On generalized Beurling algebras (Q2467954)

For \(1 < p\), let \(\Omega\) be a family of weights, i.e., non-negative measurable locally integrable functions on \(\mathbb R\) such that the function \(\omega^{\frac{1}{1-p}}\) is integrable on \(\mathbb R\). Consider the weighted space \(L_\Omega^p(\mathbb R)=\{ f :\mathbb R \rightarrow C: f\) is measurable and \(| f |^p\omega \in L^1(\mathbb R)\) for every \(\omega\in\Omega\}\). The author gives a sufficient condition on \(\Omega\) for \(L^p_\Omega(\mathbb R)\) to be a locally convex algebra with continuous multiplication. Examples of such families of weights are also given.