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Anisotropic Bourgain--Brezis inequalities

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Pith's one-line read An anisotropic Hardy–Littlewood–Sobolev endpoint inequality holds for every translation- and dilation-invariant subspace that contains no vectorial delta measure.

desk verdict Genuinely new anisotropic framework and nice corollaries, but the main theorem rests on an unproven reduction to an isotropic dichotomy, so the proof as written does not go through. read the letter →

arxiv 2608.21135 v1 pith:2SRZ2VZS submitted 2026-08-21 math.CA math.AP

Dmitriy Stolyarov This is my paper · ORCID
classification math.CAmath.AP MSC 42B2042B3546E35
keywords anisotropicHardy–Littlewood–SobolevinequalityBourgain–Brezisinequalitiesendpointp=1RieszpotentialsmultiparametricheatextensionBesov–Lorentzspacesvectorialdeltameasurestranslationanddilationinvariantsubspacescancelingdifferentialoperators
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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends the Bourgain–Brezis adjustment of the Hardy–Littlewood–Sobolev inequality, the endpoint case $p=1$, to settings in which different coordinate directions scale at different rates. Its central theorem says that if $W$ is a closed subspace of vector-valued tempered distributions on $\mathbb{R}^d$ that is invariant under translations and anisotropic dilations, and $W$ contains no vectorial delta measure $a\otimes\delta_0$, then the anisotropic Riesz potential $I_\alpha$ maps $W\cap L^1$ continuously into the sharp anisotropic homogeneous Besov space $\dot{B}^{0,1}_q$ with $q=d/(d-\alpha)$. From this, the paper derives continuity into the Lorentz space $L^{q,1}$ and into mixed-norm $L^{\vec q}$ spaces, and it shows by examples that the result recovers and extends known anisotropic Sobolev embeddings and estimates for canceling differential operators. A reader should care because the endpoint $p=1$ is where classical isoperimetric and co-area arguments fail, and the theorem supplies a harmonic-analysis route through anisotropic heat extensions.

What carries the argument

The load-bearing object is the multiparametric heat extension $H[f](x;t_1,\dots,t_d)$, obtained by heating each coordinate separately, together with a monotonicity formula for weighted $L^q$ norms of such extensions (Proposition 2.4 and Corollary 2.7). The anisotropically scaled heat extension at times $A^{-2ka_j}$ produces a Littlewood–Paley decomposition $f_k$; the proof splits the sum over scales into convex atoms, where growth is controlled by an elementary estimate, and flat atoms, where the space $W$ is used. For flat atoms the decisive mechanism is a quantitative separation from delta measures (Proposition 3.6): once an invariant heat-stable cone of measures contains no delta measure, the heat extension of any measure in the cone loses mass strictly slower than that of a delta, in the form of an exponent $\nu>0$ in estimate (3.1.8). That estimate is obtained by reducing the anisotropic monotonicity to the isotropic case of [51], and the resulting decay is propagated through a forest of flat atoms whose roots are controlled by convex neighbors, which completes the estimate.

What would settle it

Look for a concrete counterexample: a closed, translation- and dilation-invariant subspace $W\subset S'(\mathbb{R}^d;\mathbb{R}^\ell)$ with no vectorial delta measure, together with a sequence $f_n\in W\cap L^1$ such that $\|I_\alpha f_n\|_{\dot{B}^{0,1}_q}/\|f_n\|_1\to\infty$ for some $\alpha\in(0,d)$; such a sequence would refute Theorem 1.4. A less global test is to check estimate (3.1.8) directly for a heat-stable invariant cone without deltas: if no exponent $\nu>0$ exists for some such cone, the reduction in Section 3.1 collapses even if the final inequality happens to hold.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a dichotomy: in the presence of translation and dilation invariance, the only obstruction to the endpoint Hardy–Littlewood–Sobolev inequality is a vectorial delta measure. Concretely, the main theorem (Theorem 1.4) claims that for every closed, translation- and dilation-invariant subspace $W\subset S'(\mathbb{R}^d;\mathbb{R}^\ell)$, if $W$ contains no distribution of the form $a\otimes\delta_0$, then the anisotropic Riesz potential $I_\alpha$, defined as Fourier multiplication by $\rho(\xi)^{-\alpha}$, maps $W\cap L^1$ continuously to the anisotropic homogeneous Besov space $\dot{B}^{0,1}_q$, $q=d/(d-\alpha)$, for every $\alpha\in(0,d)$. The paper states that the same conclusion holds for the Lorentz space $L^{q,1}$ (Corollary 1.5) and, via the mixed-norm Hardy–Littlewood–Sobolev inequality, for $L^{\vec q}$ (Corollary 1.4). Corollaries for anisotropic Sobolev spaces, Fourier-constrained spaces, and canceling differential operators follow, including the limiting case $\alpha=d$ in the form $W\cap L^1\hookrightarrow \dot{B}^{-d,1}_\infty$.

