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

Xavier Tolsa

REVIEW 2 major objections 4 minor 111 references

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

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read Quantitative rectifiability is the common core of harmonic analysis and rough-boundary PDE

desk verdict A useful, well-written survey by the main player in the field; read it knowing that the headline Painlevé characterization is pinned to an unpublished memoir. read the letter →

arxiv 2607.16457 v2 pith:LYXUV23T submitted 2026-07-17 math.CA math.AP

Xavier Tolsa This is my paper

classification math.CAmath.AP MSC 28A7542B2031B0535J25

keywords quantitativerectifiabilityuniformRiesztransformsharmonicmeasurePainlevéproblemCarlesonε²conjectureboundaryvalueproblemsβcoefficients

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 survey argues that quantitative rectifiability—the study of how well sets approximate flat planes at every scale—has become the unifying idea behind three previously separate areas: characterizing rectifiable sets, proving boundedness of singular integrals, and solving boundary value problems for harmonic functions on rough domains. It collects the main theorems showing that β-type square functions, the L2 behavior of Riesz transforms, and harmonic measure conditions are equivalent to geometric rectifiability properties, often with explicit constants. The paper also presents recent resolutions of Carleson's ε² conjecture in the plane and higher dimensions, and describes a solution of the Painlevé problem for Lipschitz harmonic functions in terms of a metric-geometric potential. A reader should care because these results convert classical questions in analysis and PDE into quantitative geometry, yielding checkable criteria for removability, absolute continuity, and solvability.

What carries the argument

The central objects are the β coefficients, which measure how far a set or measure is from lying inside an n-plane in a given ball, and their square functions, which sum β² over scales and locations. Related machinery includes the Carleson ε² function, built from the deviation of boundary arcs from semicircles; the (d−1)-dimensional Riesz transform, whose kernel is the gradient of the Laplacian's fundamental solution; and the Alt–Caffarelli–Friedman monotonicity formula, which controls the interaction of two disjoint harmonic functions. These tools translate geometric flatness questions into L2 estimates and back, supplying the bridge between rectifiability, singular integral bounds, and bou

What would settle it

Find any Radon measure with no point masses that has polynomial growth and satisfies the β-square-function estimate (4.2) but has unbounded L2 Riesz transform: that single counterexample would falsify Theorem 4.7 and the capacity characterization of Painlevé removability in Corollaries 5.3–5.4. Alternatively, exhibit a pair of complementary corkscrew domains with Ahlfors-regular boundary for which the Carleson square-function estimate (b) of Theorem 3.10 fails while the geometric condition (a) holds.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that a single family of quantitative tools—square functions built from β coefficients measuring deviation from flatness, the L2 boundedness of Riesz transforms, and Carleson-type measure estimates—provides equivalent characterizations of rectifiability and uniform rectifiability, and that these equivalences resolve longstanding problems: the David–Semmes problem in codimension one, Carleson's ε² conjecture (now a theorem), the one-phase and two-phase free boundary problems for harmonic measure, and the geometric characterization of Lp solvability of the Dirichlet problem in rough domains. The survey presents these as a connected web of theorems

Load-bearing premise

The survey's picture of the state of the art rests on several results that are not yet published at the time of writing—a memoir announced as 'to appear' giving the Riesz-transform characterization for arbitrary measures, a preprint on the quantitative Carleson conjecture, and a preprint on Favard length and quantitative rectifiability—plus the author's own selection of advances; if any of those unverified results contains an error, the corresponding part of the survey's narr

Editorial extensions

If this is right

If the survey's account is correct, uniform rectifiability of a boundary is equivalent to the L2 boundedness of all odd singular integral operators, and in codimension one it is equivalent to the L2 boundedness of the Riesz transform alone.
The Painlevé problem for Lipschitz harmonic functions is effectively solved: compact sets are removable if and only if they carry no nonzero measure with bounded Jones–Wolff potential, and the associated capacity is comparable to a supremum over such measures.
Mutual absolute continuity of interior and exterior harmonic measure forces rectifiability of the boundary, even in domains without capacity density conditions.
In Ahlfors-regular corkscrew domains, the Lp solvability of the Dirichlet problem is characterized by big pieces of chord-arc subdomains and by the weak-A∞ property of harmonic measure, tying solvability to quantitative rectifiability plus connectivity.
The regularity problem for the Laplacian is solvable in Lp for some p>1 in two-sided chord-arc domains, while the Neumann problem remains unresolved in the same generality.

Reading between the lines

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

The success of the Riesz-transform criterion for arbitrary measures (beyond Ahlfors regularity) suggests that the same strategy might now be tried for the open David–Semmes problem in intermediate codimensions 1<n<d−1, where the obstacle is the absence of a suitable maximum principle.
The higher-dimensional form of Carleson's conjecture, via the Friedland–Hayman inequality, indicates that spectral gaps of the Laplace–Beltrami operator on spheres control free-boundary tangency; this may connect to quantitative Reifenberg flatness and to the structure of singular sets for harmonic maps.
Theorem 3.10 makes Carleson's conjecture quantitative for Ahlfors-regular boundaries; a natural next test is whether the corkscrew hypothesis can be relaxed or removed while preserving the claimed equivalence between the square-function estimates and uniform rectifiability.
Because the survey's selection is explicitly limited to the author's own line of work, the broader literature on elliptic measures, parabolic problems, and higher-codimension settings is under-represented; extending the same quantitative-rectifiability framework there is an open, plausibly fruitful direction.

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

Here's my read on the Tolsa survey. This is an expository paper, not a research announcement, so judge it as a map of the territory rather than as new results. The value is organizational: it collects the quantitative-rectifiability program, the Riesz-transform characterizations, the Painlevé problem for Lipschitz harmonic functions, and the harmonic-measure/BVP applications into one coherent narrative. For someone entering the area, this is a good first stop. The writing is clear, the theorems are stated carefully, and the dependence of the results is mostly transparent. Credit where it's due: this is not a hasty review, and the author openly says he is focusing on work tied to his own research, which is fair given how much of the subject he and his collaborators built.

