Uniformly continuous composition operators in the space of bounded \(\Phi \)-variation functions in the Schramm sense (Q2901948)

Let \(\mathcal {F}\) be te set of all convex functions \( \phi:[0,\infty)\to[0,\infty)\) such that \(\phi(0)=0\) and \(\phi(t)>0\) for \(t>0\) and choose a sequence \(\Phi=(\phi_n)\) of functions in \(\mathcal{F}\) such that \(\phi_n(x)\leq\phi_{n+1}(x)\) for \(x>0\) and such that the \(\phi_n(x)\) form a divergent series for \(x>0\). Let \(X\) be a normed space, \(C\) a closed convex subset of \(X\), and \(I=[a,b]\). A function \(f:I\to X\) is said to be of bounded \(\Phi\)-variation in the sense of Schramm in \(I\) if \(v_\Phi(f):=v_\Phi(f,I):=\sup\sum_n\phi(\|f(I_n)\|)<\infty\), where the supremum is taken over all sequences of closed nonoverlapping subintervals of \(I\) and \(f(I_n)=f(b_n)-f(a_n)\) for \(I_n=[a_n,b_n]\).NEWLINENEWLINE Denote by \(\Phi BV(I,X)\) the linear space of all functions \(f:I\to X\) such that \(v_\Phi(\lambda f)<\infty\) for some \(\lambda>0\) endowed with the norm \(\|f\|_\Phi:=|f(a)|+p_\Phi(f)\), where \(a\) is the left endpoint of \(I\) and \(p_\Phi(f):=\inf\{\epsilon>0\mid v_\Phi(f/\epsilon)\leq1\}\). Let \(Y\) be a real Banach space and \(h:I\times C\to Y\). Define the supercomposition operator \(H\) by \(H(f)(t)=h(t,f(t))\). Denote by \(I^-\) the interior of \(I\). The authors then prove the following result: If (with the notation from above) \(H\) maps \(\Phi BV(I,C)\) into \(\Phi BV(I,Y)\) and is uniformly continuous, then the left regularization of \(h\), i.e., the function \(h^-:I^-\times C\to Y\) defined by \(h^-(t,y):=\lim_{s\uparrow t}h(s,y)\), exists and \(h^-(t,y)=A(t)y+B(t)\) for some \(A:I^-\to L_c(X,Y)\), \(A(\cdot)y\in\Phi BV(I^-,X)\) and \(B\in\Phi BV(I^-,Y)\). Moreover, \(A\) and \(B\) are left-continuous in \(I^-\) (\(L_c\) denotes continuous linear maps).