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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.ϕ∗ 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=1∞g(k)pkp−2 < ∞.
2. Let G be nondiscrete with no compact open subgroup and g admissible. Then
g = f∗m a.e. for some f ∈ Lp(G) if ∫
0∞g(x)pxp−2dx < ∞.
3. Let G be an infinite discrete abelian group which contains Z,Z(r∞) or Z(r)ℵ0
as a subgroup, g admissible. Then g|(0.1) = f∗|(0.1)m a.e. for some f ∈ Lp(G) if
∫
01g(x)pxp−2dx < ∞.
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