Let G be a locally compact unimodular group with Haar measure dx and ! be the Beurling’s weight function on G (Reiter, [10]). In this paper we define a space Ap,q ! (G) and prove that Ap,q ! (G) is a translation invariant Banach space. Furthermore we discuss inclusion properties and show that if G is a locally compact abelian group then Ap,q ! (G) admits an approximate identity bounded in L1 ! (G). We also prove that the space Lp
! (G) L1!¯Lq! (G) is isometrically isomorphic to the space Ap,q ! (G) and the space of multipliers from Lp ! (G) to Lq′!−1 (G) is isometrically somorphic to the dual of the spaceAp,q! (G) iff G satisfies a property Pq p . At the end of this work we show that if G is a locally compact abelian group then the space of all multipliers from L1 ! (G) to Ap,q ! (G) is the space Ap,q! (G).