$A_1$ theory of weights for rough homogeneous singular integrals and commutators

Abstract

Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_{\Omega}$ and commutators of \BMO symbols and $T_{\Omega}$ are obtained. In particular the following estimates are proved: $$\|T_\Omega \|_{L^p(w)}\le c_{n,p}\|\Omega\|_{L^\infty} [w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)}$$ and \| [b,T_{\Omega}]f\| _{L^{p}(w)}\leq c_{n,p}\|b\|_{\BMO}\|\Omega\|_{L^{\infty}} [w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)}, for $1<p<\infty$ and $1/p+1/p'=1$.BERC 2014-2017 BCAM Severo Ochoa excellence accreditation SEV-2013-0323 MTM2014-53850-P. MTM2015-65888-C04-4-P 2017 Leonardo grant for Researchers and Cultural Creators, BBVA Foundatio