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October, 2019 Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces
Qiang Huang, Chao Zhang
Taiwanese J. Math. 23(5): 1133-1151 (October, 2019). DOI: 10.11650/tjm/181106

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.

Citation

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Qiang Huang. Chao Zhang. "Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces." Taiwanese J. Math. 23 (5) 1133 - 1151, October, 2019. https://doi.org/10.11650/tjm/181106

Information

Received: 21 June 2018; Revised: 9 November 2018; Accepted: 12 November 2018; Published: October, 2019
First available in Project Euclid: 21 November 2018

zbMATH: 07126942
MathSciNet: MR4012373
Digital Object Identifier: 10.11650/tjm/181106

Subjects:
Primary: 35J10, 42B35, 42B37, 47F05

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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Vol.23 • No. 5 • October, 2019
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