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 $\alpha \in \mathbb{R},$ denote $k_{\alpha }(z,w)=\frac{1}{(1-⟨z,w⟩{)}^{\alpha }},$ the Bergman-type integral operator $K_{\alpha }$ on $L^1({\mathbb{B}}^d,dv)$ is defined by $$K_{\alpha }f(z)={\int }_{{\mathbb{B}}^d}{k}_{\alpha }(z,w)f(w)dv(w).$$ It is an important class of operators in the holomorphic function space theory over the unit ball. We also consider the integral operator $K_{\alpha }^{+}$ on $L^1({\mathbb{B}}^d,dv)$ which is given by $$K_{\alpha }^{+}f(z)={\int }_{{\mathbb{B}}^d}|k_{\alpha }(z,w)|f(w)dv(w).$$ In this paper, we completely characterize the $L^p$-$L^q$ boundedness of $K_{\alpha },K_{\alpha }^{+}$ and $L^p$-$L^q$ compactness of $K_{\alpha }.$ 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 $K_{\alpha },K_{\alpha }^{+}$ are given.

26 pages, 1 figure

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• Schatten class Bergman-type and Szegö-type operators on bounded symmetric domains