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One-sided Rellich inequalities, Regularity problem and uniform rectifiability

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Pith's one-line read In rough planar domains, a one-sided Rellich inequality characterizes uniform rectifiability.

desk verdict A genuinely new endpoint result in rough domains, with one openly flagged but load-bearing gap in the weak-L1 regularity proof. read the letter →

arxiv 2506.03431 v1 pith:ME3GB3NK submitted 2025-06-03 math.AP math.CA

Josep M. Gallegos This is my paper
classification math.APmath.CA MSC 35J2528A7542B20
keywords one-sidedRellichinequalityuniformrectifiabilityHajłasz-SobolevgradientweakL1regularityproblemweak-no-boxesconditionfour-cornersCantorsetDirichletnontangentialmaximaloperator
verification ladder T0 review T1 audit T2 compute T3 formal

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 establishes an endpoint $L^1$ one-sided Rellich inequality: for a bounded corkscrew domain with uniformly $n$-rectifiable boundary, the weak normal derivative of the harmonic solution with Lipschitz boundary data is a Radon measure whose total variation is bounded by the $L^1$ Hajłasz gradient of the data. This holds with no connectivity or local John assumption, unlike the known $L^p$ versions tied to weak-$A_\infty$ harmonic measure. The paper also proves a converse: if the inequality holds on a corkscrew domain with $n$-Ahlfors regular boundary, the boundary satisfies the weak-no-boxes condition, which in the plane is equivalent to uniform $1$-rectifiability. Finally, it shows weak $L^1$ solvability of the regularity problem on uniformly rectifiable domains, gives a counterexample on the complement of the four-corners Cantor set, and derives that solvability of the Dirichlet problem for general elliptic operators does not imply solvability of the regularity problem.

What carries the argument

The carrying object is the endpoint one-sided Rellich inequality itself, combined with a corona decomposition of the uniformly rectifiable domain into a Carleson-controlled collection of Lipschitz subdomains: on each Lipschitz piece, an $H^1$ version of the classical Rellich inequality converts boundary integrals into $H^1$ norms of tangential derivatives, while an almost harmonic extension carries the estimate across the bad set. On the data side, the $L^1$ norm is reached by decomposing Lipschitz functions in the Hajłasz-Sobolev space $\dot{M}^{1,1}$ into atoms. In the converse direction the mechanism is the weak-no-boxes condition: a set failing it contains a non-Carleson family of empty boxes whose interiors meet the boundary in vertically separated points, and near each box the paper builds a family of uniformly Lipschitz functions with disjoint gradient supports and large normal derivatives; a random-sign sum plus Khintchine's inequality produces a single Lipschitz function for which the Rellich estimate fails. For the weak $L^1$ regularity theorem, the step from the Rellich measure bound to the nontangential maximal estimate is carried by boundedness of layer potentials on uniformly rectifiable boundaries.

What would settle it

Compute the weak-$L^1$ distribution function of $N(\nabla S\delta_y)$ on a uniformly $n$-rectifiable boundary for a Dirac mass $\delta_y$: the measure-valued layer-potential bound used in Theorem 1.6 depends on a claimed replacement of cited function-level estimates, and checking this case with constants independent of $y$ would test whether that replacement is legitimate.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: if $\Omega \subset \mathbb{R}^{n+1}$ is a bounded corkscrew domain with uniformly $n$-rectifiable boundary, then for every Lipschitz function $f$ on $\partial\Omega$, the solution $u_f$ of the Dirichlet problem for the Laplacian has a weak normal derivative $\partial_\nu u_f$ that is a Radon measure satisfying $\|\partial_\nu u_f\|_{\mathcal{M}} \lesssim \|\nabla_H f\|_{L^1(\sigma)}$. The estimate is proved directly at $p=1$ and is one-sided, requiring no comparability between normal and tangential gradients. The free-boundary converse, Theorem 1.2, says that on a corkscrew domain with $n$-Ahlfors regular boundary the validity of this inequality for every Lipschitz datum forces $\partial\Omega$ to satisfy the weak-no-boxes condition; in the planar case this is equivalent to uniform $1$-rectifiability. The paper further proves weak $L^1$ solvability of the regularity problem, $\|N(\nabla u_f)\|_{L^{1,\infty}(\sigma)} \lesssim \|\nabla_H f\|_{L^1(\sigma)}$, on the same class of domains, gives a counterexample on the complement of the four-corners Cantor set, and shows that solvability of the Dirichlet problem for general elliptic operators does not imply solvability of the regularity problem.

