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 ∈ C(Rⁿ) , we show that every such solution can be analytically continued to a sub-exponentially growing entire function on Cⁿ if, and only if, m(ξ) ≠ 0 for ξ ≠ 0 .