The soft spot is exactly where the stress-test points: the load-bearing Theorem 4.7, the full characterization of L2-bounded (d−1)-Riesz transforms for arbitrary Radon measures with no point masses, is attributed to [41], a Memoirs AMS volume that is still "To appear." Corollaries 5.3 and 5.4 and the claimed solution of the Painlevé problem for Lipschitz harmonic functions sit directly on that unpublished memoir. The survey gives no proof sketch, so the reader is asked to take the state-of-the-art claim on faith. That is a real caveat, though not a red flag: the other main results in the survey are from peer-reviewed sources, and the author is the right person to know whether the memoir is solid. I also noticed a small technical wobble in Theorem 3.2: the range 1 ≤ p < 2n/(n−2) is not meaningful for n = 1, 2, and the text later notes the equivalence holds for 1 ≤ p ≤ 2 for all n. That should be corrected in a final version, but it is minor.

The self-referential citation pattern is heavy but disclosed, and in a survey by this author it is expected. I don't see circularity: the quoted theorems are established results, not the survey proving its own assumptions.

Who is this for? Graduate students and researchers looking for a concise map of the field, and perhaps as a reference for the precise statements of the main theorems. It is not the place to verify the correctness of the newest claims. It deserves a serious referee, mainly to check that the statements match the cited sources and that the dependence on unpublished work is clearly flagged. I would accept it for peer review without hesitation, and I would probably cite it as the entry-point survey.

Referee Report

2 major / 4 minor

Summary. This survey reviews the interaction between quantitative rectifiability and three areas: (i) square functions, the traveling salesman theorem, uniform rectifiability, and Carleson's ε² conjecture; (ii) L² boundedness of Riesz transforms and the Painlevé problem for Lipschitz harmonic functions; and (iii) harmonic measure and the Lᵖ solvability of Dirichlet, regularity, and Neumann problems in rough domains. It collects many recent results, including the solution of Carleson's ε² conjecture, the David–Semmes problem in codimension one, the Dąbrowski–Tolsa characterization of L²-bounded (d−1)-Riesz transforms for arbitrary measures, the one- and two-phase free boundary problems for harmonic measure, and characterizations of Lᵖ-solvability in terms of uniform rectifiability. The paper is explicitly a partial survey centered on the author's own research and does not aim at completeness.

Significance. If the cited results are correct, this is a useful and well-organized overview of a rapidly developing area, giving precise statements and pointing to the literature. Its main strength is the coherent presentation of a large body of recent work, several items of which are not yet available in book form. The paper is transparent about its selection bias. Its usefulness is diminished, however, by the fact that a central theorem (Theorem 4.7) and several others (Theorems 3.10, 5.8) are sourced from works that are not yet peer-reviewed; the reader cannot fully verify the state of the art from the survey alone. The open questions listed (David–Semmes for intermediate dimensions, Vitushkin's conjecture, Neumann problem in chord-arc domains) are well chosen and indicate the current frontiers.

major comments (2)

[§4, Theorem 4.7 / §5, Corollaries 5.3–5.4] The paper's central narrative — the solution of the Painlevé problem for Lipschitz harmonic functions — rests on the characterization of L²-bounded (d−1)-Riesz transforms for arbitrary Radon measures (Theorem 4.7). This theorem is quoted from the Dąbrowski–Tolsa memoir [41], listed as 'To appear', with no proof sketch or indication of the main new ideas. Since the memoir is not yet peer-reviewed and the survey presents the result as established, the reader cannot independently assess the claim. Please either (i) state clearly that [41] is accepted and provide a brief outline of its proof strategy, or (ii) phrase Corollaries 5.3–5.4 as conditional on forthcoming work. This is load-bearing: a flaw in [41] would invalidate Corollary 5.3 and the claimed solution of the Painlevé problem.
[§3.5, Theorem 3.10 / §5, Theorem 5.8] These theorems are taken from the arXiv preprints [26] and [40], but in the text they are announced without any marker such as 'preprint' or 'to appear'. For a survey that aims to report the current state of the art, the provenance of each result should be unambiguous. Please add, at each such statement, a clear indication of the publication status (peer-reviewed, to appear, or preprint).

minor comments (4)

[§3.2, Theorem 3.2 and definition of β] The range 1 ≤ p < 2n/(n−2) is undefined for n=1,2 (for n=2 the denominator vanishes; for n=1 it is negative). Since the theorem is quoted from [37], please correct the statement, for instance by restricting to n≥3 and appending the separately known case 1≤p≤2 (which the paper itself notes after the theorem). Also, the definition of β^n_{p,E} for sets in the same subsection contains a typo: it should be β^n_{p,E}(x,r)=β^n_{p,H^n|E}(x,r), not β^n_{\infty,H^n|E}(x,r).
[§3.2 and §3.5] Two cross-reference errors: 'See Section 3 for more details' after Theorem 3.2 should presumably be 'Section 4'; and 'See Section 5' after the David–Jerison/Semmes remark in §3.5 should likely point to Section 6 (harmonic measure) or Section 7.
[§6] Typo: 'for alx∈F' should be 'for all x∈F'. Also, the spelling 'Dądrowski' is inconsistent in the text (see the paragraph before Theorem 5.7); please standardize.
[References] Minor typos in the bibliography: [96] has 'hamonic measure' for 'harmonic measure'; [109] uses an unusual romanization 'Vituˇskin' (standard: 'Vitushkin'). Please also check consistency of hyphens and capitalization in titles.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the survey quotes external theorems and does not derive its conclusions from its own inputs.

full rationale

This is a survey article with no new theorems and no derivation chain that transforms fitted inputs into predicted outputs. Its statements are quotations of named prior results, and the few deductions shown in the text (e.g., the Ahlfors-distortion argument implying finiteness of the Carleson square function in Theorem 3.6, or Corollary 5.3 following from Theorems 5.1 and 4.7) are logical consequences of independently stated cited theorems, not of the survey's conclusions. The author's own publications appear prominently, as the introduction openly says: 'I will focus mainly on the works that I know better and that are connected to my own research' (Section 1). That is selection bias, not circular reasoning. A possible concern is that Theorem 4.7 and Corollary 4.8 are attributed to the memoir [41], which is listed as 'To appear', and that Theorems 3.10 and 5.8 depend on preprints [26] and [40]; however, the survey does not redefine its target claims in terms of those sources, nor does it invoke a uniqueness theorem from the author's own work to forbid alternatives. Verification of unpublished results is a correctness/peer-review risk, not a circularity. No considered step exhibits self-definition, fitted-input-as-prediction, load-bearing self-citation reducing to itself, or renaming of a known result as new content.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The survey introduces no free parameters, no new postulates, and no invented entities. Its content rests on standard background in geometric measure theory and harmonic analysis (definitions of Hausdorff measure, capacity, harmonic measure, Poincaré inequalities) and on the correctness of the cited theorems, several of which are still in press or preprint at the time of writing.

