Methods and Applications of Analysis

Volume 21 (2014)

Number 3

Special issue dedicated to the 60th birthday of Stephen S.-T. Yau: Part I

Guest editors: John Erik Fornæss, Norwegian University of Science and Technology; Xiaojun Huang, Rutgers University; Song-Ying Li, University of California, Irvine; Yat Sun Poon, University of California, Riverside; Wing Shing Wong, The Chinese University of Hong Kong; and Zhouping Xin, The Institute of Mathematical Sciences, CUHK.

Heat kernels for a family of Grushin operators

Pages: 291 – 312

DOI: https://dx.doi.org/10.4310/MAA.2014.v21.n3.a2

Authors

Der-Chen Chang (Departments of Mathematics and of Computer Science, Georgetown University, Washington, D.C., U.S.A.; and Department of Mathematics, Fu Jen Catholic University, Taipei, Taiwan)

Yutian Li (Institute of Computational and Theoretical Studies, and Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong)

Abstract

We construct the heat kernel for the second-order operator $\Delta_{\mathrm{x}} = \frac{1}{2} \Sigma^n_{k=1} {\left( \frac{\partial}{\partial x_k} \right) }^2 + \frac{1}{2} \Sigma^n_{k=1} {\left( x^{m_k}_{\partial y_k} \right) }^2$ with $m_k \in \mathrm{N}$, which is a degenerate elliptic operator. Obviously, this operator is closed related to the Grushin operator $L_G = \frac{1}{2} {\left( \frac{\partial}{\partial x} \right) }^2 + \frac{1}{2} {\left( x^m \frac{\partial}{\partial y} \right) }^2$ with $m \in \mathrm{N}$. In this paper, we fist give a complete description of the geometry induced by the operator $L_G$. More precisely, given any two points in the space, the number of geodesics and the lengths of the geodesics are calculated. Then we find modified complex action functions and show that the critical values of this function will recover the lengths of the corresponding geodesics. We also find the volume element by solving a generalized transport equation. Finally, the formula for the heat kernel of the diffusion operator $\frac{\partial}{\partial t} - \Delta_{\mathrm{x}}$ is obtained. The formula involves an integral of a product between the volume function and an exponential term.

Keywords

Grushin operators, sub-Riemannian geometry, geodesics, heat kernel, volume element

2010 Mathematics Subject Classification

Primary 53C17, 53C22. Secondary 35H20.

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Published 8 October 2014