Operator-valued $(L^{p},L^{q})$ Fourier multipliers and stability theory for evolution equations

TL;DR

This paper reviews recent advances in operator-valued Fourier multipliers between different L^p spaces and their applications to stability analysis and functional calculus in evolution equations.

Contribution

It provides a nontechnical overview connecting operator-valued Fourier multipliers with stability theory and functional calculus for evolution equations.

Findings

Highlights the role of operator-valued Fourier multipliers in stability analysis.

Demonstrates applications to functional calculus.

Connects Fourier multiplier theory with evolution equations stability.

Abstract

We give an overview of some recent results on operator-valued $(L^{p},L^{q})$ Fourier multipliers and stability theory for evolution equations. The aim is to provide a relatively nontechnical introduction to the underlying ideas, emphasizing the connection between the two areas. We also indicate how operator-valued $(L^{p},L^{q})$ Fourier multipliers can be applied to functional calculus theory.