Vol. 57, No. 2, 1975

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ISSN: 0030-8730
Rearranging Fourier transforms on groups

Chung Lin

Vol. 57 (1975), No. 2, 463–473
Abstract

Let G denote an infinite locally compact abelian group and X its character group. Let 𝜃 be a suitable Haar measure on X, and 1 < p < 2. For a 𝜃-measurable function ϕ on X, we define 𝜃ϕ(t) = ({χ X : |ϕ(χ)| > t}) and ϕ(x) = inf{t > 0 : 𝜃ϕ(t) x} for x > 0 is called the nonincreasing rearrangement of ϕ. Note that even though ϕ is defined on X, the domain of ϕ is (0,). A nonnegative function g defined on (0,) is called admissible if g is nonincreasing and limx→∞g(x) = 0.

Theorems:

1. Let G be nondiscrete with a compact open subgroup and g admissible. Then g|N = f|N, where N is the set of positive integers, for some f Lp(G) if k=1g(k)pkp2 < .

2. Let G be nondiscrete with no compact open subgroup and g admissible. Then g = fm a.e. for some f Lp(G) if 0g(x)pxp2dx < .

3. Let G be an infinite discrete abelian group which contains Z,Z(r) or Z(r)0 as a subgroup, g admissible. Then g|(0.1) = f|(0.1)m a.e. for some f Lp(G) if 01g(x)pxp2dx < .

Mathematical Subject Classification 2000
Primary: 43A25
Milestones
Received: 20 February 1975
Published: 1 April 1975
Authors
Chung Lin