit a first order cocanceling annihilator. In particular, following his argument we establish

Theorem 1.1

Let $d\ge 2$ , $\alpha \in (0,d)$ , and suppose L(D) is a first order homogeneous linear partial differential operator acting on vector fields $F:{\mathbb{R}}^d\to {\mathbb{R}}^k$ . Then the estimate

$$\begin{array}{c}\hfill \Vert I_{\alpha }{F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\le C{\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^k)}\text{for}L(D)F=0\end{array}$$

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holds if and only if L(D) is cocanceling, see Definition 2.5.

We recall that the cocanceling assumption is very mild: As was observed in [[9], [18], [32]], failure of this assumption is equivalent to the existence of an unconstrained subspace of L(D)-free fields.

Beyond an intrinsic interest in the mapping properties of fractional integrals, the inequality given in Theorem 1.1 has implications for PDEs. For example, in [[10]] it was demonstrated how (1.1) implies a Lorentz space sharpening of an estimate of Bourgain and Brezis [[4]]: If $F\in L^1({\mathbb{R}}^3;{\mathbb{R}}^3)$ is divergence free, the solution of the Div-Curl system

$$\begin{array}{cc}\hfill \text{*}\mathit{curl}Z& =F\hfill \\ \hfill \text{*}\mathit{div}Z& =0\hfill \end{array}$$

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admits the estimate

$$\begin{array}{c}\hfill {\Vert Z\Vert }_{L^{3/2,1}({\mathbb{R}}^3;{\mathbb{R}}^3)}\le C{\Vert F\Vert }_{L^1({\mathbb{R}}^3;{\mathbb{R}}^3)}\end{array}$$

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for some $C>0$ .

Theorem 1.1 of course implies a similar improvement to this inequality, though in this form is useful for more general applications. For example, we immediately obtain

Corollary 1.2

Let $d\ge 2$ , $\alpha \in (0,d)$ , and $k\in \mathbb{N}\cap [0,d]$ . There exists a constant $C=C(\alpha ,d)>0$ such that for $k\le d-2$

1.14 $$\begin{array}{c}\hfill \Vert I_{\alpha }{du\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k+1}{\mathbb{R}}^d)}\le C{\Vert du\Vert }_{L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k+1}{\mathbb{R}}^d)},\end{array}$$

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while for $k\ge 2$

1.15 $$\begin{array}{c}\hfill \Vert I_{\alpha }{d}^{\ast }{u\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k-1}{\mathbb{R}}^d)}\le C{\Vert d^{\ast }u\Vert }_{L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k-1}{\mathbb{R}}^d)},\end{array}$$

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for all $u\in C_c^{\infty }({\mathbb{R}}^d;{\mathrm{\Lambda }}^k{\mathbb{R}}^d)$ .

Here, for $k\in \mathbb{N}\cap [0,d]$ , ${\mathrm{\Lambda }}^k{\mathbb{R}}^d$ denotes the vector space of k-forms, $C^{\infty }({\mathbb{R}}^d;{\mathrm{\Lambda }}^k{\mathbb{R}}^d)$ denotes the space of functions from ${\mathbb{R}}^d$ to the space of k-forms with smooth coefficients,

$$\begin{array}{cc}\hfill d& :C^{\infty }({\mathbb{R}}^d;{\mathrm{\Lambda }}^k{\mathbb{R}}^d)\to C^{\infty }({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k+1}{\mathbb{R}}^d)\hfill \\ \hfill d^{\ast }& :C^{\infty }({\mathbb{R}}^d;{\mathrm{\Lambda }}^k{\mathbb{R}}^d)\to C^{\infty }({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k-1}{\mathbb{R}}^d)\hfill \end{array}$$

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are the exterior differential and exterior co-differential, respectively, and, with an overloading of notation, $I_{\alpha }du,I_{\alpha }{d}^{\ast }u$ denote the Riesz potential acting on the $k+1,k-1$ -forms $du,d^{\ast }u$ . Precisely, for any l-form $Y\in L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^l{\mathbb{R}}^d)$ , one can express Y in global coordinates as

$$\begin{array}{c}\hfill Y=\sum _{|I|=l}{Y}_{I}d{x}_I\end{array}$$

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where $Y_I\in L^1({\mathbb{R}}^d;\mathbb{R})$ , cf. [[11], p. 237]. Then the Riesz potential of such a Y is given by the formula

1.16 $$\begin{array}{c}\hfill I_{\alpha }Y=\sum _{|I|=l}{I}_{\alpha }{Y}_{I}d{x}_I,\end{array}$$

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where $I_{\alpha }{Y}_I$ is as defined in (1.2). From this one sees $I_{\alpha }Y$ is well-defined for $Y\in L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^l{\mathbb{R}}^d)$ for any $l=1,...,d$ , and in particular that all such Y are in the domain of $I_2\equiv {(-\mathrm{\Delta })}^{-1}$ for $d\ge 3$ .

From Corollary 1.2 one not only obtains improvements to the left-hand-side of estimates for the Div-Curl system, but more generally the Hodge systems considered by Bourgain and Brezis in their paper [[5]] (see also Lanzani and Stein [[14]] for a slicing argument in the spirit of Van Schaftingen's simplification [[32]] of the original argument of Bourgain and Brezis). In particular, we give an affirmative answer to [[31], Open Problems 1 & 2], the following

Theorem 1.3

Let $d\ge 3$ and $k\in \mathbb{N}\cap [1,d-1]$ . If $F\in L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k-1}{\mathbb{R}}^d)$ and $G\in L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k+1}{\mathbb{R}}^d)$ satisfy the compatibility conditions

$$\begin{array}{c}\hfill d^{\ast }F=dG=0,\end{array}$$

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then the function $Z=d{(-\mathrm{\Delta })}^{-1}F+d^{\ast }{(-\mathrm{\Delta })}^{-1}G$ satisfies

$$\begin{array}{cc}\hfill d^{\ast }Z& =F,\hfill \\ \hfill dZ& =G,\hfill \end{array}$$

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and there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill {\Vert Z\Vert }_{{\dot{B}}_{d/(d-1),1}^{0,1}({\mathbb{R}}^d;{\mathrm{\Lambda }}^k{\mathbb{R}}^d)}\le C\left({\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k-1}{\mathbb{R}}^d)},+,{\Vert G\Vert }_{L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k+1}{\mathbb{R}}^d)}\right),\end{array}$$

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where we additionally require $F\equiv 0$ in the case $k=1$ or $G\equiv 0$ in the case $k=d-1$ .

Note that the conditions $d^{\ast }F=dG=0$ are necessitated by properties of the exterior differential and exterior co-differential, $d\circ d=0$ and $d^{\ast }\circ d^{\ast }=0$ , while the fact that the expression $Z=d{(-\mathrm{\Delta })}^{-1}F+d^{\ast }{(-\mathrm{\Delta })}^{-1}G$ is well-defined follows from the assumptions $F\in L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k-1}{\mathbb{R}}^d),G\in L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k+1}{\mathbb{R}}^d)$ and the formula (1.16).

Finally, let us record the following duality estimates, which extend [[32], Propositions 8.8 & 8.10] to the endpoint $q=\infty$ .

