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.

In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As applications, we derive the mean curvature for the Laguerre perimeter.