Sub‐Laplacians of Holomorphic L p ‐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 L p ‐functional calculi, or may be of holomorphic L p ‐type for a given p≠ 2. ‘Holomorphic L p ‐type’ means that every L p ‐spectral multiplier for L is necessarily holomorphic in a complex neighbourhood of some non‐isolated point of the L 2 ‐spectrum of L . This can in fact only arise if the group algebra L 1 ( G ) is non‐symmetric. Assume that p≠ 2. For a point ℓ in the dual g * of the Lie algebra g of G , denote by Ω(ℓ)=Ad * ( G )ℓ the corresponding coadjoint orbit. It is proved that every sub‐Laplacian on G is of holomorphic L p ‐type, provided that there exists a point ℓ∈ g * satisfying Boidol's condition (which is equivalent to the non‐symmetry of L 1 ( G )), such that the restriction of Ω(ℓ) to the nilradical of g is closed. This work improves on results in previous work by Christ and Müller and Ludwig and Müller 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(ℓ) to the nilradical of g is closed could be replaced by the weaker condition that the orbit Ω(ℓ) itself is closed. This would then prove one implication of a conjecture by Ludwig and Müller, according to which there exists a sub‐Laplacian of holomorphic L 1 (or, more generally, L p ) type on G if and only if there exists a point ℓ∈ g * whose orbit is closed and which satisfies Boidol's condition.