Load-bearing premise

The argument stands on a single quantitative dichotomy: a dilation-invariant cone of non-negative measures that contains no point mass must lose heat-extension mass strictly slower than a point mass does, and the anisotropic version of this dichotomy is taken from the isotropic result in [51] and requires the cone built from $W$ to be stable under heat extension.

Editorial extensions

If this is right

  • For every translation- and dilation-invariant subspace $W$ with no vectorial delta measures, the anisotropic Riesz potential is bounded from $W\cap L^1$ to $\dot{B}^{0,1}_q$, hence to $L^{q,1}$; this is strictly stronger than the plain $L^q$ endpoint.
  • The mixed-norm refinement follows: $I_\alpha$ maps $W\cap L^1$ boundedly into $L^{\vec q}$ whenever $\sum_j a_j/q_j = d-\alpha$, for any $1<q_j<\infty$.
  • Known anisotropic Sobolev embedding theorems and the classical div-curl estimates reappear as special cases, together with new inequalities such as $\|f\|_{L^3}\lesssim\|\partial_1 f\|_{L^1}+\|\partial_2^2 f\|_{L^1}$ for the anisotropic gradient operator.
  • For canceling constant-rank differential operators $A(\partial)$, the theorem gives $\|I_\alpha[A(\partial)u]\|_{L^{q,1}}\lesssim\|A(\partial)u\|_1$, and injectively elliptic canceling operators yield $\|\partial^s u\|_{L^{q,1}}\lesssim\|A(\partial)u\|_1$ under the homogeneity condition $\alpha=m-\langle a,s\rangle$.
  • In the limiting case $\alpha=d$, with $I_d$ defined appropriately, $W\cap L^1$ embeds continuously into $\dot{B}^{-d,1}_\infty$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the separation exponent $\nu$ in Proposition 3.6 can be made explicit, the same argument should convert Theorem 4.4 into quantitative lower Hausdorff-dimension bounds for charges in $W$; the paper currently establishes only that some positive dimension threshold exists.
  • The proof's reliance on heat stability suggests a testable extension: constructing a dilation-invariant cone of measures without delta measures that is not heat stable would identify exactly where the compactness step (Theorem 4.1) needs the assumption, and the appendix's examples indicate localization, not heat stability alone, is the fragile point.
  • Because the only property of $\rho^{-\alpha}$ used is homogeneity, the theorem should transfer to any other $(-\alpha)$-homogeneous symbol with the same dilation behaviour, although the limiting $\alpha=d$ case would then genuinely depend on the kernel's cancellation properties.
  • The flat-atom and tree mechanism appears to be a general template: any translation- and dilation-invariant subspace whose cone of measures admits a quantitative separation from delta measures should satisfy an analogous endpoint inequality, so the anisotropic setting is one instance of a broader phenomenon.
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Editorial analysis

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Referee Report

3 major / 5 minor

Summary. The paper proves an anisotropic analogue of the endpoint Hardy--Littlewood--Sobolev/Bourgain--Brezis theorem: for a closed translation-invariant subspace W of S'(R^d;R^ell) that is also invariant under the anisotropic dilations Dil_t of (1.2.1) and contains no vectorial delta measure, the anisotropic Riesz potential I_alpha maps W intersect L^1 continuously into the Besov space dot B^{0,1}_q with q=d/(d-alpha). Corollaries give L^q, L^{q,1}, and mixed-norm L^{vec q} versions, and Appendix B derives applications to anisotropic Sobolev spaces, Fourier-constrained spaces, and differential operators. The proof is structured around multiparametric heat extensions, an anisotropic Littlewood--Paley decomposition, a convex/flat atom decomposition, and a strengthened monotonicity formula (Proposition 3.6), followed by a tree-combinatorial summation argument.