Load-bearing premise

The load-bearing premise is the unproved assertion inside the proof of Theorem 1.6 that the single-layer bound for functions, $\|N(\nabla Sf)\|_{L^{1,\infty}} \lesssim \|f\|_{L^1}$, extends to all Radon measures by replacing two cited results from [HMT] with two results from [Tol]; if that replacement is invalid, the weak $L^1$ regularity estimate does not follow.

Editorial extensions

If this is right

  • The endpoint Rellich estimate holds on bounded corkscrew domains with uniformly $n$-rectifiable boundary with no local John or weak connectivity condition.
  • For the Laplacian, the regularity problem is solvable in weak $L^1$ on those domains, and by extrapolation in $L^q$ for $q \in (1-\epsilon, 1)$.
  • In the plane, among corkscrew domains with Ahlfors regular boundary, the inequality for all Lipschitz data is equivalent to uniform $1$-rectifiability.
  • There exists a corkscrew domain with Ahlfors regular boundary, the complement of the four-corners Cantor set, where the weak $L^1$ regularity estimate fails for every elliptic operator in the class.
  • Solvability of the Dirichlet problem for a general elliptic operator does not imply solvability of the regularity problem.

Reading between the lines

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

  • If the asserted replacement in Proposition 2.2 is valid, the measure-valued layer-potential bound is the key hidden ingredient; a direct proof of $\|N(\nabla S\mu)\|_{L^{1,\infty}} \lesssim \|\mu\|_{\mathcal{M}}$ would make Theorem 1.6 self-contained and likely extend the regularity estimate to domains with softer geometric hypotheses.
  • The random-sign construction suggests a quantitative stability statement not stated in the paper: for a boundary that fails the weak-no-boxes condition badly, the best constant in the Rellich inequality should grow with the number of independent box-scales, so the growth against a Carleson defect function is a testable prediction.
  • In dimensions $n \ge 2$, the weak-no-boxes condition is not equivalent to uniform rectifiability, so the paper's converse forces a strictly weaker geometric property in general; the planar corollary is the only case in which the free-boundary characterization is known to be sharp.
  • The separation of Dirichlet and regularity solvability for general elliptic operators suggests that equivalence for the Laplacian depends on special harmonic-measure structure; a natural next question is whether the counterexample can be produced with a real symmetric coefficient matrix rather than the paper's companion construction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies an L^1 endpoint version of the one-sided Rellich inequality for harmonic functions in bounded corkscrew domains with uniformly n-rectifiable boundary, without any connectivity assumption on the domain. The main positive result (Theorem 1.1) states that for every Lipschitz boundary datum f, the weak normal derivative of the harmonic extension is a Radon measure whose total variation is controlled by the L^1 norm of any Hajlasz gradient of f. The paper also proves a weak-L^1 regularity estimate (Theorem 1.6), a converse geometric statement using David and Semmes' weak-no-boxes condition (Theorem 1.2), a planar characterization of uniform 1-rectifiability (Corollary 1.4), and a counterexample on the complement of the 4-corners Cantor set showing that for general elliptic operators the Dirichlet problem does not imply the regularity problem (Propositions 1.8 and 1.9). The proofs follow the strategy of Mourgoglou and Tolsa [MT1], using a corona decomposition into Lipschitz subdomains, an almost harmonic extension, layer potential estimates, atomic decompositions of Hajlasz-Sobolev spaces, and random-sign arguments.

