Asymptotics of operators
by
Tom ter Elst
Technische Universiteit Eindhoven
Coauthors: D.W. Robinson, A. Sikora (ANU)

Let A₁, ... , A_d' be right invariant vector fields on a connected Lie group G with polynomial volume growth satisfying Hörmanders condition and K the kernel of the semigroup generated by the sublaplacian \Delta = - \sum_i=1^d' A_i². Then it is known that K satisfies good Gaussian bounds, i.e., there are b, c > 0 such that

|Kt(g)| <= c  V(t)-1/2  e-b |g|2 t-1

uniformly for all t > 0 and g in G, where |g| is the subelliptic distance of g to the identity element of G associated to the vector fields A₁, ... , A_d' and V(t) is the volume of the ball { g in G : |g| < t } . Also the derivatives of K satisfy Gaussian bounds for small t, i.e., for all n in N and i₁, ... , i_n in { 1, ... , d'} there exist b, c > 0 such that

|(Ai1 ... Ain Kt)(g)| <= c  t-n/2  V(t)-1/2  e-b |g|2 t-1 \labeleqn1

(\theequation)

uniformly for all t in <0, 1] and g in G. If n=1 then the bounds () are also valid for all large t > 0. In this talk we shall characterize the groups for which the bounds () are valid for some (all) n >= 2, uniformly for all g in G and t >= 1.

Date received: November 22, 2001