Solvability of the Poisson-Dirichlet problem in domains with boundaries of mixed dimension

L\'eo Mand\^o

REVIEW 3 major objections 4 minor 12 references

Solvability of the Poisson-Dirichlet problem in domains with boundaries of mixed dimension

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Dirichlet L^p solvability equals Poisson-Dirichlet solvability.

desk verdict Solid extension of the MPT22 equivalence to degenerate operators and mixed-dimensional boundaries, with one load-bearing unpublished dependency that the author must close. read the letter →

arxiv 2608.03373 v2 pith:LI6BU6NH submitted 2026-08-04 math.AP

Léo Mandô This is my paper · ORCID

classification math.AP MSC 35J2535J70

keywords degenerateellipticoperatorsPoisson-DirichletproblemL^psolvabilitytentspacesmixed-dimensionalboundariesmeasureGreenfunctionboundaryPoincaréinequality

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

This paper aims to prove that, for a broad class of possibly degenerate divergence-form elliptic operators $L=-\operatorname{div}(wA\nabla)$ on open sets whose boundaries need not be $(n-1)$-dimensional, the $L^p$-solvability of the homogeneous Dirichlet problem is equivalent to the solvability of the Poisson-Dirichlet problem $Lu=wf-\operatorname{div}(wF)$ with zero boundary data, and also to the Poisson-regularity problem for the adjoint operator. The setting allows boundaries of mixed dimension and drops the quantitative connectedness condition known as the Harnack chain condition, so the result covers operators such as the Caffarelli-Silvestre extension operator. A reader should care because the equivalence means one can test the hard homogeneous problem by solving the softer inhomogeneous problem, and vice versa, and the paper adds a genuinely new characterization even in the previously studied case: solvability with only the $f$-term already implies everything else.

What carries the argument

The machinery is the elliptic measure $\{\omega^x\}_{x\in\Omega}$ together with the Green function $G(x,y)$ of $L$, and the tent-space functionals $\tilde N$ (non-tangential maximal function of ball averages) and $\tilde A$ (area function) defined through cones over boundary points, normalized by the measure ratio $\rho(B)=m(B\cap\Omega)/(r\mu(B\cap\partial\Omega))$. The elliptic measure represents solutions of the homogeneous Dirichlet problem, while the Green function represents solutions of the Poisson-Dirichlet problem as $u(x)=\int_\Omega G(x,y)f\,dm(y)+\int_\Omega \nabla_y G(x,y)\cdot F(y)\,dm(y)$. The load-bearing estimate is the boundary Poincaré inequality, which yields boundary Hölder continuity, non-degeneracy of the elliptic measure, and the two Green-function estimates (3.69)-(3.70). Those estimates are what allow the proof to convert homogeneous solvability into inhomogeneous solvability by splitting $f$ or $F$ into a piece supported near the pole of the Green function, a piece near the boundary point, and a far piece controlled by Hölder decay.

What would settle it

A concrete way to test the claim is to construct an open set satisfying (H1)-(H5) for which the boundary Poincaré inequality (2.31) or the boundary Hölder estimate (2.44) fails; that would break the elliptic theory on which all the Green-function and elliptic-measure estimates rest. Alternatively, one could look for an operator on such a domain whose elliptic measure satisfies the reverse Hölder inequality (3.56) for some $p$ while the homogeneous Dirichlet estimate (1.24) fails for that $p$; Theorem 3.1 says this cannot happen, so any such example would falsify Theorem 1.5.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.5: under assumptions (H1)-(H5), for any $1<p<\infty$, the following six statements are equivalent: the $L^p$ solvability of the homogeneous Dirichlet problem $(D_p)$; the solvability of the Poisson-Dirichlet problem $(PD_p)$; $(PD_p)$ with $F=0$; $(PD_p)$ with $f=0$; the Poisson-regularity problem for the adjoint $(PR^*_{p'})$; and $(PR^*_{p'})$ with $F=0$. In other words, the non-tangential maximal estimate for solutions of $Lu=0$ with boundary data in $L^p$ is exactly as strong as the corresponding estimate for solutions of $Lu=wf-\operatorname{div}(wF)$ with zero boundary values, and exactly as strong as a gradient estimate for the adjoint problem. The proof establishes two intermediate characterizations: $(D_p)$ holds if and only if the elliptic measure satisfies a reverse Hölder inequality with respect to the boundary measure, and if and only if the Green function satisfies the estimates (3.69) and (3.70) involving $\rho\nabla G$ and $(\rho/\delta)G$. The equivalence then follows by decomposing the interior data into pieces near the pole, near the boundary, and far away.