assumptions (3)

standard math Standard analytical background: Hausdorff measures, capacities, harmonic measure, maximum principle, Riesz representation theorem (used in Section 6 to define harmonic measure via (6.1)).
Invoked throughout without proof.
standard math Friedland–Hayman eigenvalue inequality is used in Theorem 3.8 and thereafter.
Cited as [49]; the survey does not prove it.
domain assumption Correctness of cited theorems from the literature, including author's Memoirs paper [41] (To appear) and arXiv preprints [26], [40].
The survey's statements of 'recent major advances' depend on these results, which have not yet appeared in final peer-reviewed form.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions." pith.science (2026). https://pith.science/paper/LYXUV23T

@misc{pith2026260716457,
  author       = {Pith},
  title        = {Pith review of: Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LYXUV23T}},
  note         = {Machine review of arXiv:2607.16457}
}

abstract

This paper surveys different topics where the theory of quantitative rectifiability plays a central role. First, it reviews the characterization of rectifiability in terms of square functions involving $\beta$ type coefficients and the $\varepsilon^2$ conjecture of Carleson. It also discusses the deep connections between rectifiability and the $L^2$ boundedness of Riesz transforms and their application to the Painlev\'e problem for Lipschitz harmonic functions. Finally, the paper explores recent major advances in connection with harmonic measure and the $L^p$ solvability of the Dirichlet, regularity, and Neumann problems for the Laplace equation in rough domains, emphasizing the key role of quantitative rectifiability in these developments.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

111 extracted references · 63 canonical work pages

[41]

D a browski and X

D. D a browski and X. Tolsa , The measures with l^2 -bounded R iesz transform and the P ainlev\'e problem , Mem. Amer. Math. Soc., (To appear)
[26]

Casey, X

E. Casey, X. Tolsa, and M. Villa , Quantitative carleson's conjecture for ahlfors regular domains , 2025, https://arxiv.org/abs/2505.10666, https://arxiv.org/abs/2505.10666

arXiv 2025
[40]

D a browski , Favard length and quantitative rectifiability , arXiv:2408.03919, (2024)

D. D a browski , Favard length and quantitative rectifiability , arXiv:2408.03919, (2024)

arXiv 2024
[1]

L. V. Ahlfors , Bounded analytic functions , Duke Math. J., 14 (1947), pp. 1--11, http://projecteuclid.org/euclid.dmj/1077473984

arXiv 1947
[2]

Aikawa and K

H. Aikawa and K. Hirata , Doubling conditions for harmonic measure in J ohn domains , Ann. Inst. Fourier (Grenoble), 58 (2008), pp. 429--445, https://doi.org/10.5802/aif.2357, https://doi.org/10.5802/aif.2357

work page doi:10.5802/aif.2357 2008
[3]

Akman, J

M. Akman, J. Azzam, and M. Mourgoglou , Absolute continuity of harmonic measure for domains with lower regular boundaries , Adv. Math., 345 (2019), pp. 1206--1252, https://doi.org/10.1016/j.aim.2019.01.021, https://doi.org/10.1016/j.aim.2019.01.021

work page doi:10.1016/j.aim.2019.01.021 2019
[4]

Allen, D

M. Allen, D. Kriventsov, and R. Neumayer , Sharp quantitative F aber- K rahn inequalities and the A lt- C affarelli- F riedman monotonicity formula , Ars Inven. Anal., (2023), pp. Paper No. 1, 49

2023
[5]

Allen, D

M. Allen, D. Kriventsov, and R. Neumayer , Rectifiability and uniqueness of blow-ups for points with positive A lt- C affarelli- F riedman limit , Math. Ann., 391 (2025), pp. 6251--6289, https://doi.org/10.1007/s00208-024-03077-3, https://doi.org/10.1007/s00208-024-03077-3

work page doi:10.1007/s00208-024-03077-3 2025

Show all 111 references

[6]

Azzam , Poincar\'e inequalities and uniform rectifiability , Rev

J. Azzam , Poincar\'e inequalities and uniform rectifiability , Rev. Mat. Iberoam., 37 (2021), pp. 2161--2190, https://doi.org/10.4171/rmi/1258, https://doi.org/10.4171/rmi/1258

2021 doi
[7]

Azzam, S

J. Azzam, S. Hofmann, J. M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa, and A. Volberg , Rectifiability of harmonic measure , Geom. Funct. Anal., 26 (2016), pp. 703--728, https://doi.org/10.1007/s00039-016-0371-x, https://doi.org/10.1007/s00039-016-0371-x

2016 doi
[8]

Azzam, S

J. Azzam, S. Hofmann, J. M. Martell, M. Mourgoglou, and X. Tolsa , Harmonic measure and quantitative connectivity: geometric characterization of the L^p -solvability of the D irichlet problem , Invent. Math., 222 (2020), pp. 881--993, https://doi.org/10.1007/s00222-020-00984-5...

2020 doi
[9]

Azzam and M

J. Azzam and M. Mourgoglou , Tangent measures of elliptic measure and applications , Anal. PDE, 12 (2019), pp. 1891--1941, https://doi.org/10.2140/apde.2019.12.1891, https://doi.org/10.2140/apde.2019.12.1891

2019 doi
[10]

Azzam, M

J. Azzam, M. Mourgoglou, and X. Tolsa , Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability , Comm. Pure Appl. Math., 70 (2017), pp. 2121--2163, https://doi.org/10.1002/cpa.21687, https://doi.org/10.1002/cpa.21687

2017 doi
[11]

Azzam, M

J. Azzam, M. Mourgoglou, and X. Tolsa , A two-phase free boundary problem for harmonic measure and uniform rectifiability , Trans. Amer. Math. Soc., 373 (2020), pp. 4359--4388, https://doi.org/10.1090/tran/8059, https://doi.org/10.1090/tran/8059

2020 doi
[12]

