Boundedness of weighted Bergman projection with several complex variables

Abstract

In this paper, we focus on the $L^p$ regularity of weighted Bergman projection on the unit ball ${\mathbb{B}}^n$ and complex space ${\mathbb{C}}^n$ with radial weight. We prove that the weighted Bergman projection on $L^p({\mathbb{B}}^n,{\rho }_1)$ and $L^p({\mathbb{C}}^n,{\rho }_2)$ is bounded only if $p=2,$ respectively, where ${\rho }_1(z)=(1-|z{|}^2{)}^{A_1}\exp \{-\frac{B_1}{(1-|z{|}^2{)}^{{\alpha }_1}}\}$ and ${\rho }_2(z)=|z{|}^{2A_2}{\mathrm{e}}^{-B_2|z{|}^{2{\alpha }_2}}$.

Keywords:

• $L^p$ regularity
• weighted Bergman projection
• the unit ball
• complex space ${\mathbb{C}}^n$
• bounded operator

AMS Subject Classifications:

• 32A36
• 31B10
• 47B91

Acknowledgments

The authors would like to sincerely thank Prof. Tao Qian for his valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This paper is supported by the National Natural Science Foundation of China [Grant nos. 12071035 and 11971045].