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As we know, this inequality is very important in Hp-theory, it is also a main difficulty in generalizing Hp-theory of one parameter to Hp -theory of several parameters, see [2, 6, 7, 8]. The first proof of (1) is probabilistic which was given by Burkholder, Gundy and Silverstein, see [1]; Fefferman and Stein first got an analytic proof of (1) by dealing with a kind of Green’s formula on R = ∪x∈EΓα(x), see [4]; a sharpened inequality was obtained in [5] by a different approach. In the two-parameter case, Gundy and Stein set up a similar inequality to (1) by dealing with some multi-sub-linear operators like

A simple proof of an inequality dominating the area integral by the nontangential maximal function