Azzam, M

J. Azzam, M. Mourgoglou, X. Tolsa, and A. Volberg , On a two-phase problem for harmonic measure in general domains , Amer. J. Math., 141 (2019), pp. 1259--1279, https://doi.org/10.1353/ajm.2019.0032, https://doi.org/10.1353/ajm.2019.0032

2019
[13]

Azzam and R

J. Azzam and R. Schul , An analyst's traveling salesman theorem for sets of dimension larger than one , Math. Ann., 370 (2018), pp. 1389--1476, https://doi.org/10.1007/s00208-017-1609-0, https://doi.org/10.1007/s00208-017-1609-0

2018 doi
[14]

Azzam and X

J. Azzam and X. Tolsa , Characterization of n -rectifiability in terms of J ones' square function: P art II , Geom. Funct. Anal., 25 (2015), pp. 1371--1412, https://doi.org/10.1007/s00039-015-0334-7, https://doi.org/10.1007/s00039-015-0334-7

2015 doi
[15]

Badger and R

M. Badger and R. Schul , Multiscale analysis of 1-rectifiable measures II : C haracterizations , Anal. Geom. Metr. Spaces, 5 (2017), pp. 1--39, https://doi.org/10.1515/agms-2017-0001, https://doi.org/10.1515/agms-2017-0001

2017 doi
[16]

Bennewitz and J

B. Bennewitz and J. L. Lewis , On weak reverse H \"older inequalities for nondoubling harmonic measures , Complex Var. Theory Appl., 49 (2004), pp. 571--582, https://doi.org/10.1080/02781070410001731738, https://doi.org/10.1080/02781070410001731738

2004 doi
[17]

C. J. Bishop , H ARMONIC MEASURES SUPPORTED ON CURVES , ProQuest LLC, Ann Arbor, MI, 1987, http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:T-30223. Thesis (Ph.D.)--The University of Chicago

1987
[18]

C. J. Bishop , A characterization of P oissonian domains , Ark. Mat., 29 (1991), pp. 1--24, https://doi.org/10.1007/BF02384328, https://doi.org/10.1007/BF02384328

1991 doi
[19]

C. J. Bishop , Some questions concerning harmonic measure , in Partial differential equations with minimal smoothness and applications ( C hicago, IL , 1990), vol. 42 of IMA Vol. Math. Appl., Springer, New York, 1992, pp. 89--97, https://doi.org/10.1007/978-1-4612-2898-1_7, ht...

1990 doi
[20]

C. J. Bishop, L. Carleson, J. B. Garnett, and P. W. Jones , Harmonic measures supported on curves , Pacific J. Math., 138 (1989), pp. 233--236, http://projecteuclid.org/euclid.pjm/1102650148

1989
[21]

C. J. Bishop and P. W. Jones , Harmonic measure, L^2 estimates and the S chwarzian derivative , J. Anal. Math., 62 (1994), pp. 77--113, https://doi.org/10.1007/BF02835949, https://doi.org/10.1007/BF02835949

1994 doi
[22]

Bortz and S

S. Bortz and S. Hofmann , A singular integral approach to a two phase free boundary problem , Proc. Amer. Math. Soc., 144 (2016), pp. 3959--3973, https://doi.org/10.1090/proc/13035, https://doi.org/10.1090/proc/13035

2016 doi
[23]

Bortz, S

S. Bortz, S. Hofmann, J. M. Martell, and K. Nystr\"om , Solvability of the L ^p D irichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic L ipschitz graph , Invent. Math., 239 (2025), pp. 165--217, https://doi.org/10....

2025 doi
[24]

Bourgain , On the H ausdorff dimension of harmonic measure in higher dimension , Invent

J. Bourgain , On the H ausdorff dimension of harmonic measure in higher dimension , Invent. Math., 87 (1987), pp. 477--483, https://doi.org/10.1007/BF01389238, https://mathscinet.ams.org/mathscinet-getitem?mr=874032

1987 doi
[25]

Brasco, G

L. Brasco, G. De Philippis, and B. Velichkov , Faber- K rahn inequalities in sharp quantitative form , Duke Math. J., 164 (2015), pp. 1777--1831, https://doi.org/10.1215/00127094-3120167, https://doi.org/10.1215/00127094-3120167

2015 doi
[27]

Chang and X

A. Chang and X. Tolsa , Analytic capacity and projections , J. Eur. Math. Soc. (JEMS), 22 (2020), pp. 4121--4159, https://doi.org/10.4171/jems/1004, https://doi.org/10.4171/jems/1004

2020 doi
[28]

B. E. J. Dahlberg , Estimates of harmonic measure , Arch. Rational Mech. Anal., 65 (1977), pp. 275--288, https://doi.org/10.1007/BF00280445, https://doi.org/10.1007/BF00280445

1977 doi
[29]

B. E. J. Dahlberg , A note on S obolev spaces , in Harmonic analysis in E uclidean spaces ( P roc. S ympos. P ure M ath., W illiams C oll., W illiamstown, M ass., 1978), P art 1, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 183--185, http...

1978
[30]

B. E. J. Dahlberg and C. E. Kenig , Hardy spaces and the N eumann problem in L^p for L aplace's equation in L ipschitz domains , Ann. of Math. (2), 125 (1987), pp. 437--465, https://doi.org/10.2307/1971407, https://doi.org/10.2307/1971407

1987 doi
[31]

David , Unrectifiable 1 -sets have vanishing analytic capacity , Rev

G. David , Unrectifiable 1 -sets have vanishing analytic capacity , Rev. Mat. Iberoamericana, 14 (1998), pp. 369--479, https://doi.org/10.4171/RMI/242, https://doi.org/10.4171/RMI/242

1998 doi
[32]

David, M

G. David, M. Engelstein, and S. Mayboroda , Square functions, nontangential limits, and harmonic measure in codimension larger than 1 , Duke Math. J., 170 (2021), pp. 455--501, https://doi.org/10.1215/00127094-2020-0048, https://doi.org/10.1215/00127094-2020-0048

2021 doi
[33]

David, J

G. David, J. Feneuil, and S. Mayboroda , Dahlberg's theorem in higher co-dimension , J. Funct. Anal., 276 (2019), pp. 2731--2820, https://doi.org/10.1016/j.jfa.2019.02.006, https://doi.org/10.1016/j.jfa.2019.02.006

2019 doi
[34]

