Generalized Carleson Embeddings into Weighted Outer Measure Spaces

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.