MAXIMAL ESTIMATES ON GROUPS, SUBGROUPS, AND THE BOHR COMPACTIFICATION

Let G be a locally compact abelian group with character group Gamma. W e study the interplay of boundedness properties for suitably related m aximal operators defined by (weak type or strong type) multipliers for G, its subgroups, and its Bohr compactification b(G). These considera tions lead to weak type and strong type maximal estimates which genera lize fundamental theorems of de Leeuw and Saeki concerning strong type norms of single multipliers. Suppose that 1 less than or equal to p < infinity, and M is the maximal operator on L(P)(G) defined by a seque nce {psi(n)}(infinity)(n=1) of strong type Fourier multipliers which a re continuous functions on Gamma. Our main result establishes that M i s of weak type (p,p) on L(P)(G) if and only if the corresponding maxim al operator M(#) on L(P)(b(G)) is of weak type (p,p). This provides a counterpart for locally compact abelian groups of E. M. Stein's Contin uity Principle for compact groups, since the latter characterizes the weak (p,p) boundedness of M(#) when 1 less than or equal to p less tha n or equal to 2. (C) 1995 Academic Press, Inc.