Harmonic measure and quantitative connectivity: geometric characterization of the \(L^p\)-solvability of the Dirichlet problem

DOI10.1007/s00222-020-00984-5zbMath1453.31009arXiv1907.07102OpenAlexW2796791584MaRDI QIDQ2208433

Mihalis Mourgoglou, Jonas Azzam, Xavier Tolsa, José Maria Martell, Steven Hofmann

Publication date: 2 November 2020

Published in: Inventiones Mathematicae (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1907.07102

zbMATH Keywords

Dirichlet problemharmonic measuresAhlfors-David regular boundaryquantitative connectivity

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Maximal functions, Littlewood-Paley theory (42B25) Harmonic, subharmonic, superharmonic functions in higher dimensions (31B05) Harmonic analysis and PDEs (42B37)

Related Items (27)

Uniform rectifiability implies Varopoulos extensions ⋮ Generalized Carleson perturbations of elliptic operators and applications ⋮ Carleson measure estimates and \(\varepsilon\)-approximation for bounded harmonic functions, without Ahlfors regularity assumptions ⋮ Absolute continuity of the harmonic measure on low dimensional rectifiable sets ⋮ Carleson estimates for the Green function on domains with lower dimensional boundaries ⋮ A change of variable for Dahlberg-Kenig-Pipher operators ⋮ On the condition for elliptic operators in 1-sided nontangentially accessible domains satisfying the capacity density condition ⋮ Small 𝐴_{∞} results for Dahlberg-Kenig-Pipher operators in sets with uniformly rectifiable boundaries ⋮ Cantor sets with absolutely continuous harmonic measure ⋮ The Green function with pole at infinity applied to the study of the elliptic measure ⋮ Corona decompositions for parabolic uniformly rectifiable sets ⋮ Necessary condition for the L² boundedness of the Riesz transform on Heisenberg groups ⋮ Carleson perturbations for the regularity problem ⋮ A Green function characterization of uniformly rectifiable sets of any codimension ⋮ A \(d\)-dimensional analyst's travelling salesman theorem for subsets of Hilbert space ⋮ Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets ⋮ Locally uniform domains and extension of bmo functions ⋮ Small-constant uniform rectifiability ⋮ The regularity problem in domains with lower dimensional boundaries ⋮ Green function estimates on complements of low-dimensional uniformly rectifiable sets ⋮ Carleson perturbations of elliptic operators on domains with low dimensional boundaries ⋮ A weak reverse Hölder inequality for caloric measure ⋮ Good elliptic operators on Cantor sets ⋮ Square functions, nontangential limits, and harmonic measure in codimension larger than 1 ⋮ Uniform rectifiability and elliptic operators satisfying a Carleson measure condition ⋮ Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions ⋮ Carleson measure estimates for the Green function

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