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Throughout this paper, N denotes a fixed but arbitrary positive integer, T denotes the circle group, and T denotes the product of N copies of T. The normalized Lebesgue measure on T will be symbolized by P . For a measurable function f , we let ‖f‖1 = supy>0 yλf(y) where λf(y) = P ( {x ∈ T : |f(x)| > y} ) . The integers will be denoted by Z and the complex numbers by C. Let Fn = σ(e iθ1 , e2, . . . , en) denote the σ-algebra on T generated by the first n coordinate functions. For f ∈ L1(T), the conditional expectation of f with respect to Fn will be denoted E(f |Fn). Let d0(f) = E(f |F0) = ∫

On a weak type (1,1) inequality for a maximal conjugate function