Locally Lipschitz composition operators in space of the functions of bounded \(\kappa \Phi\)-variation (Q2256677)

There is a vast literature on composition operators of type \[ Hu(t):= h(u(t))\qquad (a\leq t\leq b)\tag{\(*\)} \] (autonomous case) or of more general type \[ Hu(t):= h(t,u(t))\quad (a\leq t\leq b)\tag{\(**\)} \] (non-autonomous case) in many function spaces occurring in applications. In some spaces the operator \((*)\), and even more the operator \((**)\), exhibits a rather pathological behaviour. For example, the global Lipschitz condition in norm \[ \| Hu- Hv\|\leq L\| u-v\| \] often leads to a strong degeneracy, because it holds only for affine functions \(h\). For this reason, particular emphasis has been put in the last decade to impose a local Lipschitz condition \[ \| Hu- Hv\|\leq L\| u-v\|\qquad (\| u\|,\| v\|\leq r) \] which in most cases is sufficient for applying, say, Banach's contraction mapping principle. As was shown by the reviewer et al. [Ann. Mat. Pura Appl. (4) 190, No. 1, 33--43 (2011; Zbl 1214.47058)], a local Lipschitz condition for the operator \(H\) in \((*)\) is in many function spaces equivalent to a local Lipschitz condition for the derivative \(h'\) of the underlying function \(h\). Prominent examples are spaces of bounded Jordan variation, Riesz \(p\)-variation, Medvedev \(\varphi\)-variation, or Schramm \(\Phi\)-variation. In this paper the authors show that the same result holds in the space of bounded \(\kappa\Phi\) Korenblum-Kim variation introduced in [Bull. Korean Math. Soc. 23, 171--175 (1986; Zbl 0615.26007)]. In particular, this refers to the classical space of functions of bounded Korenblum \(\kappa\)-variation [Acta Math. 135, 187--219 (1975; Zbl 0323.30030)].