Banach-Valued Modulation Invariant Carleson Embeddings and Outer-$$L^p$$ Spaces: The Walsh Case

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.