Sharp reverse Hölder inequality for $C_p$ weights and applications

We prove an appropriate sharp quantitative reverse Hölder inequality for the $C_p$ class of weights from which we obtain as a limiting case the sharp reverse Hölder inequality for the $A_\infty$ class of weights. We use this result to provide a quantitative weighted norm inequality between Calderón-Zygmund operators and the Hardy-Littlewood maximal function, precisely $\|Tf\|_{L^p(w)} \lesssim_{T,n,p,q} [w]_{C_q}(1+\log^+[w]_{C_q})\|Mf\|_{L^p(w)},$ for $w\in C_q$ and $q>p>1$, quantifying Sawyer's theorem.

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