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

### Tuomas Hytönen

University of Helsinki, Finland### Michael T. Lacey

Georgia Institute of Technology, Atlanta, USA### Ioannis Parissis

Aalto University, Finland

## Abstract

We prove a variation norm Carleson theorem for Walsh–Fourier series of functions with values in certain UMD Banach spaces, sharpening a recent result of Hytönen and Lacey. They proved the pointwise convergence of Walsh–Fourier series of $X$-valued functions under the qualitative hypothesis that $X$ has some finite tile type $q<∞$, which holds in particular if $X$ is intermediate between another UMD space and a Hilbert space. Here we relate the precise value of the tile type index to the quantitative rate of convergence: tile type $q$ of $X$ is `almost equivalent' to the $L_{p}$-boundedness of the $r$-variation of the Walsh–Fourier sums of any function $f∈L_{p}([0,1);X)$ for all $r>q$ and large enough $p$.

## Cite this article

Tuomas Hytönen, Michael T. Lacey, Ioannis Parissis, A variation norm Carleson theorem for vector-valued Walsh–Fourier series. Rev. Mat. Iberoam. 30 (2014), no. 3, pp. 979–1014

DOI 10.4171/RMI/804