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# Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators

Published:2014-11-24
Printed: Jun 2015
• Dachun Yang,
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China
• Sibei Yang,
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China
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## Abstract

Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic Schrödinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse Hölder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained.
 Keywords: Musielak-Orlicz-Hardy space, magnetic Schrödinger operator, atom, second-order Riesz transform, maximal inequality
 MSC Classifications: 42B30 - $H^p$-spaces 42B35 - Function spaces arising in harmonic analysis 42B25 - Maximal functions, Littlewood-Paley theory 35J10 - Schrodinger operator [See also 35Pxx] 42B37 - Harmonic analysis and PDE [See also 35-XX] 46E30 - Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

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