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# Revista Matemática Iberoamericana

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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 and Peijie Lin

(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 .

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