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

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 ${\epsilon }^2$ conjecture of Carleson. It also discusses the deep connections between rectifiability and the $L^2$ boundedness of Riesz transforms and their application to the 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.

Survey paper for the ICM 2026 plenary lecture of the author

References in corpus (14)

• Faber-Krahn inequalities in sharp quantitative form
• Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps
• Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality
• Rectifiability of harmonic measure
• Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem
• Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions
• The weak-$A_{\infty }$ property of harmonic and $p$-harmonic measures implies uniform rectifiability
• Uniform rectifiability from Carleson measure estimates and $\epsilon$-approximability of bounded harmonic functions
• Characterising the big pieces of Lipschitz graphs property using projections
• Plenty of big projections imply big pieces of Lipschitz graphs
• Tangent measures of elliptic harmonic measure and applications
• Discrete Reifenberg-type theorem
• Connectivity conditions and boundary Poincaré inequalities
• Analytic capacity and dimension of sets with plenty of big projections