Holomorphic L^p-type for sub-Laplacians on connected Lie groups

We study the problem of determining all connected Lie groups $G$ which have the following property (hlp): every sub-Laplacian $L$ on $G$ is of holomorphic $L^p$-type for $1\leq p<\infty, p\ne 2.$ First we show that semi-simple non-compact Lie groups with finite center have this property. We then apply an $L^p$-transference principle, essentially due to Anker, to show that every connected Lie group $G$ whose semi-simple quotient by its radical is non-compact has property (hlp). For the convenience of the reader, we give a self-contained proof of this transference principle, which generalizes the well-known Coifman-Weiss principle. One is thus reduced to studying compact extensions of solvable Lie groups. We extend previous work of Hebisch, Ludwig and M\"uller to compact extensions of certain classes of exponential solvable Lie groups.