A modulation invariant Carleson embedding theorem outside local L 2

Yen Do and Christoph Thiele developed a theory of Carleson embeddings in outer Lp spaces for the wave packet transform Fϕ(f)(u,t,η)=∫f(x)eiη(u−x)ϕ(u−xt)dxt,(u,t,η)∈R×(0,∞)×R of functions f ∈ Lp(R) in the range 2 ≤ p ≤ ∞, referred to as local L2. In this article, we formulate a suitable extension of this theory to exponents 1 < p < 2, answering a question posed by Do and Thiele. The proof of our main embedding theorem involves a refined multi-frequency Calderón-Zygmund decomposition in the vein of work by Di Plinio and Thiele and by Nazarov, Oberlin, and Thiele. We apply our embedding theorem to recover the full known range of Lp estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation.