Published online by Cambridge University Press: 25 April 2016
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
$n>4$.