Invariant weighted algebras $L_p^w(G)$

Abstract

The paper deals with weighted spaces $L_p^w(G)$ on a locally compact group G. If w is a positive measurable function on G then we define the space $L_p^w(G)$, $p\ge1$, as $L_p^w(G)=\{f:fw\in L_p(G)\}$. We consider weights such that these weighted spaces are algebras with respect to usual convolution. It is shown that for p>1 such weights exists on any sigma-compact group. We prove also a criterion known earlier in special cases: $L_1^w(G)$ is an algebra if and only if w is submultiplicative. It is proved that invariant algebras $L_p^w(G)$, $p>1$, have approximate units of standard form, but this may not be true for a non-invariant algebra.