Uniformly bounded composition operators between general Lipschitz function normed spaces (Q446235)

The author introduces the following notions of uniform boundedness and equidistant uniform boundedness of an operator (weaker than usual boundedness). An operator \(H\) mapping a metric space \(\mathcal{Y}\) into a metric space \(\mathcal{Z}\) is said to beNEWLINENEWLINE(i) uniformly bounded if, for any \(t>0\), there exists a nonnegative real number \(\gamma (t)\) such that, for any nonempty set \(B \subset \mathcal{Y}\), if \(\text{diam}B\leq t\), then \(\text{diam}H(B)\leq \gamma (t)\);NEWLINENEWLINE(ii) equidistantly uniformly bounded if this condition holds true for all two-points sets \(B\).NEWLINENEWLINEIn the paper, it is proved that the generator of any uniformly bounded or equidistantly uniformly bounded Nemytskij operator \(H\) acting between general Lipschitzian normed function spaces must be affine with respect to the function variable. This improves earlier results of the author [Funkc. Ekvacioj, Ser. Int. 25, 127--132 (1982; Zbl 0504.39008), J. Math. Anal. Appl. 359, No. 1, 56--61 (2009; Zbl 1173.47043)], where \(H\) is assumed to be Lipschitzian or uniformly continuous, respectively.