Function Theory on Quotient Domains

Speaker(s): Yuan Yuan (Syracuse University)

Time: 09:00-10:00 February 28, 2024

Venue: Online

Abstract: Let $f: D \rightarrow \Omega$ be a proper holomorphic map between two bounded domains. $\Omega$ is a quotient domain of $D$ if there exists a finite group $G$ such that $\Omega = X / G$. The function theory on $\Omega$ can be studied by transforming to $D$. In this way, we may study the Bergman projection, the Szeg\H{o} projection and the $\bar\partial$ problem on $\Omega$. In this talk, we will mainly discuss the recent work on the Szeg\H{o} projection.

We will introduce a boundary value problem for holomorphic functions on $D$ which enables us to define the Hardy space on $\Omega$ and derive a Bell type transformation formula for the Szego projection on $\Omega$. This definition of the Hardy space is different from the existing one in the literature and is a natural generalization of that on the planar domain considered by Lanzani-Stein. When $D$ is the unit ball or the polydisc, we provide a sufficient condition for the solution to the boundary value problem. We further obtain the sharp $L^p$ estimates for Szego projections on some quotient domains in $\mathbb{C}^2$.

Biography: Yuan Yuan is presently an associate professor at Syracuse University. He obtained the PhD in 2010 at Rutgers, New Brunswick with Xiaojun Huang and Jian Song. His research interests consist of rigidity problems for holomorphic maps, canonical Kahler metrics and dbar equations.

Zoom: https://us02web.zoom.us/j/81124856853?pwd=UzJRZHhNaG54ZGN6aUdEOUQzOU5odz09                                                                               Meeting ID: 811 2485 6853   Passcode: 301398