The Riesz transform for homogeneous Schrödinger operators on metric cones

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 dr²+r ²h. Let Δ be the Friedrichs Laplacian on M and let V0 be a smooth function on Y such that ΔY + V₀ + (d - 2)²/4 is a strictly positive operator on L²(Y) with lowest eigenvalue μ20 and second lowest eigenvalue μ²1 , with μ0, μ1 > 0. The operator we consider is H = Δ+V0/r², 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 Lp(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[equation presented] while if V is identically zero, then the range is [equation presented] The result in the case of an identically zero V was first obtained in a paper by H.-Q. Li [33].