Nakhlé Asmar and Stephen Montgomery-Smith, On a weak type (1,1) inequality for a maximal conjugate function.
Studia Math. 125, (1997), 13-21.
In a celebrated paper, Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of \(H^p\) spaces for \(0 < p < \infty\). 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.
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