Dyadic Triangular Hilbert Transform of Two General Functions and One Not Too General Function

Abstract

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate L p bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

On Side Lengths of Corners in Positive Density Subsets of the Euclidean Space

“…, k − 1, and ∞. However, the corresponding analogs of Theorem 3 would involve operators of complexity similar as to the so-called multilinear and simplex Hilbert transforms (see [14], [19], [21]), for which no L p -boundedness results are known at the time of writing. An encouraging sign is that the papers [19] and [21] establish estimates for the truncations of these operators with constants o(J) in the number of consecutive dyadic scales J, while [6] improves this bound to J 1−ǫ for some ǫ > 0.…”

Power-type cancellation for the simplex Hilbert transform

“…If n = 1, then the simplex Hilbert transform is the form obtained by dualization of the classical Hilbert transform. The case n = 2 was called the triangular Hilbert transform in [4].…”

“…Partial progress in the case n = 2 was made in [4] for a dyadic model and under the additional assumption that one of the functions F i takes certain special forms. The papers [5] and [6] initiated the study of growth of the bounds for the truncated simplex Hilbert transform Λ n,r,R := r≤|x 0 +···+xn|≤R n i=0 F i (x 0 , .…”

Almost-orthogonality of restricted Haar functions

“…However by Theorem 1.1 for p = 1 2 , this strategy does not result in the lower bound in (3). However by (2) the majority decision (4) at least leads to the upper bound in (3), Theorem 1.1 also improves on Corollary 1.3 below, which in turn follows from Theorem 1.2, an equivalent formulation of Corollary 6.5 in [1] by Bownik, Casazza, Marcus and Speegle.…”

“…We are also interested in the following question, because its answer could provide ideas on how to to extend the result in [3] to three general functions. (3)…”