Weighted Norm Inequalities for Rough Singular Integral Operators

Abstract

In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals TΩ with Ω ∈ L∞(Sn-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 Ap- A∞ strong and weak type inequalities for 1 < p< ∞, and A1- A∞ 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 1990s. As a corollary, we obtain the weighted A1- A∞ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function Ω ∈ Lq(Sn-1) , 1 < q< ∞, 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. (Anal PDE 10(5):1255–1284, 2017), results by the first author (Collect Math 68:129–144, 2017), suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for A∞ weights (Cruz-Uribe et al. in J Funct Anal 213:412–439, 2004, Curbera et al. in Adv Math 203:256–318, 2006), and ideas contained in previous works by Seeger (J Am Math Soc 9:95–105 1996) and Fan and Sato (Tohoku Math J 53:265–284, 2001).