Sbornik: Mathematics, 2022, Volume 213, Issue 6, Pages 844–889 DOI: https://doi.org/10.1070/SM9645 (Mi sm9645)

DOI: https://doi.org/10.1070/SM9645

Funding agency	Grant number
Russian Foundation for Basic Research	18-31-00037 ���_�
This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 18-31-00037-���_�).

Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 6, Pages 125–174
DOI: https://doi.org/10.4213/sm9645

Bibliographic databases:

§ 1. Introduction

1.1. The main theorem

Let $\mathcal{S}(\mathbb{R}^d,\mathbb{R}^\ell)$ be the Schwartz class of $\mathbb{R}^\ell$-valued functions in $d$ variables and let $\mathcal{S}'(\mathbb{R}^d,\mathbb{R}^\ell)$ be the dual space of tempered $\mathbb{R}^\ell$-valued distributions.

Our main result stated below may be thought of as a substitute for the classical Hardy-Littlewood-Sobolev inequality at the endpoint case when $p=1$ (in the present text $p$ will stand for the parameter on the left-hand side of the inequality, which is usually denoted by $q$; the summability parameter on the right-hand side is always equal to $1$ in our considerations).

Theorem 1 (and a stronger version of it, Theorem 2 below) finds many applications in the theory of inequalities involving vectorial differential operators and $L_1$ norms (so-called Bourgain-Brezis inequalities).

Our main aim is to demonstrate the strength of the martingale approach proposed in [4] in application to such type inequalities and related problems. In particular, this approach allows one to get rid of the differential or Fourier structure (though the Fourier transform is implicitly present in Theorem 1 via translation invariance); Theorem 1 applies to various versions of Hardy spaces as well. From a pure analysis point of view, the main advantage of the martingale approach lies in its sharpness: we will push to the limit the interpolation parameters on the left-hand side of such inequalities, thus solving several open problems in the field. The presentation here is slightly more condensed than in the preprint [52]: the introduction is different, and also some technical lemmas in the main body of the paper do not have proofs. An interested reader can find these proofs in the preprint. Before passing to a more detailed exposition, we give two examples.

The reader may ask: why is it so important to improve from $L_p$ to $L_{p,1}$? The difference between Lorentz and Lebesgue spaces can be emphasized via a version of Hardy’s inequality (which follows from the embedding into $L_{d/(d-\alpha),1}$ since ${|\,{\cdot}\,|^{-\alpha} \in L_{d/\alpha,\infty}}$ and does not follow directly from the embedding into $L_{d/(d-\alpha)}$ since $|\,{\cdot}\,|^{-\alpha}\notin L_{d/\alpha}$).

The author believes that the main advantage is that Theorem 1 goes beyond the differential structure described in the examples above. It shows that the phenomenon of Bourgain-Brezis inequalities belongs to the field of Harmonic Analysis rather than classical real variable analysis of differential operators. Theorem 1 says that the spaces $\mathcal{W}$ that do not contain delta measures serve as ‘Hardy spaces for fractional operators’. It is high time to introduce a Littlewood-Paley decomposition that will enable one to strengthen Theorem 1.

1.2. Littlewood-Paley decompositions and the Besov scale

We will be using Besov-Lorentz spaces since they arise naturally in our considerations. For details on Besov-Lorentz spaces, see ^1[x]1There are many reference books on the classical Besov spaces, for example, [1], [8] or [18]; Besov-Lorentz spaces seem to be omitted everywhere except [33]. The Besov-Lorentz spaces with $p \in (1,\infty)$ possess the same properties as the usual Besov spaces. Ch. 3 in [33]; we provide a brief outline only (in fact, we do not use any delicate properties of these spaces and they do not appear anywhere except the introduction, § 1). We will not need the full scale, but only the space $\dot{B}_{p,1}^{0,1}$. The norm in this space is defined by

The limit relations

One can use the embedding $\dot{B}_{d/(d-\alpha),1}^{0,1} \hookrightarrow \dot{B}_{d/(d-\alpha)}^{0,1}$ (which follows from $L_{p,1}\hookrightarrow L_p$) to obtain yet another corollary that solves Open Problem 8.2 in [42] (see also Open Problem 1 in [41]).

Theorem 2 has other applications, for example, it implies the classical Hardy inequality (going back to [19])

Theorem 2 leads to yet another corollary in the spirit of Hardy’s inequality; in the inequality below the symbol $\sigma_r$ denotes the $(d-1)$-Hausdorff measure on the sphere $\{\zeta \in \mathbb{R}^d\mid|\zeta| = r\}$.