Proposition 1.4

Let $d\ge 2$ , $\alpha \in (0,d)$ , and suppose L(D) is a first order cocanceling operator on ${\mathbb{R}}^d$ . Then the estimates for vector fields

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^d}F·\phi dx& \le {C\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^k)}{\Vert D\phi \Vert }_{L^{d,\infty }({\mathbb{R}}^d;{\mathbb{R}}^{k\times d})},\hfill \\ \hfill {\int }_{{\mathbb{R}}^d}F·\phi dx& \le {C\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^k)}{\Vert \phi \Vert }_{{\dot{B}}_{d/\alpha }^{\alpha ,\infty }({\mathbb{R}}^d;{\mathbb{R}}^k)}\hfill \end{array}$$

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hold if $L(D)F=0$ .

Here we use a slightly unusual notation for the Besov spaces ${\dot{B}}_p^{\alpha ,q}$ , which is consistent with our earlier notation. In other words, ${\dot{B}}_p^{\alpha ,q}={\dot{B}}_{p,p}^{\alpha ,q}$ .

The plan of the paper is as follows. In Sect. 2, we first recall the definition of the Lorentz spaces and several results concerning them before we prove Theorem 1.1. In Sect. 3 we prove Corollary 1.2, Theorem 1.3, and Proposition 1.4. In Sect. 4 we address an implicit claim in [[23]] that the estimate in the curl free case was optimal on the Lorentz scale. In particular, we here give a proof of this claim, which in turn, by J. Van Schaftingen's argument implies optimality of the result of the first and third named authors in [[10]] on the Lorentz scale. It is likely these results are optimal on the Besov-Lorentz scale, though we do not have an example which confirms this. Finally, in Sect. 5, we give direct proofs of several of the results for $F\in L^1({\mathbb{R}}^d;{\mathbb{R}}^d)$ such that $\text{*}\mathit{curl}F=0$ . Of course, this is not as general as the divergence free setting, though notably it does not require the surgery construction from [[10]] and therefore provides a streamlined proof for the Lorentz inequality that does not require anything beyond the coarea formula and basic interpolation of Lebesgue or Lorentz spaces.

Lorentz and Besov–Lorentz estimates

We begin by recalling some results concerning the Lorentz spaces $L^{q,r}({\mathbb{R}}^d)$ , where we follow the development of R. O'Neil in [[15]].

Definition 2.1

For f a measurable function on ${\mathbb{R}}^d$ , we define

$$\begin{array}{c}\hfill m(f,y):=|\{|f|>y\}|.\end{array}$$

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As this is a non-increasing function of y, it admits a left-continuous inverse, called the non-negative rearrangement of f, and which we denote $f^{\ast }(x)$ . Further, for $x>0$ we define

$$\begin{array}{c}\hfill f^{\ast \ast }(x):=\frac{1}{x}{\int }_0^x{f}^{\ast }(t)dt.\end{array}$$

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We can now give a definition of the Lorentz spaces $L^{q,r}({\mathbb{R}}^d)$ .

Definition 2.2

Let $1<q<+\infty$ and $1\le r<+\infty$ . We define

$$\begin{array}{c}\hfill {\Vert f\Vert }_{L^{q,r}({\mathbb{R}}^d)}:={\left({\int }_0^{\infty },{\left[t^{1/q},f^{\ast \ast },(t)\right]}^r,\frac{\mathit{dt}}{t}\right)}^{1/r},\end{array}$$

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and for $1\le q\le +\infty$ and $r=+\infty$

$$\begin{array}{c}\hfill {\Vert f\Vert }_{L^{q,\infty }({\mathbb{R}}^d)}:=\underset{t>0}{sup}{t}^{1/q}{f}^{\ast \ast }(t).\end{array}$$

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The Lorentz space $L^{q,r}({\mathbb{R}}^d)$ is defined as

$$\begin{array}{c}\hfill L^{q,r}({\mathbb{R}}^d):=\left\{{f\text{measurable}:\Vert f\Vert }_{L^{q,r}({\mathbb{R}}^d)},<,+,\infty \right\}.\end{array}$$

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For such parameters q, r, these functionals can be shown to be norms and the associated spaces $L^{q,r}({\mathbb{R}}^d)$ Banach spaces (see, e.g., [[27], Chapter V]). Concerning estimates involving the norm for functions in these spaces, a simpler quantity for our purposes is a quasi-norm which does not involve rearrangements:

2.1 $$\begin{array}{c}\hfill {\left|\left|\left|f\right|\right|\right|}_{L^{q,r}({\mathbb{R}}^d)}\equiv q^{1/r}{\left({\int }_0^{\infty },{\left({t|\{|f|>t\}|}^{1/q}\right)}^r,\frac{\mathit{dt}}{t}\right)}^{1/r}.\end{array}$$

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In particular, one can show this is equivalent to the norm on ${\Vert f\Vert }_{L^{q,r}({\mathbb{R}}^d)}$ (see, e.g. [[27], Theorem 3.21 on p. 204]):

Proposition 2.3

Let $1<q<+\infty$ and $1\le r\le +\infty$ . Then

$$\begin{array}{c}\hfill {\left|\left|\left|f\right|\right|\right|}_{L^{q,r}({\mathbb{R}}^d)}\le {\Vert f\Vert }_{L^{q,r}({\mathbb{R}}^d)}\le q^{\prime }{\left|\left|\left|f\right|\right|\right|}_{L^{q,r}({\mathbb{R}}^d)}.\end{array}$$

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Finally, we recall that these spaces support an analogue of Young's convolution inequality, see [[15], Theorem 3.1]:

Theorem 2.4

Let $f\in L^{q_1,r_1}({\mathbb{R}}^d)$ and $g\in L^{q_2,r_2}({\mathbb{R}}^d)$ , and suppose $1<q<+\infty$ and $1\le r\le +\infty$ satisfy

$$\begin{array}{cc}\hfill \frac{1}{q_1}+\frac{1}{q_2}-1& =\frac{1}{q}\hfill \\ \hfill \frac{1}{r_1}+\frac{1}{r_2}& \ge \frac{1}{r}.\hfill \end{array}$$

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Then

$$\begin{array}{c}\hfill {\Vert f\ast g\Vert }_{L^{q,r}({\mathbb{R}}^d)}\le {3q\Vert f\Vert }_{L^{q_1,r_1}({\mathbb{R}}^d)}{\Vert g\Vert }_{L^{q_2,r_2}({\mathbb{R}}^d)}.\end{array}$$

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We next give a few more details of the argument of Theorem 1.1 in the base case $L(D)=\text{*}\mathit{div}$ . Toward the inequality (1.1), as discussed in the introduction, the reduction argument given in [[10]] implies that it suffices to prove (1.7) for every oriented piecewise $C^1$ closed loop $\mathrm{\Gamma }$ that satisfies (1.5). This inequality, in turn, will follow if we can establish the convolution inequalities (1.8), (1.9), (1.10), and (1.11).

