Notes on Transference of Continuity from Maximal Fourier Multiplier Operators on R^n to Those on T^n

Given a sequence {φj} 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 φj of the given sequence. On the other hand, the restrictions φj|Λ of φj 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 {φj} to the counterpart maximal operator corresponding to {φj|Λ}. For the study, it is a powerful tool that, if k∈L^{1}(Γ), then the maximal operator corresponding to {k*φj} inherits the strong or weak typeness (p,q) from the one associated with {φj}. 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