Maximal Inequalities of Weak Type

for f(x) e LV(X). Applications are given, and a generalization to where the domain of {T.} is an F-space. Inequality (1) occurs in many different areas in analysis. Indeed, the proof of (1) is often a key step in a convergence proof; the two conditions on {TJ} are in many cases equivalent (see Corollary 1.2). In particular, one can often prove the almost everywhere divergence of some limit (2) by calculating (1) to be false. By a recent theorem of E. M. Stein [19], (1) follows from the almost everywhere convergence of (2) in many situations from Fourier analysis. For example, let {TJ} be a sequence of translation-invariant linear operators on LP(G), where G is a compact group or its homogeneous space and 1 ? p 0 and f(x) e LD(G). Another example of a theorem of this type is contained in [4]. As an example, let