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28 March 2024
 
  » arxiv » 1209.3383

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A variation norm Carleson theorem for vector-valued Walsh-Fourier series
Tuomas P. Hytönen ; Michael T. Lacey ; Ioannis Parissis ;
Date 15 Sep 2012
AbstractWe 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 Hyt"onen 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<r leq infty and (r/(q-1))’ < p < infty. We also show that if this conclusion is true for a variant of the variational Carleson operator then the space X necessarily has tile-type t for all t >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)’.
Source arXiv, 1209.3383
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