# Banach-valued modulation invariant Carleson embeddings and outer-$L^p$ spaces: the Walsh case

## TL;DR

This paper establishes modulation invariant embeddings from Banach-valued Bochner spaces on the Walsh group to outer-$L^p$ spaces, leading to bounds and sparse domination results for a model of the bilinear Hilbert transform.

## Contribution

It introduces new embedding bounds for Banach-valued functions in the Walsh setting, extending the theory to UMD spaces close to Hilbert spaces.

## Key findings

- Proved modulation invariant embedding bounds for Banach-valued functions.
- Derived $L^p$ bounds and sparse domination for the Banach-valued tritile operator.
- Extended the analysis of bilinear Hilbert transform models to Banach spaces.

## Abstract

We prove modulation invariant embedding bounds from Bochner spaces $L^p(\mathbb{W};X)$ on the Walsh group to outer-$L^p$ spaces on the Walsh extended phase plane. The Banach space $X$ is assumed to be UMD and sufficiently close to a Hilbert space in an interpolative sense. Our embedding bounds imply $L^p$ bounds and sparse domination for the Banach-valued tritile operator, a discrete model of the Banach-valued bilinear Hilbert transform.