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

Abstract

The Fock-Bargmann-Hartogs domain $D_{n,m}(\hat{I}¼)$ is defined by $$D_{n,m}(\hat{I}¼):=\{(z,w)\in {\mathbb{C}}^n\times {\mathbb{C}}^m:\Vert w{\Vert }^2<e^{-\hat{I}¼\Vert z{\Vert }^2}\},$$ where $\hat{I}¼>0.$ The Fock-Bargmann-Hartogs domain $D_{n,m}(\hat{I}¼)$ is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. In this paper, we first compute the weighted Bergman kernel of $D_{n,m}(\hat{I}¼)$ with respect to the weight $(-\ddot{I}{)}^{\hat{I}}\pm$, where $\ddot{I}(z,w):=\Vert w{\Vert }^2-e^{-\hat{I}¼\Vert z{\Vert }^2}$ is a defining function for $D_{n,m}(\hat{I}¼)$ and $\hat{I}\pm >-1$. Then, for $p\in [1,\infty ),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\hat{I}¼),(-\ddot{I}{)}^{\hat{I}}\pm }$ is unbounded on $L^p(D_{n,m}(\hat{I}¼),(-\ddot{I}{)}^{\hat{I}}\pm )$, 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 $p\in [1,\infty )\setminus \{2\}$. This result turns out to be completely different from the well-known positive $L^p$ regularity result on bounded strongly pseudoconvex domain.