arXiv 1206.29972012-06-14math.AP

度量锥上齐次薛定谔算子的里斯变换

The Riesz transform for homogeneous Schrödinger operators on metric cones

Andrew Hassell, Peijie Lin

备注：: 36 pages, 3 figures

摘要：

我们考虑度量锥上的Schroedinger算子，其截面是一个维数为$d-1 \geq 2$的闭黎曼流形$(Y, h)$。因此锥$M = (0, \infty)_r \times Y$上的度量是$dr^2 + r^2 h$。设$Δ$是$M$上的Friedrichs Laplacian算子，$V_0$是$Y$上的光滑函数，使得$Δ_Y + V_0 + (d-2)^2/4$是$L^2(Y)$上的严格正算子，其最低特征值为$μ^2_0 $，次低特征值为$μ^2_1$，其最低特征值为$μ_0, μ_1 > 0$ .我们考虑的算子是$H = Δ+ V_0/r^2$，一个在$M$上具有平方反比势的薛定谔算子;注意$H$是-2次齐次的。本文研究了里斯变换$T = \nabla H^{-1/2}$，并确定了$T = \nabla H^{-1/2}$在$L^p(M)$上有界的$p$的精确范围.这是通过对算子$(H + 1)^{-1}$进行精确分析并确定其积分核的完全渐近性来实现的。本文证明了当$V$不恒为零时，$p$对$L^p$有界性的值域为$$ d/ \Big(min(1+d/2+μ_0, d \Big) < p < d / \Big(max(d/2-μ_0, 0) \Big),$$，而当$V$恒为零时，值域为$$ 1 < p < d / \Big(max(d/2-μ_1, 0 \Big).$$。Q.李

英文摘要：

We consider Schroedinger operators on metric cones 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 $Δ$ be the Friedrichs Laplacian on $M$ and $V_0$ be a smooth function on $Y$, such that $Δ_Y + V_0 + (d-2)^2/4$ is a strictly positive operator on $L^2(Y)$, with lowest eigenvalue $μ^2_0 $ and second lowest eigenvalue $μ^2_1$, with $μ_0, μ_1 > 0$. The operator we consider is $H = Δ+ V_0/r^2$, 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 $$ d/ \Big(min(1+d/2+μ_0, d \Big) < p < d / \Big(max(d/2-μ_0, 0) \Big),$$ while if $V$ is identically zero, then the range is $$ 1 < p < d / \Big(max(d/2-μ_1, 0 \Big).$$ The result in the case $V$ identically zero was first obtained in a paper by H.-Q. Li.

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