The subalgebra of $L\sp 1(AN)$ generated by the Laplacian

Waldemar Hebisch
1993 Proceedings of the American Mathematical Society  
We prove that for the Iwasawa AN groups corresponding to complex semisimple Lie groups, the subalgebra of L[{AN) associated to a distinguished laplacian is isomorphic with the algebra of integrable radial functions on R" . In [1] Cowling et al. derived a formula for the heat semigroup generated by a distinguished laplacian on a large class of Iwasawa AN groups and proved that the maximal function constructed from the semigroup is of weak type (1, 1). In this paper we show that in the case of
more » ... t in the case of the AN groups corresponding to complex semisimple Lie groups the results of [1] can be strengthened once we notice that the subalgebra of LX(AN) associated to a distinguished laplacian is isomorphic with the algebra of integrable radial functions on Rn . This implies that the maximal operators associated to the Riesz means are of weak type (1, 1). Also the functional calculus for the laplacian on 7?" can be transferred to the distinguished laplacian on AN. This seems to be the first construction of a nonanalytic functional calculus on groups of exponential growth. Let G denote a connected, complex semisimple Lie group and g its Lie algebra. Denote by 0 a Cartan involution of g, and write for the associated Cartan decomposition. Fix a maximal abelian subspace a of p; this determines a root space decomposition 0 = 00 © Yl5a' qGA A denoting the set of roots of the pair (g, a). Corresponding to a choice of the ordering of the roots, we have an Iwasawa decomposition g = aenet. Let G = ANK be the corresponding Iwasawa decomposition of Cr. A distinguished laplacian on AN can be constructed as follows. Let n: g -> p be the projection (defined by the Cartan decomposition). We define a positive definite form B on o©n setting B(X, Y) = B(nX, nY) where B is the Killing form
doi:10.1090/s0002-9939-1993-1111218-3 fatcat:nkomrvw37zat3pzutierb644s4