Boundedness of square functions related with fractional Schrödinger semigroups on stratified Lie groups

In this paper, we consider a Schrödinger operator $L=-{\mathrm{\Delta }}_{\mathbb{H}}+V$ on the stratified Lie group $\mathbb{H}$. First, we establish fractional heat kernel estimates related to $L^{\beta }$, $\beta \in (0,1)$. By utilizing kernel estimations and the fractional Carleson measure, we are able to derive a characterization of the Campanato type space $BM{O}_L^v(\mathbb{H})$. Second, we demonstrate that both Littlewood-Paley $\mathbf{g}$-functions and area functions are bounded on $BM{O}_L^v(\mathbb{H})$. Finally, we also obtain that the above square functions are bounded on the Morrey space $L_{p,\kappa }^{\gamma ,\theta }(\mathbb{H})$ and the weak Morrey space $W{L}_{1,\kappa }^{\gamma ,\theta }(\mathbb{H})$, respectively.