Beurling algebra analogues of theorems of Wiener–Lévy–Żelazko and Żelazko
Tom 195 / 2009
Streszczenie
Let $0< p\leq 1$, let $\omega :\mathbb Z\rightarrow [1,\infty)$ be a weight on $\mathbb{Z}$ and let $f$ be a nowhere vanishing continuous function on the unit circle $\mit\Gamma$ whose Fourier series satisfies $\sum _{n\in \mathbb{Z}}{|\widehat f(n)|^p \omega(n)} < \infty$. Then there exists a weight $\nu$ on $\mathbb{Z}$ such that $\sum _{n\in \mathbb{Z}}{|\widehat{(1/f)}(n)|^p \nu(n)} < \infty$. Further, $\nu$ is non-constant if and only if $\omega$ is non-constant; and $\nu=\omega$ if $\omega$ is non-quasianalytic. This includes the classical Wiener theorem ($p=1$, $\omega= 1$), Domar theorem ($p=1$, $\omega$ is non-quasianalytic), Żelazko theorem ($\omega=1$) and a recent result of Bhatt and Dedania ($p=1$). An analogue of the Lévy theorem at the present level of generality is also developed. Given a locally compact group $G$ with a continuous weight $\omega$ and $0 < p < 1$, the locally bounded space $L^p(G,\omega)$ is closed under convolution if and only if $G$ is discrete if and only if $G$ admits an atom. This generalizes and refines another result of Żelazko.
