On the growth of vector-valued Fourier series

TL;DR

This paper extends classical Fourier series growth results to vector-valued functions in UMD Banach spaces, establishing a version of the little Carleson theorem for such functions.

Contribution

It proves the little Carleson theorem for vector-valued Fourier series in UMD Banach spaces, a significant generalization of scalar results.

Findings

Establishment of growth bounds for vector-valued Fourier series

Extension of Carleson theorem to Banach space-valued functions

Validation of the theorem in the context of UMD Banach spaces

Abstract

We prove the 'little Carleson theorem' on the growth of Fourier series for functions taking values in a UMD Banach space.