In their celebrated paper [3], Burkholder, Gundy, and Silverstein used
Brownian motion to derive a maximal function characterization of H-p
spaces for 0 < p < infinity. In the present paper, are show that the m
ethods in [3] extend to higher dimensions and yield a dimension-free w
eak type (1,1) estimate for a conjugate function on the N-dimensional
torus.