Two stars theorems for traces of the Zygmund space

Abstract:

For a Banach space $X$ defined in terms of a big-$O$ condition and its subspace x defined by the corresponding little-$o$ condition, the biduality property (generalizing the concept of reflexivity) asserts that the bidual of x is naturally isometrically isomorphic to $X$. The property is known for pairs of many classical function spaces (such as $(\ell _\infty , c_0)$, (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 $S\subset \mathbb {R}^n$ of a generalized Zygmund space $Z^\omega (\mathbb {R}^n)$. 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 $Z^\omega (\mathbb {R}^n)|_S$.