A variation norm Carleson theorem for vector-valued Walsh-Fourier series

TL;DR

This paper establishes a variation norm Carleson theorem for Walsh-Fourier series of functions valued in UMD Banach spaces with finite tile-type, extending classical results to vector-valued settings and intermediate spaces.

Contribution

It introduces a new variation norm Carleson theorem for Walsh-Fourier series in UMD Banach spaces with finite tile-type, linking tile-type to variational bounds.

Findings

Proves the variation norm Carleson theorem for Walsh-Fourier series in UMD Banach spaces.

Shows the necessity of tile-type conditions for the theorem to hold.

Extends results to intermediate spaces via complex interpolation.

Abstract

We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hyt\"onen and Lacey. Given q \geq 2 we show that, if the space X has tile-type t for all t>q, then the r-variation of the Walsh-Fourier sums of any function f \in L^p ([0,1) ; X) belongs to L^p, whenever q<r \leq \infty and (r/(q-1))' < p < \infty. We also show that if this conclusion is true for a variant of the variational Carleson operator then the space X necessarily has tile-type t for all t >q. For intermediate spaces, i.e. spaces X= [Y,H]_s which are complex interpolation spaces between some UMD space Y and a Hilbert space H, the tile-type is q=2/s. We show that in this case the variation norm Carleson theorem remains true for all r > q in the larger range p > (2r/q)'.