Fractional time-dependent Schrödinger equation on the Heisenberg group (Q957910)

Let \(\Delta\) be the Kohn sublaplacian on the Heisenberg group \(\mathbb{H}^n\), \(L=(-\Delta)^{\alpha}\), \(2/3<\alpha<1\). The operator \(iL\) generates the one-parameter group \(\{V_t\}_{t\in \mathbb{R}}\) of unitary operators on \(L^2(\mathbb{H}^n)\). For a fixed \(\varphi\in C_c^{\infty}(\mathbb{H}^n)\), the local maximal function of the group \(V_t\) is defined by \[ Mf(z,u)=\varphi(z,u)\underset{0\leq t\leq 1}\sup| V_tf(z,u)| . \] The main result of the paper is the following theorem. Theorem 1.1 There exists a \(\gamma>0\) such that for every \(\varepsilon>0\) we have \[ | Mf| _{L^2}\leq C | f|_{W_{\gamma,\varepsilon}}. \] Here \(| f| _{W_{\gamma,\varepsilon}}\) is a Sobolev anisotropic type norm. Theorem 1.1 is the analogue of the results of \textit{J. Zienkiewicz} for the fractional powers of \(\Delta\) [Stud. Math. 122, No. 1, 15--37 (1997; Zbl 0869.22004); Stud. Math. 161, No. 2, 99--111 (2004; Zbl 1056.22004)]. The proof of the present result uses the approach of J. Zienkiewicz, as well as ideas developed in works of \textit{J. Bourgain} [Isr. J. Math. 77, No. 1--2, 1--16 (1992; Zbl 0798.35131)] and \textit{A. Carbery, M. Christ} and \textit{J. Wright} [J. Am. Math. Soc. 12, No. 4, 981--1015 (1999; Zbl 0938.42008)].