Composition operators on Gelfand-Shilov classes

TL;DR

This paper investigates the properties of composition operators on Gelfand-Shilov classes of ultradifferentiable functions, establishing necessary conditions and optimal bounds for their boundedness and inclusion relations.

Contribution

It provides new necessary conditions for the boundedness of composition operators on Gelfand-Shilov classes and determines the optimal index for polynomial symbols.

Findings

Boundedness of $oldsymbol{ ext{psi}}'$ is necessary for well-defined composition operators.

Identifies the optimal index $oldsymbol{d'}$ for polynomial $oldsymbol{ ext{psi}}$ ensuring inclusion.

Establishes conditions for composition operators on classical Gelfand-Shilov classes.

Abstract

We study composition operators on global classes of ultradifferentiable functions of Beurling type invariant under Fourier transform. In particular, for the classical Gelfand-Shilov classes $\Sigma_d,\ d > 1,$ we prove that a necessary condition for the composition operator $f\mapsto f\circ \psi$ to be well defined is the boundedness of $\psi'.$ We find the optimal index $d'$ for which $C_\psi(\Sigma_d({\mathbb R}))\subset \Sigma_{d'}({\mathbb R})$ holds for any non-constant polynomial $\psi.$