Notes on Transference of Continuity from Maximam Fourier Multiplier Operators on Rn to Those on Tn.

Abstract

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 ∈L1(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.