On Functions With Fourier Transforms in Alipp(G)

Abstract

Let G be a metrizable locally compact Abelian group with dual group Ĝ. For 1 ≤ p < ∞, A<inf>p</inf>(G) denotes the vector space of all complex-valued functions in L1(G) whose Fourier transforms f belong to Lp (Ĝ). Research on the spaces A<inf>p</inf>(G) was initiated by Warner in [14] and Larsen, Liu and Wang in [7], Martin and Yap in [8]. Let Lip(α,p) and lip(α,p) denote the Lipschitz spaces defined on G. In the present paper, the space Alip<inf>p</inf>(G) consisting of all complex-valued functions f ∈ lip(α, 1) whose Fourier transforms fbelong to Lp(Ĝ) is investigated. In the first section invariant properties and asymptotic estimates for the translation and modulation operators axe given. Furthermore it is showed that space Alip<inf>p</inf>(G) is homogeneous Banach space. At the end of this work, it is proved that the space of all multipliers from L1 (G) to Alip<inf>p</inf>(G) is the space Alip<inf>p</inf>(G). © 2008 Warsaw University.