解析帐篷空间上的两个正则性问题
Two Regularity Problems on Analytic Tent Spaces
浏览论文内容
中文总结 AI 辅助
该研究针对Hardy型解析帐篷空间，完全刻画了三类分数次积分算子的有界性与紧性、随机泰勒级数几乎必然属于该空间的条件，还解决了相关嵌入问题，推广了经典分析定理。
中文摘要 AI 辅助
我们研究单位圆盘上Hardy型解析帐篷空间$\boldsymbol{\textit{AT}}^p_{q,\boldsymbol{\textit{\u03b1}}}$的两个正则性问题：分数次积分与泰勒系数的随机化。对于分数次积分，我们完全刻画了Hadamard、Flett和Riemann–Liouville算子在解析帐篷空间之间的有界性与紧性，并得到了解析Triebel–Lizorkin空间的平行结果；特别地，当$t=0$时，该情形完全解决了解析帐篷空间对应的嵌入问题。对于随机化，我们完全刻画了当$f \boldsymbol{\u2208} \boldsymbol{\textit{AT}}^p_{q,\boldsymbol{\textit{\u03b1}}}$时，随机泰勒级数$\boldsymbol{\textit{R}}f$几乎必然属于解析帐篷空间的条件，同时也得到了其Triebel–Lizorkin对应结果。作为证明的一部分，我们确定了与$\boldsymbol{\textit{AT}}^p_{q,\boldsymbol{\textit{\u03b1}}}$相关的随机符号空间，并解决了解析帐篷空间到混合范数空间的嵌入问题。这些结果推广了Hardy–Littlewood、Littlewood的经典定理，以及它们后续在Bergman空间和混合范数空间中的类似结果。
英文摘要
We study two regularity problems on Hardy-type analytic tent spaces $\mathcal{AT}^p_{q,α}$ on the unit disk: fractional integration and randomization of Taylor coefficients. For fractional integration, we characterize completely the boundedness and compactness of the Hadamard, Flett, and Riemann--Liouville operators between analytic tent spaces, and obtain parallel results for analytic Triebel--Lizorkin spaces. In particular, the case $t=0$ yields a complete solution to the corresponding embedding problem for analytic tent spaces. For randomization, we characterize completely when the random Taylor series $\mathcal{R}f$ belongs almost surely to an analytic tent space whenever $f\in \mathcal{AT}^p_{q,α}$, and we also obtain the Triebel--Lizorkin counterpart. As part of the proof, we identify the random symbol space associated with $\mathcal{AT}^p_{q,α}$ and solve the embedding problem from analytic tent spaces into mixed norm spaces. These results extend classical theorems of Hardy--Littlewood and Littlewood, as well as their later analogues for Bergman and mixed norm spaces.
