Publikation: Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus
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2016
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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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BARRAZA 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/s00041-015-9437-7BibTex
@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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<dcterms:abstract xml:lang="eng">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<sup>s</sup><sub>pq</sub>(T<sup>n</sup>,E) and on the Sobolev space W<sup>k</sup><sub>p</sub>(T<sup>n</sup>,E), where E is an arbitrary Banach space, 1≤p,q≤∞, s∈R and k∈N<sub>0</sub>. 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.</dcterms:abstract>
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