JournalsrmiVol. 30, No. 2pp. 477–522

The Riesz transform for homogeneous Schrödinger operators on metric cones

  • Andrew Hassell

    Australian National University, Canberra, Australia
  • Peijie Lin

    Australian National University, Canberra, Australia
The Riesz transform for homogeneous Schrödinger operators on metric cones cover

Abstract

We consider Schrödinger operators on a metric cone whose cross section is a closed Riemannian manifold (Y,h)(Y, h) of dimension d12d-1 \geq 2. Thus the metric on the cone M=(0,)r×YM = (0, \infty)_r \times Y is dr2+r2hdr^2 + r^2 h. Let Δ\Delta be the Friedrichs Laplacian on MM and let V0V_0 be a smooth function on YY such that ΔY+V0+(d2)2/4\Delta_Y + V_0 + (d-2)^2/4 is a strictly positive operator on L2(Y)L^2(Y) with lowest eigenvalue μ02\mu^2_0 and second lowest eigenvalue μ12\mu^2_1, with μ0,μ1>0\mu_0, \mu_1 > 0. The operator we consider is H=Δ+V0/r2H = \Delta + V_0/r^2, a Schrödinger operator with inverse square potential on MM; notice that HH is homogeneous of degree 2-2. We study the Riesz transform T=H1/2T = \nabla H^{-1/2} and determine the precise range of pp for which TT is bounded on Lp(M)L^p(M). This is achieved by making a precise analysis of the operator (H+1)1(H + 1)^{-1} and determining the complete asymptotics of its integral kernel. We prove that if VV is not identically zero, then the range of pp for LpL^p boundedness is

(dmin(1+d/2+μ0,d),dmax(d/2μ0,0)),\Big(\frac{d}{\min(1+{d}/{2}+\mu_0, d)} , \frac{d}{\max({d}/{2}-\mu_0, 0)}\Big),

while if VV is identically zero, then the range is

(1dmax(d/2μ1,0)).\Big(1 \frac{d}{\max({d}/{2}-\mu_1, 0)}\Big).

The result in the case of an identically zero VV was first obtained in a paper by H.-Q. Li [33].

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. 477–522

DOI 10.4171/RMI/790