1 results
THE
$L^{q}$ ESTIMATES OF RIESZ TRANSFORMS ASSOCIATED TO SCHRÖDINGER OPERATORS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 101 / Issue 3 / December 2016
- Published online by Cambridge University Press:
- 25 April 2016, pp. 290-309
- Print publication:
- December 2016
-
- Article
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- You have access
- Export citation
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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 
$n>4$.