Significance. If the results are correct, they are significant. Theorem 1.1 provides the first one-sided Rellich inequality at the L^1 endpoint that is independent of any connectivity or weak-A-infinity assumptions, and Theorem 1.2 gives a new free-boundary rigidity statement: validity of this endpoint inequality forces the boundary to satisfy the weak-no-boxes condition, which in the plane is equivalent to uniform 1-rectifiability. The weak-L^1 regularity theorem (Theorem 1.6) and its extrapolation to p<1 extend the range of the regularity problem beyond the Hardy-space result of [GMT]. The counterexample on the 4-corners Cantor set and its use to separate Dirichlet and regularity solvability for general elliptic operators answers a question of Kenig and Pipher. The paper is clearly written, uses no fitted parameters, and builds on published black-box results. However, several endpoint estimates involving layer potentials with measure data are asserted rather than proved, and these assertions are load-bearing for the main regularity theorem.

major comments (3)
  1. [Section 2.3, Proposition 2.2] The second estimate, ||N(grad S mu)||_{L^{1,infinity}(sigma)} <= C ||mu||_M for Radon measures mu, is explicitly stated to be unproved in [HMT], and the author only says that two substitutions are needed: [HMT, Prop. 3.19] should be replaced by [Tol, Theorem 2.21] and [HMT, (2.1.4)] by [Tol, Theorem 2.5]. This is a load-bearing step: the proof of Theorem 1.6 applies this estimate to the measure partial_nu u|_partial_Omega, which is a priori only a Radon measure, not an L^1 function. The cited results do not obviously imply the claimed weak-(1,1) estimate for the nontangential maximal function of grad S mu: Tolsa's Theorem 2.21 is an L^2 boundedness statement for non-homogeneous Calderon-Zygmund operators, and Theorem 2.5 is a weak-(1,1) result that would need to be combined with a nontangential maximal function estimate and a vector-valued argument. The manuscript does not supply this argument, so Theorem 1.6 is not justified as written. The author should either prove the estimate in detail or identify a precise published statement that contains it.
  2. [Section 2.3, Proposition 2.3] The proposition states that ||N(d_j D f)||_{L^{1,infinity}(sigma)} <= C ||grad_H f||_{L^1(sigma)} for f in Lip(partial Omega) cap M^{1,1}(sigma), citing [HMT, Prop. 3.37]. The author notes that the result in [HMT] is written for p>1 and in terms of tangential derivatives, and that the tangential derivatives are bounded above by grad_H f. This is not sufficient to justify the endpoint L^{1,infinity} bound: boundedness on L^p for p>1 does not automatically imply weak-type (1,1) for a sublinear operator. Since this estimate is needed for the double-layer contribution in Theorem 1.6, the present derivation is a gap. A direct proof or a precise citation of a weak-(1,1) endpoint result for the gradient of the double layer potential on uniformly rectifiable boundaries is required.
  3. [Section 3.2, Lemma 3.1] The proof claims that for each Lipschitz subdomain Omega_R, the function grad_t ef|_partial Omega_R is a multiple of a Lipschitz H^1 atom, based on the assertions that it is supported in 2B0 cap partial Omega_R and has integral zero over partial Omega_R. The integral identity is not generally true for closed hypersurfaces in dimensions n >= 2: for a smooth compact hypersurface S, one has int_S grad_S phi dH^n = -int_S phi H dH^n, where H is the mean curvature vector, which is not identically zero. Thus grad_t ef typically does not have zero mean, even for compactly supported ef. This claim is used to control the sum of H^1 norms of grad_t ef over the corona pieces, which is essential for Lemma 3.1 and hence for Theorem 1.1. The gap appears repairable by decomposing grad_t ef into its mean-zero part (an atom) and a constant part on each Omega_R, and then controlling the constant part by the same Carleson packing arguments, but this must be written out carefully.
minor comments (5)
  1. [Lemma 2.1] The citation is to '[MT]', which does not appear in the reference list; it should presumably be '[MT1]'.
  2. [Theorem 1.6 and Section 1.1] Theorem 1.6 is stated with the nontangential maximal operator N, while the definition of solvability of the regularity problem in (1.11) uses the modified nontangential maximal operator eN. For harmonic functions N and eN are comparable, but this equivalence is not stated in the theorem or its proof; the terminology should be aligned.
  3. [Proof of Corollary 1.7] The proof says the argument is analogous to [GMT, Theorem 1.3] by substituting Kolmogorov's inequality for Holder's inequality and the weak-L^1 estimate for (R^Delta_p). Since the extrapolation below p=1 is delicate, the author should spell out the localization and extrapolation steps or point to the precise statements in [GMT] that are being adapted.
  4. [Theorem 2.5] The author states that the corona decomposition from [MT1, Section 3] works for n=1 although the original proof is written for n>=2. A brief indication of why the arguments carry over to n=1 would be helpful.
  5. [References] There are numerous typographical errors and inconsistent formatting in the reference list, e.g., 'Laplace's equation', 'el lyptic', and inconsistent use of accents in authors' names. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the main estimates are proved from cited black-box theorems, and the paper's one clearly load-bearing unproved assertion (Proposition 2.2 for measures) is a proof gap rather than a circular reduction.

