Ap-A∞ estimates for multilinear maximal and sparse operators

Abstract

We obtain mixed A p -A ∞ estimates for a large family of multilinear maximal and sparse operators. Operators from this family are known to control for instance multilinear Calderón-Zygmund operators, square functions, fractional integrals, and the bilinear Hilbert transform. Our results feature a new multilinear version of the Fujii-Wilson A ∞ characteristic that allows us to recover the best known estimates in terms of the A p characteristic for dependent weights as a special case of the mixed characteristic estimates for general tuples of weights.

Sparse bounds for maximal rough singular integrals via the Fourier transform

“…Estimating the sparse norm(s) of a sublinear or multisublinear operator entails a sharp control over the behavior of such operator in weighted L pspaces; this theme has been recently pursued by several authors, see for instance [1,8,19,20,21,33]. This sharp control is exemplified in the following proposition, which is a collection of known facts from the indicated references.…”

Singular Brascamp–Lieb inequalities with cubical structure

“…It follows from a well known paradigm used in [2] in the context of the bilinear Hilbert transform, another prominent singular Brascamp-Lieb form. In our instance, the reduction is essentially a consequence of Theorem 1.11 in [21]. Indeed, one obtains in [21] more general bounds in weighted L p spaces, with good dependence on the weight classes.…”

“…In our instance, the reduction is essentially a consequence of Theorem 1.11 in [21]. Indeed, one obtains in [21] more general bounds in weighted L p spaces, with good dependence on the weight classes. Further, one can deduce vector valued bounds as in [18].…”

“…where in the last inequality we used that (Q ′ ) * ⊆ Q * implies that Q ′ is contained in one of the 3 m cubes of the same size as Q that are contained in Q * and we applied the assumption of sparsity of S on each of these cubes. Applying Theorem 1.11 in [21] with ρ i := 0, r i := 1, t i := p i , and w i := 1 to (2.1) for each 1 ≤ i ≤ 3 m yields Theorem 1.1. Alternatively, the reduction of Theorem 1.1 to Theorem 1.2 might possibly be accomplished with estimates for sparse operators rather than sparse forms as in [15].…”

Off-diagonal sharp two-weight estimates for sparse operators

“…The necessity of the A α pq -condition, and the lower bound in (1.3), follows simply by substituting f = 1 Q for any Q ∈ S and estimating A r,α S (f σ) ≥ A r,α {Q} (f σ) = |Q| −α σ(Q)1 Q , so the main point of the theorem is the other estimate. Theorem 1.1 includes several known cases: (The "Sobolev" case 1/p−1/q = 1−α of these results, together with multilinear extensions, can also be recovered from the recent general framework of [Zor16]. )…”