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

Abstract

In a celebrated paper, Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of Hp spaces for 0 < p < ∞. In this paper, we show that their method extends to higher dimensions and yields a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.