Analysis Seminar: Weighted Estimates for the Bergman Projection on Planar Domains

Speaker: Nathan Wagner, GMU

Title: Weighted Estimates for the Bergman Projection on Planar Domains

Abstract:   Motivated by real variable harmonic analysis, weighted inequalities for the Bergman projection operator have been studied since the late 1970s. They were first proven on the unit disk, which was quickly followed by a several variable generalization to the unit ball. Recent work has focused on more general domains in several complex variables (e.g.  smoothly bounded strongly pseudoconvex domains), but comparatively little work was done on rougher planar domains. In this talk, we discuss joint work with A. Walton Green on this topic. We investigate weighted Lebesgue space estimates for the Bergman projection on a simply connected planar domain via the domain's  Riemann map. We extend the bounds which follow from a standard change-of-variable argument in two ways. First, we provide a regularity condition on the Riemann map, which turns out to be necessary in the case of uniform domains, in order to obtain the full range of weighted estimates for the Bergman projection for weights in a Békollè-Bonami-type class. Second, by slightly strengthening our condition on the Riemann map, we obtain the weighted weak-type (1,1) estimate as well. Our proofs draw on techniques from both conformal mapping and dyadic harmonic analysis. This talk will build off concepts and theorems we discussed in analysis seminars the previous semester, and will also include a survey of some unweighted regularity results for the Bergman projection on planar domains to better situate our work in context. 

Date/Time: Friday, February 6, 11:30am

Location: Exploratory Hall, Room 4106 or Zoom