Characterization of temperatures associated to Schrödinger operators with initial data in Morrey spaces

Abstract

Let $\mathcal{L}$ be a Schrödinger operator of the form $\mathcal{L}=-\Delta +V$ acting on $L^2({\mathbb{R}}^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q\ge n.$ 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^{-λ}\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.

Minor corrections. To be appeared in Taiwanese Journal of Mathematics. arXiv admin note: substantial text overlap with arXiv:1710.01160