For such curves, (1.8) and (1.9) follow from standard convolution inequalities, while we now explain in more detail the inequalities (1.10) and (1.11). The inequality (1.10) follows from the fact that in Euclidean space oriented piecewise $C^1$ closed loops can be identified with integral currents which admit spanning surfaces. In particular, by [[7], 4.2.10], given $T={\mu }_{\mathrm{\Gamma }}$ , there exists a (generalized) surface S which satisfies (in a generalized sense)

2.2 $$\begin{array}{cc}\hfill \partial S& =T\hfill \end{array}$$

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2.3 $$\begin{array}{cc}\hfill ||S||({\mathbb{R}}^d)& \le c||T||{({\mathbb{R}}^d)}^2.\hfill \end{array}$$

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From this, one easily argues the estimate (1.10) by the computation

$$\begin{array}{cc}\hfill \Vert p_t\ast {\mu }_{\mathrm{\Gamma }}{\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}& =\Vert p_t{\ast T\Vert }_{L^1({\mathbb{R}}^d)}\hfill \\ \hfill & =\Vert p_t{\ast \partial S\Vert }_{L^1({\mathbb{R}}^d)}\hfill \\ \hfill & \le \frac{\Vert t^{1/2}|\mathrm{\nabla }{p}_t{|\Vert }_{L^1({\mathbb{R}}^d)}||S||({\mathbb{R}}^d)}{t^{1/2}},\hfill \end{array}$$

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the identity

$$\begin{array}{c}\hfill \Vert t^{1/2}|\mathrm{\nabla }{p}_t{|\Vert }_{L^1({\mathbb{R}}^d)}=c^{\prime }\end{array}$$

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and the isoperimetric inequality (2.3).

Concerning the estimate (1.11), it can be argued even simpler than the ${\mathcal{H}}^1-BMO$ duality utilized to estimate an analogous quantity in [[10]], as it follows from a simple expansion of the convolution on dyadic annuli, using (1.5): In particular,

$$\begin{array}{cc}\hfill |p_t\ast {\mu }_{\mathrm{\Gamma }}(x)|& \le {\int }_{{\mathbb{R}}^d}{p}_t(x-y)d||{\mu }_{\mathrm{\Gamma }}||(y)\hfill \\ \hfill & =\sum _{n\in \mathbb{Z}}{\int }_{B(x,2^n\sqrt{t})\backslash B(x,2^{n-1}\sqrt{t})}{p}_t(x-y)d||{\mu }_{\mathrm{\Gamma }}||(y)\hfill \\ \hfill & \le \sum _{n\in \mathbb{Z}}\frac{1}{{(4\pi t)}^{d/2}}exp(-2^{2n-2}/4){\int }_{B(x,2^n\sqrt{t})\backslash B(x,2^{n-1}\sqrt{t})}d||{\mu }_{\mathrm{\Gamma }}||(y)\hfill \\ \hfill & \le \sum _{n\in \mathbb{Z}}\frac{1}{{(4\pi t)}^{d/2}}exp(-2^{2n-2}/4)||{\mu }_{\mathrm{\Gamma }}||(B(x,2^n\sqrt{t}))\hfill \\ \hfill & \le \sum _{n\in \mathbb{Z}}\frac{C}{{(4\pi t)}^{d/2}}{2}^n\sqrt{t}exp(-2^{2n-2}/4)\hfill \\ \hfill & =\frac{C^{\prime }}{t^{(d-1)/2}},\hfill \end{array}$$

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where C is as in (1.5) and

$$\begin{array}{c}\hfill C^{\prime }:=\sum _{n\in \mathbb{Z}}\frac{C}{{(4\pi )}^{d/2}}{2}^{n}exp(-2^{2n-2}/4).\end{array}$$

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This and the argument of the introduction completes the proof of the Lorentz inequality in the case $L(D)=\text{*}\mathit{div}$ .

Concerning the Besov-Lorentz inequality, we follow the work of Stolyarov [[28]] with a definition of the space ${\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)$ through a minor modification of that for Besov spaces:

$$\begin{array}{c}\hfill {\Vert F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}:=\sum _{n\in \mathbb{Z}}{\Vert F\ast ({\psi }_{2^{n+1}}-{\psi }_{2^n})\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)},\end{array}$$

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where

2.4 $$\begin{array}{c}\hfill {\psi }_r(x)=r^d\psi \left(r,x\right)\end{array}$$

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are dilates of some function $\psi \in \mathcal{S}({\mathbb{R}}^d)$ which satisfies

$$\begin{array}{cc}\hfill \text{*}\mathit{supp}\widehat{\psi }& \subset B(0,1),\hfill \\ \hfill \widehat{\psi }\equiv 1& \text{on}B(0,1/2).\hfill \end{array}$$

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One familiar with Besov spaces [[1], Definition 4.1.2] observes that Besov-Lorentz spaces are defined analogously, only with the replacement of Lebesgue norms in the definition with Lorentz norms. These spaces arise in the real interpolation of Besov spaces [[16]], and have also been called Lorentz-Besov spaces in the monograph of J. Peetre [[17]] (see Example 6 on p. 57 as well as p. 106, 232). While these references are classical, a systematic treatment of these spaces as well various relationships with Triebel-Lizorkin analogues seems to be a recent development: In [[20], equation (1) on p. 1018], A. Seeger and W. Trebels define an inhomogeneous version denoted by $B_1^0[L^{d/(d-\alpha ),1}]$ , while it is remarked in the comments at the end of the introduction there that the results in the paper hold for their homogeneous counterpart, which is denoted by ${\dot{B}}_1^0[L^{d/(d-\alpha ),1}]$ . One can check that the only difference between the definition in [[20]] and ours is the choice of Littlewood-Paley decomposition, provided one utilizes the same norm on the Lorentz space $L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)$ .

When presented with the Lorentz embedding proved in [[10]] and the Besov-Lorentz embedding proved here, a natural question is whether one can deduce one from the other. From the results in [[20]] one understands that the latter is indeed stronger: First, one has the embedding

2.5 $$\begin{array}{c}\hfill {\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)\hookrightarrow L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k).\end{array}$$

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Indeed, with $s=0$ in equation (3) on p. 1018 one finds the identification

$$\begin{array}{c}\hfill {\dot{F}}_2^0[L^{d/(d-\alpha ),1}]\equiv L^{d/(d-\alpha ),1},\end{array}$$

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so that the claim follows by an application of the inhomogeneous variant of Theorem 1.1 (iv) with the choices

$$\begin{array}{cc}\hfill s_0& =s_1=0,\hfill \\ \hfill q_0& =1,\hfill \\ \hfill q_1& =2,\hfill \\ \hfill p_0& =p_1=\frac{d}{d-\alpha },\hfill \\ \hfill r_0& =r_1=1,\hfill \end{array}$$

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as one can check that they satisfy

$$\begin{array}{c}\hfill s_0=s_1,p_0=p_1\ne q_1,r_0\le r_1,q_0\le min\{p_1,q_1,r_1\}.\end{array}$$

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Second, since $s_0=s_1$ and $p_0=p_1\ne q_0$ , Theorem 1.2 (iv) asserts the reverse inclusion can only hold if

$$\begin{array}{c}\hfill q_1=1\ge max\{p_0=\frac{d}{d-\alpha },q_0=2,r_0=1\},\end{array}$$

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which is not valid (and note here the subscripts are opposite those immediately preceding in the invocation of Theorem 1.1 (iv)).

