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

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 $${\int }_{E}w\le C{\left(\frac{\mathrm{\mid }E\mathrm{\mid }}{(c-b)}\right)}^{\epsilon }{\int }_{\mathbb{R}}{\left(M^{+}{\chi }_{(a,c)}\right)}^{p}w<\mathrm{\infty }\mathrm{.}$$ 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 $1<p<q<\mathrm{\infty }$ if $w\in C_q^{+}$ then $$\mathrm{\parallel }{M}^{+}f{\mathrm{\parallel }}_{L^p(w)}\lesssim \mathrm{\parallel }{M}^{\mathrm{♯},+}f{\mathrm{\parallel }}_{L^p(w)}$$ and conversely if such an inequality holds, then $w\in C_p^{+}\mathrm{.}$