Weighted mixed weak-type inequalities for multilinear operators

Abstract

In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let w = (w1, ..., wm) and ν = w 1 m 1 ...w 1 m m , the main result of the paper sentences that under different conditions on the weights we can obtainwhere T is a multilinear Calderón-Zygmund operator. To obtain this result we first prove it for the m-fold product of the Hardy-Littlewood maximal operator M , and also for M( f )(x): the multi(sub)linear maximal function introduced in [13].As an application we also prove a vector-valued extension to the mixed weighted weak-type inequalities of multilinear Calderón-Zygmund operators.where T is Hardy-Littlewood maximal function or any Calderón-Zygmund operator.Quantitative estimates of these mixed weighted results can be found in [16]. Moreover, it was conjectured in [7] that the conclusion of the previous theorem still holds if a weaker and more general hypothesis is satisfied. That is, if we have the following conditions on the weights, u ∈ A 1 and v ∈ A ∞ . Recently, in [14] the first two authors and C. Pérez solved such conjecture. Namely, the following theorem was proved, which constitutes the more difficult case of this class of mixed weighted inequalities.

An Endpoint Weak-Type Estimate for Multilinear Calderón–Zygmund Operators

“…The first proof uses the Calderón-Zygmund decomposition and is a weighted version of the proof in [16]; the second proof uses the Nazarov-Treil-Volberg method and is a weighted version of the proof in [20]. See [10] for a related result that is deduced using multilinear extrapolation. Remark 1.…”

Sawyer-type inequalities for Lorentz spaces

“…There is no reason to restrict ourselves to the study of one-variable Sawyer-type inequalities. Quite recently, the bound (1.1) has been extended to the multi-variable setting in [40]. More precisely, for weights w 1 , .…”

“…Motivated by the conjecture of E. Sawyer in [54], we can ask ourselves if it is possible to obtain bounds like (1.8) for multi-linear Calderón-Zygmund operators T . Once again, the endpoint case p 1 = • • • = p m = 1 has already been considered and extensively investigated in [40]. There, it was shown that for weights w 1 , .…”

“…To achieve this, we build upon (1.8), but unlike in [40], we manage to avoid the use of extrapolation arguments like the ones in [49]. Instead, we present in Theorem 5 a novel technique that allows us to replace M by T exploiting the fine structure of the Lorentz space L p,∞ , the A R p condition, and the recent advances in sparse domination.…”

“…Instead, we present in Theorem 5 a novel technique that allows us to replace M by T exploiting the fine structure of the Lorentz space L p,∞ , the A R p condition, and the recent advances in sparse domination. One can even go further and consider inequalities like (1.10) assuming multivariable conditions on the weights involved, as it was done in [40] with the endpoint case p 1 = • • • = p m = 1 and weights in A 1 . In Sect.…”