With the definition we have introduced above, we find that we must estimate

$$\begin{array}{c}\hfill \Vert I_{\alpha }{F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}=\sum _{n\in \mathbb{Z}}{\Vert I_{\alpha }F\ast ({\psi }_{2^{n+1}}-{\psi }_{2^n})\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}.\end{array}$$

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In this form, an observation analogous to that of Stolyarov is that if we define the multiplier

$$\begin{array}{c}\hfill \widehat{m}(\xi ):=\frac{\widehat{\psi }(\xi )-\widehat{\psi }(2^{-1}\xi )}{{(2\pi |\xi |)}^{\alpha }exp(-4{\pi }^2{|\xi |}^2)},\end{array}$$

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then, with the use of the notation for scaling introduced in (2.4), one has

$$\begin{array}{c}\hfill I_{\alpha }F\ast ({\psi }_{2^{n+1}}-{\psi }_{2^n})=2^{-n\alpha }{p}_{2^{-2n}}\ast F\ast m_{2^n}.\end{array}$$

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Here we use the fact that $\widehat{m}$ is a Schwartz function to write the expression as a convolution. In particular, the fact that $m\in L^1({\mathbb{R}}^d)$ and the invariance of the space $L^1({\mathbb{R}}^d)$ with respect to the scaling (2.4) implies $m_{2^n}\in L^1({\mathbb{R}}^d)$ with

$$\begin{array}{c}\hfill \Vert m_{2^n}{\Vert }_{L^1({\mathbb{R}}^d)}={\Vert m\Vert }_{L^1({\mathbb{R}}^d)}=:c.\end{array}$$

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By Young's inequality on the Lorentz scale we obtain the bound

$$\begin{array}{c}\hfill \Vert I_{\alpha }F\ast ({\psi }_{2^{n+1}}-{\psi }_{2^n}){\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\le 3\frac{d}{d-\alpha }c{2}^{-n\alpha }{\Vert p_{2^{-2n}}\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)},\end{array}$$

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so that summation over $n\in \mathbb{Z}$ gives the inequality

2.6 $$\begin{array}{cc}\hfill \Vert I_{\alpha }{F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}& \le \frac{3cd}{d-\alpha }\sum _{n\in \mathbb{Z}}{2}^{-n\alpha }{\Vert p_{2^{-2n}}\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}.\hfill \end{array}$$

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The right-hand-side of this inequality is (a constant multiple of) the discrete quantity that Stolyarov obtains an upper bound for in his paper to prove the Besov-Lorentz bound for the general class of subspaces. To pass to the continuous version, we use the semi-group property of the heat kernel and another application of Young's inequality on the Lorentz scale: For each $n\in \mathbb{Z}$ and all $s\in (2^{-2n-2},2^{-2n})$ ,

$$\begin{array}{cc}\hfill \Vert p_{2^{-2n}}{\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}& =\Vert p_{2^{-2n}-s}\ast p_s{\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\hfill \\ \hfill & \le \Vert p_{2^{-2n}-s}\ast |p_s\ast F{|\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\hfill \\ \hfill & \le \frac{3d}{d-\alpha }{\Vert p_s\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}.\hfill \end{array}$$

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In particular, integration from $s=2^{-2n-2}$ to $2^{-2n}$ with respect to the measure ds/s gives the inequality

$$\begin{array}{cc}\hfill \Vert p_{2^{-2n}}{\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}& \le \frac{1}{ln(4)}\frac{3d}{d-\alpha }{\int }_{2^{-2n-2}}^{2^{-2n}}{\Vert p_s\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\frac{\mathit{ds}}{s},\hfill \end{array}$$

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which in combination with (2.6) yields

$$\begin{array}{cc}\hfill \Vert I_{\alpha }{F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}& \le {\left(\frac{3d}{d-\alpha }\right)}^2\frac{c}{ln(4)}\sum _{n\in \mathbb{Z}}{2}^{-n\alpha }{\int }_{2^{-2n-2}}^{2^{-2n}}{\Vert p_s\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\frac{\mathit{ds}}{s}.\hfill \end{array}$$

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By further manipulation we obtain

$$\begin{array}{cc}\hfill \sum _{n\in \mathbb{Z}}& 2^{-n\alpha }{\int }_{2^{-2n-2}}^{2^{-2n}}{\Vert p_s\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\frac{\mathit{ds}}{s}\hfill \\ \hfill & \le \sum _{n\in \mathbb{Z}}{2}^{\alpha }{\int }_{2^{-2n-2}}^{2^{-2n}}{s}^{\alpha /2-1}{\Vert p_s\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}ds,\hfill \end{array}$$

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and thus

2.7 $$\begin{array}{c}\hfill \Vert I_{\alpha }{F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\le \overline{c}{\int }_0^{\infty }{s}^{\alpha /2-1}{\Vert p_s\ast F\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}ds\end{array}$$

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for

$$\begin{array}{c}\hfill \overline{c}:=2^{\alpha }{\left(\frac{3d}{d-\alpha }\right)}^2\frac{c}{ln(4)}.\end{array}$$

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The inequality (2.7) is exactly the control of the Besov-Lorentz norm by the quantity (1.6) claimed in the introduction. In particular, by the argument of the introduction and that preceding in this Section, we have established the estimate claimed in Theorem 1.1 for $F\in L^1({\mathbb{R}}^d;{\mathbb{R}}^d)$ such that $\text{*}\mathit{div}F=0$ .

To conclude the proof of Theorem 1.1, we follow the argument of J. Van Schaftingen in [[32]] that the general case follows by an algebraic reduction. To do this, we first recall a few facts on differential operators. We will work with first order homogeneous linear differential operators with constant coefficients, which can be written as

$$\begin{array}{c}\hfill L(D)F=\sum _{i=1}^d{L}_i{\partial }_{i}F=\sum _{i=1}^d{\partial }_i(L_{i}F),\end{array}$$

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where $L_i\in \text{Lin}({\mathbb{R}}^k;{\mathbb{R}}^l)\simeq {\mathbb{R}}^{l\times k}$ . Due to the usefulness of Fourier transform for linear equations, it is natural to look at the symbol map

$$\begin{array}{c}\hfill L(\xi )=\sum _{i=1}^d{\xi }_i{L}_i\in \text{Lin}({\mathbb{R}}^k;{\mathbb{R}}^l)\text{for}\xi \in {\mathbb{R}}^d.\end{array}$$

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We make the simple observation that we can write

$$\begin{array}{c}\hfill L(D)F=\text{div}(TF),\text{where}TF=(L_{1}F|L_{2}F|...|L_{d}F),\end{array}$$

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so $T\in \text{Lin}({\mathbb{R}}^k,{\mathbb{R}}^{l\times d})$ . The divergence of a matrix field is considered row wise.

We recall the definition of cocancellation:

Definition 2.5

An operator L(D) as above is said to be cocanceling if and only if

$$\begin{array}{c}\hfill \bigcap _{\xi \in {\mathbb{R}}^d}kerL(\xi )=\{0\}.\end{array}$$

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We will show that cocancellation is equivalent with injectivity of the map T defined above. The following lemma should be compared with [[32], Prop. 2.5] and the proof of [[9], lem. 3.11].

Lemma 2.6

We have that

$$\begin{array}{c}\hfill \bigcap _{\xi \in {\mathbb{R}}^d}kerL(\xi )=kerT.\end{array}$$

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To prove this, note that a vector $F\in {\mathbb{R}}^k$ lies in the left hand side if and only if

$$\begin{array}{c}\hfill (TF)\xi =0\text{for all}\xi \in {\mathbb{R}}^d⟺TF=0,\end{array}$$

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which yields the conclusion.

