Resumen de Sub-laplacians of holomorphic -type on exponential solvable groups

Waldemar Hebisch, Jean Ludwig, Detlef Müller

• Let denote a right-invariant sub-Laplacian on an exponential, hence solvable Lie group , endowed with a left-invariant Haar measure. Depending on the structure of , and possibly also that of may admit differentiable -functional calculi, or may be of holomorphic -type for a given . 'Holomorphic -type' means that every -spectral multiplier for is necessarily holomorphic in a complex neighbourhood of some non-isolated point of the -spectrum of . This can in fact only arise if the group algebra is non-symmetric.

  Assume that . For a point in the dual of the Lie algebra of , denote by the corresponding coadjoint orbit. It is proved that every sub-Laplacian on is of holomorphic -type, provided that there exists a point satisfying Boidol's condition (which is equivalent to the non-symmetry of ), such that the restriction of to the nilradical of is closed. This work improves on results in previous work by Christ and M?ller and Ludwig and M?ller in twofold ways: on the one hand, no restriction is imposed on the structure of the exponential group , and on the other hand, for the case , 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 to the nilradical of is closed could be replaced by the weaker condition that the orbit itself is closed. This would then prove one implication of a conjecture by Ludwig and M?ller, according to which there exists a sub-Laplacian of holomorphic (or, more generally, ) type on if and only if there exists a point whose orbit is closed and which satisfies Boidol's condition.