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 \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].

Locations

• Revista Matemática Iberoamericana - View - PDF
• arXiv (Cornell University) - View - PDF