In particular, L(D) is cocanceling if and only if T is left invertible. If this is the case, we can write an explicit left inverse in terms of the adjoint $T^{\ast }$ of T,

$$\begin{array}{c}\hfill T^{†}={(T^{\ast }T)}^{-1}{T}^{\ast }.\end{array}$$

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We can thus proceed with the proof of the main result.

Conclusion of the proof of Theorem 1.1

The necessity of cocancellation follows from by plugging in a Dirac mass in the estimate and noting that $I_{\alpha }\notin L^{d/(d-\alpha )}$ .

Conversely, we already proved the desired estimate for $L(D)=\text{div}$ . We note that if $L(D)F=0$ , we can write $\text{div}(TF)=0$ and $F=T^{†}TF$ , so that

$$\begin{array}{c}\hfill I_{\alpha }F=I_{\alpha }(T^{†}TF)=T^{†}{I}_{\alpha }(TF),\end{array}$$

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and using the inequality for divergence free measures and the fact that $T^{†},T$ are bounded maps on finite dimensional spaces we obtain

$$\begin{array}{cc}\hfill \Vert I_{\alpha }{F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}& =\Vert T^{†}{I}_{\alpha }{(TF)\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\hfill \\ \hfill & \le C\Vert I_{\alpha }{(TF)\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{0,1}({\mathbb{R}}^d;{\mathbb{R}}^{l\times d})}\le C{\Vert TF\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^{l\times d})}\hfill \\ \hfill & \le {C\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^k)},\hfill \end{array}$$

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which completes the proof. $\square$

Hodge systems and duality estimates

We first note that Corollary 1.2 follows from Theorem 1.1 since the $L^1$ vector fields du and $d^{\ast }u$ satisfy the first order conditions $d(du)=0$ and $d^{\ast }(d^{\ast }u)=0$ , which are cocanceling for the claimed ranges of k. This algebraic fact is elementary to check, see also [[32], Prop. 3.3] where the cocancellation of d on $\ell$ -forms is proved, $\ell \le d-1$ . The claim for $d^{\ast }$ follows by duality.

Proof of Theorem 1.3

We first infer from Corollary 1.2 with $\alpha =1$ that

3.1 $$\begin{array}{c}\hfill \Vert I_1{F\Vert }_{{\dot{B}}_{d/(d-1),1}^{0,1}({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k-1}{\mathbb{R}}^d)}\le C{\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k-1}{\mathbb{R}}^d)},\end{array}$$

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3.2 $$\begin{array}{c}\hfill \Vert I_1{G\Vert }_{{\dot{B}}_{d/(d-1),1}^{0,1}({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k+1}{\mathbb{R}}^d)}\le C{\Vert G\Vert }_{L^1({\mathbb{R}}^d;{\mathrm{\Lambda }}^{k+1}{\mathbb{R}}^d)},\end{array}$$

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where $F\equiv 0$ if $k=1$ and $G\equiv 0$ if $k=d-1$ . Next, we note that since the Hodge Laplacian coincides with the real variable Laplacian, we can express

3.3 $$\begin{array}{c}\hfill Z=d{I}_{2}F+d^{\ast }{I}_{2}G=(d{I}_1)I_{1}F+(d^{\ast }{I}_1)I_{1}G,\end{array}$$

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where we used ${(-\mathrm{\Delta })}^{-1}=I_2$ and the semigroup property of Riesz potentials. Note that $d{I}_1,d^{\ast }{I}_1$ give rise to a zero-homogeneous Fourier multiplier, hence can be represented as Calderón–Zygmund operators, and are therefore bounded on the Besov-Lorentz spaces (Here one should be careful to note that these operators are mappings from one exterior algebra into another.). It follows from (3.1), (3.2), and (3.3) that (and for the convenience of display we remove the notation in the norm which details the images of each map)

$$\begin{array}{cc}\hfill {\Vert Z\Vert }_{{\dot{B}}_{d/(d-1),1}^{0,1}({\mathbb{R}}^d)}& \le C\left(\Vert ,(d{I}_1),I_1,{F\Vert }_{{\dot{B}}_{d/(d-1),1}^{0,1}({\mathbb{R}}^d)},+,{\Vert (d^{\ast }{I}_1)I_{1}G\Vert }_{{\dot{B}}_{d/(d-1),1}^{0,1}({\mathbb{R}}^d)}\right)\hfill \\ \hfill & \le C\left(\Vert ,I_1,{F\Vert }_{{\dot{B}}_{d/(d-1),1}^{0,1}({\mathbb{R}}^d)},+,{\Vert I_{1}G\Vert }_{{\dot{B}}_{d/(d-1),1}^{0,1}({\mathbb{R}}^d)}\right)\hfill \\ \hfill & \le C\left({\Vert F\Vert }_{L^1({\mathbb{R}}^d)},+,{\Vert G\Vert }_{L^1({\mathbb{R}}^d)}\right),\hfill \end{array}$$

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which completes the proof. $\square$

Proof of Proposition 1.4

Both inequalities follow from our main result, Theorem 1.1. To prove the first estimate, we observe that the semi-group property of the Riesz potentials and Hölder's inequality on the Lorentz scale implies

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^d}F·\phi dx& ={\int }_{{\mathbb{R}}^d}{I}_{1}F·{(-\mathrm{\Delta })}^{1/2}\phi dx\hfill \\ \hfill & \le C\Vert I_1{F\Vert }_{L^{d/(d-1),1}({\mathbb{R}}^d;{\mathbb{R}}^k)}{\Vert R^{\ast }D\phi \Vert }_{L^{d,\infty }({\mathbb{R}}^d;{\mathbb{R}}^k)},\hfill \end{array}$$

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where we denote by $R^{\ast }$ the adjoint of the Riesz transforms, $R^{\ast }=-\text{*}\mathit{div}{I}_1$ , which satisfies the identity $R^{\ast }D\phi ={(-\mathrm{\Delta })}^{1/2}\phi$ . This inequality, Theorem 1.1, and the bound

$$\begin{array}{c}\hfill \Vert R^{\ast }{D\phi \Vert }_{L^{d,\infty }({\mathbb{R}}^d;{\mathbb{R}}^k)}\le C{\Vert D\phi \Vert }_{L^{d,\infty }({\mathbb{R}}^d;{\mathbb{R}}^{k\times d})}\end{array}$$

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then implies the desired result, the last inequality following from the fact that $R^{\ast }$ is bounded on the Lorentz spaces.

