## Taiwanese Journal of Mathematics

### Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces

This article is in its final form and can be cited using the date of online publication and the DOI.

#### 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 $q \geq n$. Let $L^{p,\lambda}(\mathbb{R}^{n})$, $0 \leq \lambda \lt n$ 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 $\sup_{x_B,r_B} r_B^{-\lambda} \int_0^{r_B^2} \!\! \int_{B(x_B,r_B)} |\nabla u(x,t)|^2 \, dx dt \leq C \lt \infty.$ 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 $0 \leq \lambda \lt n$. This result extends the analogous characterization found by Fabes and Neri in [8] for the classical BMO space of John and Nirenberg.

#### Article information

Source
Taiwanese J. Math., Advance publication (2019), 19 pages.

Dates
First available in Project Euclid: 21 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1542790913

Digital Object Identifier
doi:10.11650/tjm/181106

#### Citation

Huang, Qiang; Zhang, Chao. Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces. Taiwanese J. Math., advance publication, 21 November 2018. doi:10.11650/tjm/181106. https://projecteuclid.org/euclid.twjm/1542790913

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