Differential transforms related to Caputo time-fractional derivatives and semigroups generated by fractional Schrödinger operators

Let \(\{e^{-t{\mathcal {L}}^{\alpha }}\}_{t>0}\) be the heat semigroup related to the fractional Schrödinger operator \(\mathcal {L}^{\alpha }:=(-\varDelta +V)^{\alpha }\) with \(\alpha \in (0,1)\), where V is a non-negative potential belonging to the reverse Hölder class. In this paper, we analyze the convergence of the following type of the series

$$\begin{aligned} T_{N,t}^{\alpha ,\beta }(f)=\sum _{j=N_{1}}^{N_{2}}v_{j}\Big (t^{\beta }\partial _{t}^{\beta }e^{-t{\mathcal {L}}^{\alpha }}(f)\Big |_{t=t_{j+1}}- t^{\beta }\partial _{t}^{\beta }e^{-t{\mathcal {L}}^{\alpha }}(f)\Big |_{t=t_{j}}\Big ) \end{aligned}$$

for \(\beta >0\) and for any \(N=(N_{1},N_{2})\in \mathbb {Z}^{2}\) with \(N_{1}<N_{2}\), where \(\{t_{j}\}_{j\in \mathbb {Z}}\) is an increasing sequence in \((0,\infty )\) and \(\{v_{j}\}_{j\in \mathbb {Z}}\) is a bounded sequence of real numbers. The symbol \(\partial _{t}^{\beta }\) denotes the Caputo time-fractional derivative. We prove that the maximal operator \(T_{*,t}^{\alpha ,\beta }(f)=\sup _{\begin{array}{c} N\in \mathbb {Z}^{2} N_{1}<N_{2} \end{array}}|T_{N,t}^{\alpha ,\beta }(f)|\) is bounded on weighted Lebesgue spaces \(L^{p}_{w}({\mathbb {R}}^{n})\), and is a bounded operator from \(BMO_{{\mathcal {L}},w}^{\gamma }({\mathbb {R}}^{n})\) into \(BLO_{{\mathcal {L}},w}^{\gamma }({\mathbb {R}}^{n})\), where \(\gamma \in [0,1)\) and w belongs to the class of weights associated with the auxiliary function \(\rho (x,V)\).