Corollary 3, in particular, leads to the inequality

The appearance of the Hardy space in these considerations is not accidental. In fact, Sobolev-type and Hardy spaces are two examples of Fourier constrained spaces we survey in the forthcoming subsection.

1.3. Fourier constrained spaces

Let $l$ and $d$ be natural numbers. We will work with functions that map $\mathbb{R}^d$ to $\mathbb{C}^l$. We equip the latter space with the standard Euclidean norm on $\mathbb{R}^{2l}$:

Let $k \leqslant l$ be a natural number. Consider a smooth mapping $\Omega\colon S^{d-1}\to G(l,k)$. The notation $S^{d-1}$ and $G(l,k)$ is used for the unit sphere in $\mathbb{R}^d$ and the (complex) Grassmannian, that is, the set of all (complex) linear $k$-dimensional subspaces of $\mathbb{C}^l$. The map $\Omega$ gives rise to a generalization of the Sobolev space

One can prove that the Schwartz functions are dense in $W_1^\Omega$ (see the preprint [52]). A good definition of the distributional space $\mathcal{W}$ related to $\Omega$ was introduced by Ayoush and Wojciechowski in [5]. For any $L \in G(l,k)$, we denote by $\pi_L$ the orthogonal projection of $\mathbb{C}^l$ onto $L$.

This condition was introduced by Roginskaya and Wojciechowski in [38] and Van Schaftingen in [42] independently.

Thus, Theorem 2 implies the theorem below in this case.

1.4. Historical remarks

The Hardy-Littlewood-Sobolev inequality was invented by Sobolev in [45] as a tool to prove what is now called the Sobolev embedding theorem; unfortunately, his method did not work for the limit summability exponent $1$. See Ch. 5 in [50] for the classical Hardy-Littlewood-Sobolev inequality and its applications. The simplest and most natural case mentioned in Example 2 is equivalent via the Calderón-Zygmund theory to the limiting Sobolev embedding for the space $\dot{W}_1^1(\mathbb{R}^d)$. If $\alpha = 1$ and we embed into the Lebesgue space $L_{d/(d-1)}$, we get the classical Gagliardo-Nirenberg inequality obtained by Gagliardo [16] and Nirenberg [32]; see the book [31] for more information about this classical case. The embedding into the best possible Lorentz space $L_{d/(d-1),1}$ was first proved by Alvino in [2] and then rediscovered by Poornima [35] and Tartar (see [58] for more historical remarks on this question). For higher-order smoothnesses and Besov spaces, the complete result was obtained by Kolyada in [25]; see the papers [7] and [46] for earlier results and [48] for a different approach. We note that most of these results are formulated in the more general anisotropic setting, that is, when derivates with respect to different coordinates can have different orders. We refer the reader to the book [8] for details on anisotropic theory.

The inequality

The case of general differential operators as in Example 6, $\alpha = 1$, and the Lebesgue space instead of a Besov-Lorentz one was considered by Van Schaftingen in [42]. See earlier papers [55] (symmetric gradient), [27] (Hodge differentials), [29] (sharp constants in some of these inequalities), [12] (higher order derivatives and related approximation problems) for particular cases and the surveys [43] and [47] for more historical details. The differential operators satisfying (1.8) (see (1.6)) are called cancelling; the ellipticity condition can be replaced by a weaker constant rank condition: see [37]. For results on Hardy’s inequalities as in Corollary 1, see [30] and [14]. For results on Lorentz spaces $L_{d/(d-1),1}$ in the case of first-order operators $A$, see [49]. For generalizations to metric spaces other than $\mathbb{R}^d$, see [15].

There is also a related problem about sharp estimates for the singularities of charges in $\operatorname{BV}^\Omega$. The question is: what is the best possible bound from below for the lower Hausdorff dimension of $\mu \in \operatorname{BV}^\Omega$? It was raised in [38]. We note that if $\Omega$ is purely antisymmetric ((1.7) holds true for any $\zeta \in S^{d-1}$ as in Example 7), then both $\operatorname{BV}^\Omega$ and $W_1^\Omega$ are contained in the real Hardy class $\mathcal{H}_1(\mathbb{R}^d,\mathbb{C}^l)$ by the celebrated Uchiyama theorem [59]. Therefore, any measure $\mu \in \operatorname{BV}^\Omega$ is absolutely continuous with respect to the Lebesgue measure. The question for general $\Omega$ seems to be open. Partial results were obtained in [3], [5], [38] and [54]. The author supposes that the methods in the present text can also help in this related problem; see the preprint [51].

