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Journal of Fourier Analysis and Applications

, Volume 22, Issue 4, pp 823–853 | Cite as

Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus

  • B. Barraza Martínez
  • R. Denk
  • J. Hernández Monzón
  • T. Nau
Article

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. Here, we restrict ourselves to pseudodifferential operators with x-independent symbols (Fourier multipliers). We show that a parabolic toroidal pseudodifferential operator generates an analytic semigroup on the Besov space \(B_{pq}^s({\mathbb T}^n,E)\) and on the Sobolev space \(W_p^k({\mathbb T}^n,E)\), where E is an arbitrary Banach space, \(1\le p,q\le \infty \), \(s\in {\mathbb R}\) and \(k\in {\mathbb N}_0\). 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.

Keywords

Pseudodifferential operators Vector-valued Sobolev spaces Toroidal Fourier transform Generation of analytic semigroup 

Mathematics Subject Classification

35S05 47D06 35R20 

Notes

Acknowledgments

The authors would like to thank COLCIENCIAS (Project 121556933488) and DAAD for the financial support.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • B. Barraza Martínez
    • 1
  • R. Denk
    • 2
  • J. Hernández Monzón
    • 1
  • T. Nau
    • 2
  1. 1.Departamento de MatemáticasUniversidad del NorteBarranquillaColombia
  2. 2.Fachbereich für Mathematik und StatistikUniversität KonstanzKonstanzGermany

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