Asymptotics of subcoercive semigroups on nilpotent Lie groups

One Can associate asymptotic approximates G(infinity) and H-infinity with e ach nilpotent Lie group G and pure m-th order weighted subcoercive operator H by a scaling limit. Then the semigroups S and S-(infinity) generated by H and H-infinity, on the spaces L-p(G), p is an element of [1, infinity], s atisfy (lim)(t-->infinity) parallel to St - S(t)((infinity))parallel to (p- ->p) = 0 if, and only if, G = G(infinity). If G not equal G(infinity) then (lim)(t-->infinity) parallel toM(f)(St -.S-t((infinity)))parallel to (p-->p ) = 0 on the spaces L-p(g), where g denotes the Lie algebra of G, and M-f d enotes the operator of multiplication by any bounded function which vanishe s at infinity.