For a related problem on trace inequalities, see [17] (the classical case of the first gradient can be found in [31]). Theorem 2 implies (via trace inequalities for Riesz potentials; see [1]) the following ‘trace’ theorem.

We have already said that the classical embedding theorems (that is, the ones for classical Sobolev spaces) allow anisotropic generalizations (see [25] for the strongest possible result in the classical setting of pure derivatives). Some partial results in the anisotropic setting and spaces of functions in the style of $W_1^\Omega$ were obtained in [23], [24] and [53]; see [24] for applications to questions in Banach space theory (the idea that inequalities of the type we discuss can deliver interesting information about the isomorphic type of Banach spaces goes back to [22] and [28]).

The techniques we use are different from those usually used in the field; however, there is some similarity to [10] and [12]. We rely upon Harmonic Analysis tools such as the time-frequency decomposition and Harnack’s inequality. This allows us to get rid of the differential structure and work in a more general setting of Fourier restrictions. We finish the introduction with the description of our methods.

1.5. Plan of the proof

The paper [4] suggested a discrete model for the problems mentioned above: the spaces $\operatorname{BV}^\Omega$ have relatives in the world of discrete time martingales over regular filtrations. In the discrete model problems are simpler, and the paper [4] contains solutions to them. The approach is based upon four ingredients: the monotonicity formula (in the world of discrete martingales it reduces to a simple form of convexity in $L_p$), splitting into convex and flat atoms, an improvement of the monotonicity formula in the case when the corresponding cancellation condition holds true, and a combinatorial argument.

Our plan is to transfer the approach of [4] to the Euclidean setting, finding appropriate translations for the notions of a martingale, an atom, the monotonicity formula, and other objects in that paper. The translation appears to be not quite literal, so there will be several new entities (the main of these are horizontal graphs in § 6 below: there was no horizontal interaction in the discrete world; this also forces us to introduce another classification of atoms: there will be saturated and nonsaturated atoms).

Section 2 contains our interpretation of the words ‘martingale’ and ‘monotonicity formula’. The first notion is replaced with the heat extension; more specifically, we consider a discrete-time martingale

Section 3 provides an improvement of the Bennett-Carbery-Tao monotonicity formula for rank-one measures in $\mathcal{W}$ when the latter space does not contain delta measures. The idea is that the inequality expressing the monotonicity formula turns to equality only when $f$ is a delta measure. Theorem 4 says that the measures $\mu$ such that $a\otimes \mu \in \mathcal{W}$ are somehow separated from delta-measures, and thus one can improve the monotonicity formula in Proposition 1 for these measures $\mu$. The separation is expressed through the notion of an invariant cone of measures from [36] (we adjust this notion to fit our Schwartz-class approach). Though we do not use the notions of tangent cone or tangent measure, the material in § 3 is reminiscent of some parts of the paper [36].

Section 4 contains our interpretation of the notion ‘atom’. We use a version of a time-frequency decomposition. However, we do not decompose the function over the space, but rather consider its norms in weighted $L_1$-spaces, where the weight is localized in a neighbourhood of an atom. The Uncertainty Principle says that the function $f_k = \operatorname{H}[f](\,\cdot\,, A^{-2k})$ behaves like a function on the lattice $A^{-k}\mathbb{Z}^d$. We exploit this principle by considering the values $\|f_k\|_{L_1(w_{k,j})}$, where the weight $w_{k,j}$ is concentrated in a neighbourhood of the point $A^{-k}j$, $j \in\mathbb{Z}^d$; the quantity $\|f_k\|_{L_1(w_{k,j})}$ is then treated as the value of the martingale $f$ on the atom $(k,j)$. We define convex and flat atoms similarly to [4] and prove several useful lemmas about our weights. This allows us to estimate the sum over convex atoms in the manner similar to [4]; this estimate is proved in Proposition 3.

