Boundary regularity of planar domains and the Carleson epsilon-function

A longstanding conjecture of Carleson stated that the tangent points of the boundaries of certain planar domains can be characterized by the behavior of the Carleson \(\varepsilon\)-function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that having some Dini type control of the Carleson \(\varepsilon\)-function implied the existence of tangents, up to a set of \(H^1\)-measure zero. A natural question is whether quantitative control on this function gives quantitative information about the regularity of the boundary. In this talk, we will present results that give a positive answer to this question.

Note: this event is supported by PIMS as part of the, Emerging Leaders Lecture Series.