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 =2. ‘Holomorphic Lp -type’ means that every Lp -spectral multiplier for L is necessarily holomorphic in a complex neighbourhood 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 =2. For a point in the dual g∗ of the Lie algebra g of G, denote by Ω( )=Ad ∗ (G) the corresponding coadjoint orbit. It is proved that every sub-Laplacian on G is of holomorphic Lp -type, provided that there exists a point ∈ g∗ satisfying Boidol’s condition (which is equivalent to the non-symmetry of L1(G)), such that the restriction of Ω( ) to the nilradical of g is closed. This work improves on results in previous work by Christ and M¨uller and Ludwig and M¨uller in twofold ways: on the one hand, no restriction is imposed on the structure of the exponential group G, and on the other hand, for the case p> 1, the conditions need to hold for a single coadjoint orbit only, and not for an open set of orbits. It seems likely that the condition that the restriction of Ω( ) to the nilradical of g is closed could be replaced by the weaker condition that the orbit Ω( ) itself is closed. This would then prove one implication of a conjecture by Ludwig and M¨uller, according to which there exists a sub-Laplacian of holomorphic L1 (or, more generally, Lp) type on G if and only if there exists a point ∈ g∗ whose orbit is closed and which satisfies Boidol’s condition.