Load-bearing premise

The load-bearing premise is that the standard elliptic theory—the Green function representation, boundary oscillation and Hölder estimates, the elliptic measure and its non-degeneracy, and the maximum principle—is valid in this no-Harnack-chain, mixed-dimensional setting even though the paper only cites it from an unpublished book in preparation and says the arguments adapt from a prior framework because they rely only on a boundary Poincaré inequality. If that adaptation is wrong, or if the cited book never appears, the main theorem lacks its foundation.

Editorial extensions

If this is right

If Theorem 1.5 is right, then for this class of degenerate operators on mixed-dimensional boundaries, verifying the homogeneous Dirichlet problem automatically gives $L^p$ estimates for the inhomogeneous equation $Lu=wf-\operatorname{div}(wF)$ with zero boundary data.
The new characterization (iii) means that divergence-form data $F$ is redundant for testing solvability: it is enough to test the Poisson-Dirichlet problem with only the $f$-term.
By duality, solvability of the homogeneous Dirichlet problem for $L$ is equivalent to the Poisson-regularity problem for the adjoint $L^*$, so a gradient estimate for the adjoint is the same as a non-tangential estimate for $L$.
Corollaries 4.1 and 4.2 upgrade the equivalence to an existence theorem: under the same conditions, weak solutions exist for general boundary data $g\in L^p(\partial\Omega,\mu)$ and interior data in the weighted tent spaces, with the expected non-tangential and gradient estimates.

Reading between the lines

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

If the foundational elliptic theory holds as claimed, the same equivalence is likely to extend to other boundary conditions, such as Neumann or Robin problems, in this mixed-dimensional setting without connectedness, paralleling recent results in chord-arc domains.
The redundancy of $F$ in characterization (iii) suggests a practical shortcut: to test $L^p$ solvability of an operator numerically or probabilistically, one can probe it with scalar weights $f$ only and ignore vector-valued divergence data.
The measure ratio $\rho$ and its growth condition (H4) look like the right quantitative replacement for the usual $(n-1)$-dimensional normalization; a natural testable question is whether the theorem survives under a two-sided bound on $\rho$ or under an interior-only Poincaré inequality, as the paper itself flags.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Core result is a real extension: Theorem 1.5 moves the Dirichlet/Poisson-Dirichlet/regularity equivalence from the uniformly elliptic, (n-1)-Ahlfors-regular setting of [MPT22] to degenerate w-weighted operators on mixed-dimensional boundaries, and it drops the Harnack chain condition. The new implication (iii) is genuinely new even in the old setting. The proof strategy—tent-space duality plus a boundary Poincaré inequality—is coherent, and Lemma 3.2 is a substantial piece of work: removing quantitative connectedness is exactly where the old arguments needed the most care. The appendix is a genuine service: it makes the tent-space machinery self-contained for this geometry. This is not a paper that hides its argument behind 'by standard methods'; the main line is visible.

The soft spots are real but curable. The foundational elliptic theory—boundary oscillation, boundary Hölder continuity, Green function representation, elliptic measure construction, maximum principle—is imported from the unpublished book [FM] 'in preparation', with a paragraph asserting that the [DFM23] arguments adapt because they rely on a boundary Poincaré inequality. That adaptation is load-bearing: Lemma 3.3, (i)⇒(iii) and (iii)⇒(i) all invoke the Green representation or boundary Hölder estimates. If the book never appears or the adaptation is wrong, Theorem 1.5 has no verifiable foundation. But the stated basis is plausible, and the boundary Poincaré inequality that supposedly replaces the missing ingredient is actually proved in this paper as Theorem 2.5, so I do not read this as a dealbreaker—just a serious verifiability gap the author needs to close, either by proving the needed statements or by making a draft of [FM] available. The other concrete problem is minor but real: Corollaries 4.1 and 4.2 assume (H6), which is never defined; the statements as printed are formally incomplete. There are also scattered typos ('de solvability', 'definitions an main result') that a copyedit will fix.

