Boundedness of composition operator on several variable Paley-Wiener space

In this paper we have considered composition operators on the several variable Paley-Wiener space L2 pi(Cn). We have proved that for any continuous function phi : Cn -> Cn the composition operator C phi on L2 pi(Cn) is bounded iff phi has the following form: phi(z) = Az + b, z is an element of Cn, where A is an element of GL(n,R) with ||A||op <= 1 and b is an element of Cn. Our proof is different from the single variable case and mainly depends on a theorem of Malgrange, some techniques from harmonic analysis and certain asymptotic behaviour of Bessel's function. (c) 2022 Elsevier Inc. All rights reserved.