Geometric measure theory, harmonic analysis, and PDEs

This project explores several questions in harmonic analysis, geometric measure theory, and elliptic partial differential equations (PDEs). Below are the primary topics of focus. Rectifiability is a fundamental concept in geometric measure theory, playing a key role in many analytical problems. These include the L2-boundedness of singular integral operators and square functions, the removability of singularities for Lipschitz harmonic functions, and boundary value problems for elliptic PDEs in irregular domains. A central objective of this project is to investigate the relationship between the boundedness of a suitable conical square function, rectifiability, and Favard lengtha measure quantifying the orthogonal projection of sets in all directions on the plane. Progress on this objective could contribute to resolving the open direction of Vitushkin's conjecture on removable singularities for bounded analytic functions. Harmonic measure is another critical topic, especially in the context of solving the Dirichlet problem for the Laplace equation. A key open question in this area involves determining the sharp bound for the dimension of harmonic measure in Euclidean space. This project aims to address problems related to this question, along with free boundary problems involving harmonic measure. We also investigate boundary value problems for harmonic functions and elliptic PDE solutions. One focus is on the Lp solvability of the Neumann problem in chord-arc domains, an open question first raised in the early 1990s. Other related questions include the regularity problem on rough domains and its connection to rectifiability. Additionally, we plan to address a question posed by Fang-Hua Lin concerning the unique continuation of harmonic functions that vanish continuously on a relatively open subset V of a Lipschitz domain's boundary, with gradients vanishing on a subset of V with positive surface measure. Quasiconformal mappings naturally arise in the study of elliptic PDEs in the plane and are instrumental in deducing properties of elliptic measures from harmonic measures. A critical aspect of this project is understanding the regularity of quasiconformal mappings in terms of the Beltrami coefficient, particularly within the Sobolev and Triebel-Lizorkin scale of spaces. Finally, the project includes a parabolic analysis component, focusing on the removability properties of solutions to the heat equation and the fractional heat equation. Specifically, we aim to explore these properties in terms of caloric capacities, which can be considered as analogs to Newtonian and Lipschitz harmonic capacities in the Euclidean framework.