Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

Abstract

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W k p (R n , E) with k ∈ N0, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve nonautonomous parabolic pseudodifferential equations in Sobolev spaces.

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

“…Proof. Using Theorem 4.9, this follows from the abstract result on Cauchy problems, Theorem 2.5.1 of Chapter IV in [2] analogously to the proof of Theorem 4.4 in [10].…”

“…The inverse continuous Fourier transform is then given by (F −1 R n g)(x) = (2π) −n/2 R n e ix·ξ g(ξ)dξ (x ∈ R n ). We will use the following result from [10], Lemma 2.3.…”

“…Remark 4.7. In the proof of the following lemma, we will use the elementary inequality (4)(5)(6)(7)(8)(9)(10)(11)(12) for N ∈ N, where we have set (e −iη − 1) γ := (e −iη 1 − 1) γ 1 · . .…”

“…= lim εց0 T n K ε (η, λ)(∂ α u)(x − η)dη (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) with K ε from Lemma 4.8. From (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), Lemma 4.8 and dominated convergence, we get…”

Pseudodifferential operators with non-regular operator-valued symbols

“…Our approach is based on oscillatory integrals and careful kernel estimates for them, the technical key result being Lemma 3.2. For the case of constant coefficients, results on vector-valued pseudodifferential operators were obtained in our paper [BDH12].…”