AI 中文总结
该论文围绕定量可求长性展开，回顾其与可求长性刻画、里斯变换L²有界性的联系及应用，还探讨了其在粗糙区域中拉普拉斯方程相关问题的L^p可解性方面的关键作用。
AI 中文摘要
本文综述了定量可求长性理论起核心作用的不同主题。首先回顾了用涉及β型系数的平方函数对可求长性的刻画以及卡尔森的ε²猜想。还讨论了可求长性与里斯变换的L²有界性之间的深层联系及其在利普希茨调和函数的Painlevé问题中的应用。最后探讨了与调和测度以及粗糙区域中拉普拉斯方程的狄利克雷、正则性和诺伊曼问题的L^p可解性相关的近期重大进展，强调了定量可求长性在这些进展中的关键作用。
英文摘要
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 $\varepsilon^2$ conjecture of Carleson. It also discusses the deep connections between rectifiability and the $L^2$ boundedness of Riesz transforms and their application to the 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.
CommentsSurvey paper for the ICM 2026 plenary lecture of the author
Journal refProceedings of the International Congress of Mathematicians 2026 - Volume 2: Plenary Lectures. 2026, 313-340 10.1137/25M1804054
