Operator-valued Fourier multipliers on toroidal Besov spaces

Abstract

We prove in this paper that a sequence M : Z n → L(E) of bounded variation is a Fourier multiplier on the Besov spaceand E a Banach space, if and only if E is a UMD-space. This extends in some sense the Theorem 4.2 in [AB04] to the n−dimensional case. The result is used to obtain existence and uniqueness of solution for some Cauchy problems with periodic boundary conditions.

The nuclear trace of periodic vector‐valued pseudo‐differential operators with applications to index theory

“…The analytical and spectral properties for periodic operators have been treated in the work of Ruzhansky and Turunen [50, 51], Delgado [26], Molahajloo and Wong [42–44], and in the works of the authors [15–17, 20, 23, 25, 38, 39]. For the nuclearity of multilinear periodic operators in ‐spaces we refer the reader to the works of the authors [21, 22] and to Delgado and Wong [31] for the linear case. If , Amann [2], Arendt and Bu [4, 5], Barraza, Gonzalez and Hernández [6], Barraza, Denk, Hernández and Nau [7], Rabinovich [46], Bu and Kim [11–13] and Bu [14] investigated the mapping properties of the vector‐valued pseudo‐differential operators on ‐spaces and Besov spaces. The applications to PDE's studied by Denk and Nau [32], Keyantuo, Lizama, and Poblete [36] and references therein.…”

Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaces

“…The above references deal with the scalar-valued case. In the situation where the considered functions have values in some Banach space E, the situation depends on the geometric properties of E. If E is a UMD space (and hence in particular reflexive), then Mikhlin-type results yield L pboundedness, see Arendt-Bu [3], Keyantuo-Lizama-Poblete [12], Barraza-González-Hernández [5]. The case of general Banach spaces was studied by Amann [2] on R n and by Denk-Barraza-Hernández-Nau [4] on T n .…”