The bilinear Hilbert transform in UMD spaces

Abstract

We prove $L^p$-bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces $L^p(\mathbb{R};X)$ into outer Lebesgue spaces on the time-frequency-scale space ${\mathbb{R}}_{+}^3$.

Major revision: the proof of the single-tree estimate has been corrected, resulting in various changes throughout the paper. 69 pages, 5 figures