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

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 $q<\infty$, 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 $p$.