Interactions between Quantitative Rectifiability, Singular Integrals, and Boundary Value Problems for Harmonic Functions

2026/07/01 by Xavier Tolsa
Mathematics · #Spectral Theory in Mathematical Physics #Differential Equations and Boundary Problems #Analytic and geometric function theory

paper · pdf · doi:10.1137/25m1804054

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 β type coefficients and the ε² conjecture of Carleson. It also discusses the deep connections between rectifiability and the L² boundedness of Riesz transforms and their application to the Painlevé 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.

Citations

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

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