# The Riesz transform for homogeneous Schrödinger operators on metric cones

### Andrew Hassell

Australian National University, Canberra, Australia### Peijie Lin

Australian National University, Canberra, Australia

## Abstract

We consider Schrödinger operators on a metric cone whose cross section is a closed Riemannian manifold $(Y,h)$ of dimension $d−1≥2$. Thus the metric on the cone $M=(0,∞)_{r}×Y$ is $dr_{2}+r_{2}h$. Let $Δ$ be the Friedrichs Laplacian on $M$ and let $V_{0}$ be a smooth function on $Y$ such that $Δ_{Y}+V_{0}+(d−2)_{2}/4$ is a strictly positive operator on $L_{2}(Y)$ with lowest eigenvalue $μ_{0}$ and second lowest eigenvalue $μ_{1}$, with $μ_{0},μ_{1}>0$. The operator we consider is $H=Δ+V_{0}/r_{2}$, a Schrödinger operator with inverse square potential on $M$; notice that $H$ is homogeneous of degree $−2$. We study the Riesz transform $T=∇H_{−1/2}$ 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].

## Cite this article

Andrew Hassell, Peijie Lin, The Riesz transform for homogeneous Schrödinger operators on metric cones. Rev. Mat. Iberoam. 30 (2014), no. 2, pp. 477–522

DOI 10.4171/RMI/790