# Revista Matemática Iberoamericana

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**Volume 30, Issue 3, 2014, pp. 979–1014**

**DOI: 10.4171/RMI/804**

Published online: 2014-08-27

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

Tuomas Hytönen^{[1]}, Michael T. Lacey

^{[2]}and Ioannis Parissis

^{[3]}(1) University of Helsinki, Finland

(2) Georgia Institute of Technology, Atlanta, USA

(3) Aalto University, Finland

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 $q<\infty$, 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 $f\in L^p([0,1);X)$ for all $r>q$ and large enough $p$.

*Keywords: *Pointwise convergence, variational norm, Walsh–Fourier series

Hytönen Tuomas, Lacey Michael, Parissis Ioannis: A variation norm Carleson theorem for vector-valued Walsh–Fourier series. *Rev. Mat. Iberoam.* 30 (2014), 979-1014. doi: 10.4171/RMI/804