Generation of semigroups for vector-valued pseudodifferential operators on the torus

TL;DR

This paper studies vector-valued pseudodifferential operators on the torus, proving they generate analytic semigroups on various function spaces without restrictions on the Banach space, and applies this to solve periodic pseudodifferential Cauchy problems.

Contribution

It establishes the generation of analytic semigroups by parabolic toroidal pseudodifferential operators on Banach space-valued Besov and Sobolev spaces without restrictions.

Findings

Generates analytic semigroups on Banach space-valued Besov spaces.

Generates analytic semigroups on Banach space-valued Sobolev spaces.

Provides existence and uniqueness results for periodic pseudodifferential Cauchy problems.

Abstract

We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. We show that a parabolic toroiodal pseudodifferential operator generates an analytic semigroup both on Banach space-valued Besov spaces and on Banach space-valued Sobolev spaces. Here, no condition on the Banach space is imposed. For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.