Heat flow and the fundamental solution of the Baouendi-Grushin operator

Abstract

For the Baouendi-Grushin operator ${\mathcal{L}}_{\gamma }={\Delta }_x+|x{|}^{2\gamma }{\Delta }_y$ in ${\mathbb{R}}^{m+k}$, with $\gamma >0$, we prove that the time integral of the heat kernel with pole at the origin equals $C_{\gamma ,Q}{N}_{\gamma }^{2-Q}$, where $N_{\gamma }$ is the gauge and $Q$ the homogeneous dimension. When $\gamma \ne 1$ no closed form of the heat kernel is known. After a partial Fourier transform in $y$, the time integral of the heat kernel becomes the Green function of an anharmonic oscillator, which solves a generalized modified Bessel equation, and the gauge appears when the inverse Fourier transform is computed as a Hankel transform. In particular, the heat flow provides a new proof that $C_{\gamma ,Q}{N}_{\gamma }^{2-Q}$ is a fundamental solution of $-{\mathcal{L}}_{\gamma }$ with pole at the origin.