full rationale

I walked the derivation chain. Theorem 1.1 is proved from Proposition 3.2, which shows the weak normal derivative is a Radon measure with mass controlled by Lip(f) times sigma(B0). Proposition 3.2 rests on Lemma 3.1, whose proof uses the corona decomposition (Theorem 2.5, quoted from [MT1]) and the Hardy-space Rellich inequality in Lipschitz domains (Theorem 2.4, quoted from [DK1]); these are external results, not reformulations of (1.12). The passage from localized estimates to the global L1 estimate uses the atomic decomposition Theorem 2.7 from [GMT]; although [GMT] shares an author, the theorem is a published statement about Hajlasz-Sobolev spaces on Ahlfors-regular metric spaces and does not assume the one-sided Rellich inequality or the regularity conclusion. Theorem 1.6 uses Theorem 1.1 (already proved) together with the layer-potential bounds in Propositions 2.2 and 2.3. The second part of Proposition 2.2, asserting the weak-L1 bound for N(grad S mu) for Radon measures, is explicitly said not to be proved in [HMT] and is justified only by a sketch of substitutions from Tolsa; this is a load-bearing proof gap that should be filled, but it is not a circular step because the asserted bound is not an input to Theorem 1.1 or Theorem 1.6 under another name. The converse direction (Theorem 1.2) constructs bad Lipschitz functions from the failure of WNB and uses Khintchine and Paley-Zygmund; no fitted parameter is renamed as a prediction. I found no equation where a claimed output is identical, by construction, to an input or to a self-citation. Thus the appropriate circularity score is 0; the correctness risk lies in the unproved measure-data estimate in Proposition 2.2, not in circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims are universal estimates with no free parameters. The main burden rests on external deep theorems in geometric measure theory and on one explicitly admitted unproved step (Proposition 2.2, second part). The four-corners Cantor set and the boxes are geometric constructions from prior literature, not new invented entities.

assumptions (7)
  • domain assumption Corona decomposition of Ω into Lipschitz subdomains with Carleson packing (Theorem 2.5, from [MT1]).
    Used in Section 3 to build the almost harmonic extension. The author asserts the n=1 case follows from [MT1]'s n≥2 argument without carrying out the proof.
  • standard math Boundedness of Riesz transforms and layer potentials on uniformly rectifiable sets ([DS1], [NTV], [HMT]).
    Propositions 2.2 and 2.3 are used in the proof of Theorem 1.6.
  • standard math Atomic decomposition of M^{1,1} (Theorem 2.7, from [GMT]).
    Used to reduce Theorem 1.1 to atoms. This is a self-cited published theorem, treated as independent support.
  • ad hoc to paper Asserted modification of [HMT] for N(∇Sμ) with measure data (Proposition 2.2, second part).
    The author states this is not proved in [HMT] and only sketches the changes needed. It is load-bearing for Theorem 1.6.
  • domain assumption Boundary Harnack principle and Green function comparability in uniform domains (Theorems 2.10 and 2.11).
    Used in Lemma 4.2 for the four-corners Cantor set counterexample.
  • standard math David-Semmes theorem: for 1-Ahlfors regular sets, the weak-no-boxes condition is equivalent to uniform 1-rectifiability (Theorem 5.4).
    Used to obtain Corollary 1.4.
  • standard math Existence of subcube families with the properties in Section 5.4 via [NTV, Section 16].
    Used to build the random bad functions in the proof of Theorem 1.2.

