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Spaces of Lipschitz and Hölder functions and their applications. (English) Zbl 1069.46004

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 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 [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.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
46T20 Continuous and differentiable maps in nonlinear functional analysis
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