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 $p\mathrm{\ne }2$. By holomorphic $L^p$-type'' we mean that every $L^p$-spectral multiplier for $L$ is necessarily holomorphic in a complex neighborhood 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 $l$ in the dual $\frak g ^*$ of the Lie algebra $\frak g$ 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 $L^p$-type, provided there exists a point $l\in \frak g ^*$ satisfying Boidol’s condition’’ (which is equivalent to the non-symmetry of $L^1(G)$), such that the restriction of $\mathrm{\Omega }(l)$ to the nilradical of $\mathfrak{g}$ is closed.