David, J

G. David, J. Feneuil, and S. Mayboroda , Green function estimates on complements of low-dimensional uniformly rectifiable sets , Math. Ann., 385 (2023), pp. 1797--1821, https://doi.org/10.1007/s00208-022-02379-8, https://doi.org/10.1007/s00208-022-02379-8

2023 doi
[35]

David and D

G. David and D. Jerison , Lipschitz approximation to hypersurfaces, harmonic measure, and singular-integrals , Indiana University Mathematics Journal, 39 (1990), pp. 831--845

1990
[36]

David and P

G. David and P. Mattila , Removable sets for lipschitz harmonic functions in the plane , Revista Matematica Iberoamericana, 16 (2000), pp. 137--215

2000
[37]

David and S

G. David and S. Semmes , Singular integrals and rectifiable sets in R ^n : B eyond L ipschitz graphs , Ast\'erisque, (1991), p. 152

1991
[38]

David and S

G. David and S. Semmes , Analysis of and on uniformly rectifiable sets , American Mathematical Soc., 1993

1993
[39]

David and T

G. David and T. Toro , Reifenberg parameterizations for sets with holes , American Mathematical Soc., 2012

2012
[42]

D a browski and M

D. D a browski and M. Villa , Analytic capacity and dimension of sets with plenty of big projections , Trans. Amer. Math. Soc., 378 (2025), pp. 3897--3950, https://doi.org/10.1090/tran/9265, https://doi.org/10.1090/tran/9265

2025 doi
[43]

Edelen, A

N. Edelen, A. Naber, and D. Valtorta , Quantitative R eifenberg theorem for measures , Math. Z., 310 (2025), pp. Paper No. 45, 70, https://doi.org/10.1007/s00209-025-03743-5, https://doi.org/10.1007/s00209-025-03743-5

2025 doi
[44]

Eiderman, F

V. Eiderman, F. Nazarov, and A. Volberg , The s - R iesz transform of an s -dimensional measure in R ^2 is unbounded for 1<s<2 , J. Anal. Math., 122 (2014), pp. 1--23, https://doi.org/10.1007/s11854-014-0001-1, https://doi.org/10.1007/s11854-014-0001-1

2014 doi
[45]

E. B. Fabes, M. Jodeit, Jr., and N. M. Rivi\`ere , Potential techniques for boundary value problems on C 1 -domains , Acta Math., 141 (1978), pp. 165--186, https://doi.org/10.1007/BF02545747, https://doi.org/10.1007/BF02545747

1978 doi
[46]

Feneuil and L

J. Feneuil and L. Li , The L ^p P oisson- N eumann problem and its relation to the N eumann problem , 2024, https://arxiv.org/abs/2406.16735, https://arxiv.org/abs/2406.16735

2024 arXiv
[47]

Fleschler, X

I. Fleschler, X. Tolsa, and M. Villa , Faber- K rahn inequalities, the A lt- C affarelli- F riedman formula, and C arleson's ^2 conjecture in higher dimensions , 2024, https://arxiv.org/abs/2306.06187, https://arxiv.org/abs/2306.06187

2024 arXiv
[48]

Fleschler, X

I. Fleschler, X. Tolsa, and M. Villa , Carleson's ^2 conjecture in higher dimensions , Invent. Math., 241 (2025), pp. 207--307, https://doi.org/10.1007/s00222-025-01337-w, https://doi.org/10.1007/s00222-025-01337-w

2025 doi
[49]

Friedland and W

S. Friedland and W. K. Hayman , Eigenvalue inequalities for the D irichlet problem on spheres and the growth of subharmonic functions , Comment. Math. Helv., 51 (1976), pp. 133--161, https://doi.org/10.1007/BF02568147, https://doi.org/10.1007/BF02568147

1976 doi
[50]

J. M. Gallegos , One-sided R ellich inequalities, regularity problem and uniform rectifiability , 2025, https://arxiv.org/abs/2506.03431, https://arxiv.org/abs/2506.03431

2025 arXiv
[51]

Garnett, M

J. Garnett, M. Mourgoglou, and X. Tolsa , Uniform rectifiability from C arleson measure estimates and -approximability of bounded harmonic functions , Duke Math. J., 167 (2018), pp. 1473--1524, https://doi.org/10.1215/00127094-2017-0057, https://doi.org/10.1215/00127094-2017-0057

2018 doi
[52]

J. B. Garnett and D. E. Marshall , Harmonic measure , vol. 2, Cambridge University Press, 2005

2005
[53]

Girela-Sarri\'on , Geometric conditions for the L^2 -boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in R^d , J

D. Girela-Sarri\'on , Geometric conditions for the L^2 -boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in R^d , J. Anal. Math., 137 (2019), pp. 339--372, https://doi.org/10.1007/s11854-018-0075-2, https://doi.org/10....

2019 doi
[54]

Girela-Sarri\' o n and X

D. Girela-Sarri\' o n and X. Tolsa , The R iesz transform and quantitative rectifiability for general R adon measures , Calc. Var. Partial Differential Equations, 57 (2018), pp. Art. 16, 63, https://doi.org/10.1007/s00526-017-1294-6, https://mathscinet.ams.org/mathscinet-getit...

2018 doi
[55]

Haj asz , Sobolev spaces on an arbitrary metric space , Potential Anal., 5 (1996), pp

P. Haj asz , Sobolev spaces on an arbitrary metric space , Potential Anal., 5 (1996), pp. 403--415, https://doi.org/10.1007/BF00275475, https://doi.org/10.1007/BF00275475

1996 doi
[56]

Hofmann , Quantitative absolute continuity of harmonic measure and the D irichlet problem: a survey of recent progress , Acta Math

S. Hofmann , Quantitative absolute continuity of harmonic measure and the D irichlet problem: a survey of recent progress , Acta Math. Sin. (Engl. Ser.), 35 (2019), pp. 1011--1026, https://doi.org/10.1007/s10114-019-8444-z, https://doi.org/10.1007/s10114-019-8444-z

2019 doi
[57]

Hofmann, P

S. Hofmann, P. Le, J. M. Martell, and K. Nystr\"om , The weak- A_ property of harmonic and p -harmonic measures implies uniform rectifiability , Anal. PDE, 10 (2017), pp. 513--558, https://doi.org/10.2140/apde.2017.10.513, https://doi.org/10.2140/apde.2017.10.513

