Operator-valued Fourier multipliers on periodic Besov spaces

TL;DR

This paper characterizes when operator-valued Fourier multipliers act on periodic Besov spaces, linking the property to the UMD condition of the Banach space, and applies this to solve certain periodic Cauchy problems.

Contribution

It extends previous one-dimensional results to n-dimensional cases, establishing a characterization of Fourier multipliers on Besov spaces with applications to PDEs.

Findings

Fourier multipliers are characterized by the UMD property of the Banach space.

The result generalizes previous theorems to higher dimensions.

Applications include existence and uniqueness of solutions to periodic Cauchy problems.

Abstract

We prove in this paper that a sequence $M:\mathbb{Z}^{n}\to\mathcal{L}(E)$ of bounded variation is a Fourier multiplier on the Besov space $B_{p,q}^{s}(\mathbb{T}^{n},E)$ for $s\in\mathbb{R}$, $1<p<\infty$, $1\leq q\leq\infty$ and $E$ a Banach space, if and only if $E$ is a UMD-space. This extends in some sense the Theorem 4.2 in [AB04] to the $n-$dimensional case. The result is used to obtain existence and uniqueness of solution for some Cauchy problems with periodic boundary conditions.