Variations around a general quantum operator (Q829669)

The authors take the quantum operator \[ D_{\beta}[f](t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t} \] with suitable function \(\beta\), and study the basic properties of this operator, and also of its operator inverse, from function space viewpoint. They define the function space \[ \mathscr{L}_{\beta}^{p}[a, b]=\left\{f:\left.I \rightarrow \mathbb{C}\left|\int_{a}^{b}\right| f\right|^{p} \mathrm{~d}_{\beta}<\infty\right\}. \] It turns out that this space is a Banach space. Among other things, Hölder's inequality is also proven.