01-3   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## Sub-Laplacians of holomorphic Lp-type on exponential solvable groups

Let L denote a right-invariant sub-Laplacian on an exponential, hence solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G, and possibly also that of L, L may admit differentiable Lp-functional calculi, or may be of holomorphic Lp-type for a given p not equal 2. By ``holomorphic Lp-type''we mean that every Lp-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some non-isolated point of the L2-spectrum of L. This can in fact only arise if the group algebra L1(G) is non-symmetric.

Assume that p not equal 2. For a point l in the dual g* of the Lie algebra g of G, we denote by Omega(l)=Ad*(G)l the corresponding coadjoint orbit. We prove that every sub-Laplacian on G is of holomorphic Lp-type, provided there exists a point l in g* satisfying ``Boidol's condition'' (which, by \cite{Poguntke} is equivalent to the non-symmetry of L1(G)), such that the restriction of Omega(l) to the nilradical of g is closed. This work improves on the results in \cite{Ludwig-Mueller} in twofold ways: On the one hand, we no longer impose any restriction on the structure of the exponential group G, and on the other hand, for the case p greater 1, our conditions need to hold for a single coadjoint orbit only, and not for an open set of orbits, in contrast to \cite{Ludwig-Mueller}.

It seems likely that the condition that the restriction of Omega(l) to the nilradical of g is closed could be replaced by the weaker condition that the orbit Omega(l) itself is closed. This would then prove one implication of a conjecture made in \cite{Ludwig-Mueller}, according to which there exists a sub-Laplacian of holomorphic L1(G) (or, more generally, Lp(G))-type on G if and only if there exists a point l in g* whose orbit is closed and which satisfies Boidol's condition.

Mathematics Subject Classification (1991): 22E30, 22E27, 43A20

Bibliographical note: erschienen in J. London Math. Soc. 72 (2) (2005), 364-390

Keywords: exponential solvable Lie group, sub-Laplacian, functional calculus, Lp-spectral multiplier, symmetry, unitary representation, holomorphic Lp-type, heat kernel, transference, perturbation