Pointwise convergence of Walsh--Fourier series of vector-valued functions

Abstract

We prove a version of Carleson's Theorem in the Walsh model for vector-valued functions: For $1<p<\infty$, and a UMD space $Y$, the Walsh-Fourier series of $f\in L^p(0,1;Y)$ converges pointwise, provided that $Y$ is a complex interpolation space $Y=[X,H{]}_{\theta }$ between another UMD space $X$ and a Hilbert space $H$, for some $\theta \in (0,1)$. Apparently, all known examples of UMD spaces satisfy this condition.

15 pages

References in corpus (1)

• On the growth of vector-valued Fourier series

Cited by in corpus (3)

• On weighted norm inequalities for the Carleson and Walsh-Carleson operators
• A variation norm Carleson theorem for vector-valued Walsh-Fourier series
• The Vector Valued Quartile Operator