Analysis on Lie groups

Nick Varopoulos
1996 Revista matemática iberoamericana  
Dedicated to the memory of my mother 0. Introduction. Statement of the theorems. In what follows G will denote a real connected Lie group and = ; P n j=1 X 2 j + X 0 will denote some subelliptic left invariant Laplacian (cf. 1]). This, for us here, will mean that X 0 X 1 : : : X n are left invariant elds on G (i.e. X f g = (X f ) g , f g (x) = f(gx)) and that X 1 : : : X n are generators of the Lie algebra of G (i.e. together with all their successive brackets they span the Lie algebra of G
more » ... ie algebra of G (cf. 2])). I shall denote by dg = d'g the left Haar measure of G and by d r g = d'(g ;1 ) = m(g) d'g the right Haar measure and by m(g) = m G (g) the modular function. We can then construct T t = e ;t (t > 0) the Heat di usion semigroup and t (g) the corresponding Heat di usion kernel that is de ned by T t f(x) = Z G f(y) t (y ;1 x) dy t > 0 x 2 G f 2 C 1 0 (G) : When X 0 = 0 w e s a y t h a t = 0 is driftless. A driftless Laplacian 0 is formally self adjoint with respect to d r g. It follows that the modi ed Laplacian~ = m 1=2 0 m ;1=2 is formally self adjoint with respect to dg. It is then more convenient to consider the modi ed semigroup T t = m 1=2 e ;t 0 m ;1=2 and it is very easy to see (cf. 3]) that the 791
doi:10.4171/rmi/215 fatcat:aheshajx65cbdgf4py3e2wn4ve