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

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 $w\in C_{p}^{+}.$