TY - JOUR

T1 - The Riesz transform for homogeneous Schrödinger operators on metric cones

AU - Hassell, Andrew

AU - Lin, Peijie

PY - 2014

Y1 - 2014

N2 - 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 Lp 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].

AB - 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 Lp 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].

KW - Inverse square potential

KW - Metric cone

KW - Resolvent

KW - Riesz transform

KW - Schrödinger operator

UR - http://www.scopus.com/inward/record.url?scp=84905583426&partnerID=8YFLogxK

U2 - 10.4171/rmi/790

DO - 10.4171/rmi/790

M3 - Article

SN - 0213-2230

VL - 30

SP - 477

EP - 522

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

IS - 2

ER -