Quantitative estimates for the dimension drop of harmonic measure on Ahlfors regular boundaries

Speaker: Dr. Yingying Cai，Universitat Autònoma de Barcelona

Title: Quantitative estimates for the dimension drop of harmonic measure on Ahlfors regular boundaries

Language: Chinese

Time & Venue: 2026.07.27   10:00-11:00    MCM110

Abstract: In the first part of this two-hour series, I will introduce the fundamental concepts and recent developments regarding the dimension drop of harmonic measure. We consider domains of the form $\Omega = \mathbb{R}^{n+1} \setminus E$, where the boundary $E$ is an $s$-Ahlfors regular compact set. A central question in geometric measure theory is understanding how the geometric properties of the boundary, specifically non-flatness, influence the dimension of the associated harmonic measure. I will begin by reviewing classical results and providing the necessary background. Then, based on joint work with Xavier Tolsa, I will state our main quantitative estimates for the dimension drop. In the remainder of this first talk, we will focus on the case of planar domains. I will explain how we establish a quantitative threshold $s_0 = 1 - c\delta_0^2$ under Azzam's uniform non-flatness condition ($\beta_\infty + \beta_{\text{hole}} \ge \delta_0$), highlighting the key geometric intuitions in $\mathbb{R}^2$.