2017 doi
[58]

Hofmann, J

S. Hofmann, J. M. Martell, and S. Mayboroda , Uniform rectifiability, C arleson measure estimates, and approximation of harmonic functions , Duke Math. J., 165 (2016), pp. 2331--2389, https://doi.org/10.1215/00127094-3477128, https://doi.org/10.1215/00127094-3477128

2016 doi
[59]

Hofmann, J

S. Hofmann, J. M. Martell, S. Mayboroda, T. Toro, and Z. Zhao , Uniform rectifiability and elliptic operators satisfying a C arleson measure condition , Geom. Funct. Anal., 31 (2021), pp. 325--401, https://doi.org/10.1007/s00039-021-00566-4, https://doi.org/10.1007/s00039-021-00566-4

2021 doi
[60]

Hofmann, M

S. Hofmann, M. Mitrea, and M. Taylor , Singular integrals and elliptic boundary problems on regular S emmes- K enig- T oro domains , Int. Math. Res. Not. IMRN, (2010), pp. 2567--2865, https://doi.org/10.1093/imrn/rnp214, https://doi.org/10.1093/imrn/rnp214

2010 doi
[61]

Jaye and F

B. Jaye and F. Nazarov , Reflectionless measures and the M attila- M elnikov- V erdera uniform rectifiability theorem , in Geometric aspects of functional analysis, vol. 2116 of Lecture Notes in Math., Springer, Cham, 2014, pp. 199--229, https://doi.org/10.1007/978-3-319-09477...

2014 doi
[62]

Jaye and F

B. Jaye and F. Nazarov , Reflectionless measures for C alder\'on- Z ygmund operators I : general theory , J. Anal. Math., 135 (2018), pp. 599--638, https://doi.org/10.1007/s11854-018-0047-6, https://doi.org/10.1007/s11854-018-0047-6

2018 doi
[63]

Jaye and F

B. Jaye and F. Nazarov , Reflectionless measures for C alder\'on- Z ygmund operators II : W olff potentials and rectifiability , J. Eur. Math. Soc. (JEMS), 21 (2019), pp. 549--583, https://doi.org/10.4171/JEMS/844, https://doi.org/10.4171/JEMS/844

2019 doi
[64]

B. Jaye, X. Tolsa, and M. Villa , A proof of C arleson's ^2 -conjecture , Ann. of Math. (2), 194 (2021), pp. 97--161, https://doi.org/10.4007/annals.2021.194.1.2, https://doi.org/10.4007/annals.2021.194.1.2

2021 doi
[65]

D. S. Jerison and C. E. Kenig , Boundary behavior of harmonic functions in non-tangentially accessible domains , Advances in Mathematics, 46 (1982), pp. 80--147

1982
[66]

P. W. Jones , Rectifiable sets and the traveling salesman problem , Invent. Math., 102 (1990), pp. 1--15

1990
[67]

P. W. Jones , On scaling properties of harmonic measure , in Perspectives in analysis, vol. 27 of Math. Phys. Stud., Springer, Berlin, 2005, pp. 73--81, https://doi.org/10.1007/3-540-30434-7\_7, https://doi.org/10.1007/3-540-30434-7_7

2005 doi
[68]

P. W. Jones and T. Murai , Positive analytic capacity but zero B uffon needle probability , Pacific J. Math., 133 (1988), pp. 99--114, http://projecteuclid.org/euclid.pjm/1102689569

1988
[69]

P. W. Jones and T. H. Wolff , Hausdorff dimension of harmonic measures in the plane , Acta Mathematica, 161 (1988), pp. 131--144

1988
[70]

Kenig, D

C. Kenig, D. Preiss, and T. Toro , Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions , J. Amer. Math. Soc., 22 (2009), pp. 771--796, https://doi.org/10.1090/S0894-0347-08-00601-2, https://doi.org/10.1090/S0894-0347-08-00601-2

2009 doi
[71]

Kenig and T

C. Kenig and T. Toro , Free boundary regularity below the continuous threshold: 2-phase problems , J. Reine Angew. Math., 596 (2006), pp. 1--44, https://doi.org/10.1515/CRELLE.2006.050, https://doi.org/10.1515/CRELLE.2006.050

2006 doi
[72]

C. E. Kenig , Harmonic analysis techniques for second order elliptic boundary value problems , vol. 83, American Mathematical Soc., 1994

1994
[73]

J. C. L\'eger , Menger curvature and rectifiability , Ann. of Math. (2), 149 (1999), pp. 831--869, https://doi.org/10.2307/121074, https://doi.org/10.2307/121074

1999 doi
[74]

Martikainen and T

H. Martikainen and T. Orponen , Characterising the big pieces of L ipschitz graphs property using projections , J. Eur. Math. Soc. (JEMS), 20 (2018), pp. 1055--1073, https://doi.org/10.4171/JEMS/782, https://doi.org/10.4171/JEMS/782

2018 doi
[75]

Mattila , Smooth maps, null-sets for integralgeometric measure and analytic capacity , Ann

P. Mattila , Smooth maps, null-sets for integralgeometric measure and analytic capacity , Ann. of Math. (2), 123 (1986), pp. 303--309, https://doi.org/10.2307/1971273, https://doi.org/10.2307/1971273

1986 doi
[76]

Mattila , Geometry of sets and measures in euclidean spaces: fractals and rectifiability , vol

P. Mattila , Geometry of sets and measures in euclidean spaces: fractals and rectifiability , vol. 44, Cambridge University Press, 1995

1995
[77]

Mattila, M

P. Mattila, M. S. Melnikov, and J. Verdera , The C auchy integral, analytic capacity, and uniform rectifiability , Ann. of Math. (2), 144 (1996), pp. 127--136, https://doi.org/10.2307/2118585, https://doi.org/10.2307/2118585

1996 doi
[78]

Mattila and P

P. Mattila and P. V. Paramonov , On geometric properties of harmonic Lip _1 -capacity , Pacific J. Math., 171 (1995), pp. 469--491, http://projecteuclid.org/euclid.pjm/1102368927

1995
[79]

Mattila and D

P. Mattila and D. Preiss , Rectifiable measures in R ^n and existence of principal values for singular integrals , J. London Math. Soc. (2), 52 (1995), pp. 482--496, https://doi.org/10.1112/jlms/52.3.482, https://doi.org/10.1112/jlms/52.3.482

