Quantitative $C_p$ estimates for Calderón-Zygmund operators

Javier Canto

Published 2018-11-13Version 1

We prove an appropriate 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 $q>p>1$ improving Sawyer's theorem.