Revista Matemática Iberoamericana

Volume 30, Issue 2, 2014, pp. 477–522
DOI: 10.4171/RMI/790

Published online: 2014-07-08

The Riesz transform for homogeneous Schrödinger operators on metric cones

Andrew Hassell[1] and Peijie Lin[2]

(1) Australian National University, Canberra, Australia
(2) Australian National University, Canberra, Australia

We consider Schrödinger operators on a metric cone whose cross section is a closed Riemannian manifold $(Y, h)$ of dimension $d-1 \geq 2$. Thus the metric on the cone $M = (0, \infty)_r \times Y$ is $dr^2 + r^2 h$. 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 $\mu^2_0$ and second lowest eigenvalue $\mu^2_1$, 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 $$\Big(\frac{d}{\min(1+{d}/{2}+\mu_0, d)} , \frac{d}{\max({d}/{2}-\mu_0, 0)}\Big),$$ while if $V$ is identically zero, then the range is $$\Big(1 \frac{d}{\max({d}/{2}-\mu_1, 0)}\Big).$$ The result in the case of an identically zero $V$ was first obtained in a paper by H.-Q. Li [33].

Keywords: Metric cone, Schrödinger operator, Riesz transform, inverse square potential, resolvent

Hassell Andrew, Lin Peijie: The Riesz transform for homogeneous Schrödinger operators on metric cones. Rev. Mat. Iberoam. 30 (2014), 477-522. doi: 10.4171/RMI/790