Regularity of the Weighted Bergman Projection on the Fock–bargmann–hartogs Domain

Abstract

The Fock-Bargmann-Hartogs domain D n,m (µ) is defined bywhere µ > 0. The Fock-Bargmann-Hartogs domain D n,m (µ) is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. In this paper, we first compute the weighted Bergman kernel of D n,m (µ) with respect to the weight (−ρ) α , where ρ(z, w) := w 2 − e −µ z 2 is a defining function for D n,m (µ) and α > −1. Then, for p ∈ [1, ∞), we show that the corresponding weighted Bergman projection P Dn,m(µ),(−ρ) α is unbounded on L p (D n,m (µ), (−ρ) α ), except for the trivial case p = 2. In particular, this paper gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is L p irregular when p ∈ [1, ∞) \ {2}. This result turns out to be completely different from the well-known positive L p regularity result on bounded strongly pseudoconvex domain.