Generalized Carleson Embeddings into Weighted Outer Measure Spaces

TL;DR

This paper establishes generalized Carleson embeddings for the wave packet transform in weighted outer measure spaces, extending previous Lebesgue results and utilizing geometric $L^2$ restriction estimates.

Contribution

It introduces a weighted extension of Carleson embeddings into outer $L^p$ spaces for $2<p< infty$, generalizing prior unweighted results and employing novel geometric restriction estimates.

Findings

Proves weighted Carleson embeddings for wave packet transform

Extends Lebesgue results to weighted outer measure spaces

Uses geometric $L^2$ restriction estimates that may be of independent interest

Abstract

We prove generalized Carleson embeddings for the continuous wave packet transform from $L^p(\mathbb{R},w)$ into an outer $L^p$ space for $2< p < \infty$ and weight $w \in A_{p/2}$. This work is a weighted extension of the corresponding Lebesgue result in arXiv:1309.0945v3 and generalizes a similar result in arXiv:1207.1150. The proof in this article relies on $L^2$ restriction estimates for the wave packet transform which are geometric and may be of independent interest.