On the citation pattern: it is heavy on the author's group, but the central equivalence is not assumed, and the self-citation points to the very results being extended. I do not see circularity. The reader's stress-test concern about [FM] holds up; the rest of the reader's assessment is fair.

This paper is for people working on non-smooth boundary value problems, tent spaces, and degenerate elliptic operators; they will get real value from the equivalence and from the boundary Poincaré technique. My recommendation: send it to peer review, with a clear request that the author address the [FM] dependency and define (H6).

Referee Report

3 major / 4 minor

Summary. The paper proves, under assumptions (H1)-(H5), an equivalence between the L^p-solvability of the homogeneous Dirichlet problem, the solvability of the Poisson-Dirichlet problem (separately for data f and F), and the solvability of the Poisson-regularity problem for the adjoint, for degenerate elliptic operators L=-div(wA∇) on domains whose boundaries may have mixed dimension and without a Harnack chain condition. The main result is Theorem 1.5, with item (iii) claimed as new even in the setting of [MPT22]. The paper also develops tent-space duality results in the appendix and obtains existence of solutions with general data in Corollaries 4.1 and 4.2.

Significance. If correct, Theorem 1.5 gives a substantial generalization of the Mourgoglou-Poggi-Tolsa equivalence to degenerate operators and mixed-dimensional boundaries, and item (iii) is indeed new. The tent-space framework and the self-contained appendix on tent-space duality are valuable and clearly presented. The strength of the paper lies in the clean duality arguments that assemble Theorem 3.1 and Lemma 3.3 into a global equivalence. However, the correctness of the result is currently conditional on two load-bearing issues: the imported elliptic theory from the unpublished book [FM], and an apparent algebraic error in Lemma 2.6 that undermines the proof of the boundary Poincaré inequality.

major comments (3)

[Section 2.2, pp. 12-17] The foundational elliptic theory is asserted to hold in the present setting by citing [FM], a book in preparation. Specifically, Proposition 2.14 (boundary oscillation), Proposition 2.15 (boundary Hölder continuity), Theorems 2.16 and 2.18 (Green function and representation), Theorem 2.20 (elliptic measure), and Lemma 2.24 (maximum principle) are stated without proof in the no-Harnack-chain, mixed-dimensional setting. This is load-bearing: Lemma 3.3, the proofs of (i)⇒(iii), (iii)⇒(i), and (i)⇒(iv) all invoke Theorem 2.18 or Proposition 2.15. The manuscript states that the arguments of [DFM23] adapt because they rely only on a boundary Poincaré inequality, but those arguments are not provided. As written, the main theorem is conditional on an unavailable reference; the author should either include the missing proofs or restrict the statement to a setting where the cited results are available.
[Lemma 2.6, Eq. (2.33)] The identity in (2.33) is algebraically incorrect. From the definition ρ(B)=m(B∩Ω)/(r μ(B∩∂Ω)), one obtains ∫_{B(ξ,λr)∩Ω} g^2 dm = m(B(ξ,λr)) · avg_m = λr ρ(B(ξ,λr)) μ(B(ξ,λr)) · avg_m, so that (∫ g^2 dm)^{1/2} = (λr ρ μ)^{1/2} (avg_m)^{1/2} = μ^{1/2} (avg_μ)^{1/2}. The displayed formula (∫ g^2 dm)^{1/2} = (λrρ)^{-1/2} ( (1/μ) ∫ g^2 dm )^{1/2} would require λrρ μ = 1, which is not an identity. Consequently the inequality (2.32) is not justified and, as stated, fails in the classical case Ω=R^n_+, m=dx, μ=H^{n-1}|_{∂Ω}, g=1, λ=1: the left side behaves like r^{n-1} · r^{n/2}, while the right side behaves like r^{n/2}. Since Lemma 2.6 is used in the proof of the boundary Poincaré inequality (Theorem 2.5), which underlies Theorem 2.8 and much of the subsequent elliptic theory, this is a load-bearing error.
[Section 4.1, proof of (i)⇒(iii) and (i)⇒(iv)] In the estimate of the local term u0, the paper uses ∫_{Bx/8} |u0|^2 dm ≲ δ(x)^2 ∫_Ω |∇u0|^2 dm and attributes this to the boundary Poincaré inequality (2.31). However, (2.31) is an L^{2*}-Poincaré inequality; converting it to an L^2 estimate introduces a factor m(B)^{1-2/2*}, which is not uniformly bounded under (H1)-(H5). An L^2-Poincaré estimate with constant r for u∈W0(B∩Ω) is not proved in the paper. The same step is used in the proof of (i)⇒(iv). The argument must be repaired, for example by proving the needed L^2-Poincaré inequality or by replacing the local estimate with a different one.

