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 $p{\ne} 2$. ‘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 $p{\ne} 2$. for a point $\ell$ in the dual $\frak{g}^*$ of the lie algebra $\frak{g}$ of $g$, denote by $\omega(\ell){=}\ad^*(g)\ell$ 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 $\ell{\in} \frak{g}^*$ satisfying boidol's condition (which is equivalent to the non-symmetry of $l^1(g)$), such that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ 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 $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed could be replaced by the weaker condition that the orbit $\omega(\ell)$ 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 $\ell{\in} \frak{g}^*$ whose orbit is closed and which satisfies boidol's condition.