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

TL;DR

This paper characterizes the traces of solutions to Schr"odinger operator equations with Morrey space boundary data, establishing a link between Morrey and Campanato spaces through a Carleson measure condition.

Contribution

It introduces a new characterization of boundary traces of Schr"odinger operator solutions using Carleson conditions and relates Morrey and Campanato spaces in this context.

Findings

Trace characterization of Schr"odinger solutions with Morrey boundary data.

Equivalence between Morrey and Campanato spaces for the operator.

Extension of previous boundary value problem results.

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].