| Summary: | In this paper, we consider a Schrödinger operator $ L = -\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 $ BMO_{L}^{v}(\mathbb{H}) $. Second, we demonstrate that both Littlewood-Paley $ {\bf g} $-functions and area functions are bounded on $ BMO^{v}_{L}(\mathbb{H}) $. Finally, we also obtain that the above square functions are bounded on the Morrey space $ L^{\gamma, \theta}_{p, \kappa}(\mathbb{H}) $ and the weak Morrey space $ WL^{\gamma, \theta}_{1, \kappa}(\mathbb{H}) $, respectively.
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