$L^p$-Boundedness of Forelli-Rudin Type Operators on Rational Hartogs Triangles

Let $H_{m\mathrm{/}n}=\{(z_1,z_2)\in {\text{C}}^2:\mathrm{\mid }{z}_1{\mathrm{\mid }}^m<\mathrm{\mid }{z}_2{\mathrm{\mid }}^n<1\},\gcd (m,n)=1,$ be a rational Hartogs triangle. We characterize the $L^p$-boundedness of Forelli–Rudin type operators associated with its Bergman kernel. For the operators with kernel $\mathrm{\mid }{B}_{m\mathrm{/}n}(z,w){\mathrm{\mid }}^{c\mathrm{/}2}$, the characterization holds for all $a,b\in \mathbb{R}$ and $c>0$; for the operators with kernel $B_{m\mathrm{/}n}(z,w{)}^N$, it holds for every $N\in {\mathbb{Z}}_{+}$. The conditions are necessary and sufficient and recover the sharp $L^p$-ranges of the Bergman projection and the Berezin transform.