论文标题
Bergman型操作员的LP-LQ问题
The Lp-Lq problems of Bergman-type operators
论文作者
论文摘要
令$ \ mathbb {b}^d $为复杂空间上的单位球$ \ mathbb {c}^d $，并带有标准化的lebesgue量$ dv。 $ l^1（\ Mathbb {b}^d，dv）$上的Bergman型积分运算符$k_α$由$$k_αf（z）= \ int _ {\ int _ {\ mathbb {b}^d}k_α（z，z，w）f（z，w）f（w）dv（w）dv（w）dv（w）。我们还考虑了$ l^1上的积分运算符$k_α^+ $（\ mathbb {b}^d，dv）$，由$$k_α^+ f（z）= \ int _ {\ int _ {\ mathbb {b}^d}^d} \ vertk_α（z，z，z，w）\ vert f（w） $ l^p $ - $ l^q $ $k_α，k_α^+$和$ l^p $ - $ l^q $ $k_α的紧凑性。阿米尔。数学。 Soc。 （2017年），MR3710638]在有限域$ \ mathbb {b}^d。$的情况下，给出了痕量公式和一些尖锐的规范估计值为$k_α，k_α^+$。
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.
