The Riesz transform for homogeneous Schrödinger operators on metric cones

Abstract

We consider Schrödinger operators on a metric cone whose cross section is a closed Riemannian manifold $(Y,h)$ of dimension $d-1\ge 2$. Thus the metric on the cone $M=(0,\infty {)}_r\times Y$ is $d{r}^2+r^{2}h$. Let $\Delta$ be the Friedrichs Laplacian on $M$ and let $V_0$ be a smooth function on $Y$ such that ${\Delta }_Y+V_0+(d-2{)}^2/4$ is a strictly positive operator on $L^2(Y)$ with lowest eigenvalue ${\mu }_0^2$ and second lowest eigenvalue ${\mu }_1^2$, with ${\mu }_0,{\mu }_1>0$. The operator we consider is $H=\Delta +V_0/r^2$, a Schrödinger operator with inverse square potential on $M$; notice that $H$ is homogeneous of degree $-2$. We study the Riesz transform $T=\nabla H^{-1/2}$ and determine the precise range of $p$ for which $T$ is bounded on $L^p(M)$. This is achieved by making a precise analysis of the operator $(H+1{)}^{-1}$ and determining the complete asymptotics of its integral kernel. We prove that if $V$ is not identically zero, then the range of $p$ for $L^p$ boundedness is

while if $V$ is identically zero, then the range is

The result in the case of an identically zero $V$ was first obtained in a paper by H.-Q. Li [33].