A local T(b) theorem for perfect multilinear Calderón–Zygmund operators

Abstract

We prove a multilinear local T (b) theorem that differs from previously considered multilinear local T (b) theorems in using exclusively general testing functions b as opposed to a mix of general testing functions and indicator functions. The main new feature is a set of relations between the various testing functions b that, to our knowledge has not been observed in the literature and is necessitated by our approach. For simplicity we restrict attention to the perfect dyadic model.

Singular Brascamp–Lieb inequalities with cubical structure

“…It would be desirable to have a more complete understanding of the analogs of results in [20] in the continuous setting. In the broader context, there are a number of similar results in the dyadic setting which await a transfer into the continuous setting, such as for example [17], [19], [11].…”

The bilinear Hilbert transform in UMD spaces

“…It remains to prove the weakened claim with (υ k ) k∈Z satisfying (34). Let υ ∈ C N (R) be even, smooth on B 2b/3 , with 0 < υ( z) ≤ 1, and…”

“…where the limit is taken over any increasing sequence of compact sets covering R 3 + . We proceed using the framework of outer Lebesgue spaces, as introduced by Do and Thiele [21] and successfully utilised in a number of papers (see for example [13][14][15][18][19][20]34,44,[46][47][48]). The idea of this method is to construct quasinorms F i L p i…”

Duality for Outer $$L^p_\mu (\ell ^r)$$ Spaces and Relation to Tent Spaces

“…This is for example the case of the bilinear Hilbert transform in [3,11,12], the variational Carleson operator in [8,21], the variational bilinear iterated Fourier inversion operator in [13], a family of trilinear multiplier forms with singularity over a one-dimensional subspace in [7], and the uniform bilinear Hilbert transform in [23]. Analogous applications of the outer L p spaces framework in other settings with different geometries can be found in [2,9,10,12,17,20].…”