One-sided $C_p$ estimates via $M^{♯}$ function

Abstract

We recall that $w\in C_p^{+}$ if there exist $\epsilon >0$ 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 $w\in C_p^{+}.$