Constructions of regular algebras $\mathscr L_p^w(G)$

A criterion for (Shilov) regularity of weighted algebras ${\mathscr L}_1^w(G)$ on a locally compact Abelian group $G$ is known from works of Beurling (1949) and Domar (1956). In the present paper this criterion is extended to translation-invariant weighted algebras $\mathscr L_p^w(G)$ with $p>1$. Regular algebras $\mathscr L_p^w(G)$ are constructed on any $\sigma$-compact Abelian group $G$. It was proved earlier by the author that $\sigma$-compactness is necessary (in the Abelian case) for the existence of weighted algebras $\mathscr L_p^w(G)$ with $p>1$.
Bibliography: 11 titles.