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

Abstract

Let $ H_{m/n}=\{(z_1,z_2)\in\C^2:|z_1|^m<|z_2|^n<1\}, \qquad \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 $|B_{m/n}(z,w)|^{c/2}$, the characterization holds for all $a,b\in\R$ and $c>0$; for the operators with kernel $B_{m/n}(z,w)^N$, it holds for every $N\in\Z_+$. The conditions are necessary and sufficient and recover the sharp $L^p$-ranges of the Bergman projection and the Berezin transform.