  Criterion of (Shilov) regularity for weighted algebras $L_1^w(G)$ on a
locally compact abelian group $G$ is known by works of Beurling (1949) and
Domar (1956). In the present paper this criterion is extended to translation
invariant weighted algebras $L_p^w(G)$ with $p>1$. Regular algebras $L_p^w(G)$
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 $L_p^w(G)$ with $p>1$.
