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arXiv · 1209.3383

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

Tuomas P. Hytönen·Michael T. Lacey·Ioannis Parissis
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Abstract

We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hytönen and Lacey. Given q \geq 2 we show that, if the space X has tile-type t for all t>q, then the r-variation of the Walsh-Fourier sums of any function f \in L^p ([0,1) ; X) belongs to L^p, whenever q q. For intermediate spaces, i.e. spaces X= [Y,H]_s which are complex interpolation spaces between some UMD space Y and a Hilbert space H, the tile-type is q=2/s. We show that in this case the variation norm Carleson theorem remains true for all r > q in the larger range p > (2r/q)'.

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Tuomas P. Hytönen, Michael T. Lacey, Ioannis Parissis. 2013-10-14. A variation norm Carleson theorem for vector-valued Walsh-Fourier series. https://doi.org/10.4171/rmi%2F804

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