Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus
B. Barraza Martínez, R. Denk, J. Hernández Monzón, and T. Nau
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 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.
J. Fourier Anal. Appl. 22 (2016), 823-853.
doi:10.1007/s00041-015-9437-7
The paper is available here (preprint version).