On Functions with Fourier Transforms in Aplip (G)

Abstract

Let G be a metrizable locally compact Abelian group with dual group (G) over cap. For 1 <= p < infinity, A(p) (G) denotes the vector space of all complex-valued functions in L-1 (G) whose Fourier transforms <(f)over cap> belong to L-p <((G))over cap>. Research on the spaces A(p) (G) was initiated by Warner in [14] and Larsen, Liu and Wang in [7], Martin and Yap in [8]. Let Lip (alpha; p) and lip (alpha; p) denote the Lipschitz spaces defined on G. In the present paper, the space A(p)(lip) (G) consisting of all complex-valued functions f is an element of lip (alpha; 1) whose Fourier transforms (f) over cap belong to L-p <((G))over cap> is investigated. In the first section invariant properties and asymptotic estimates for the translation and modulation operators are given. Furthermore it is showed that space A(p)(lip) (G) is homogeneous Banach space. At the end of this work, it is proved that the space of all multipliers from L-1 (G) to A(p)(lip) (G) is the space Alip (G).