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Geometric analysis on a step 2 Grusin operator
by Chin-Huei Chang   Der-Chen Chang   Bernard Gaveau   Peter C. Greiner   Hsuan-Pei Lee

Vol. 4 No. 2 (2009) P.119~P.188

ABSTRACT

The Grusin operator $\Delta G \ = \ \frac 12 (\partial^2_x + x^2\partial^2_y), \ x,y \ \in \ \mathbb{R}$, is studied by Hamilton-Jacobi theory. In particular, we find all the geodesics of $\Delta G$ of the induced nonholonomic geometry, construct a modified complex action $f$ which allows us to obtain the heat kernel $P_t$ of $\Delta G$. The small time asymptotics of $P_t$ at all critical points of $f$ are computed. Finally we discuss the connection between $\Delta G$ and the subLaplacian of the 1-dimensional Heisenberg group.

KEYWORDS
Grusin operator, geodesic, heat kernel, small time asymptotics

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 53C17, 34K10, 25H20

MILESTONES