Carleson measures associated with families of multilinear operators

Abstract

In this work we investigate the construction of Carleson measures from families of multilinear integral operators applied to tuples of L ∞ and BM O functions. We show that if the family R t of multilinear operators possesses cancellation in each variable, then for BM O functions b 1 ,. .. , b m , the measure |R t (b 1 ,. .. , b m)(x)| 2 dxdt/t is Carleson. However, if the family of multilinear operators has cancellation in all variables combined this result is still valid if b j are L ∞ functions, but it may fail if b j are unbounded BM O functions, as we indicate via an example. As an application of our results we obtain a multilinear quadratic T (1) type theorem and a multilinear version of a quadratic T (b) theorem analogous to those in Semmes [23].

A new proof of the bilinear T(1) Theorem

“…In fact, in [16] this theorem is proved for a more general accretive setting. Similar results to Theorem 2.4 were proved independently by Grafakos-Oliveira [13] and by Grafakos-Liu-Maldonado-Yang [12] for a certain range of exponents and under a variety of regularity and cancellation assumptions. Here we will need the specific version of Theorem 2.4 as stated above.…”

Multilinear local Tb theorem for square functions

“…We now state a multilinear version of the T (1) theorem for square functions due to [15], [11] and [12]. [15] to extend the theorem above to the complete quasi-Banach case, that is, with 1/m < p ≤ 1.…”

Hardy space estimates for bilinear square functions and Calderon-Zygmund operators

“…This work also provides bilinear versions of the linear results in [18,50,22,20,35]; in particular there is a close parallel with the Hardy space results in [35]. There has also been a considerable development of bilinear Littlewood-Paley-Stein theory in recent years, see for example [42,43,32,29,25,5,24]. All of these articles deal with Littlewood-Paley-Stein operator mapping properties from L p 1 × L p 2 into L p for p 1 , p 2 > 1.…”

“…, which is an object that is closely related to boundedness properties of S Θ , see for example [42,43,32,25,29,24]. Our main square function boundedness result is the following theorem.…”