Asymptotic expansion of the heat kernel for a class of hypoelliptic operators

Abstract

A result of Hormander [7, 14] states that this operator is hypoelliptic if the vector fields X1, . . . , Xl and all their commutators [Xi1 , [Xi2 . . . [Xis−1 , Xis ] . . .], s≤r, up to length r span the tangent space to M at each point. We recall that an operator A is said to be hypoelliptic on M if for any open set U ⊂M and distributions u, f on U satisfying Au = f , f ∈ C∞(U) implies u ∈ C∞(U). In [17] for the operator (2) it was shown (with m = 2) that ‖u‖m/r ≤ C(‖Au‖0 + ‖u‖0) , for all u ∈ C∞(M), where ‖ · ‖s denotes the norm in the usual Sobolev space Hs(M). For the operator (1) this estimate and hypoellipticity were proved in [6, 16]. We assume that the operator (1) is formally selfadjoint and positive, that is, (Au, v) = (u,Av) and (Au, u) ≥ 0 for all u, v ∈ C∞(M). It is easy to show that

Locations

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