We construct Ahlfors regular Cantor sets of small dimension in the plane, such that the Hausdorff measure on is equivalent to the harmonic measure associated to its complement. In particular Green’s function in satisfies whenever and is far from .
Nous construisons des ensembles de Cantor , Ahlfors-réguliers de petite dimension dans le plan, tels que la mesure de Hausdorff sur est équivalente à la mesure harmonique associée à son complémentaire. En particulier, la fonction de Green dans satisfait lorsque et est loin de .
Accepted:
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Keywords: Harmonic measure, Cantor set, Hausdorff measure
Mot clés : Mesure harmonique, Ensemble de Cantor, mesure de Hausdorff
Guy David 1; Cole Jeznach 2; Antoine Julia 1
@article{JEP_2023__10__1277_0, author = {Guy David and Cole Jeznach and Antoine Julia}, title = {Cantor sets with absolutely continuous harmonic measure}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1277--1298}, publisher = {\'Ecole polytechnique}, volume = {10}, year = {2023}, doi = {10.5802/jep.245}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.245/} }
TY - JOUR AU - Guy David AU - Cole Jeznach AU - Antoine Julia TI - Cantor sets with absolutely continuous harmonic measure JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 1277 EP - 1298 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.245/ DO - 10.5802/jep.245 LA - en ID - JEP_2023__10__1277_0 ER -
%0 Journal Article %A Guy David %A Cole Jeznach %A Antoine Julia %T Cantor sets with absolutely continuous harmonic measure %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 1277-1298 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.245/ %R 10.5802/jep.245 %G en %F JEP_2023__10__1277_0
Guy David; Cole Jeznach; Antoine Julia. Cantor sets with absolutely continuous harmonic measure. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1277-1298. doi : 10.5802/jep.245. https://jep.centre-mersenne.org/articles/10.5802/jep.245/
[ABHM19] - “Rectifiability, interior approximation and harmonic measure”, Ark. Mat. 57 (2019) no. 1, p. 1-22 | DOI | MR | Zbl
[AHM + 16] - “Rectifiability of harmonic measure”, Geom. Funct. Anal. 26 (2016) no. 3, p. 703-728 | DOI | MR | Zbl
[AHM + 20] - “Harmonic measure and quantitative connectivity: geometric characterization of the -solvability of the Dirichlet problem”, Invent. Math. 222 (2020) no. 3, p. 881-993 | DOI | MR | Zbl
[Aik01] - “Boundary Harnack principle and Martin boundary for a uniform domain”, J. Math. Soc. Japan 53 (2001) no. 1, p. 119-145 | DOI | MR | Zbl
[Anc86] - “On strong barriers and an inequality of Hardy for domains in ”, J. London Math. Soc. (2) 34 (1986) no. 2, p. 274-290 | DOI | MR | Zbl
[Azz20] - “Dimension drop for harmonic measure on Ahlfors regular boundaries”, Potential Anal. 53 (2020) no. 3, p. 1025-1041 | DOI | MR | Zbl
[Azz21] - “Semi-uniform domains and the property for harmonic measure”, Internat. Math. Res. Notices (2021) no. 9, p. 6717-6771 | DOI | MR | Zbl
[Bat96] - “Harmonic measure of some Cantor type sets”, Ann. Acad. Sci. Fenn. Math. 21 (1996) no. 2, p. 255-270 | MR | Zbl
[BZ15] - “Hausdorff and harmonic measures on non-homogeneous Cantor sets”, Ann. Acad. Sci. Fenn. Math. 40 (2015) no. 1, p. 279-303 | DOI | MR | Zbl
[Car85] - “On the support of harmonic measure for sets of Cantor type”, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), p. 113-123 | DOI | MR | Zbl
[Dah77] - “Estimates of harmonic measure”, Arch. Rational Mech. Anal. 65 (1977) no. 3, p. 275-288 | DOI | MR | Zbl
[DEM21] - “Square functions, nontangential limits, and harmonic measure in codimension larger than 1”, Duke Math. J. 170 (2021) no. 3, p. 455-501 | DOI | MR | Zbl
[DFM21] - Elliptic theory for sets with higher co-dimensional boundaries, Mem. Amer. Math. Soc., vol. 274, no. 1346, American Mathematical Society, Providence, RI, 2021 | DOI
[DFM23] - Elliptic theory in domains with boundaries of mixed dimension, Astérisque, vol. 442, Société Mathématique de France, Paris, 2023
[DJ90] - “Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals”, Indiana Univ. Math. J. 39 (1990) no. 3, p. 831-845 | DOI | MR | Zbl
[DM21] - “Good elliptic operators on Cantor sets”, Adv. Math. 383 (2021), article ID 107687, 21 pages | DOI | MR | Zbl
[DM22] - “Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions”, Adv. Math. 410 (2022), article ID 108717, 52 pages | DOI | MR | Zbl
[GM05] - Harmonic measure, New Math. Monogr., vol. 2, Cambridge University Press, Cambridge, 2005 | DOI
[JW88] - “Hausdorff dimension of harmonic measures in the plane”, Acta Math. 161 (1988) no. 1-2, p. 131-144 | DOI | MR | Zbl
[MV86] - “On the harmonic measure of discontinuous fractals”, 1986, LOMI preprints
[Tol23] - “The dimension of harmonic measure on some AD-regular flat sets of fractional dimension”, 2023 | arXiv
[Vol92] - “On the harmonic measure of self-similar sets on the plane”, in Harmonic analysis and discrete potential theory (Frascati, 1991), Plenum, New York, 1992, p. 267-280 | DOI | MR
[Vol93] - “On the dimension of harmonic measure of Cantor repellers”, Michigan Math. J. 40 (1993) no. 2, p. 239-258 | DOI | MR | Zbl
[Vol22] - “One phase problem for two positive harmonic functions: below the codimension 1 threshold”, 2022 | arXiv
[Zdu97] - “Harmonic measure on the Julia set for polynomial-like maps”, Invent. Math. 128 (1997) no. 2, p. 303-327 | DOI | MR | Zbl
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