The bilinear Hilbert transform in UMD spaces

TL;DR

This paper establishes $L^p$ bounds for the bilinear Hilbert transform on functions valued in intermediate UMD spaces, extending known results to a broader class of Banach spaces using advanced embedding techniques.

Contribution

It introduces novel $L^p$ bounds for the bilinear Hilbert transform in UMD spaces beyond Banach lattices, utilizing embeddings into outer Lebesgue spaces.

Findings

Proved $L^p$ bounds for the bilinear Hilbert transform in intermediate UMD spaces.

Extended the scope of known bounds to non-lattice UMD spaces.

Developed embedding techniques from Bochner to outer Lebesgue 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 $\mathbb{R}^3_+$.