Uniform rectifiability from Carleson measure estimates and {\(\epsilon\)}-approximability of bounded harmonic functions

DOI10.1215/00127094-2017-0057zbMath1396.28005arXiv1611.00264OpenAlexW3103811225MaRDI QIDQ1645024

Mihalis Mourgoglou, Xavier Tolsa, John B. Garnett

Publication date: 28 June 2018

Published in: Duke Mathematical Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1611.00264

zbMATH Keywords

Carleson measuresuniform rectifiability\(\epsilon\)-approximation

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Geometric measure and integration theory, integral and normal currents in optimization (49Q15) Harmonic, subharmonic, superharmonic functions in higher dimensions (31B05) Length, area, volume, other geometric measure theory (28A75) Integration of real functions of several variables: length, area, volume (26B15) Hausdorff and packing measures (28A78) Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions (31A15)

Related Items (21)

Uniform rectifiability implies Varopoulos extensions ⋮ Carleson measure estimates and \(\varepsilon\)-approximation for bounded harmonic functions, without Ahlfors regularity assumptions ⋮ BMO solvability and absolute continuity of caloric measure ⋮ The two-phase problem for harmonic measure in VMO ⋮ Corona decompositions for parabolic uniformly rectifiable sets ⋮ Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets ⋮ A two-phase free boundary problem for harmonic measure and uniform rectifiability ⋮ \(L^2\)-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type ⋮ BMO solvability and absolute continuity of harmonic measure ⋮ $\varepsilon $-approximability of harmonic functions in $L^p$ implies uniform rectifiability ⋮ Failure of \(L^2\) boundedness of gradients of single layer potentials for measures with zero low density ⋮ Uniform rectifiability and \(\varepsilon\)-approximability of harmonic functions in \(L^p\) ⋮ Uniform rectifiability and elliptic operators satisfying a Carleson measure condition ⋮ Plenty of big projections imply big pieces of Lipschitz graphs ⋮ On big pieces approximations of parabolic hypersurfaces ⋮ A new elliptic measure on lower dimensional sets ⋮ Quantitative absolute continuity of harmonic measure and the Dirichlet problem: a survey of recent progress ⋮ Quantitative Fatou theorems and uniform rectifiability ⋮ Square function estimates, the BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets ⋮ Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions ⋮ Elliptic Theory for Sets with Higher Co-dimensional Boundaries

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