Geometry and Analysis of the Grusin Operator

Abstract In this paper we introduce a new technique for analyzing the heat kernel associated to the Grusin Laplacian. We prove pointwise estimates for arbitrary derivatives of the Grusin heat kernel and the resolvent in terms of the control distance associated with the Grusin operator. Our technique is based on the global $$L^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -Gevrey method.