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

TL;DR

This paper establishes a relationship between one-sided $C_{p}$ weight conditions and bounds involving the $M^{lat}$ function, extending the understanding of weighted inequalities in harmonic analysis.

Contribution

It proves that for $1<p<q< obreak\infty$, $w otin C_{q}^{+}$ implies a bound between $M^{+}f$ and $M^{lat,+}f$, and vice versa, linking weight classes to maximal function estimates.

Findings

If $w otin C_{q}^{+}$, then $ orm{M^{+}f}_{L^{p}(w)}$ cannot be controlled by $ orm{M^{lat,+}f}_{L^{p}(w)$.

For $w otin C_{p}^{+}$, the inequality $ orm{M^{+}f}_{L^{p}(w)} ot o orm{M^{lat,+}f}_{L^{p}(w)}$ fails.

The paper characterizes $C_{p}^{+}$ weights via inequalities involving $M^{+}$ and $M^{lat,+}$ functions.

Abstract

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 \[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^{+}\chi_{(a,c)}\right)^{p}w<\infty. \] 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 $w\in C_{p}^{+}.$