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

• DOI: 10.48550/arxiv.1712.03952
• Publication Date: 2017-01-01
• Authors: Qiang Huang; Chao Zhang
• Keywords: Nabla symbol; Characterization; TRACE (psycholinguistics)

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