AI 中文总结
本文研究单位球加权Bergman空间上由[0,1)测度诱导的加权Cesàro型算子，在0<p,q≤∞全范围给出有界性刻画，部分结果在一维或单位圆盘情形下也为新结论。
AI 中文摘要
本文研究由[0,1)上的测度诱导的加权Cesàro型算子\\[ \mathcal C_{μ,β}^ξ:A_α^{p}(\mathbb B_n) \longrightarrow A_α^{q}(\mathbb B_n) \\]，并在0<p,q≤∞的全范围内得到其有界性刻画。首先，我们将单位圆盘上1≤p≤q<∞范围内的相关结果推广到单位球情形。借助原子分解，我们进一步处理了0<p<1且p≤q<∞的情况，从而得到0<p≤q<∞范围内的完整刻画，后一结果即使在一维情形下也是全新的。对于0<q<p≤∞的范围，我们通过涉及诱导测度尾函数的积分条件来刻画其有界性，该结果即使在单位圆盘上也属新结论。最后，我们刻画了当目标空间为\\(H^\infty(\mathbb B_n)\\)时对应算子的有界性。
英文摘要
In this paper, we study weighted Cesàro type operators \[ \mathcal C_{μ,β}^ξ:A_α^{p}(\mathbb B_n) \longrightarrow A_α^{q}(\mathbb B_n) \] induced by measures on $[0,1)$, and obtain boundedness characterizations throughout the range $0<p,q\le\infty$. We first extend the corresponding results on the unit disk to the unit ball in the range $1\le p\le q<\infty$. By means of atomic decomposition, we further treat the case $0<p<1$ and $p\le q<\infty$, thereby obtaining a complete characterization for $0<p\le q<\infty$. The latter result is new even in one dimension. For the range $0<q<p\le\infty$, we characterize the boundedness in terms of an integral condition involving the tail function of the inducing measure. This result is also new even on the unit disk. Finally, we characterize the boundedness of the corresponding operators when the target space is $H^\infty(\mathbb B_n)$.
