Extension of the Hilbert-valued Lipschitz mappings (Q1596181)

\textit{W. B. Johnson} and \textit{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 \(\widetilde f: X\to Y_0\), \(\widetilde f|_M=f\).