The Lp-Lq problems of Bergman-type operators

Abstract

Let $\mathbb{B}^d$ be the unit ball on the complex space $\mathbb{C}^d$ with normalized Lebesgue measure $dv.$ For $α\in\mathbb{R},$ denote $k_α(z,w)=\frac{1}{(1-\langle z,w\rangle)^α},$ the Bergman-type integral operator $K_α$ on $L^1(\mathbb{B}^d,dv)$ is defined by $$ K_αf(z)=\int_{\mathbb{B}^d}k_α(z,w)f(w)dv(w).$$ It is an important class of operators in the holomorphic function space theory over the unit ball. We also consider the integral operator $K_α^+$ on $L^1(\mathbb{B}^d,dv)$ which is given by $$ K_α^+ f(z)=\int_{\mathbb{B}^d}\vert k_α(z,w)\vert f(w)dv(w).$$ In this paper, we completely characterize the $L^p$-$L^q$ boundedness of $K_α,K_α^+$ and $L^p$-$L^q$ compactness of $K_α.$ The results of boundedness are in fact the Hardy-Littlewood-Sobolev theorem but also prove the conjecture of G. Cheng et al [Trans. Amer. Math. Soc. (2017), MR3710638 ] in the case of bounded domain $\mathbb{B}^d.$ Meanwhile, a trace formula and some sharp norm estimates of $K_α,K_α^+$ are given.