Fefferman--Stein type inequalities via area and maximal functions for Schr\"odinger operators with applications

Abstract

In this paper, we establish a Fefferman--Stein inequality in terms of area function and non-tangential maximal function associated with the Schr\"odinger operator $\mathcal{L} = -\Delta + V$ on stratified Lie groups $\mathcal G$, where $\Delta$ 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 Schr\"odinger operator for functions in the Orlicz space $L\log^{+}L(\mathcal G_1 \times \mathcal G_2).$ Using these atomic decompositions, we further prove weak-type endpoint estimates for the area integral operator and the Riesz transforms associated with the Schr\"odinger 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.