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Linear and nonlinear extensions of Lipschitz functions from subsets of metric spaces. (English. Russian original) Zbl 1213.54040

St. Petersbg. Math. J. 19, No. 3, 397-406 (2008); translation from Algebra Anal. 19, No. 3, 106-118 (2007).
The paper under review establishes a relationship between the linear and nonlinear extension constants for Lipschitz functions defined on subsets of metric spaces. Let \(M\) be a metric space and \(B\) a Banach space. Denote by \(\text{Lip}(M, B)\) the space of \(B\)-valued Lipschitz functions on \(M\). In the case \(B=\mathbb R\) the corresponding space is denoted by \(\text{Lip}(M)\). A subset \(S\) of \(M\) is said to admit a simultaneous Lipschitz extension if there exists a linear bounded operator \(T: \text{Lip}(S) \to \text{Lip}(M)\) such that \(Tf_{|_{S}}=f\) for every \(f \in \text{Lip}(S)\). The set of all such operators is denoted by \(\text{Ext}(S, M)\), and the optimal extension constant is given by \(\lambda(S, M)=\text{inf}\{\|T\|: T \in \text{Ext}(S, M) \}.\) The global linear Lipschitz extension is given by
\[ \lambda(M)=\text{sup} \{\lambda(S, M): S \subset M \}. \]
Similarly the authors define the nonlinear Lipschitz extension constant \(\nu(S, M, B)\) as the infimum of the constants \(C\) such that every \(f \in \text{Lip}(S, B)\) admits an extension to a function \(\tilde{f} \in \text{Lip}(M, B)\) satisfying \( |\tilde{f}|_{\text{Lip(M, B)}} \leq C |f|_{\text{Lip(S, B)}}.\) Then they define \(\nu(M, B)= \text{sup} \;\{\nu(S, M, B): S \subset M \}\). Denote by \({\mathcal B}_{fin}\) the category of all finite-dimensional Banach spaces. Finally, they set
\[ \nu(M)= \text{sup} \;\{\nu(M, B): B \in {\mathcal B}_{fin} \}. \]
The main result of the paper states that \(\lambda(M)= \nu(M)\) for every metric space \(M\). Its proof is based on some results from A. Brudnyi and Y. Brudnyi [Am. J. Math. 129, No. 1, 217–314 (2007; Zbl 1162.46042)]. The paper also contains several consequences of this equality.

MSC:

54E40 Special maps on metric spaces
26A16 Lipschitz (Hölder) classes
46E15 Banach spaces of continuous, differentiable or analytic functions
54C20 Extension of maps

Citations:

Zbl 1162.46042
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References:

[1] A. Brudnyĭ and Yu. Brudnyĭ, Simultaneous extensions of Lipschitz functions, Uspekhi Mat. Nauk 60 (2005), no. 6(366), 53 – 72 (Russian, with Russian summary); English transl., Russian Math. Surveys 60 (2005), no. 6, 1057 – 1076. · Zbl 1133.26300
[2] Alexander Brudnyi and Yuri Brudnyi, Metric spaces with linear extensions preserving Lipschitz condition, Amer. J. Math. 129 (2007), no. 1, 217 – 314. · Zbl 1162.46042
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