Taiwanese Journal of Mathematics

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

Qiang Huang and Chao Zhang

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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., Volume 23, Number 5 (2019), 1133-1151.

Dates
Received: 21 June 2018
Revised: 9 November 2018
Accepted: 12 November 2018
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

Mathematical Reviews number (MathSciNet)
MR4012373

Zentralblatt MATH identifier
07126942

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis 42B37: Harmonic analysis and PDE [See also 35-XX] 35J10: Schrödinger operator [See also 35Pxx] 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Keywords
Dirichlet problem heat equation Schrödinger operators Morrey space Carleson measure reverse Hölder inequality

Citation

Huang, Qiang; Zhang, Chao. Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces. Taiwanese J. Math. 23 (2019), no. 5, 1133--1151. doi:10.11650/tjm/181106. https://projecteuclid.org/euclid.twjm/1542790913


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