Significance. If the proof is completed, the result is a substantial extension of [51] to non-isotropic homogeneities, a regime where isoperimetric and coarea methods are not available; the sharper Besov--Lorentz targets and the applications to anisotropic differential operators are valuable. The elementary proof of the multiparametric heat monotonicity (Proposition 2.4 and Corollary 2.7) is a useful contribution in its own right, and the paper is honest about the points where it relies on [51]. However, the central reduction in Section 3 currently rests on an under-verified transfer of an isotropic dichotomy to anisotropic-only dilation-invariant cones, and one lemma used in the compactness argument is stated without proof. These are load-bearing gaps rather than presentation issues, so the paper needs a major revision before it can be considered self-contained.

major comments (3)
  1. [Section 3.1, Eqs. (3.1.34)--(3.1.36)] The proof of Proposition 3.6 is the only quantitative separation of the invariant cone M from delta measures, and it closes by asserting that "Proposition 3.10 and the proof of Theorem 3.1 in [51]" convert a violation of (3.1.36) into the existence of a delta measure in M. This transfer is not justified as written. The dichotomy in [51] concerns cones invariant under isotropic (Euclidean) dilations, whereas M in Definition 3.1 is invariant only under the anisotropic Dil_t of (1.2.1). The comparability estimate (3.1.35) shows that the anisotropic derivative functional is comparable to an isotropic one, but it does not make M closed under isotropic blow-ups, which is what the tangent-measure argument in [51] requires. No replacement proof is supplied for cones with only anisotropic dilation invariance. Since Proposition 3.6 is used in Proposition 3.12, Theorem 4.1, Corollary 4.5, and hence in Theorem 1.4, this is a load-bearing gap. The citation is also internally unresolved: no Proposition 3.10 appears in the present manuscript.
  2. [Section 4.1, Lemma 4.1 and Theorem 4.1] Lemma 4.1 is stated with "We omit the tedious proof," yet it is exactly the lemma that provides the uniform Lipschitz bound used in Theorem 4.1 to conclude precompactness of the sequence {f^n_K} in the weighted L^1 space (paragraph around (4.1.13)--(4.1.16)). The reference to Lemma 12 in [51] covers a related but not identical statement, and the hypotheses here involve the anisotropic weights and multiparametric heat extension. Since this compactness step is the bridge from the assumed failure of (4.1.9) to the limiting rank-one object h in M_W, the proof as written is not self-contained at a critical point. Please supply a proof of Lemma 4.1, or a precise statement and verification of the quoted lemma from [51].
  3. [Section 4.1, Proposition 4.2 and Remark 4.3] Proposition 4.2, which is used in Corollary 4.5 and Theorem 4.3, is derived entirely from the "second part of Theorem 5 in [51]" without stating that theorem. Remark 4.3 asserts, on the basis of a five-line inspection, that this part needs neither delta_0 notin M_W nor f in W. This is a load-bearing import: if the quoted theorem's hypotheses differ in the vector-valued, weighted, or flatness conditions, the estimate (4.1.34) collapses. Please state the quoted theorem fully and check each hypothesis after the rescaling (4.1.26)--(4.1.29), or give a direct proof.
minor comments (5)
  1. [Section 3.1, line after (3.1.36)] The reference to "Proposition 3.10" is ambiguous because no Proposition 3.10 occurs in this manuscript; if it is a proposition in [51], the citation should be written as [51, Proposition 3.10].
  2. [Appendix A.3, Eq. (A.3.2)] The Besov-Lorentz norm dot B^{beta,1}_{q,r} is defined in (A.3.2), but the statements of Theorem 1.4 and Corollary 1.5 use dot B^{0,1}_q without specifying that r=q; this should be made explicit at the statement of the theorem.
  3. [Definition 2.14] When an atom's parallelepiped is not contained in a single (k-1)-generation parallelepiped, the parent is chosen arbitrarily among intersecting ones; the paper should state that the subsequent tree estimates and the final bound are independent of this choice, or fix a canonical choice.
  4. [Lemma 4.10] The proof of Lemma 4.10 is only sketched, and the sketch uses the notation i -> 0 -> j without first fixing the maximal index in the definition of the graph Gamma_k; the argument would be clearer if the full proof from Lemma 13 of [51] were reproduced in the appendix.
  5. [Lemma 2.6] The proof of the second part of Lemma 2.6 is dismissed as an exercise in standard calculus techniques; for a published proof, a sentence indicating the dominated convergence argument and the local integrability used would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

The anisotropic Bourgain–Brezis theorem is derived by reducing to an independent isotropic result; no step makes the conclusion an input by definition or by construction.

