In this paper, we prove an abstract homomorphism theorem for bilinear multipliers in the setting of locally compact Abelian (LCA) groups. We also provide some applications. In particular, we obtain a bilinear abstract version of de Leeuw’s theorem for bilinear multipliers of strong and weak types. We also obtain necessary conditions on bilinear multipliers on non-compact LCA groups, yielding boundedness for the corresponding operators on products of rearrangement invariant spaces. Our investigations extend some existing results in ?~n to the framework of general LCA groups, and yield new boundedness results for bilinear multipliers in quasi-Banach spaces.
展开▼