Section 5 proposes a compactness argument that allows us to perturb (that is, add some flexibility to) the improvement of the Bennett-Carbery-Tao monotonicity formula obtained in § 3, that is, to prove a similar monotonicity formula for functions $f \in \mathcal{W}$ that are somehow close to rank-one measures; the precise formulation is in Theorem 5. In [4] it was shown that the growth of the $L_p$-norm of a martingale on a flat atom is smaller than the growth of the same quantity for the martingale generated by a delta measure. In the Euclidean case the situation is more complicated since we do not have perfect localization in the space variable (due to the Uncertainty Principle). So we introduce an additional concentration assumption. We prove that if this assumption holds for some atom, and the atom is flat, then the function $f$ is close to being a positive rank-one measure, and its $L_p$-norm in a certain weighted space grows slower than that of a delta measure.

In our reasonings graphs will indicate the subordination of atoms: if there is an arrow from $\mathfrak{A}$ to $\mathfrak{B}$, then some quantity of $\mathfrak{B}$ can be estimated by a similar quantity of $\mathfrak{A}$ uniformly. Section 6 introduces graphs that mark the horizontal subordination of atoms. We study simple combinatorial properties of these graphs and introduce the second classification of atoms (the first being flat/convex). Atoms can be saturated or nonsaturated. For saturated atoms we prove good bounds for the growth of the $L_p$-norm (this follows from Theorem 5 since saturated atoms fulfill the concentration condition in that theorem). What is more, if a nonsaturated atom is subordinate to a saturated one, then there is a certain control of the growth of the $L_p$-norm as the weight drifts from the former atom to the latter one; the precise formulation is in Theorem 6.

The final section, § 7, concludes the proof. We introduce the graph $\Gamma$ similar to the graph in [4]. Here its structure is more complicated, and it is not uniform. In fact, there can be vertices with infinitely many kids. However, $\Gamma$ is still a forest, that is, a union of trees. The improved monotonicity formula allows us to run induction over an individual tree (Proposition 4). Then a combinatorial argument similar to [4] reduces an estimate for a sum over flat atoms to the sum over convex atoms which we have already estimated.

§ 2. Gaussians and monotonicity formulae

Consider the heat extension of a function $f\in L_1(\mathbb{R}^d,\mathbb{R}^\ell)$, that is,

Let $A$ be a large natural parameter to be chosen later. It will be convenient for further considerations to assume that $A$ is odd. We will use the notation

The next lemma and corollary are standard; we provide the proofs since we will use formulae from them.

The unweighted case of this proposition was proved in [6], this is a particular case of Proposition 3.1 in that paper (see Proposition 5 in the blog post [57]). Proposition 1 can be thought of as interpolation between the two extreme cases of $p=1$ and $p=\infty$. It is not quite clear how to interpolate in this context. Throughout the rest of this section we use the notation

§ 3. Improving the monotonicity formula

We will measure the regularity of our weights using a version of the modulus of continuity (or smoothness at infinity).

The next definition is inspired by [36].

This theorem can be thought of as a quantification of Remark 7. The proof is quite lengthy (though fairly straightforward) and occupies the rest of § 3. First, we would like to get rid of time. We recall the function $Q_p = Q_p[\mu,G]$ defined in (2.10).

Before we pass to the proof, we note that if $Q_p(1)$ is finite, then $Q_p(t)$ is finite for any $t\in (0,1)$ by Proposition 1. Therefore, we will always work with finite quantities in the proofs below.

Now we present the proof of Proposition 2, thus completing the proof of Theorem 4.

We need to perturb Theorem 4 slightly.

§ 4. Time-frequency decomposition and control of convex atoms

Recall that our main target is to prove (2.4). We rewrite the right-hand side as a telescopic sum:

Let $\theta_1 > d$ be a number to be specified in what follows. Consider the weight $w$ defined by

Now consider the partition of $\mathbb{R}^d$ into $A$-adic cubes. The cubes $\{Q_{0,j}\}_{j}$ have centres in the lattice $\mathbb{Z}^d$ and tile the whole space (up to a set of measure zero):

We adjust the partition of unity (4.5) to each scale:

The weights just introduced allow us to split the sum (4.2) further:

Let $\varepsilon$ be a small parameter to be specified in what follows.

We will also simply say ‘flat’ and ‘convex’, suppressing the dependence on $\varepsilon$.

Convex atoms are easier to deal with, and in Proposition 3 we establish the ‘half’ of inequality (2.4) that corresponding to convex atoms.

The proof of Proposition 3 needs some preparation and is based upon several useful lemmas. It is presented at the end of § 4.

