On characterization of Poisson integrals of Schr\"odinger operators with Morrey traces

Let $L$ be a Schr"odinger operator of the form $L=-\mathrm{\Delta }+V$ acting on $L^2({\mathbb{R}}^n)$ where the nonnegative potential $V$ belongs to the reverse H"older class $B_q$ for some $q\ge n\mathrm{.}$ In this article we will show that a function $f\in L^{2,\lambda }({\mathbb{R}}^n),0<\lambda <n$ is the trace of the solution of $\mathbb{L}u=-u_{tt}+Lu=0,u(x,0)=f(x),$ where $u$ satisfies a Carleson type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-\lambda}\int_0^{r_B}\int_{B(x_B, r_B)} t|\nabla u(x,t)|^2 {dx dt} \leq C <\infty. \end{eqnarray*} Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces ${\mathcal{L}}_L^{2,\lambda }({\mathbb{R}}^n)$ associated to the operator $L$, i.e. $${\mathcal{L}}_L^{2,\lambda }({\mathbb{R}}^n)=L^{2,\lambda }({\mathbb{R}}^n)\mathrm{.}$$ Conversely, this Carleson type condition characterizes all the $\mathbb{L}$-harmonic functions whose traces belong to the space $L^{2,\lambda }({\mathbb{R}}^n)$ for all $0<\lambda <n$. This extends the previous results of [FJN, DYZ, JXY].