Vol. 104, No. 1, 1983

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On the search for weighted norm inequalities for the Fourier transform

Nestor Edgardo Aguilera and Eleonor Ofelia Harboure de Aguilera

Vol. 104 (1983), No. 1, 1–14
Abstract

B. Muckenhoupt posed in [1] the problem of characterizing those non-negative functions u and v, which for some p, 1 p < , the inequality

∫                    ∫
+∞  ˆ   p            +∞     p
−∞  |f(x)|u (x)dx ≤ C  −∞  |f(x)|v(x)dx

holds for any f, where f denotes the Fourier transform of f. In this paper we deal only with the case where either u 1 or v 1, finding that when v 1, 1 < p < 2, a necessary condition is that for any r > 0,

⌊                     ⌋
+∑∞  (∫ r(k+1)      )b  1∕b
⌈             u(x)dx  ⌉   ≤  Crp−1
k=−∞   rk

where b = 2(2 p), and that a sufficient condition (v 1,1 p) is that for any measurable set E,

∫
u(x)dx ≤ C |E |p−1.
E

Similar conditions are obtained for the case u 1. 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 any conclusions on our necessary condition.

Mathematical Subject Classification 2000
Primary: 42A38
Milestones
Received: 24 September 1980
Published: 1 January 1983
Authors
Nestor Edgardo Aguilera
Eleonor Ofelia Harboure de Aguilera