Two stars theorems for traces of the Zygmund space

Abstract

For a Banach space X X defined in terms of a big- O O condition and its subspace x defined by the corresponding little- o o condition, the biduality property (generalizing the concept of reflexivity) asserts that the bidual of x is naturally isometrically isomorphic to X X . The property is known for pairs of many classical function spaces (such as ( ℓ ∞ , c 0 ) (\ell _\infty , c_0) , (BMO, VMO), (Lip, lip), etc.) and plays an important role in the study of their geometric structure. The present paper is devoted to the biduality property for traces to closed subsets S ⊂ R n S\subset \mathbb {R}^n of a generalized Zygmund space Z ω ( R n ) Z^\omega (\mathbb {R}^n) . The method of the proof is based on a careful analysis of the structure of geometric preduals of the trace spaces along with a powerful finiteness theorem for the trace spaces Z ω ( R n ) | S Z^\omega (\mathbb {R}^n)|_S .