L^p regularity of weighted Bergman projection on Fock-Bargmann-Hartogs domain

2019/10/14 by He, Le, Tang, Yanyan, Tu, Zhenhan
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1910.05892

Abstract

The Fock-Bargmann-Hartogs domain D_{n, m}(μ) is defined by D_{n, m}(μ):=\(z, w)∈ℂⁿ×ℂ^m:\Vert w \Vert²0. 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_D_{n, 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.

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• Estimates for some Weighted Bergman Projections
• Estimates of the L^p norms of the Bergman projection on strongly pseudoconvex domains
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• L^p regularity of the Bergman projection on generalizations of the Hartogs triangle in ℂⁿ⁺¹