Strong asymptotic expansions in a multidirection

In this paper we prove that, for asymptotically bounded holomorphic functions defined in a polysector in ${\mathbb C}^n$, the existence of a strong asymptotic expansion in Majima's sense following a single multidirection towards the vertex entails (global) asymptotic expansion in the whole polysector. Moreover, we specialize this result for Gevrey strong asymptotic expansions. This is a generalization of a result proved by A. Fruchard and C. Zhang for asymptotic expansions in one variable, but the proof, mainly in the Gevrey case, involves different techniques of a functional-analytic nature.