Regularities of time-fractional derivatives of semigroups related to Schrödinger operators with application to Hardy-Sobolev spaces on Heisenberg groups (Q2211426)

Summary: In this paper, assume that \(L=- \Delta_{\mathbb{H}^n}+V\) is a Schrödinger operator on the Heisenberg group \(\mathbb{H}^n\), where the nonnegative potential \(V\) belongs to the reverse Hölder class \(B_{\mathcal{Q}/2}\). By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups \(\{e^{-tL}\}_{t>0}\) and \(\{ e^{-t\sqrt{L}}\}_{t>0}\), respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space \(H_L^{1,\alpha} (\mathbb{H}^n)\) associated with \(L\). Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with \(L\).