1995 doi
[80]

J. E. McMillan , Boundary behavior of a conformal mapping , Acta Math., 123 (1969), pp. 43--67, https://doi.org/10.1007/BF02392384, https://mathscinet.ams.org/mathscinet-getitem?mr=257330

1969 doi
[81]

Melnikov, A

M. Melnikov, A. Poltoratski, and A. Volberg , Uniqueness theorems for C auchy integrals , Publ. Mat., 52 (2008), pp. 289--314, https://doi.org/10.5565/PUBLMAT\_52208\_03, https://doi.org/10.5565/PUBLMAT_52208_03

2008 doi
[82]

Merlo, M

A. Merlo, M. Mourgoglou, and C. Puliatti , Layer potentials for elliptic operators with DMO -type coefficients: big pieces tb theorem, quantitative rectifiability, and free boundary problems , 2025, https://arxiv.org/abs/2505.23478, https://arxiv.org/abs/2505.23478

2025 arXiv
[83]

Mi\'skiewicz , Discrete R eifenberg-type theorem , Ann

M. Mi\'skiewicz , Discrete R eifenberg-type theorem , Ann. Acad. Sci. Fenn. Math., 43 (2018), pp. 3--19, https://doi.org/10.5186/aasfm.2018.4301, https://doi.org/10.5186/aasfm.2018.4301

2018
[84]

Molero, M

A. Molero, M. Mourgoglou, C. Puliatti, and X. Tolsa , L^2 -boundedness of gradients of single layer potentials for elliptic operators with coefficients of D ini mean oscillation-type , Arch. Ration. Mech. Anal., 247 (2023), pp. Paper No. 38, 59, https://doi.org/10.1007/s00205-...

2023 doi
[85]

Mourgoglou, B

M. Mourgoglou, B. Poggi, and X. Tolsa , Solvability of the P oisson-- Di richlet problem with interior data in L ^ p' -carleson spaces and its applications to the L ^p -regularity problem , J. Eur. Math. Soc. (2025), published online first, (2025), https://doi.org/10.4171/JEMS/1660

2025 doi
[86]

Mourgoglou and X

M. Mourgoglou and X. Tolsa , Harmonic measure and R iesz transform in uniform and general domains , J. Reine Angew. Math., 758 (2020), pp. 183--221, https://doi.org/10.1515/crelle-2017-0037, https://doi.org/10.1515/crelle-2017-0037

2020 doi
[87]

Mourgoglou and X

M. Mourgoglou and X. Tolsa , The regularity problem for the L aplace equation in rough domains , Duke Math. J., 173 (2024), pp. 1731--1837, https://doi.org/10.1215/00127094-2023-0044, https://doi.org/10.1215/00127094-2023-0044

2024 doi
[88]

Mourgoglou and X

M. Mourgoglou and X. Tolsa , Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains , 2024, https://arxiv.org/abs/2407.20385, https://arxiv.org/abs/2407.20385

2024 arXiv
[89]

Naber and D

A. Naber and D. Valtorta , Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps , Annals of Mathematics, (2017), pp. 131--227

2017
[90]

Nazarov, X

F. Nazarov, X. Tolsa, and A. Volberg , On the uniform rectifiability of AD -regular measures with bounded R iesz transform operator: the case of codimension 1 , Acta Math., 213 (2014), pp. 237--321, https://doi.org/10.1007/s11511-014-0120-7, https://doi.org/10.1007/s11511-014-0120-7

2014 doi
[91]

Nazarov, X

F. Nazarov, X. Tolsa, and A. Volberg , The R iesz transform, rectifiability, and removability for L ipschitz harmonic functions , Publ. Mat., 58 (2014), pp. 517--532, http://projecteuclid.org/euclid.pm/1405949331

2014
[92]

Okikiolu , Characterization of subsets of rectifiable curves in R ^n , J

K. Okikiolu , Characterization of subsets of rectifiable curves in R ^n , J. London Math. Soc.(2), 46 (1992), pp. 336--348

1992
[93]

Orponen , Plenty of big projections imply big pieces of L ipschitz graphs , Invent

T. Orponen , Plenty of big projections imply big pieces of L ipschitz graphs , Invent. Math., 226 (2021), pp. 653--709, https://doi.org/10.1007/s00222-021-01055-z, https://doi.org/10.1007/s00222-021-01055-z

2021 doi
[94]

P. V. Paramonov , Harmonic approximations in the C^1 -norm , Mat. Sb., 181 (1990), pp. 1341--1365, https://doi.org/10.1070/SM1992v071n01ABEH002129, https://doi.org/10.1070/SM1992v071n01ABEH002129

1990 doi
[95]

Prats and X

M. Prats and X. Tolsa , The two-phase problem for harmonic measure in VMO , Calc. Var. Partial Differential Equations, 59 (2020), pp. Paper No. 102, 58, https://doi.org/10.1007/s00526-020-01760-2, https://doi.org/10.1007/s00526-020-01760-2

2020 doi
[96]

Prats and X

M. Prats and X. Tolsa , Notes on hamonic measure , to appear, 2025, https://mat.uab.es/ xtolsa/mesuraharmonica.pdf

2025
[97]

Riesz and M

F. Riesz and M. Riesz , On the boundary values of an analytic function . Quatri \`e me congr \`e s des math. scand. 1916, 27-44 (1916)., 1916

1916
[98]

Semmes , Analysis vs.\ geometry on a class of rectifiable hypersurfaces in R ^n , Indiana Univ

S. Semmes , Analysis vs.\ geometry on a class of rectifiable hypersurfaces in R ^n , Indiana Univ. Math. J., 39 (1990), pp. 1005--1035, https://doi.org/10.1512/iumj.1990.39.39048, https://doi.org/10.1512/iumj.1990.39.39048

1990 doi
[99]

Tapiola and X

O. Tapiola and X. Tolsa , Connectivity conditions and boundary P oincar\'e inequalities , Anal. PDE, 17 (2024), pp. 1831--1870, https://doi.org/10.2140/apde.2024.17.1831, https://doi.org/10.2140/apde.2024.17.1831

2024 doi
[100]

