$L^{p}$ REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN. Issue 2 (8th October 2020)

• Record Type: Journal Article
• Authors: HE, LE; TANG, YANYAN; TU, ZHENHAN
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 102:Issue 2(2020)
• Page Start: 282
• Page End: 292
• Publication Date: 2020-10-08
• Subjects: 32A36, -- 32A25, -- 32A07; Fock–Bargmann–Hartogs domain, -- Lp regularity, -- weighted Bergman kernel, -- weighted Bergman projection; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972719001424
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : The Fock–Bargmann–Hartogs domain $D_{n, m}(\, \unicode[STIX]{x1D707}):=\{(z, w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$, where $\unicode[STIX]{x1D707}>0$, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of $D_{n, m}(\, \unicode[STIX]{x1D707})$ with respect to the weight $(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$, where $\unicode[STIX]{x1D70C}(z, w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ and $\unicode[STIX]{x1D6FC}>-1$ . Then, for $p\in [1, \infty ), $ we show that the corresponding weighted Bergman projection $P_{D_{n, m}(\, \unicode[STIX]{x1D707}), (-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$ is unbounded on $L^{p}(D_{n, m}(\, \unicode[STIX]{x1D707}), (-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$, except for the trivial case $p=2$ . This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^{p}$ irregular when $p\in [1, \infty )\setminus \{2\}$, in contrast to the well-known positive $L^{p}$ regularity result on a bounded strongly pseudoconvex domain.