Asymptotics of subcoercive semigroups on nilpotent Lie groups

Research Report MRR98-022

Asymptotics of subcoercive semigroups on nilpotent Lie groups

N. Dungey , A.F.M. ter Elst, Derek W. Robinson , Adam Sikora

Abstract: Let G be a nilpotent Lie group and H a pure m-th order subcoercive operator constructed from a weighted basis of the Lie algebra g of G. We construct asymptotic approximates G_\infty and H_\infty of G and H by a scaling limit which ensures that g=g_\infty as vector spaces and that G_\infty and H_\infty are automatically scale invariant. We then compare the asymptotic orbits of the semigroup S generated by H with those of the corresponding semigroup S^(\infty) generated by H_\infty. In the simplest case, G=G_\infty, we prove that on the spaces L_p(G) one has
lim_t\to\infty |S_t-S^(\infty)_t|_p\to
p=0
for all p\in[1,\infty]. But if G\neq G_\infty then we show that the analogous result fails for all p \in [1,\infty]. Nevertheless, on the spaces L_p(g) one has
\lim_t\to\infty|M_f(S_t-S^(\infty)_t)|_p\to
p=0
for all p\in[1,\infty] where M_f denotes the operator of multiplication by any bounded function which vanishes at infinity.

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