Tasso , Rectifiability of a class of integralgeometric measures and applications , 2025, https://arxiv.org/abs/2206.14044, https://arxiv.org/abs/2206.14044

E. Tasso , Rectifiability of a class of integralgeometric measures and applications , 2025, https://arxiv.org/abs/2206.14044, https://arxiv.org/abs/2206.14044

2025 arXiv
[101]

Tolsa , The semiadditivity of continuous analytic capacity and the inner boundary conjecture , Amer

X. Tolsa , The semiadditivity of continuous analytic capacity and the inner boundary conjecture , Amer. J. Math., 126 (2004), pp. 523--567, https://mathscinet.ams.org/mathscinet-getitem?mr=2058383

2004
[102]

Tolsa , Bilipschitz maps, analytic capacity, and the C auchy integral , Ann

X. Tolsa , Bilipschitz maps, analytic capacity, and the C auchy integral , Ann. of Math. (2), 162 (2005), pp. 1243--1304, https://doi.org/10.4007/annals.2005.162.1243, https://doi.org/10.4007/annals.2005.162.1243

2005 doi
[103]

Tolsa , Principal values for R iesz transforms and rectifiability , J

X. Tolsa , Principal values for R iesz transforms and rectifiability , J. Funct. Anal., 254 (2008), pp. 1811--1863, https://doi.org/10.1016/j.jfa.2007.07.020, https://doi.org/10.1016/j.jfa.2007.07.020

2008 doi
[104]

Tolsa , Uniform rectifiability, C alder\'on- Z ygmund operators with odd kernel, and quasiorthogonality , Proc

X. Tolsa , Uniform rectifiability, C alder\'on- Z ygmund operators with odd kernel, and quasiorthogonality , Proc. Lond. Math. Soc. (3), 98 (2009), pp. 393--426, https://doi.org/10.1112/plms/pdn035, https://doi.org/10.1112/plms/pdn035

2009 doi
[105]

Tolsa , Characterization of n -rectifiability in terms of J ones' square function: part I , Calc

X. Tolsa , Characterization of n -rectifiability in terms of J ones' square function: part I , Calc. Var. Partial Differential Equations, 54 (2015), pp. 3643--3665, https://doi.org/10.1007/s00526-015-0917-z, https://doi.org/10.1007/s00526-015-0917-z

2015 doi
[106]

Tolsa , Rectifiability of measures and the _p coefficients , Publ

X. Tolsa , Rectifiability of measures and the _p coefficients , Publ. Mat., 63 (2019), pp. 491--519, https://doi.org/10.5565/PUBLMAT6321904, https://doi.org/10.5565/PUBLMAT6321904

2019 doi
[107]

Tolsa and T

X. Tolsa and T. Toro , The two-phase problem for harmonic measure in VMO and the chord-arc condition , Trans. Amer. Math. Soc. Ser. B, 11 (2024), pp. 1294--1315, https://doi.org/10.1090/btran/197, https://doi.org/10.1090/btran/197

2024 doi
[108]

Verchota , Layer potentials and regularity for the Dirichlet problem for Laplace's equation in lipschitz domains , J

G. Verchota , Layer potentials and regularity for the Dirichlet problem for Laplace's equation in lipschitz domains , J. Funct. Anal., 59 (1984), pp. 572--611

1984
[109]

A. G. Vitu skin , Analytic capacity of sets in problems of approximation theory , Uspehi Mat. Nauk, 22 (1967), pp. 141--199

1967
[110]

Volberg , Calder\' o n- Z ygmund capacities and operators on nonhomogeneous spaces , vol

A. Volberg , Calder\' o n- Z ygmund capacities and operators on nonhomogeneous spaces , vol. 100 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence...

2003 doi
[111]

T. H. Wolff , Counterexamples with harmonic gradients in R ^3 , in Essays on F ourier analysis in honor of E lias M . S tein ( P rinceton, NJ , 1991), vol. 42 of Princeton Math. Ser., Princeton Univ. Press, Princeton, NJ, 1995, pp. 321--384

1991

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [41]

D a browski and X

D. D a browski and X. Tolsa , The measures with l^2 -bounded R iesz transform and the P ainlev\'e problem , Mem. Amer. Math. Soc., (To appear)

[2] [26]

Casey, X

E. Casey, X. Tolsa, and M. Villa , Quantitative carleson's conjecture for ahlfors regular domains , 2025, https://arxiv.org/abs/2505.10666, https://arxiv.org/abs/2505.10666

arXiv 2025

[3] [40]

D a browski , Favard length and quantitative rectifiability , arXiv:2408.03919, (2024)

D. D a browski , Favard length and quantitative rectifiability , arXiv:2408.03919, (2024)

arXiv 2024

[4] [1]

L. V. Ahlfors , Bounded analytic functions , Duke Math. J., 14 (1947), pp. 1--11, http://projecteuclid.org/euclid.dmj/1077473984

arXiv 1947

[5] [2]

Aikawa and K

H. Aikawa and K. Hirata , Doubling conditions for harmonic measure in J ohn domains , Ann. Inst. Fourier (Grenoble), 58 (2008), pp. 429--445, https://doi.org/10.5802/aif.2357, https://doi.org/10.5802/aif.2357

work page doi:10.5802/aif.2357 2008

[6] [3]

Akman, J

M. Akman, J. Azzam, and M. Mourgoglou , Absolute continuity of harmonic measure for domains with lower regular boundaries , Adv. Math., 345 (2019), pp. 1206--1252, https://doi.org/10.1016/j.aim.2019.01.021, https://doi.org/10.1016/j.aim.2019.01.021

work page doi:10.1016/j.aim.2019.01.021 2019

[7] [4]

Allen, D

M. Allen, D. Kriventsov, and R. Neumayer , Sharp quantitative F aber- K rahn inequalities and the A lt- C affarelli- F riedman monotonicity formula , Ars Inven. Anal., (2023), pp. Paper No. 1, 49

2023

[8] [5]

Allen, D

M. Allen, D. Kriventsov, and R. Neumayer , Rectifiability and uniqueness of blow-ups for points with positive A lt- C affarelli- F riedman limit , Math. Ann., 391 (2025), pp. 6251--6289, https://doi.org/10.1007/s00208-024-03077-3, https://doi.org/10.1007/s00208-024-03077-3

work page doi:10.1007/s00208-024-03077-3 2025