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THE $L^{q}$ ESTIMATES OF RIESZ TRANSFORMS ASSOCIATED TO SCHRÖDINGER OPERATORS

Part of: Elliptic equations and systems Partial differential operators Harmonic analysis in several variables

Published online by Cambridge University Press:  25 April 2016

QINGQUAN DENG ,
YONG DING  and
XIAOHUA YAO
QINGQUAN DENG*
Affiliation:
School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Science, Central China Normal University, Wuhan 430079, PR China email dengq@mail.ccnu.edu.cn
YONG DING
Affiliation:
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing 100875, PR China email dingy@bnu.edu.cn
XIAOHUA YAO
Affiliation:
School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Science, Central China Normal University, Wuhan 430079, PR China email yaoxiaohua@mail.ccnu.edu.cn
*
✉dengq@mail.ccnu.edu.cn
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Abstract

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Let $H=-\unicode[STIX]{x1D6E5}+V$ be a Schrödinger operator with some general signed potential $V$. This paper is mainly devoted to establishing the $L^{q}$-boundedness of the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ for $q>2$. We mainly prove that under certain conditions on $V$, the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,p_{0})$ with a given $2<p_{0}<n$. As an application, the main result can be applied to the operator $H=-\unicode[STIX]{x1D6E5}+V_{+}-V_{-}$, where $V_{+}$ belongs to the reverse Hölder class $B_{\unicode[STIX]{x1D703}}$ and $V_{-}\in L^{n/2,\infty }$ with a small norm. In particular, if $V_{-}=-\unicode[STIX]{x1D6FE}|x|^{-2}$ for some positive number $\unicode[STIX]{x1D6FE}$, $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,n/2)$ and $n>4$.

Keywords

Lp estimatesRiesz transformSchrödinger operatoroff-diagonal estimatereverse Hölder class

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 47F05: Partial differential operators (should also be assigned at least one other classification number in section 47) 35J10: Schrödinger operator
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 3 , December 2016 , pp. 290 - 309
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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