Spaces of Lipschitz and Hölder functions and their applications
For a metric space \((M,d)\), the author considers derived metrics of the form \({\omega \circ d}\) and the associated space Lip\(_\omega(M)\) of \({\omega \circ d}\)-Lipschitz functions that vanish at a distinguished point of~\(M\). If \(\omega(t)=t^\alpha\), one obtains the Hölder space Lip\(^{(\alpha)}(M)\). The main issue of the paper is to study these Banach spaces and their canonical preduals, denoted \({\mathcal F}_\omega(M)\) or \({\mathcal F}^{(\alpha)}(M)\), see \textit{G.~Godefroy, N. J. Kalton}, [Stud.\ Math.\ 159, No.1, 121--141 (2003; Zbl 1059.46058)]. For example, it is shown that \({\mathcal F}_\omega(M)\) has the Schur property whenever \(\omega(t)/t\to \infty\) for \(t\to 0\), and for uniformly discrete metric spaces it has the Radon-Nikodym property and the approximation property. The core of the paper deals with the question, raised in [\textit{N.~Weaver}, `Lipschitz Algebras', World Scientific (1999; Zbl 0936.46002)], whether the ``little'' Lipschitz space lip\(^{(\alpha)}(K)\) over a compact metric space is necessarily isomorphic to \(c_0\). This is known to be true for compact subsets in \({\mathbb R}^n\). The main result says that there are a wealth of counterexamples: If \(K\) is a compact convex subset of \(\ell_2\), then lip\(^{(\alpha)}(K)\) is isomorphic to \(c_0\) if and only if \(K\) is finite-dimensional. The conclusion persists for subsets of Banach spaces with nontrivial type or for subsets of arbitrary Banach spaces if \(0<\alpha\leq 1/2\); but the author conjectures the theorem to hold also in the remaining range \(1/2<\alpha <1\). On the other hand, he shows that lip\((K)\) always embeds almost isomorphically into~\(c_0\). The key to prove these results is to use the natural quotient map \(\beta: {\mathcal F}^{(\alpha)}(B_X) \to X\), which is known to admit a section that is uniformly continuous on \(B_X\), and to study the question for which Banach spaces \(Y\) there are quotient maps \(Q:Y\to X\) having such sections. Arguments that are local in character are then applied to show for instance that there is no such quotient map from an \({\mathcal L}_1\)-space onto \(\ell_2\). This proves the above assertions on subsets of~\(\ell_2\). In the other direction, the author shows that a quotient map from \(Y\) to \(Y/E\) admits uniformly continuous sections on \(B_{Y/E}\) if \(E\) is superreflexive. There are numerous further results on the structure of \({\mathcal F}^{(\alpha)}(M)\) or Lip\(^{(\alpha)}(M)\) in this very interesting paper.
scientific article; zbMATH DE number 2123817