Let a\,..., ad be a basis of the Lie algebra g of a connected Lie group G and let M be a Lie subgroup of,G. If dx is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space X = G/M and S : X x G -> C i s a continuous cocycle, then under a rather weak condition on dx and S there exists in a natural way a (weakly*) continuous representation U of G in L p (X\dx) for all P e [l,oo]. Let A, be the infinitesimal generator with respect to U and the direction a, for all i 6 {

more »

... 1 , . . . , d). We consider n-th order strongly elliptic operators W = 5Z C «-A" with complex coefficients c a . We show that the semigroup S generated by the closure of H has a reduced heat kernel *: and we derive upper bounds for K and all its derivatives. 2000 Mathematics subject classification: primary 43A85, 22D30, 22E25, 22E45, 35KO5.