A modulation invariant Carleson embedding theorem outside local $L^2$

TL;DR

This paper extends the theory of Carleson embeddings in outer L^p spaces for wave packet transforms to the range 1<p<2, using a refined multi-frequency Calderón-Zygmund decomposition, and applies it to bilinear Hilbert transforms.

Contribution

It formulates a new extension of Carleson embedding theory to lower p values (1<p<2), answering a previously open question.

Findings

Established a full range of L^p estimates for bilinear Hilbert transforms.

Developed a refined multi-frequency Calderón-Zygmund decomposition.

Extended Carleson embedding theory outside local L^2 range.

Abstract

The article arXiv:1309.0945 by Do and Thiele develops a theory of Carleson embeddings in outer $L^p$ spaces for the wave packet transform of functions in $ L^p(\mathbb R)$, in the $2\leq p\leq \infty$ range referred to as local $L^2$. In this article, we formulate a suitable extension of this theory to exponents $1<p<2$, answering a question posed in arXiv:1309.0945. The proof of our main embedding theorem involves a refined multi-frequency Calder\'on-Zygmund decomposition. We apply our embedding theorem to recover the full known range of $L^p$ estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation.