On Some Properties of the Spaces Awp(X)(R n)

Abstract

For 1 ≤ p < ∞, A<inf>p</inf> (Rn) denotes the space of all complex-valued functions in L1 (Rn) whose Fourier transforms f̌ belong to Lp(Rn). A number of authors such as Larsen, Liu and Wang [12], Martin and Yap [14], Lai [11] worked on this space. Some generalizations to the weighted case was given by Gurkanli [7], Feichtinger and Gurkanli [4], Fischer, Gurkanli and Liu [5]. In the present paper we give another generalization of A<inf>p</inf> (Rn) to the generalized Lebesgue space Lp(x)(Rn). We define A p(x)<inf>w</inf> (Rn) to be the space of all complex-valued functions in L1<inf>w</inf> (Rn) whose Fourier transforms f̌ belong to the generalized Lebesgue space L p(x)(Rn). We endow it with a sum norm and show that A p(x)<inf>w</inf> (Rn) is an S<inf>w</inf>(Rn) space [2]. Further we discuss the multipliers of Ap(x)<inf>w</inf> (Rn).