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

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.