full rationale

No circular step is identifiable. The main claim, Theorem 1.4, is proved by a multiparametric heat-extension argument that ultimately reduces the decisive estimate (Proposition 3.6) to an isotropic dichotomy already proved in the author's earlier work [51]. That cited result is a published theorem whose stated hypotheses are translation and Euclidean-dilation invariant subspaces without vectorial delta measures; it does not include the anisotropic conclusion of the present paper, so citing it is a reduction to an independent external result rather than a renamed or self-defined input. The new multiparametric heat machinery (Proposition 2.4, Lemma 3.8, Propositions 3.12 and 4.4) is developed in the paper and is not defined in terms of the target inequality, and Corollaries 1.4 and 1.5 follow from Theorem 1.4 by the semigroup property and standard embeddings rather than by reusing the conclusion. Appendix B derives examples from the main theorem rather than using them as assumptions, and Appendix C explicitly constructs counterexamples showing that the concentration and tree-splitting hypotheses cannot be dropped. The only potentially fragile point is the transfer, in Proposition 3.6, of the isotropic dichotomy to a cone invariant only under anisotropic dilations; that is a correctness or robustness concern about the cited theorem's applicability, not a circularity, because the cited result is not the same as the anisotropic target and the paper does not fit a parameter to the quantity it later claims to predict.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirical constants or data-dependent parameters appear. The proof introduces technical constants A, epsilon, and theta_i, but these are existentially quantified proof parameters and do not enter the statement as fitted values. The main additional assumptions are the structural invariance of W and the cited isotropic results from [51].

assumptions (5)
  • domain assumption W is a closed translation and dilation invariant subspace of S'(R^d;R^l) not containing vectorial delta measures
    This is the main hypothesis of Theorem 1.4; without it the inequality fails by Lemma 1.2 and the divergence in (1.2.17).
  • standard math The isotropic version of the strengthened monotonicity formula, specifically Proposition 3.10 and Theorem 3.1 of [51]
    Used as a black box in Section 3.1 to prove Proposition 3.6, which provides the quantitative separation of a cone from delta measures.
  • domain assumption The cone M_W is heat stable under the multiparametric heat semigroup
    Stated as Proposition A.1 and proved by convolution approximation; needed to apply Proposition 3.12 to the vector-valued space.
  • standard math Widder's representation theorem for non-negative solutions of the heat equation (Theorem 5 in [62])
    Used in Lemma 2.6 and Corollary 2.7 to translate the monotonicity estimates into PDE form.
  • standard math Combinatorial lemmas 4.10 and 4.12, cited as identical to Lemmas 13 and 15 of [51]
    Their proofs are not reproduced here; the tree decomposition of flat atoms in Section 4.2 relies on them.

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Pith. "Pith review of Anisotropic Bourgain--Brezis inequalities." pith.science (2026). https://pith.science/paper/2SRZ2VZS

@misc{pith2026260821135,
  author       = {Pith},
  title        = {Pith review of: Anisotropic Bourgain--Brezis inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SRZ2VZS}},
  note         = {Machine review of arXiv:2608.21135}
}
read the original abstract

We provide an adjustment of the Hardy--Littlewood--Sobolev inequality for p=1 to the anisotropic setting. Several examples of anisotropic Bourgain--Brezis inequalities are obtained as corollaries of the main theorem.

Figures

Figures reproduced from arXiv: 2608.21135 by the authors.

Figure 1
Figure 1. A function f with its dilations Dil1/2 f and Dil1/2 f. We will be using the following normalization of the Fourier transform: ˆf(ξ) = Z Rd f(x)e −2πi⟨x,ξ⟩ dx, ξ ∈ R d , f ∈ L1(R d ). (1.2.9) Consider an anisotropic version of the Riesz potential Iα, α ∈ (0, d), defined as the Fourier multiplier Iα f = h (ρ(·))−α ˆf(·) i ˇ , f ∈ L1(R d ). (1.2.10) For the properties of the classical isotropic Riesz potentials, see Se… view at source ↗
Figure 2
Figure 2. Classical 3-adic squares, a = (1, 1), A = 3, and anisotropic rectangles, a = (2/3, 4/3), A = 33/4 ; each rectangle Qk,j is tiled by the parallelepipeds of (k+2)th generation in this case. If we choose A = 33/2 for the second anisotropy, the A-adic parallelepipeds form a tree-like structure. We will still need the tree structure and define it in the most natural way. Definition 2.14. Let (k, j) be an atom and let k ≥… view at source ↗
Figure 3
Figure 3. Illustration to the proof of Proposition 3.12. Here [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

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