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arXiv · 2607.16457

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

Xavier Tolsa
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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.

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Xavier Tolsa. 2026-07-17. Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions. https://doi.org/10.1137/25m1804054

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