The Boundedness of Forelli–Rudin Type Operators on the Siegel Upper Half Space

Abstract

We completely characterize $L^p$-$L^q$ boundedness of two classes of Forelli–Rudin type operators associated to parameters (a, b, c) on the Siegel upper half space for all $(p, q)\in [1, \infty ]\times [1, \infty ]$. The characterization is by descriptions of triple parameters (a, b, c), in which the description of the parameter c makes the results different from the corresponding results on the unit ball of $\mathbb {C}^n$. These results not only generalize and extend some known results on the upper half plane (Cheng et al., Trans. Amer. Math. Soc. (12)2017, 8643-8662, Theorem 5) and on the Siegel upper space (Liu et al., Complex Anal. Oper. Theory, 13(2019), 685-701, Theorem 1), but also improved the related results in (Wang and Liu, Czechoslovak Math. J. 71(146)(2021), 475-490) by removing the assumption of the parameters. In particular, we answer Conjecture 1.4 of Wang and Liu for the case $p=1, q=\infty$ by an improved version. Furthermore, we establish the relationship of the $L^p$-$L^q$ boundedenss of Forelli–Rudin type operators between the ball $\mathbb {B}_n$ and the Siegel upper half space ${\mathcal {U}}$ based on the boundedness of embedding maps induced by the Cayley transforms between $L^p(\mathbb {B}_n)$ and $L^p({\mathcal {U}})$.