$L^p$-boundedness of Berezin transforms on generalized Hartogs triangles

Abstract

Let $m,l\in\N$ be relatively prime and let \[ Ω_{m/l}^{n+1}=\bigl\{(z,w)\in\C^n\times\C:\ \norm{z}^{m}<\abs{w}^{l}<1\bigr\} \] be the rational generalized Hartogs triangle of exponent $m/l$ in $\C^{n+1}$. In this paper, we study the Berezin transform $\BB_{m/l,n}$ associated with the Bergman kernel of $Ω_{m/l}^{n+1}$, and prove that $\BB_{m/l,n}$ is bounded on $L^p(Ω_{m/l}^{n+1})$ if and only if $p>m+nl$.