Given a sequence { φ
j } of bounded functions on the dual group
Γ of a locally compact abelian group
G, we have a family of Fourier multiplier operators each element of which is made from a component φ
j of the given sequence. On the other hand, the restrictions φ
j |
Λ of φ
j to a subgroup
Λ of
Γ build Fourier multiplier operators on
G ⁄
Λ⊥. We are interested in the transference of continuity from the maximal operator constructed by the family of Fourier multiplier operators composed of { φ
j } to the counterpart maximal operator corresponding to { φ
j |
Λ }. For the study, it is a powerful tool that, if
k ∈
L1(
Γ), then the maximal operator corresponding to {
k *φ
j } inherits the strong or weak typeness (
p,
q ) from the one associated with { φ
j }. First we give a method of showing it. Our result contains the case
p =
q =1 and our proof is simpler and more straightforward than the one in [2]. Next we consider the case of
G =
R n and
Λ =
Z n, and develop arguments over Lorentz spaces and Hardy spaces.
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