Hausdorff–Choquet角空间与粗糙极大算子的弱型(1,1)界
Hausdorff-Choquet Angular Spaces and Weak-Type $(1,1)$ Bounds for Rough Maximal Operators
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中文总结 AI 辅助
本文针对E.M. Stein的公开猜想，研究带粗糙核的极大算子，证明当核属于与Hausdorff–Choquet角空间相关的空间时该算子满足弱型(1,1)界，部分解决猜想并改进了已有结果，还确定了Hausdorff–Choquet尺度下一致弱型(1,1)估计的最优指数。
中文摘要 AI 辅助
本文研究极大算子$\u200b\ud835\udc40_\u03a9 f(x):= \u200b\ud835\udc5b_{r>0}\u200b\frac{1}{r^n}\u200b\ud835\udc46_{|y|<r}\u200b|f(x-y)|\u200b|\u03a9\ud835\udc5c\frac{y}{|y|}\ud835\udc5d|\u200b dy\u200b。E.~M.~Stein提出的长期公开猜想询问：当$\u03a9$仅属于$L^1(\u03a3^{n-1})$时，极大算子$\u200b\ud835\udc40_\u03a9$是否为弱型(1,1)。我们通过证明核函数$\u03a9\u2208\u214b(\u03a3^{n-1})$时$\u200b\ud835\udc40_\u03a9$的弱型(1,1)界，部分解决了该问题，相比M.~Christ与Rubio de Francia的工作有显著改进。此处$\u214b(\u03a3^{n-1})$是与Hausdorff–Choquet角空间相关的空间，且$L\ud835\udc5f^+\u200b L(\u03a3^{n-1})\u2282neq\u214b(\u03a3^{n-1})\u2282 L^1(\u03a3^{n-1})$。最后，对一般Hausdorff–Choquet尺度$\u214b\u214b_\u03b1$，我们证明$\u03b1=(n-1)/2$是一致弱型(1,1)估计的最优指数。
英文摘要
In the present paper, we consider the maximal operator \[ \mathcal M_Ωf(x) := \sup_{r>0}\frac{1}{r^n} \int_{|y|<r} |f(x-y)| \left| Ω\!\left(\frac{y}{|y|}\right) \right|\,dy. \] A longstanding open conjecture raised by E.M.Stein asks whether the maximal operator $\mathcal M_Ω$ is of weak type $(1,1)$ when $Ω$ is merely in $L^1(\mathbb S^{n-1})$. We partially settle this problem by proving weak type $(1,1)$ bounds of $\mathcal M_Ω$ with kernel $Ω\in\mathcal X(\mathbb S^{n-1})$, yielding a significant improvement over the work of M.~Christ and Rubio de Francia. Here$\mathcal X(\mathbb S^{n-1})$ is the space related to the Hausdorff--Choquet angular space and $$ L\log^+\!L(\mathbb S^{n-1}) \subsetneq \mathcal X(\mathbb S^{n-1})\subset L^1(\mathbb S^{n-1}). $$ Prior to our work, the condition $Ω\in L\log^+\!L(\mathbb S^{n-1}) $ remained the best known size restriction guaranteeing the weak-type$ (1,1)$ bounds for all dimensions $n > 2$. To our knowledge, the present paper is the first to relax this threshold for $n>2$. Finally, for the general Hausdorff--Choquet scale $\mathcal H\mathcal C_α$, we show that $α=(n-1)/2$ is the sharp exponent for uniform weak type $(1,1)$ estimates.
发表机构
- Northeastern University（东北大学）
- State Key Laboratory of Synthetical Automation for Process Industries（流程工业综合自动化国家重点实验室）
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