An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions

We study the equation $m(D)f=0$ in a large class of sub-exponentially growing functions. Under appropriate restrictions on $m\in C({\mathbb{R}}^n)$, we show that every such solution can be analytically continued to a sub-exponentially growing entire function on ${\mathbb{C}}^n$ if and only if $m(\xi )≠0$ for $\xi ≠0$.