End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

Abstract

In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing that, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding "finite" case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calderón-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm class. The same occurs with the multilinear Calderón-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases.

Multilinear singular integrals on non-commutative $$L^p$$ spaces

“…Vector-valued extensions of multilinear Calderón-Zygmund operators have mostly been studied within the more restrictive framework of ℓ p spaces and function lattices. Boundedness of these extensions is classically obtained through weighted norm inequalities, more recently in connection with localized techniques such as sparse domination: see [16] and the more recent [6,37,42] for a non-exhaustive overview of their interplay. The paper [10], by Y. Ou and one of us, contains a bilinear multiplier theorem which applies to certain non-lattice UMD spaces.…”

“…We send to Subsection 3.3 and to the references [7,8] for more details on sparse bounds and to [37,38] for a survey of the weighted inequalities that may be derived as a consequence. Theorem 1.3 is obtained as a corollary of Theorem 3.31 using Example 3.21.…”

The bilinear Hilbert transform in UMD spaces

“…For p 3 ≤ 1 it follows from the sparse domination for BHF Π established in Theorem 8.4: this implies weighted estimates which, by extrapolation, yield the claimed bounds (97) (plus further weighted estimates that we do not discuss here). The details of this weighted multilinear extrapolation are worked out in [38,Section 4.1] (see also [29,30]).…”

Sparse domination implies vector-valued sparse domination

“…From the point of view of weights, it was made clear by Li, Martell, and Ombrosi in [31] that rather than assuming a condition on each individual weight, it is more appropriate to consider the multilinear weight classes characterized by the multisublinear analogue of the Hardy Littlewood maximal operator, introduced by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González in [29]. Subsequently, through the extrapolation theorems of the second author [40] and Li, Martell, Martikainen, Ombrosi, and Vuorinen [30], it was shown that these weight classes allow one to handle vector-valued extensions with Banach function spaces outside of the class of UMD spaces, such as ∞ . However, these methods do not exceed the example of iterated L q -spaces.…”