Characterization of temperatures associated to Schr\"odinger operators with initial data in Morrey spaces

Let $\mathcal{L}$ be a Schr\"odinger operator of the form $\mathcal{L} = -\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let $L^{p,\lambda}(\mathbb{R}^{n})$, $0\le \lambda<n$ denote the Morrey space on $\mathbb{R}^{n}$. In this paper, we will show that a function $f\in L^{2,\lambda}(\mathbb{R}^{n})$ is the trace of the solution of ${\mathbb L}u=u_{t}+{\mathcal{L}}u=0, u(x,0)= f(x),$ where $u$ satisfies a Carleson-type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-\lambda}\int_0^{r_B^2}\int_{B(x_B, r_B)} |\nabla u(x,t)|^2 {dx dt} \leq C <\infty. \end{eqnarray*} Conversely, this Carleson-type condition characterizes all the ${\mathbb L}$-carolic functions whose traces belong to the Morrey space $L^{2,\lambda}(\mathbb{R}^{n})$ for all $0\le \lambda<n$. This result extends the analogous characterization founded by Fabes and Neri for the classical BMO space of John and Nirenberg.