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arXiv 2207.01355math.CA
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通过$M^{\sharp}$函数的单侧$C_{p}$估计值

One-sided $C_{p}$ estimates via $M^{\sharp}$ function

María Lorente, Francisco J. Martín-Reyes, Israel P. Rivera-Ríos

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摘要：

我们记得$w\in C_{p}^{+}$如果存在$\varepsilon>0$和$C>0$，使得对于任何具有$c-b<b-a$和任何可测集$E\subset(a,b)$的$a<b<c$，以下成立\[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^{+}χ_{(a,c)}\right)^{p}w<\infty. \]这个条件是由Riveros和de la Torre引入的，作为Muckenhoupt和Sawyer首先研究的$C_{p}$条件的单侧对应。本文证明了给定$1<p<q<\infty$如果$w\in C_{q}^{+}$则\[ \|M^{+}f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)} \]，反之如果这样的不等式成立，则$w\in C_{p}^{+}.$

英文摘要：

We recall that $w\in C_{p}^{+}$ if there exist $\varepsilon>0$ and $C>0$ such that for any $a<b<c$ with $c-b<b-a$ and any measurable set $E\subset(a,b)$, the following holds \[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^{+}χ_{(a,c)}\right)^{p}w<\infty. \] This condition was introduced by Riveros and de la Torre as a one-sided counterpart of the $C_{p}$ condition studied first by Muckenhoupt and Sawyer. In this paper we show that given $1<p<q<\infty$ if $w\in C_{q}^{+}$ then \[ \|M^{+}f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)} \] and conversely if such an inequality holds, then $w\in C_{p}^{+}.$

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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