Traces of vector-valued Sobolev Spaces

The aim of the paper is to characterize the trace space of vector-valued Sobolev spaces W p (R , E) , where E is an arbitrary Banach space. In particular, we do not assume that the underlying Banach space E has the UMD property. Vector-valued Sobolev and Besov spaces are widely used in abstract evolution equations, cf. e.g. Amann [1, 2, 4], Veraar and Weis [57] or Denk, Hieber, Prüss, Saal, and Seiler [9, 10, 44, 11], and in the theory of integral operators, cf. e.g. König [31], Pietsch [41], and Hytönen and Veraar [25]. Let us also mention that traces for certain vector-valued Sobolev spaces on domains associated with parabolic problems have been investigated by P. Weidemaier [58, 59]. For certain applications it is quite inconvenient to assume that the Banach space E has the UMD property. There are simple examples of Banach spaces which do not have this property. The UMD property implies reflexivity of E. So it rules out Hölder spaces and L1 and variants of those, cf. Amann [1]. On the other hand the UMD property is essential for the validity of Michlin-Hörmander type Fourier multiplier assertions in corresponding Lp-spaces. So, there is a battle in that field which properties of the distribution spaces really depend on the UMD property and which do not. Hence one has to avoid techniques which are based on such multipliers. A way out is the use of vector-valued spaces of Besovand Lizorkin-Triebel type, where multiplier theorems are available and their Fourier-analytic definition uses Littlewood-Paley type decompositions. These scales of spaces seem to be of interest also for themselves. However, in contrast to the scalar case they do not contain (fractional) Sobolev spaces as special cases. Nevertheless, it turns out that some assertions can be derived via rather elementary embeddings within the above scale of spaces. This concerns inequalities of GagliardoNirenberg-type and related Sobolev type embeddings (see [50]) as well as limiting cases of embeddings (see [32]). The paper is organized as follows. In Section 2 we introduce vector-valued Besov and Lizorkin-Triebel spaces. Section 3 deals with atomic and subatomic decompositions of vector-valued Besov and Lizorkin-Triebel spaces. These decompositions will be used as a main tool to prove our trace theorems in Section 4. The paper is based on earlier manuscripts of the authors, see [49] and [47]. There one can find an extended treatment in particular a characterization of Lizorkin-Triebel spaces by means of differences, characterizations