L^p boundedness for the Bergman projections over n-dimensional generalized Hartogs triangles

Abstract

The n-dimensional generalized Hartogs triangles are domains defined by ${\mathbb{H}}_{\mathbf{p}}^n:=\left\{\right(z_1,\dots ,z_n)\in {\mathbb{C}}^n:|z_1{|}^{p_1}<\cdots <|z_n{|}^{p_n}<1\}$ with $\mathbf{p}:=(p_1,\dots ,p_n)\in ({\mathbb{Z}}^{+}{)}^n$ and $n\ge 2$. In this paper, we first obtain an estimate for the Bergman kernel of ${\mathbb{H}}_{\mathbf{p}}^n$ and then use it to establish the $L^p$ boundedness of the associated Bergman projections. Our result generalizes the $L^p$ boundedness result for two-dimensional generalized Hartogs triangles obtained by L.D. Edholm and J.D. McNeal in [Bergman subspaces and subkernels: degenerate Lp mapping and zeroes. J Geom Anal. 2017;27:2658–2683] to n-dimensional settings.

Keywords:

• Bergman projection
• $L^p$ boundedness
• generalized Hartogs triangles

AMS SUBJECT CLASSIFICATIONS:

• 32A36
• 32A25

Acknowledgements

The author thanks his Ph.D. advisor Prof. Feng Rong for helpful comments and suggestions to this manuscript. The author also wishes to thank the anonymous referee for their useful comments.

Disclosure statement

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

Additional information

Funding

The author is partially supported by the National Natural Science Foundation of China [Grant No. 11871333].