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. [CACDPO], results by the first author in [L], suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for $A_{\infty }$ weights [CMP,CGMP] and ideas contained in previous works by A. Seeger in [S] and D. Fan and S. Sato [FS].

29 pages, v3: minor changes

References in corpus (3)

• A sparse domination principle for rough singular integrals
• Sparse domination theorem for multilinear singular integral operators with $L^r$-Hörmander condition
• On weak weighted estimates of martingale transform

Cited by in corpus (6)

• Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates
• Bloom weighted bounds for sparse forms associated to commutators
• Sharp reverse Hölder inequality for $C_p$ weights and applications
• From weak type weighted inequality to pointwise estimate for the decreasing rearrangement
• Two-weight extrapolation on function spaces and applications
• Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators