On the weighted Wiener-L\'evy theorem: analogue on Euclidean space, strong converse on LCA group and applications to modulation spaces

Abstract

We consider the space (weighted Fourier algebra) of Banach algebra valued functions $A^q_{\omega}(\Gamma,\cX),$ which consists of all Fourier transforms of functions in $L^q_\omega(G,\cX)$. Here $\omega$ is a Beurling-Domar type weight on a discrete abelian group $G$, $\Gamma$ is the dual of $G$, and $\cX$ is a unital commutative Banach algebra. We shall prove a strong converse of the Wiener-L\'evy theorem in vector valued weighted setting. Specifically, we proved that if $F$ is a $\cX-$valued function defined on $\mathbb{C}$ such that the composition $F\circ f:\Gamma\to\cX$ is in $A^q(\Gamma,\cX)$ ($1\leq q <2$) for all $f\in A^1_\omega(\Gamma,\mathbb{C})$, then $F$ must be real analytic on $\mathbb R^2$. Here the range of $q$ is sharp. Further, its multivariate analogue and analogue for locally compact abelian $G$ are also established. This is the first result which generalizes the classic theorems of Helson, Kahane, Katznelson and Rudin \cite{helson,rud} in the presence of proposed weight. On the other hand, we established the analogue of Wiener-L\'evy theorem in Euclidean Fourier algebra $A_{\omega}^q(\mathbb R^d)$ for a weight $\omega$ of regular growth. As an application, we establish similar results for weighted modulation, Wiener amalgam and Fourier amalgam spaces. This complements the work of Bhimani-Ratnakumar \cite{bhimani2016functions} and Feichtinger-Kobayashi-Sato \cite{HGWL1, HGWL2}; and enables us to shed light on nonlinearity while understanding the dynamics of dispersive PDE in these spaces.