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Cite this review

Pith. "Pith review of One-sided Rellich inequalities, Regularity problem and uniform rectifiability." pith.science (2026). https://pith.science/paper/ME3GB3NK

@misc{pith2026250603431,
  author       = {Pith},
  title        = {Pith review of: One-sided Rellich inequalities, Regularity problem and uniform rectifiability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ME3GB3NK}},
  note         = {Machine review of arXiv:2506.03431}
}
abstract

Let $\Omega\subset \mathbb R^{n+1}$, $n\geq1$, be a bounded open set satisfying the interior corkscrew condition with a uniformly $n$-rectifiable boundary but without any connectivity assumptions. We establish the estimate $$ \Vert \partial_\nu u_f \Vert_{M} \lesssim \Vert \nabla_H f \Vert_{L^1(\partial\Omega)}, \quad \mbox{for all $f\in\operatorname{Lip}(\partial\Omega)$} $$ where $u_f$ is the solution to the Dirichlet problem with boundary data $f$, $\partial_\nu u_f$ is the normal derivative of $u_f$ at the boundary in the weak sense, $\Vert \cdot \Vert_{M}$ denotes the total variation norm and $\nabla_H f$ is the Haj{\l}asz-Sobolev gradient of $f$. Conversely, if $\Omega\subset \mathbb R^{n+1}$ is a corkscrew domain with $n$-Ahlfors regular boundary and the previous inequality holds for solutions to the Dirichlet problem on $\Omega$, then $\partial\Omega$ must satisfy the weak-no-boxes condition introduced by David and Semmes. Hence, in the planar case, the one-sided Rellich inequality characterizes the uniform rectifiability of $\partial\Omega$. We also show solvability of the regularity problem in weak $L^1$ for bounded corkscrew domains with a uniformly $n$-rectifiable boundary, that is $$\Vert N(\nabla u_f) \Vert_{L^{1,\infty}(\partial\Omega)} \lesssim \Vert \nabla_H f\Vert_{L^1(\partial\Omega)},\quad \mbox{for all $f\in\operatorname{Lip}(\partial\Omega)$}$$ where $N$ is the nontangential maximal operator. As an application of our results, we prove that for general elliptic operators, the solvability of the Dirichlet problem does not imply the solvability of the regularity problem.

Figures

Figures reproduced from arXiv: 2506.03431 by the authors.

Figure 1
Figure 1. Graphical representation of the standard box M(ϵ) in the plane with ϵ = 1/7. The box M has width 2 and height 10ϵ. Let E be an n-Ahlfors regular set in R n+1. For each ϵ > 0, let G(ϵ) denote the set of points (x, t) ∈ E × R+ for which it is impossible to find a translation and dilation of the standard ϵ-box M contained in B(x, t) such that ( diam M ≥ ϵt, M ∩ E = ∅, but E intersects the content of M. Definition 5.1. … view at source ↗
Figure 2
Figure 2. Example of the set PQ in the first case. Here ∂Ω is the black set and PQ the grey set. Let uQ be the solution of the Dirichlet problem on Ω with boundary data fQ [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figure 3
Figure 3. Example of the set PQ formed by the ball BQ and the two cylinders C1 and Cn+1 in the third case. Let F = Sk i=1 Fi ∪ {Q0} and consider the construction of fQ’s for Q ∈ F from Section 5.2. For the sake of contradiction, we also assume that the one-sided Rellich inequality (1.12) holds for all f Lipschitz. In particular, we suppose ∂νufQ is a Radon measure. Lemma 5.11. Fix j ∈ {1, . . . , k}. For a random choice of si… view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions

    math.CA 2026-07 unverdicted novelty 1.0 of 10

    A field survey showing that quantitative rectifiability underpins results on Riesz transforms, removable sets, harmonic measure, and L^p boundary value problems for the Laplacian.

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