Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus |
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Authors: | B Barraza Martínez R Denk J Hernández Monzón T Nau |
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Institution: | 1.Departamento de Matemáticas,Universidad del Norte,Barranquilla,Colombia;2.Fachbereich für Mathematik und Statistik,Universit?t Konstanz,Konstanz,Germany |
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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. |
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