Abstract. Let $G$ be a locally compact abelian group with Haar measure $dx.$ Here we obtain a concrete dual space characterisation for the space $M(S(G),A^p(G)),$ where $S(G)$ is a Segal algebra contained in $A^p(G)$, $1< p< \infty$. Further, we define $\Lambda(A^p)$ sets for each $1< p< \infty $ and show that when $G$ is compact and $F$, a $\Lambda(A^p)$ set, every element of $L^1_F(G)$ is $L^p$-improving.

AMS Subject Classification (1991): 43A22

Received January 25, 1994 and in revised form April 12, 1994. (Registered under 5609/2009.)