minor comments (4)

[Corollaries 4.1 and 4.2] The statements assume (H1)-(H6), but (H6) is never defined in the paper; presumably (H1)-(H5) is intended.
[Throughout] There are several typos: 'wich implies' in the proof of Proposition 2.2, 'de solvability' in Definition 1.3, 'phipher' in reference [KP95], and 'article do not plan' in Section 1.2. These should be corrected.
[Lemma 2.6] The statement says g∈L^p(2B) but the proof and statement use L^2 norms; the exponent should be 2.
[Assumption (H4)] The choice ε = C_ρ^{-1} in (1.15) is unusual; the paper would benefit from a remark on why this exponent is natural and how the final constants depend on C_ρ.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.5 is proved from stated hypotheses and cited background elliptic theory; the main equivalence is derived, not assumed.

full rationale

The central equivalence in Theorem 1.5 is not built into the definitions or fitted parameters. Each implication is argued: (i) implies (iii) and (iv) by decomposing the solution and using the Green-function estimates from Lemma 3.3; (iii) and (iv) imply (i) through the Green-function characterization of solvability; (vi) implies (iv) and (iv) implies (v) by duality in the style of [MPT22]. None of these steps equates (D_p) with (PD_p) by construction, and no fitted quantity is renamed as a prediction. The hypotheses (H1)-(H5) are assumptions used to obtain the boundary Poincaré inequality, not conclusions fitted to the theorem. The main external dependency is the foundational elliptic theory in Section 2.2: boundary Hölder estimates, Green function representation, elliptic measure, and maximum principle are cited to [DFM23] and to the unpublished book [FM] by Feneuil and Mayboroda. The paper itself notes in Section 2.2 that 'there is no work in the literature that perfectly matches our setting' and refers the reader to [FM] 'in preparation' for the exact arguments. This is a genuine completeness and verifiability concern, and it is load-bearing for the proof of Theorem 1.5, but it is not circularity: the cited background theory concerns the elliptic operator and measure, not the equivalence theorem that is the paper's contribution. The result is a derivation from stated inputs, not a restatement of them.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The theorem is conditional on structural assumptions (H1)-(H5), which are domain hypotheses, not empirical fits. The only hand-chosen parameter is the exponent epsilon=C_rho^{-1} in (H4). The foundational elliptic theory is imported from [DFM23] and the unpublished [FM]; no new entities are introduced.

free parameters (1)

Exponent epsilon = C_rho^{-1} in assumption (H4) = C_rho^{-1}
The growth bound rho(Lambda B) <= C_rho Lambda^{1-epsilon} rho(B) uses the specific power 1-epsilon with epsilon=C_rho^{-1}; this choice makes the dyadic sums in Lemma 2.6 and Theorem 2.5 converge and is chosen by hand rather than derived.

