Abstract view
Secondorder Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators


Published:20141124
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
Abstract
Let $A:=(\nablai\vec{a})\cdot(\nablai\vec{a})+V$ 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
secondorder Riesz transforms $VA^{1}$ and
$(\nablai\vec{a})^2A^{1}$ are bounded from the
MusielakOrliczHardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the MusielakOrlicz 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.
MSC Classifications: 
42B30, 42B35, 42B25, 35J10, 42B37, 46E30 show english descriptions
$H^p$spaces Function spaces arising in harmonic analysis Maximal functions, LittlewoodPaley theory Schrodinger operator [See also 35Pxx] Harmonic analysis and PDE [See also 35XX] Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B30  $H^p$spaces 42B35  Function spaces arising in harmonic analysis 42B25  Maximal functions, LittlewoodPaley theory 35J10  Schrodinger operator [See also 35Pxx] 42B37  Harmonic analysis and PDE [See also 35XX] 46E30  Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