In a similar manner we argue the second inequality of the Proposition. First, by duality we have

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^d}F·\phi dx& \le {C\Vert F\Vert }_{{\dot{B}}_{d/(d-\alpha )}^{-\alpha ,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}{\Vert \phi \Vert }_{{\dot{B}}_{d/\alpha }^{\alpha ,\infty }({\mathbb{R}}^d;{\mathbb{R}}^k)}.\hfill \end{array}$$

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Next, we observe that the definition of ${\dot{B}}_{d/(d-\alpha )}^{-\alpha ,1}({\mathbb{R}}^d;{\mathbb{R}}^k)$ , in analogy with that of the Besov-Lorentz space utilized in Sect. 2, is

$$\begin{array}{c}\hfill {\Vert F\Vert }_{{\dot{B}}_{d/(d-\alpha )}^{-\alpha ,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}=\sum _{n\in \mathbb{Z}}{2}^{-\alpha n}{\Vert F\ast ({\psi }_{2^{n+1}}-{\psi }_{2^n})\Vert }_{L^{d/(d-\alpha )}({\mathbb{R}}^d;{\mathbb{R}}^k)}.\end{array}$$

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Therefore a slight modification of the argument of Theorem 1.1 in Sect. 2 leads to the estimate

$$\begin{array}{c}\hfill {\Vert F\Vert }_{{\dot{B}}_{d/(d-\alpha )}^{-\alpha ,1}({\mathbb{R}}^d;{\mathbb{R}}^k)}\le C^{\prime }{\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^k)},\end{array}$$

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which completes the proof of our second claim. $\square$

Optimality on the Lorentz scale

In a now classical paper on Sobolev embeddings, Alvino [[2]] proved (with sharp constant) that one has

$$\begin{array}{c}\hfill {\Vert u\Vert }_{L^{d/(d-1),1}({\mathbb{R}}^d)}\le C{\Vert \mathrm{\nabla }u\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}.\end{array}$$

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Such an inequality extends to the case Du is a Radon measure, that is, $Du\in M_b({\mathbb{R}}^d;{\mathbb{R}}^d)$ by approximation.

The first result of this Section is a construction which will show the optimality of Alvino's result. Here we should clarify our meaning of optimality. This is the endpoint of where the Lorentz spaces are normable, so that on the scale of normable spaces it is clear this is optimal. We will now show that the result cannot hold for a smaller choice of second parameter in the quasi-norm introduced in (2.1), which is what we intend by the phrase optimality.

Lemma 4.1

For every $q<1$ , there exists a sequence ${\{u_N\}}_{N\in \mathbb{N}}\subset BV({\mathbb{R}}^d)$ with

$$\begin{array}{c}\hfill ||D{u}_N||({\mathbb{R}}^d)\le C\end{array}$$

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independent of $N\in \mathbb{N}$ and

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}{\left|\left|\left|u_N\right|\right|\right|}_{L^{d/(d-1),q}({\mathbb{R}}^d)}=+\infty .\end{array}$$

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Proof

Define

4.1 $$\begin{array}{c}\hfill u_N=\sum _{i=1}^N{h}_i{\chi }_{B(0,r_i)}(x)\end{array}$$

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where $h_i\ge 0$ and $r_i\searrow 0$ will be chosen later such that

$$\begin{array}{c}\hfill ||D{u}_N||({\mathbb{R}}^d)=\sum _{i=1}^N{h}_i{\omega }_d{r}_i^{d-1}\le C\end{array}$$

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independent of N and

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}{\left|\left|\left|u_N\right|\right|\right|}_{L^{d/(d-1),q}({\mathbb{R}}^d)}=+\infty .\end{array}$$

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To this end, we define $H_0:=0$ ,

$$\begin{array}{c}\hfill H_i:=\sum _{j=1}^i{h}_j\end{array}$$

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and compute

$$\begin{array}{cc}\hfill {\left|\left|\left|u_N\right|\right|\right|}_{L^{d/(d-1),q}({\mathbb{R}}^d)}^q& =\frac{d}{d-1}{\int }_0^{H_N}{\left({t|\{|u|>t\}|}^{(d-1)/d}\right)}^q\frac{\mathit{dt}}{t}\hfill \\ \hfill & =\frac{d}{d-1}\sum _{i=0}^{N-1}{\int }_{H_i}^{H_{i+1}}{\left(t,{\omega }_d^{(d-1)/d},r_i^{d-1}\right)}^q\frac{\mathit{dt}}{t}\hfill \\ \hfill & =\frac{d}{d-1}{\omega }_d^{q(d-1)/d}\sum _{i=0}^{N-1}{r}_i^{(d-1)q}{\int }_{H_i}^{H_{i+1}}{t}^{q-1}dt.\hfill \end{array}$$

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Since $q<1$ , $t\mapsto t^{q-1}$ is decreasing and therefore

$$\begin{array}{c}\hfill {\int }_{H_i}^{H_{i+1}}{t}^{q-1}dt\ge H_{i+1}^{q-1}(H_{i+1}-H_i)=H_{i+1}^{q-1}{h}_{i+1}.\end{array}$$

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In particular, with a shift of indices we find

$$\begin{array}{c}\hfill {\left|\left|\left|u_N\right|\right|\right|}_{L^{d/(d-1),q}({\mathbb{R}}^d)}^q\ge \frac{d}{d-1}{\omega }_d^{q(d-1)/d}\sum _{i=1}^N{r}_i^{(n-1)q}{H}_i^{q-1}{h}_i.\end{array}$$

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Therefore it remains to choose $h_i,r_i$ such that

$$\begin{array}{c}\hfill \sum _{i=1}^{\infty }{r}_i^{(d-1)q}{H}_i^q\frac{h_i}{H_i}=+\infty \end{array}$$

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and recall we must do so in a way the ensures

$$\begin{array}{c}\hfill \sum _{i=1}^{\infty }{h}_i{\omega }_d{r}_i^{d-1}\le C.\end{array}$$

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Choose $h_i=2^i$ , so that $H_i=2^{i+1}$ . Thus we now are left to choose $r_i$ such that

$$\begin{array}{c}\hfill \sum _{i=1}^{\infty }{r}_i^{(d-1)q}{H}_i^q\frac{h_i}{H_i}=2^{q-1}\sum _{i=1}^{\infty }{\left(2^i,r_i^{(d-1)}\right)}^q=+\infty \end{array}$$

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and

$$\begin{array}{c}\hfill \sum _{i=1}^{\infty }{2}^i{r}_i^{(d-1)}<+\infty .\end{array}$$

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But then the choice $2^i{r}_i^{(d-1)}=\frac{1}{i^{1/q}}$ is sufficient, as $q<1$ . $\square$

Observe that Lemma 4.1 implies the optimality on the Lorentz scale of Alvino's result [[2]], that the second parameter in the Lorentz estimate cannot be taken less than 1. In fact, taking $N\to \infty$ in the previous proof, we can prove the non inclusion of BV in subcritical Lorentz spaces:

Corollary 4.2

For every $q<1$ , there exists $u\in BV({\mathbb{R}}^d)\backslash L^{d/(d-1),q}({\mathbb{R}}^d)$ .

Proof

Let $u_N$ be the sequence defined in (4.1). Note that

$$\begin{array}{c}\hfill u_N\nearrow u=\sum _{i=1}^{\infty }{h}_i{\chi }_{B(0,r_i)}\text{a.e.}\end{array}$$

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This immediately implies

$$\begin{array}{c}\hfill {\left|\left|\left|u\right|\right|\right|}_{L^{d/(d-1),q}({\mathbb{R}}^d)}\ge {\left|\left|\left|u_N\right|\right|\right|}_{L^{d/(d-1),q}({\mathbb{R}}^d)}\nearrow \infty ,\end{array}$$

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so $u\notin L^{d/(d-1),q}({\mathbb{R}}^d)$ .

