Thirdly, quasi-normal inequalities of a-variation maximal operator and a-conditional variation maximal operator of scalar predictable tree martingales areidentified by the use of martingale transforms and by the construction of convex or concave function method; on this basis and with the help of previsiblity or regu-larity, Burkholder-Davis-Gundy's inequality of a-variation maximal operator and a-conditional variation maximal operator of scalar predictable tree martingales are iden-tified by the application of Hardy-Lorentz interpolation theory. At the same time, bythe use of G.
再次,应用鞅变换和构造凸或凹函数方法证明了标量值可料树鞅的α-方极大算子和α-条件方极大算子的拟范数不等式;然后,在这些拟范数不等式的基础上,应用Hardy-Lorentz空间插值方法证明了当树鞅是可控或正规树鞅时关于标量值可料树鞅α-方极大算子和α-条件方极大算子的Burkholder-Davis-Gundy`s不等式成立。