Sharp reverse Hölder inequality for $C_p$ weights and applications

Abstract: We prove an appropriate sharp quantitative reverse H\"older inequality for the $C_p$ class of weights from which we obtain as a limiting case the sharp reverse H\"older inequality for the $A_\infty$ class of weights. We use this result to provide a quantitative weighted norm inequality between Calder\'on-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.