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 $$\mathcal{L}$ = -\Delta+V$actingon$L^2(\mathbb R^n)$wherethenonnegativepotential$V$belongstothereverseH\ddot{o}lderclass$B_q$forsome$q\geq n.$Let$L^{p,\lambda}($\mathbb{R}$^{n})$,$0\le \lambda<n$denotetheMorreyspaceon$$\mathbb{R}$^{n}$\mathrm{.}Inthispaper,wewillshowthatafunction$f\in L^{2,\lambda}($\mathbb{R}$^{n})$isthetraceofthesolutionof${\mathbb L}u=u_{t}+{$\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.