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


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 XX-valued functions under the qualitative hypothesis that XX has some finite tile type q<q<\infty, which holds in particular if XX 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 qq of XX is `almost equivalent' to the LpL^p-boundedness of the rr-variation of the Walsh–Fourier sums of any function fLp([0,1);X)f\in L^p([0,1);X) for all r>qr>q and large enough pp.

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