关于 Hausdorff 容量意义下 Capacity Muckenhoupt 权的一个注记
A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content
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中文总结 AI 辅助
本文证明在固定指标 p 时，Capacity Muckenhoupt 权类关于指标 δ 不满足自改进性质，补充了该类权性质的研究。
中文摘要 AI 辅助
设 $p\in[1,\infty)$ 且 $\delta\in(0,n]$。近期，为了刻画基于 Hausdorff 容量 $\mathcal H_\infty^\delta$ 的 Choquet 积分上极大算子的加权有界性，引入了一类 Capacity Muckenhoupt 权，记为 $\mathcal A_{p,\delta}$，并且对于任意固定的 $\delta\in(0,n]$，已证明这类新权类关于指标 $p\in(1,\infty)$ 满足自改进性质。在本注记中，我们证明：对于每个固定的 $p\in[1,\infty)$，Capacity Muckenhoupt 权类 $\mathcal A_{p,\delta}$ 关于指标 $\delta\in(0,n]$ 不满足自改进性质。
英文摘要
Let $p\in[1,\infty)$ and $δ\in(0,n]$. Recently, to characterize the weighted boundedness of maximal operator on Choquet integrals based on Hausdorff content $\mathcal H_\infty^δ$, a class of Capacity Muckenhoupt weights is introduced, denoted by $\mathcal A_{p,δ}$, and, for any fixed $δ\in(0,n]$, this new class of weights is proved to satisfy the self-improving property with respect to the index $p\in(1,\infty)$. In this note, we show that, for every fixed $p\in[1,\infty)$, the class of capacity Muckenhoupt weights $\mathcal A_{p,δ}$ fails to satisfy the self-improving property with respect to the index $δ\in(0,n]$.
