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Extension of the Hilbert-valued Lipschitz mappings. (English. Russian original) Zbl 0965.46047

Mosc. Univ. Math. Bull. 54, No. 6, 7-14 (1999); translation from Vestn. Mosk. Univ., Ser I 1999, No. 6, 9-16 (1999).
W. B. Johnson and J. Lindenstrauss [see Contemp. Math. 26, 189-206 (1984; Zbl 0539.46017)] posed the question: is it true that for \(p\in(2,\infty)\) there exists a constant \(C = C(p)>0\) such that any \(K\)-Lipschitzian mapping \(f\) from the subset \(M\subset L_p[0,1]\) in \(\ell_2\) can be extended up to the \(KC\)-Lipschitzian mapping \(\widetilde f: L_p(]0,1[)\to \ell_2\)?
The author gives a positive answer to this question by proving the following assertion:
Let \(X\in(\mathcal K)\) and \(Y_0\) be a separable Hilbert space. Then there exists a \(K>0\) such that for any sets \(M\subset X\) and \(K_1\)-Lipschitzian mapping \(f: M\to Y_0\) there exists a \((KK_1)\)-Lipschitzian mapping \(\widetilde f: X\to Y_0\), \(\widetilde f|_M=f\).

MSC:

46T20 Continuous and differentiable maps in nonlinear functional analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0539.46017
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