The Fourier multiplier problem for spaces of continuous functions with P-summable transforms

Abstract

In this paper we consider spaces Ap, p ∈ [1, 2], and multipliers (Ap, Aq), p ∈ [1, 2], q ∈ [1, 2]. In 4.4 and 6.1 we identify (Ap, Aq) for p ∈ [1, 2], q ∈ [p, 2], and in 7.3 we identify (A2, A1). In 7.1 we give a sufficient condition, and in 7.5 a necessary condition, for membership of (Ap, Aq), p ∈ [1, 2], q ∈ [1, 2]. We give, in 7.2, a necessary condition for membership of (A2, Aq), q ∈[l,2). We include constructive proofs of some strict inclusion results for Ap, p ∈ [1,2], (3.1 and 3.2), and also, in 5.3, for (AP,AP), p∈[l,2].