Pointwise convergence of vector-valued Fourier series

TL;DR

This paper extends Carleson's theorem to vector-valued functions in interpolation spaces between UMD spaces and Hilbert spaces, proving pointwise convergence of Fourier series for these functions.

Contribution

It establishes a vector-valued version of Carleson's theorem for a broad class of interpolation spaces, including Schatten class valued functions.

Findings

Proves pointwise convergence of Fourier series in vector-valued L^p spaces.

Answers affirmatively a question of Rubio de Francia on Schatten class functions.

Identifies a class of spaces where classical Fourier analysis results hold in the vector-valued setting.

Abstract

We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.