We consider Schrödinger operators on a metric cone whose cross section is a closed Riemannian manifold of dimension . Thus the metric on the cone is . Let be the Friedrichs Laplacian on and let be a smooth function on such that is a strictly positive operator on with lowest eigenvalue and second lowest eigenvalue , with . The operator we consider is , a Schrödinger operator with inverse square potential on ; notice that is homogeneous of degree . We study the Riesz transform and determine the precise range of for which is bounded on . This is achieved by making a precise analysis of the operator and determining the complete asymptotics of its integral kernel. We prove that if is not identically zero, then the range of for boundedness is
while if is identically zero, then the range is
The result in the case of an identically zero was first obtained in a paper by H.-Q. Li .
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. 477–522DOI 10.4171/RMI/790