Note that since $||D{u}_N||({\mathbb{R}}^d)\le C$ , weak-* compactness in BV implies that, on a subsequence, $u_N\stackrel{\ast }{\rightharpoonup }u$ in $BV({\mathbb{R}}^d)$ . It follows that $u\in BV({\mathbb{R}}^d)$ , which completes the proof. $\square$

For our purposes here it will be useful to observe another consequence of this construction, the following

Lemma 4.3

For every $r<1$ , there exists a sequence ${\{u_N\}}_{N\in \mathbb{N}}\subset BV({\mathbb{R}}^d)$ with

$$\begin{array}{c}\hfill ||D{u}_N||({\mathbb{R}}^d)\le C\end{array}$$

Graph

independent of $N\in \mathbb{N}$ and

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}{\left|\left|\left|I_{\alpha },D,u_N\right|\right|\right|}_{L^{d/(d-\alpha ),r}({\mathbb{R}}^n;{\mathbb{R}}^n)}=+\infty .\end{array}$$

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Proof

We begin with an the inequality for u in terms of potentials and its gradient

$$\begin{array}{c}\hfill |u(x)|\le c{I}_{1-\alpha }|I_{\alpha }Du(x)|.\end{array}$$

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We will estimate u by a standard potential estimate for $I_{1-\alpha }$ . To this end, we recall an estimate of Hedberg, see e.g. [[1], Proposition 3.1.2 (a)] which asserts

$$\begin{array}{c}\hfill |I_{\beta }{f|\le \mathcal{M}(f)}^{1-\beta p/d}{\Vert f\Vert }_{L^p({\mathbb{R}}^d)}^{\beta p/d}.\end{array}$$

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The choice $f=|I_{\alpha }Du(x)|$ , $\beta =1-\alpha$ , and $p=d/(d-\alpha )$ yields

$$\begin{array}{c}\hfill I_{1-\alpha }|I_{\alpha }Du(x)|\le \mathcal{M}(|I_{\alpha }{Du(x)|)}^{(n-1)/(n-\alpha )}\Vert |I_{\alpha }Du{|\Vert }_{L^{d/(d-\alpha )}({\mathbb{R}}^d)}^{1-(d-1)/(d-\alpha )}.\end{array}$$

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By the boundedness of the maximal function (see [[8], Theorem 1.4.19] for the case $q(d-1)/(d-\alpha )<1$ ) and properties of the Lorentz spaces, this shows that

$$\begin{array}{c}\hfill {\left|\left|\left|u\right|\right|\right|}_{L^{d/(d-1),q}({\mathbb{R}}^d)}\le {\left|\left|\left|I_{\alpha },D,u\right|\right|\right|}_{L^{d/(d-\alpha ),q(d-1)/(d-\alpha )}({\mathbb{R}}^n)}^{(d-1)/(d-\alpha )}{\left|\left|\left|I_{\alpha },D,u\right|\right|\right|}_{L^{d/(d-\alpha )}({\mathbb{R}}^d)}^{1-(d-1)/(d-\alpha )}.\end{array}$$

Graph

By the embedding proved in [[19], [23]], this implies

$$\begin{array}{c}\hfill {\left|\left|\left|u\right|\right|\right|}_{L^{d/(d-1),q}({\mathbb{R}}^d)}\le {\left|\left|\left|I_{\alpha },D,u\right|\right|\right|}_{L^{d/(d-\alpha ),q(d-1)/(d-\alpha )}({\mathbb{R}}^d;{\mathbb{R}}^d)}^{(d-1)/(d-\alpha )}||Du||{({\mathbb{R}}^d)}^{1-(d-1)/(d-\alpha )}.\end{array}$$

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But then for any $r<1$ we may choose

$$\begin{array}{c}\hfill q=r\times \frac{d-\alpha }{d-1}<r<1,\end{array}$$

Graph

and the construction from Lemma 4.1 with this choice of q yields the desired sequence.

$\square$

From this we obtain the optimality of Theorem 1.1 in [[23]]. Here we remark that while compactness properties of bounded sequences in BV again allows one to write down the limit

$$\begin{array}{c}\hfill u=\sum _{i=1}^{\infty }{h}_i{\chi }_{B(0,r_i)}(x)\in BV({\mathbb{R}}^d),\end{array}$$

Graph

the fact that $I_{\alpha }Du\notin L^{d/(d-\alpha ),r}({\mathbb{R}}^d;{\mathbb{R}}^d)$ is not obvious in this case.

Simplifications in the curl free case

That one has Lorentz and Besov-Lorentz embeddings for $F\in L^1({\mathbb{R}}^d;{\mathbb{R}}^d)$ such that $\text{*}\mathit{curl}F=0$ is contained in Theorem 1.1. We here will give an even more direct proof, which improves upon those given in [[12], [22]] and in this less general setting even simplifies some of the argument from Sect. 2 above. In particular, the goal of this Section is to establish

Theorem 5.1

Let $d\ge 2$ and $\alpha \in (0,d)$ . There exists a constant $C=C(\alpha ,d)>0$ such that

$$\begin{array}{c}\hfill {\Vert F\Vert }_{{\dot{B}}_{d/(d-\alpha ),1}^{-\alpha ,1}({\mathbb{R}}^d;{\mathbb{R}}^d)}\le C{\Vert F\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}\end{array}$$

Graph

for all $F\in L^1({\mathbb{R}}^d;{\mathbb{R}}^d)$ such that $\text{*}\mathit{curl}F=0$ .

Theorem 5.1 has an interesting history, as it was pointed out to us by D. Stolyarov that one can deduce it from Theorem 4 of V.I. Kolyada's paper [[12], Theorem 4]. The third named author was not aware of this during the writing of [[23]], which gives a different proof of what is unfortunately a slightly weaker result. We can here rectify this in giving a simpler proof of the same result.

Proof of Theorem 5.1

First we claim that it suffices to prove the estimate

$$\begin{array}{c}\hfill {\int }_0^{\infty }{t}^{\alpha /2-1}\Vert p_t\ast D{\chi }_E{\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}dt\le C||D{\chi }_E||({\mathbb{R}}^d)\end{array}$$

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for all ${\chi }_E\in BV({\mathbb{R}}^d)$ , the space of functions of bounded variation. Indeed, in analogy with the argument in [[23]], for general $u\in BV({\mathbb{R}}^d)$ one begins with the representation

$$\begin{array}{c}\hfill Du={\int }_{-\infty }^{\infty }D{\chi }_{E_s}ds\end{array}$$

Graph

where $E_s:=\{u>s\}$ . With this representation, an application of Minkowski's inequality for integrals and Fubini's theorem yields

$$\begin{array}{cc}& {\int }_0^{\infty }{t}^{\alpha /2-1}{\Vert p_t\ast Du\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}dt\hfill \\ \hfill & \le {\int }_{-\infty }^{\infty }{\int }_0^{\infty }{t}^{\alpha /2-1}{\Vert p_t\ast D{\chi }_{E_s}\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}dtds.\hfill \end{array}$$

Graph

Therefore, if one has established the desired inequality for ${\chi }_E\in BV({\mathbb{R}}^d)$ , the result for general $u\in BV({\mathbb{R}}^d)$ follows from this chain of inequalities and the coarea formula

$$\begin{array}{c}\hfill ||Du||({\mathbb{R}}^d)={\int }_{-\infty }^{\infty }||D{\chi }_{E_s}||({\mathbb{R}}^d)ds.\end{array}$$

Graph

Notice that if one only wants to prove the Lorentz inequality (1.1), or the weaker Lebesgue inequality, the essential ingredients at the point are only Minkowski's inequality for integrals and the coarea formula.

