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 Ω with Ω ∈ L ∞ (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 ∞ strong and weak type inequalities for 1 < p < ∞, and A 1 − 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 90's. As a corollary, we obtain the weighted A 1 − A ∞ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function Ω ∈ L q (S n−1 ), 1 < q < ∞, and provide Fefferman-Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by , results by the first author in [37], suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for A ∞ weights [13,16] and ideas contained in previous works by A. Seeger in [50] and D. Fan and S. Sato [22].

A sparse approach to mixed weak type inequalities

“…The purpose of the following sections we will be settling (3.1) for the operators in the main theorems. To achieve in that task we will rely upon sparse domination results, and more in particular we will use suitable splittings of the sparse families involved in the spirit of [8,23,25].…”

“…where g ≃ χ G . Then for s = 1 + 1 2τn[uv] A∞ , notice that, arguing as in [25] (uv) s (G ∩ Q) (uv) s (Q)…”

“…

In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that uv ∈ A∞ for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in [8] and extended in [23] and [25]Sawyer also conjectured that (1.1) should hold as well for the Hilbert transform. Cruz-Uribe, Martell and Pérez [7] generalized (1.1) to higher dimensions and actually proved that Sawyer's conjecture holds for Calderón-Zygmund operators via the following extrapolation argument.Theorem C. Assume that for every w ∈ A ∞ and some 0 < p < ∞,Then for every u ∈ A 1 and everyThe conditions on the weights in that extrapolation result lead them to conjecture that (1.1), and consequently the corresponding estimate for Calderón-Zygmund operators should hold as well with u ∈ A 1 and v ∈ A ∞ .

…”

“…In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that uv ∈ A∞ for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in [8] and extended in [23] and [25]…”

On the Composition of Rough Singular Integral Operators

“…Suppose that for any r ∈ (1, 2), and any bounded function f with compact support, there exists a sparse family of cubes S, such that for any function g ∈ L 1 (R d ), Added in Proof. After this paper was prepared, we learned that Li et al [31] established the quantitative weighted bounds for linear operators satisfying the assumptions in Theorem 4.5 with β = 0 (where the bilinear sparse domination is given in the form (3.13)), which coincides the conclusion in Theorem 4.5 for β = 0. The argument in [31] is different from the argument in the proof of Theorem 4.5 and is of independent interest.…”