Subadditivity of Homogeneous Norms on Certain Nilpotent Lie Groups

Abstract

Let N be a Lie group with its Lie algebra generated by the leftinvariant vector fields X¡,.. .,Xk on N.An explicit fundamental solution for the (hypoelliptic) operator L = Xx + ■ ■ ■ + Xk on N has been obtained for the Heisenberg group by Folland [1] and for the nilpotent (Iwasawa) groups of isometries of rank-one symmetric spaces by Kaplan and Putz [2].Recently Kaplan [3] introduced a (still larger) class of step-2 nilpotent groups N arising from Clifford modules for which similar explicit solutions exist.As in the case of L being the ordinary Laplacian on N = R*, these solutions are of the form g t-» const|J ^|j2-m, g e N, where the "norm" function || || satisfies a certain homogeneity condition.We prove that the above norm is also subadditive.Let u, b be real finite-dimensional vector spaces each equipped with a positive definite quadratic form | \2.Let u: uXMobea composition of these quadratic forms [3, p. 148] normalized in the sense that n(u0, v) = v for some u0 G u. Define <p: b X to -* u by demanding <m, <b(v, ©')> = < p(u, v), v'}, u G u; v, v' G to, relative to the inner products < , ) induced by the given quadratic forms.Let 3 denote the orthogonal complement to Rm0 in u and it: u -» 3 the orthogonal projection.Now set n = b X 3 and define a bracket on n by [(v, z), (v\ z')] = (0, tr° <p(u, v')).On the simply connected analytic group N, corresponding to the Lie algebra n (i.e. on Kaplan's type H group) we define a norm function by ||n|| = (|u|4 + 16|z|2)1/4, where n = exp(t) + z), v G b, z G 3; n = b ® 3. We now prove Theorem.The norm function || || is subadditive, i.e.Hwi'H < \\n\\ + lln'll, n,n'(EN.Proof.We have llnn'H4 = ||exp(t) + v' + z + z' +1[v, t/])||4 -[©+ t/|4+ 16|z + z' +\[v, v']\2.Now \v + vf = |ü|4 + |t/|4 + 4<o, v')2 + 4|tf O, t/> + 4|ü'|2<ü,ü'> + 2|ü|2|t;f, (1)