assumptions (6)

domain assumption Assumption (H1): corkscrew point condition: for any ball B centered on ∂Ω, there exists x∈B with B(x, C_crk^{-1} r) ⊂ Ω.
Quantitative accessibility of Ω from every boundary ball; used throughout for the boundary Poincaré inequality and elliptic measure estimates.
domain assumption Assumption (H2): boundary measure µ is doubling.
Needed for maximal function estimates and tent space equivalences.
domain assumption Assumption (H3): m|Ω = w dx with w>0, and m is doubling in the whole space.
Defines the weighted measure and gives m(2B)≲m(B), used in all averaging arguments.
domain assumption Assumption (H4): rho(Lambda B) <= C_rho Lambda^{1-epsilon} rho(B) for Lambda > 1 with epsilon = C_rho^{-1}.
Controls growth of the measure-ratio rho; the specific exponent is chosen so dyadic sums converge in Lemma 2.6 and Theorem 2.5.
domain assumption Assumption (H5): Poincaré-Sobolev inequality (1.16) on all balls centered in Ω̄ and uniqueness of L^2-gradient limits.
Provides the Sobolev machinery that replaces the interior-only Poincaré inequality of [DFM23].
domain assumption Elliptic theory from [FM] in preparation: Green function, elliptic measure, boundary Hölder estimates, maximum principle in the exact setting.
Section 2.2 states these results hold in the present Harnack-chain-free setting by citing the unpublished book [FM]; the adaptations from [DFM23] are asserted but not shown.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Solvability of the Poisson-Dirichlet problem in domains with boundaries of mixed dimension." pith.science (2026). https://pith.science/paper/LI6BU6NH

@misc{pith2026260803373,
  author       = {Pith},
  title        = {Pith review of: Solvability of the Poisson-Dirichlet problem in domains with boundaries of mixed dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LI6BU6NH}},
  note         = {Machine review of arXiv:2608.03373}
}

abstract

We extend several characterizations of the $L^p$-solvability of the homogeneous Dirichlet problem to (possibly degenerate) elliptic operators $L=-\textrm{div}(wA\nabla)$ defined on a large class of open sets $\Omega$ in $\mathbb{R}^n$. This framework encompasses not only uniformly elliptic operators in Lipschitz domains, but also Caffarelli-Sylvestre-type operators, and boundaries $\partial\Omega$ that are not $(n-1)$-dimensional, for instance. We prove that, for $p\in (1,\infty)$, the $L^p$-solvability of the homogeneous Dirichlet problem is equivalent to the solvability of the Poisson-Dirichlet problem $Lu=wf-\textrm{div}(wF)$, \emph{i.e.} to the existence of a solution $u$ satisfying a $L^p$ non-tangential estimate whenever $f$ and $F$ belong to suitable weighted $L^p$ tent spaces. Furthermore, it is also equivalent to the solvability of the Poisson-Regularity problem for data in appropriate weighted $L^{p'}$ tent spaces, where estimates are obtained for $\nabla u$ rather than $u$.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages

[1]

Harmonic measure and quantitative connectivity: geometric characterization of theL p-solvability of the Dirichlet problem

[Azz+20] Jonas Azzam, Steve Hofmann, José María Martell, Mihalis Mourgoglou, and Xavier Tolsa. “Harmonic measure and quantitative connectivity: geometric characterization of theL p-solvability of the Dirichlet problem”.Invent. Math.222.3 (2020), pp. 881–993. [Azz21] Jonas Azzam. “Semi-uniform domains and theA ∞ property for harmonic measure”. Int. Math. R...

work page 2020
[2]

The Wiener test for degenerate elliptic equations

1983, pp. 577–589. [FJK83b] E. B. Fabes, D. S. Jerison, and C. E. Kenig. “The Wiener test for degenerate elliptic equations”.Ann. Inst. Fourier32.3 (1983), pp. 151–182. 39 [FKS82] Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni. “The local regularity of solutions of degenerate elliptic equations”.Commun. Partial Differ. Equations7 (1982), pp. 77–1...

arXiv 1983
[65]

