On the weighted Wiener-Lévy theorem: analogue on Euclidean space, strong converse on LCA group and applications to modulation spaces
Divyang G. Bhimani, Karishman B. Solanki
Abstract
We consider the space (weighted Fourier algebra) of Banach algebra valued functions $A^q_ω(Γ,\cX),$ which consists of all Fourier transforms of functions in $L^q_ω(G,\cX)$. Here $ω$ is a Beurling-Domar type weight on a discrete abelian group $G$, $Γ$ is the dual of $G$, and $\cX$ is a unital commutative Banach algebra. We shall prove a strong converse of the Wiener-Lévy 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 $F\circ f:Γ\to\cX$ is in $A^q(Γ,\cX)$ ($1\leq q <2$) for all $f\in A^1_ω(Γ,\mathbb{C})$, 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-Lévy theorem in Euclidean Fourier algebra $A_ω^q(\mathbb R^d)$ for a weight $ω$ 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.