Let \({\mathcal {L}}=-\Delta _{{\mathbb {G}}}+\Upsilon \) be a Schrödinger operator on the stratified Lie group \({\mathbb {G}}\) with the nonnegative potential \(\Upsilon \) belonging to the reverse Hölder class \(B_{Q/2}\) , where Q is the homogeneous dimension of \({\mathbb {G}}\) . In this paper, we initiate the investigation of the Dirichlet problem: \(\begin{aligned} {{\mathbb {L}}}u(g,s):={\mathcal {L}} u(g,s)-\partial _s^2u(g,s)=0\,,\quad \forall (g,s)\in {\mathbb {G}}\times {\mathbb {R}}^+ \end{aligned}\) with \(L^p\ (1\le p<\infty )\) data on the stratified Lie group \({\mathbb {G}}\) , which is new even in the case of the upper-half \((n+1)\) -dimensional Euclidean space \({\mathbb {R}}^{n+1}_+\) since the previous known results in this case require that the potential function \(\Upsilon \in B_{n}\) . Moreover, we obtain the \(BMO_{{\mathcal {L}}}({\mathbb {G}})\) -boundedness of two nontangential maximal functions related to heat and Poisson kernels, respectively. Finally, the end-point estimates of the fractional integral \({{\mathcal {L}}}^{-\alpha /2}\) and its generalization \(\Upsilon ^\alpha {\mathcal {L}}^{-\beta }\) from the Hardy type space \(H^1_{{\mathcal {L}}}({\mathbb {G}})\) into \(L^{Q/(Q-\alpha )}({\mathbb {G}})\) and \(L^{Q/(Q-2(\beta -\alpha ))}({\mathbb {G}})\) , respectively, will be obtained, which extend the corresponding work of Krantz [Math. Ann. , 1979] where the classical Hardy space related to the sub-Laplacian on Heisenberg group was investigated .