Estimates of harmonic measure

[Dah77] Björn E. J. Dahlberg. “Estimates of harmonic measure”.Arch. Ration. Mech. Anal.65 (1977), pp. 275–288. [DFM19] Guy David, Joseph Feneuil, and Svitlana Mayboroda. “Dahlberg’s theorem in higher co-dimension”.J. Funct. Anal.276.9 (2019), pp. 2731–2820. [DFM21] G.David,J.Feneuil,andS.Mayboroda.Elliptic theory for sets with higher co-dimensional bounda...

work page 1977
[1983]

Sobolev meets Poincaré

[HK95] Piotr Hajłasz and Pekka Koskela. “Sobolev meets Poincaré”.C. R. Acad. Sci., Paris, Sér. I320.10 (1995), pp. 1211–1215. [HKM06] Juha Heinonen, Tero Kilpeläinen, and Olli Martio.Nonlinear potential theory of degen- erate elliptic equations. Unabridged republication of the 1993 original. Mineola, NY: Dover Publications,

work page 1995
[1993]

LayerpotentialsandregularityfortheDirichletproblemforLaplace’s equation in Lipschitz domains

[Ver84] GregoryVerchota.“LayerpotentialsandregularityfortheDirichletproblemforLaplace’s equation in Lipschitz domains”.J. Funct. Anal.59 (1984), pp. 572–611. 41

work page 1984
[1995]

Construction of a singular elliptic-harmonic measure

40 [MM80] Luciano Modica and Stefano Mortola. “Construction of a singular elliptic-harmonic measure”.Manuscr. Math.33 (1980), pp. 81–98. [MPT22] Mihalis Mourgoglou, Bruno Poggi, and Xavier Tolsa.Solvability of the Poisson-Dirichlet problem with interior data inLp′ -Carleson spaces and its applications to theLp-regularity problem. Preprint, arXiv:2207.10554

arXiv 1980
[2006]

With a new preface, corrigenda, and epilogue consisting of other new material

[HKM18] Juha Heinonen, Tero Kilpeläinen, and Olli Martio.Nonlinear potential theory of de- generate elliptic equations. With a new preface, corrigenda, and epilogue consisting of other new material. Reprint of the 2006 edition. Dover Books Math. Mineola, NY: Dover Publications,

work page 2006
[2018]

Squarefunction/non- tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators

[Hof+15] SteveHofmann,CarlosKenig,SvitlanaMayboroda,andJillPipher.“Squarefunction/non- tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators”.J. Am. Math. Soc.28.2 (2015), pp. 483–529. [Hof+17] Steve Hofmann, Phi Le, José María Martell, and Kaj Nyström. “The weak-A∞ property of harmonic andp-harmonic measures...

work page 2015

Show all 12 references

[2022]

The regularity problem for the Laplace equation in rough domains

[MT24] Mihalis Mourgoglou and Xavier Tolsa. “The regularity problem for the Laplace equation in rough domains”.Duke Math. J.173.9 (2024), pp. 1731–1837. [MZ19] Svitlana Mayboroda and Zihui Zhao. “Square function estimates, the BMO Dirichlet problem, and absolute continuity of ...

2024
[2023]

Lipschitz approximation to hypersurfaces, harmonic mea- sure, and singular integrals

[DJ90] Guy David and D. Jerison. “Lipschitz approximation to hypersurfaces, harmonic mea- sure, and singular integrals”.Indiana Univ. Math. J.39.3 (1990), pp. 831–845. [DM23] G. David and S. Mayboroda. “Harmonic measure is absolutely continuous with respect to the Hausdorff me...

1990
[2024]

Generalized Carleson perturbations of elliptic oper- ators and applications

[FM] Joseph Feneuil and Svitlana Mayboroda.Analysis, Geometry, and PDE in a lower dimensional world. In preparation. [FP22] Joseph Feneuil and Bruno Poggi. “Generalized Carleson perturbations of elliptic oper- ators and applications”.Trans. Am. Math. Soc.375.11 (2022), pp. 755...

