arXiv 2003.004792020-03-01math.FAmath.CV

Bergman型算子的Lp-Lq问题

The Lp-Lq problems of Bergman-type operators

Lijia Ding, Kai Wang

备注：: 26 pages, 1 figure

摘要：

设$\mathbb{B}^d$是复空间$\mathbb{C}^d$上的单位球，具有正规化Lebesgue测度$dv.$。当$α\in\mathbb{R},$表示$k_α(z,w)=\frac{1}{(1-\langle z,w\rangle)^α},$时，定义了$L^1(\mathbb{B}^d,dv)$上的Bergman型积分算子$K_α$为$$ K_αf(z)=\int_{\mathbb{B}^d}k_α(z,w)f(w)dv(w).$$。我们还考虑了由$$ K_α^+ f(z)=\int_{\mathbb{B}^d}\vert k_α(z,w)\vert f(w)dv(w).$$给出的$L^1(\mathbb{B}^d,dv)$上的积分算子$K_α^+$。本文完整地刻画了$K_α,K_α^+$的$L^p$-$L^q$有界性和$K_α.$的$L^p$-$L^q$紧性。Cheng et al [Trans. Amer. Math. Soc.（2017），MR 3710638]在有界域$\mathbb{B}^d.$的情况下同时给出了$K_α,K_α^+$的迹公式和一些尖锐的模估计.

英文摘要：

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.

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