Banach-valued multilinear singular integrals with modulation invariance

TL;DR

This paper extends the boundedness of certain multilinear singular integrals, including the bilinear Hilbert transform, to Banach-valued functions within UMD spaces, using advanced phase-space and tree estimate techniques.

Contribution

It introduces a novel extension of phase-space projection techniques to UMD-valued functions and establishes boundedness results for Banach-valued multilinear singular integrals.

Findings

Boundedness of Banach-valued multilinear singular integrals including the bilinear Hilbert transform.

Extension of phase-space projection technique to UMD-valued setting.

Application of recent UMD-valued bounds for bilinear Calderón-Zygmund operators.

Abstract

We prove that the class of trilinear multiplier forms with singularity over a one dimensional subspace, including the bilinear Hilbert transform, admit bounded $L^p$-extension to triples of intermediate $\mathrm{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\mathrm{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\mathrm{UMD}$-valued setting. This is then employed to obtain appropriate single tree estimates by appealing to the $\mathrm{UMD}$-valued bound for bilinear Calder\'on-Zygmund operators recently obtained by the same authors.