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

Canadian mathematical bulletin

Sibei Yang²  Dachun Yang¹ 

[1] School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China;School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China

关键词: Musielak-Orlicz-Hardy space;    magnetic Schrödinger operator;    atom;    second-order Riesz transform;    maximal inequality;   

DOI  :  10.4153/CMB-2014-060-x

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $A:=-(abla-ivec{a})cdot(abla-ivec{a})+V$ be amagnetic 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 $0le Vin L^1_{mathrm{loc}}(mathbb{R}^n)$ satisfy some reverseHölder conditions. Let $varphicolon mathbb{R}^nimes[0,infty)o[0,infty)$ be such that$varphi(x,cdot)$ for any given $xinmathbb{R}^n$ is an Orlicz function,$varphi(cdot,t)in {mathbb A}_{infty}(mathbb{R}^n)$ for all $tin (0,infty)$(the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index$I(varphi)in(0,1]$. In this article, the authors prove thatsecond-order Riesz transforms $VA^{-1}$ and$(abla-ivec{a})^2A^{-1}$ are bounded from theMusielak-Orlicz-Hardy space $H_{varphi,,A}(mathbb{R}^n)$, associated with $A$,to the Musielak-Orlicz space $L^{varphi}(mathbb{R}^n)$. Moreover, the authorsestablish the boundedness of $VA^{-1}$ on $H_{varphi, A}(mathbb{R}^n)$. As applications, somemaximal inequalities associated with $A$ in the scale of $H_{varphi,A}(mathbb{R}^n)$ are obtained.

【 授权许可】

Unknown