Maximal Estimates on Groups, Subgroups, and the Bohr Compactification

Let G be a locally compact abelian group with character group Γ. We study the interplay of boundedness properties for suitably related maximal operators defined by (weak type or strong type) multipliers for G, its subgroups, and its Bohr compactification b(G). These considerations lead to weak type and strong type maximal estimates which generalize fundamental theorems of de Leeuw and Saeki concerning strong type norms of single multipliers. Suppose that 1 ≤ p < ∞, and M is the maximal operator on Lp(G) defined by a sequence {ψn}∞n = 1 of strong type Fourier multipliers which are continuous functions on γ. Our main result establishes that M is of weak type (p, p) on Lp(G) if and only if the corresponding maximal operator M# on Lp(b(G)) is of weak type (p, p). This provides a counterpart for locally compact abelian groups of E. M. Stein′s Continuity Principle for compact groups, since the latter characterizes the weak (p, p) boundedness of M# when 1 ≤ p ≤ 2.