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

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2016
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Barraza Martínez, Bienvenido
Hernández Monzón, Jairo
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Journal of Fourier Analysis and Applications. 2016, 22(4), pp. 823-853. ISSN 1069-5869. eISSN 1531-5851. Available under: doi: 10.1007/s00041-015-9437-7
Zusammenfassung

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 Bspq(Tn,E) and on the Sobolev space Wkp(Tn,E), where E is an arbitrary Banach space, 1≤p,q≤∞, s∈R and k∈N0. 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.

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510 Mathematik
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Pseudodifferential operators ; Vector-valued Sobolev spaces ; Toroidal Fourier transform ; Generation of analytic semigroup
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ISO 690BARRAZA MARTÍNEZ, Bienvenido, Robert DENK, Jairo HERNÁNDEZ MONZÓN, Tobias NAU, 2016. Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus. In: Journal of Fourier Analysis and Applications. 2016, 22(4), pp. 823-853. ISSN 1069-5869. eISSN 1531-5851. Available under: doi: 10.1007/s00041-015-9437-7
BibTex
@article{BarrazaMartinez2016-08Gener-30781,
  year={2016},
  doi={10.1007/s00041-015-9437-7},
  title={Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus},
  number={4},
  volume={22},
  issn={1069-5869},
  journal={Journal of Fourier Analysis and Applications},
  pages={823--853},
  author={Barraza Martínez, Bienvenido and Denk, Robert and Hernández Monzón, Jairo and Nau, Tobias}
}
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