On the search for weighted norm inequalities for the Fourier transform

B. Muckenhoupt posed in [1] the problem of characterizing those non-negative functions u and v, which for some p, 1 </? < oo, the inequality f+X\f(x) \"u(x) dx < Cf+°°\f(x) |M*) dx holds for any/, where/denotes the Fourier transform of/. In this paper we deal only with the case where either u = \ or υ = 1, finding that when v = \, \ <p <2, a necessary condition is that for any r> 0, k =-<x> where b = 2/(2 — p), and that a sufficient condition (υ = 1, 1 <p) is that for any measurable set E, ί ' u(x)dx<C\E\P-\ JE Similar conditions are obtained for the case u=\. Although we will show that the sufficient condition is not necessary (in §4, Corollary 1 and again in §6, Corollary 3 and Remark 4), we were unable to obtain an