Multiplication and transformation of derivatives (Q1063710)

The main result is following: Let \({\mathcal D}\) be the class of all derivatives on \(J=<0,1>\), \({\mathcal L}\) be the class of all Lebesgue functions on J, \({\mathcal H}\) be the class of all increasing homeomorphisms of J onto J, \(T({\mathcal L})=\{h\in {\mathcal H}:\) \(f\circ h\in {\mathcal D}\) for each \(f\in {\mathcal L}\}\) and \(T({\mathcal D})=\{h\in {\mathcal H}:\) \(f\circ h\in {\mathcal D}\) for each \(f\in {\mathcal D}\}.\) Let \(h\in {\mathcal H}\), g be the inverse \(h^{-1}\) of h and \(\gamma\) be a mapping of J onto \(<0,\infty >\) such that \(\underline{D}g\leq \gamma \leq \bar Dg,\) where \(\underline{D}g(\bar Dg)\) denotes the lower (the upper) derivative of g. Then \(f\in T({\mathcal L})\) iff \[ \limsup_{x\to a,x\in J}(1/(g(x)- g(a)))\int^{x}_{a}\sup \gamma (<\min (t,x),\quad \max (t,x)>)dt<\infty \] for each \(a\in J\) and \(h\in T({\mathcal D})\) iff \[ \limsup_{x\to a,x\in J}(1/(g(x)-g(a)))\int^{x}_{a}var(\gamma;I_ t)dt \] for each \(a\in J\). There \(var(\gamma;I_ t)\) is the variation of \(\gamma\) on \(I_ t=<\min (t,x),\max (t,x)>\) if \(\infty \not\in \gamma (I_ t)\) and \(var(\gamma;I_ t)=\infty\) if \(\infty \in \gamma (I_ t)\).