Gabor multipliers for weighted Banach spaces on locally compact abelian groups

Abstract

We use a projective groups representation ρ of the unimodular group G× ˆ G on L 2 ( G ) to define Gabor wavelet transform of a function f with respect to a window function g , where G is a locally compact abelian group and ˆ G its dual group. Using these transforms, we define a weighted Banach H 1 , ρ w ( G ) and its antidual space H 1 ∼ , ρ w ( G ) , w being a moderate weight function on G × ˆ G . These spaces reduce to the well known Feichtinger algebra S 0 ( G ) and Banach space of Feichtinger distribution S (cid:2) 0 ( G ) respectively for w ≡ 1. We obtain an atomic decomposition of H 1 , ρ w ( G ) and study some properties of Gabor multipliers on the spaces L 2 ( G ) , H 1 , ρ w ( G ) and H 1 ∼ , ρ w ( G ). Finally, we prove a theorem on the compactness of Gabor multiplier operators on L 2 ( G ) and H 1 , ρ w ( G ), which reduces to an earlier result of Feichtinger [Fei 02, Theorem 5.15 (iv)] for w = 1 and G = R d .