We need to specify our choice of the Lorentz quasi-norm:

We need to introduce the space $L_{p,1}(v)$. The norm in this space is defined by

Sometimes we will need to keep track of the constants in our inequalities. In fact, only the dependence on $A$ is of crucial importance (a small exception only occurs in § 7 below). We will write $\lesssim_A$ to indicate that the constant implicit in the symbol $\lesssim$ is independent of $A$. To avoid ambiguity we will usually comment on this type of independence.

The last corollary will be needed only in § 6. It is more convenient to present it here. Recall that $\alpha =d(p-1)/p$.

§ 5. Compactness argument

The informal meaning of the flat/convex classification of atoms is that $\varepsilon$-flat atoms mark places in $\mathbb{R}^d$ where the function $f_k$ is close to a positive rank-one function. Recall Lemma 2, which says that the presence of a $0$-flat atom ensures that ${f_{k+3} = a\otimes h}$ with $h \geqslant 0$. So we are looking for a compactness argument which will allow us to replace $0$ with $\varepsilon$, obtain that $f_k$ is somehow close to being a positive rank-one function, and then apply Corollary 6 to it. Unfortunately, this principle seems to fail in general. Imagine a function $f_k$ concentrated far away from the cube $Q_{k,j}$ and taking arbitrary values in this cube. It seems that we can deduce nothing from the flatness of $(k,j)$ since the behavior of $f_k$ in a neighbourhood of $Q_{k,j}$ is not very much related to the weighted norms under consideration.

This hints that our searched-for compactness argument should involve the assumption that $f_k$ is concentrated in a neighbourhood of $Q_{k,j}$. Fix some number $\theta_2$ to be specified below and assume that $\theta_2 < \theta_1$. Consider the weight

The proof of Theorem 5 occupies the rest of § 5. We start with several useful lemmas. Let $\Omega \subset \mathbb{R}^d$ be an open convex set. We define the Lipschitz semi-norm by the formula

Till the end of § 5 we use the notation

In particular, if $\gamma =1/2$, the concentration condition (5.2) implies (5.9) with $f:= f_2$ and $t:=A^{-4}$.

§ 6. Horizontal interaction

Let $(k,j)$ be an atom. It is convenient to introduce the notation

Similar smoothing maximal function were also used in [12]. This particular definition has been borrowed from [56].

We fix $k$, $\theta_4$ and $f$ provisionally. We suppress these parameters in our notation if this does not lead to ambiguity and simply write $f^*_j$ and $\operatorname{M}_j$. The maximal operator we have introduced generates an interesting oriented graph.

Note that each vertex has at most one incoming arrow. Informally, an arrow $\vec j \to j$ signifies that the point $\vec j$ dominates $j$ in the sense that we can estimate the quantity $f_j^*$ uniformly in terms of $f_{\vec j}^*$.

In particular, there are no pairs of arrows $i\to j$ and $j \to i$ in $\widetilde\Gamma_k$, so this is indeed an oriented graph. We fix

Lemmas 13 and 14 imply the following statement.

Recall the weight $u$ defined in (5.1).

§ 7. Vertical interaction and control of flat atoms

Now we introduce a graph $\Gamma$ that expresses the vertical domination of atoms. In this graph arrows always go down, that is, from an atom $(k,j)$ to $(k+1,j')$.

It will be important for our considerations that any two cubes $Q_{k,j}$ and $Q_{k',j'}$ are either disjoint up to a set of measure zero, or one of them contains the other (this follows from the assumption that $A$ is odd).

In Lemma 17 and Corollary 12 we assume that $\varepsilon$ is sufficiently small (depending on $A$).

Let $\delta^{**} = \delta^*/2$. The next two corollaries are immediate consequences of Lemma 17 and do not need proofs.

Now we fix $A$ so large that (7.1) holds true. We fix $\varepsilon$ to be as small as prescribed by Theorems 5 and 6 for our particular choice of $A$.

Let us investigate the roots of our trees.

Recall that, since we have already fixed $A$, all the constants in our inequalities are now allowed to depend on $A$. However, we still care about the uniformity with respect to $N$.

The next corollary follows immediately from Proposition 5: one needs to compute the sum of a geometric series.

Acknowledgements

I would like to express my gratitude to Rami Ayoush and Michal Wojciechowski for long and fruitful collaboration and for sharing their ideas with me. I also wish to thank Daniel Spector for discussions concerning this work.

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Citation in format AMSBIB

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