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

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