2022 arXiv
[2025]

Gallegos.One-sided Rellich inequalities, Regularity problem and uniform rec- tifiability

[Gal25] Josep M. Gallegos.One-sided Rellich inequalities, Regularity problem and uniform rec- tifiability. Preprint, arXiv:2506.03431

arXiv

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Harmonic measure and quantitative connectivity: geometric characterization of theL p-solvability of the Dirichlet problem

[Azz+20] Jonas Azzam, Steve Hofmann, José María Martell, Mihalis Mourgoglou, and Xavier Tolsa. “Harmonic measure and quantitative connectivity: geometric characterization of theL p-solvability of the Dirichlet problem”.Invent. Math.222.3 (2020), pp. 881–993. [Azz21] Jonas Azzam. “Semi-uniform domains and theA ∞ property for harmonic measure”. Int. Math. R...

work page 2020

[2] [2]

The Wiener test for degenerate elliptic equations

1983, pp. 577–589. [FJK83b] E. B. Fabes, D. S. Jerison, and C. E. Kenig. “The Wiener test for degenerate elliptic equations”.Ann. Inst. Fourier32.3 (1983), pp. 151–182. 39 [FKS82] Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni. “The local regularity of solutions of degenerate elliptic equations”.Commun. Partial Differ. Equations7 (1982), pp. 77–1...

arXiv 1983

[3] [65]

Estimates of harmonic measure

[Dah77] Björn E. J. Dahlberg. “Estimates of harmonic measure”.Arch. Ration. Mech. Anal.65 (1977), pp. 275–288. [DFM19] Guy David, Joseph Feneuil, and Svitlana Mayboroda. “Dahlberg’s theorem in higher co-dimension”.J. Funct. Anal.276.9 (2019), pp. 2731–2820. [DFM21] G.David,J.Feneuil,andS.Mayboroda.Elliptic theory for sets with higher co-dimensional bounda...

work page 1977

[4] [1983]

Sobolev meets Poincaré

[HK95] Piotr Hajłasz and Pekka Koskela. “Sobolev meets Poincaré”.C. R. Acad. Sci., Paris, Sér. I320.10 (1995), pp. 1211–1215. [HKM06] Juha Heinonen, Tero Kilpeläinen, and Olli Martio.Nonlinear potential theory of degen- erate elliptic equations. Unabridged republication of the 1993 original. Mineola, NY: Dover Publications,

work page 1995

[5] [1993]

LayerpotentialsandregularityfortheDirichletproblemforLaplace’s equation in Lipschitz domains

[Ver84] GregoryVerchota.“LayerpotentialsandregularityfortheDirichletproblemforLaplace’s equation in Lipschitz domains”.J. Funct. Anal.59 (1984), pp. 572–611. 41

work page 1984

[6] [1995]

Construction of a singular elliptic-harmonic measure

40 [MM80] Luciano Modica and Stefano Mortola. “Construction of a singular elliptic-harmonic measure”.Manuscr. Math.33 (1980), pp. 81–98. [MPT22] Mihalis Mourgoglou, Bruno Poggi, and Xavier Tolsa.Solvability of the Poisson-Dirichlet problem with interior data inLp′ -Carleson spaces and its applications to theLp-regularity problem. Preprint, arXiv:2207.10554

arXiv 1980

[7] [2006]

With a new preface, corrigenda, and epilogue consisting of other new material

[HKM18] Juha Heinonen, Tero Kilpeläinen, and Olli Martio.Nonlinear potential theory of de- generate elliptic equations. With a new preface, corrigenda, and epilogue consisting of other new material. Reprint of the 2006 edition. Dover Books Math. Mineola, NY: Dover Publications,

work page 2006

[8] [2018]

Squarefunction/non- tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators

[Hof+15] SteveHofmann,CarlosKenig,SvitlanaMayboroda,andJillPipher.“Squarefunction/non- tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators”.J. Am. Math. Soc.28.2 (2015), pp. 483–529. [Hof+17] Steve Hofmann, Phi Le, José María Martell, and Kaj Nyström. “The weak-A∞ property of harmonic andp-harmonic measures...

work page 2015