Fefferman--Stein type inequalities via area and maximal functions for Schrödinger operators with applications

In this paper, we establish a Fefferman–Stein inequality in terms of area function and non-tangential maximal function associated with the Schrödinger operator $\mathcal{L}=-\mathrm{\Delta }+V$ on stratified Lie groups $\mathcal{G}$, where $\mathrm{\Delta }$ denotes the sub-Laplacian on $\mathcal{G}$ and $V$ is a nonnegative locally integrable function. As an application, we extend this inequality to the tensor product ${\mathcal{G}}_1\times {\mathcal{G}}_2$ of two stratified Lie groups and develop atomic decompositions associated with the Schrödinger operator for functions in the Orlicz space $L{\log }^{+}L({\mathcal{G}}_1\times {\mathcal{G}}_2)\mathrm{.}$ Using these atomic decompositions, we further prove weak-type endpoint estimates for the area integral operator and the Riesz transforms associated with the Schrödinger operator on $L{\log }^{+}L({\mathcal{G}}_1\times {\mathcal{G}}_2),$ thereby extending the celebrated result of R.,Fefferman and E.M.,Stein \cite{FSt1982} to the setting of singular integrals with non-smooth kernels.