On the primary ideal structure at infinity for analytic Beurling algebras (Q1064518)

For continuous sumbmultiplicative weight functions w, we consider the Beurling algebra \(L^ 1_ w({\mathbb{R}})\) consisting of all locally integrable functions that satisfy \[ \| f\|_ w=\int^{\infty}_{-\infty}| f(t)| w(t)dt<\infty, \] supplied with convolution multiplication. We are interested in weights of the form \[ w(t)=\exp (\frac{\pi}{2}| t| +\psi (t)),\quad t\in {\mathbb{R}}, \] where \(\psi (t)=o(| t|)\) as \(| t| \to \infty.\) The Fourier transform \[ \hat f(z)=\int^{\infty}_{-\infty}f(t)e^{- itz}dt \] of a function \(f\in L^ 1_ w({\mathbb{R}})\) is continuous on the strip \(S=\{z\in {\mathbb{C}}:| Im z| \leq \pi /2\}\), analytic in the interior \(S^ 0\), and vanishes at infinity. An ideal I in \(L^ 1_ w({\mathbb{R}})\) is said to be primary at infinity if it is closed and \(\{z\in S:\hat f(z)=0\) for all \(f\in I\}=\emptyset\). Put \[ \delta_+(f)=- \limsup_{\xi \to +\infty}e^{-\xi}\log | \hat f(\xi)| \geq 0\quad and\quad \delta_-(f)=- \limsup_{\xi \to -\infty}e^{\xi}\log | \hat f(\xi)| \geq 0. \] For \(\alpha\),\(\beta\geq 0\), introduce the ideals \[ I^+_{\alpha}=\{f\in L^ 1_ w({\mathbb{R}}):\delta_+(f)\geq \alpha \}\quad and\quad I^-_{\beta}=\{f\in L^ 1_ w({\mathbb{R}}):\delta_-(f)\geq \beta \}. \] The main result states that if \(\psi\) is decreasing on the interval \((-\infty,0)\), increasing on (0,\(\infty)\), and satisfies the non-quasianalyticity condition \[ \int^{\infty}_{-\infty}(1+t^ 2)^{-1}\psi (t)dt<\infty, \] \(I^+_{\alpha}\cap I^-_{\beta}\) is a primary ideal at infinity for all \(\alpha,\beta\geq 0\), and that in fact, every primary ideal at infinity has this form. This generalizes a result of \textit{B. I. Korenbljum} [Tr. Moskov. Mat. Obšč. 7, 121-148 (1958; Zbl 0085.093)] for the case \(\psi\equiv 0\).