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

Abstract

Let $L$ be a Schr\"odinger operator of the form $L=-\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ 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 ${\mathbb L}u=-u_{tt}+L u=0, u(x,0)= f(x),$ 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 $\mathscr{L}_L^{2,\lambda}({\mathbb{R}^n})$ associated to the operator $L$, i.e. $$\mathscr{L}_L^{2,\lambda}(\mathbb{R}^n)= {L}^{2,\lambda}(\mathbb{R}^n). $$ 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].