Weighted norm inequalities for rough singular integral operators

Abstract

In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals $T_\Omega$ with $\Omega\in L^\infty(\mathbb{S}^{n-1})$ and the Bochner--Riesz multiplier at the critical index $B_{(n-1)/2}$. More precisely, we prove qualitative and quantitative versions of Coifman--Fefferman type inequalities and their vector-valued extensions, weighted $A_p-A_\infty$ strong and weak type inequalities for $1<p<\infty$, and $A_1-A_\infty$ type weak $(1,1)$ estimates. Moreover, Fefferman--Stein type inequalities are obtained, proving in this way a conjecture raised by the second-named author in the 90's. As a corollary, we obtain the weighted $A_1-A_\infty$ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function $\Omega\in L^q(\mathbb{S}^{n-1})$, $1<q<\infty$, and provide Fefferman--Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by Conde--Alonso et al. \cite{CACDPO}, results by the first author in \cite{L}, suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for $A_{\infty}$ weights \cite{CMP,CGMP} and ideas contained in previous works by A. Seeger in \cite{S} and D. Fan and S. Sato \cite{FS}.Juan de la Cierva - Formaci\'on 2015 FJCI-2015-24547. BBVA Foundatio