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 -valued functions under the qualitative hypothesis that has some finite tile type , which holds in particular if 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 of is `almost equivalent' to the -boundedness of the -variation of the Walsh–Fourier sums of any function for all and large enough .
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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. 979–1014DOI 10.4171/RMI/804