Toward establishing the desired inequality for sets of finite perimeter, we recall again the classical convolution inequalities

5.1 $$\begin{array}{cc}\hfill \Vert p_t\ast D{\chi }_E{\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le \Vert p_t{\Vert }_{L^1({\mathbb{R}}^d)}|D{\chi }_E|({\mathbb{R}}^d),\hfill \end{array}$$

Graph

5.2 $$\begin{array}{cc}\hfill \Vert p_t\ast D{\chi }_E{\Vert }_{L^{\infty }({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le \Vert p_t{\Vert }_{L^{\infty }({\mathbb{R}}^d)}|D{\chi }_E|({\mathbb{R}}^d).\hfill \end{array}$$

Graph

Interpolation of these inequalities alone would not suffice, and so we require two additional inequalities which are special to characteristic functions of sets. It is here that the curl free case is much simpler than the divergence free case, as these two inequalities follow immediately from integration by parts and classical convolution estimates:

5.3 $$\begin{array}{cc}\hfill \Vert p_t\ast D{\chi }_E{\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le \Vert D{p}_t{\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}{\Vert {\chi }_E\Vert }_{L^1({\mathbb{R}}^d)},\hfill \end{array}$$

Graph

5.4 $$\begin{array}{cc}\hfill \Vert p_t\ast D{\chi }_E{\Vert }_{L^{\infty }({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le \Vert D{p}_t{\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}{\Vert {\chi }_E\Vert }_{L^{\infty }({\mathbb{R}}^d)}.\hfill \end{array}$$

Graph

In particular, one does not need to perform surgery, consider a ball growth condition, maximal function estimates, or generalized minimal surfaces.

From here, one interpolates (5.1) and (5.4) to obtain

5.5 $$\begin{array}{c}\hfill \Vert p_t\ast D{\chi }_E{\Vert }_{L^{p,1}({\mathbb{R}}^d;{\mathbb{R}}^d)}\le \frac{C_3}{t^{1/2p^{\prime }}}|D{\chi }_E|{({\mathbb{R}}^d)}^{1/p},\end{array}$$

Graph

where we have used

$$\begin{array}{c}\hfill \Vert p_t{\Vert }_{L^1({\mathbb{R}}^d)}=1,\Vert {\chi }_E{\Vert }_{L^{\infty }({\mathbb{R}}^d)}=1,\text{and}{\Vert D{p}_t\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}=\frac{c}{t^{1/2}}.\end{array}$$

Graph

In a similar manner, interpolation of (5.3) and (5.2) yields

5.6 $$\begin{array}{cc}\hfill \Vert p_t\ast D{\chi }_E{\Vert }_{L^{p,1}({\mathbb{R}}^d;{\mathbb{R}}^d)}& \le \frac{C_4}{t^{1/2p+d/2p^{\prime }}}\Vert {\chi }_E{\Vert }_{L^1({\mathbb{R}}^d)}^{1/p}|D{\chi }_E|{({\mathbb{R}}^d)}^{1-1/p}\hfill \end{array}$$

Graph

where we have used

$$\begin{array}{c}\hfill \Vert p_t{\Vert }_{L^{\infty }({\mathbb{R}}^d)}=\frac{1}{{(4\pi t)}^{d/2}}\text{and}{\Vert D{p}_t\Vert }_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}=\frac{c}{t^{1/2}}.\end{array}$$

Graph

As before, we use the fact that in interpolation of Lebesgue spaces, one can improve the second parameter in the interpolation on the Lorentz scale. One can do even better here than one, though below one the estimate is no longer linear and so the result for the curl free case does not follow from the inequality for characteristics functions of sets. Notice that if one only wants the Lebesgue scale inequality, the interpolation is an exercise in Real Analysis.

Finally, we can make the estimate by splitting the integral in two pieces

$$\begin{array}{cc}\hfill {\int }_0^{\infty }& t^{\alpha /2-1}{\Vert p_t\ast D{\chi }_E\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}dt\hfill \\ \hfill & ={\int }_0^r{t}^{\alpha /2-1}{\Vert p_t\ast D{\chi }_E\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}dt\hfill \\ \hfill & +{\int }_r^{\infty }{t}^{\alpha /2-1}{\Vert p_t\ast D{\chi }_E\Vert }_{L^{d/(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}dt\hfill \\ \hfill & =:I+II.\hfill \end{array}$$

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For I, we use the interpolated inequality (5.5) with the choice of $p=\frac{d}{d-\alpha }$ to obtain

$$\begin{array}{cc}\hfill I& \le \frac{C_3}{\mathrm{\Gamma }\left(\frac{\alpha }{2}\right)}|D{\chi }_E|{({\mathbb{R}}^d)}^{(d-\alpha )/d}{\int }_0^r{t}^{\alpha /2-1-\alpha /2d}dt\hfill \\ \hfill & =C_3^{\prime }|D{\chi }_E|{({\mathbb{R}}^d)}^{(d-\alpha )/d}{r}^{\alpha /2-\alpha /2d},\hfill \end{array}$$

Graph

while for II we use the interpolated inequality (5.6), with the same choice of p to obtain

$$\begin{array}{cc}\hfill \mathit{II}& \le \frac{C_4}{\mathrm{\Gamma }\left(\frac{\alpha }{2}\right)}\Vert {\chi }_E{\Vert }_{L^1({\mathbb{R}}^d)}^{(d-\alpha )/d}|D{\chi }_E|{({\mathbb{R}}^d)}^{\alpha /d}{\int }_r^{\infty }{t}^{\alpha /2-1-1/2p-d/2p^{\prime }}dt\hfill \\ \hfill & =C_4^{\prime }\Vert {\chi }_E{\Vert }_{M_b({\mathbb{R}}^d)}^{(d-\alpha )/d}|D{\chi }_E|{({\mathbb{R}}^d)}^{\alpha /d}{r}^{-1/2+\alpha /2d}.\hfill \end{array}$$

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The desired inequality then follows from optimizing in r and the isoperimetric inequality

$$\begin{array}{c}\hfill {|E|}^{1-1/d}\le c_d|D{\chi }_E|({\mathbb{R}}^d).\end{array}$$

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This concludes the proof, the Section, and the paper. $\square$

Acknowledgements

The authors would like to thank Dima Stolyarov for his comments on an early draft of the manuscript, as well as the referees for their careful reading and comments. F.H. is supported by the Fannie & John Hertz Foundation. D. S. is supported by the Taiwan Ministry of Science and Technology under research Grant number 110-2115-M-003-020-MY3 and the Taiwan Ministry of Education under the Yushan Fellow Program. Part of this work was undertaken while D. S. was visiting the National Center for Theoretical Sciences in Taiwan. He would like to thank the NCTS for its support and warm hospitality during the visit.

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By Felipe Hernandez; Bogdan Raiță and Daniel Spector

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