Clifford martingale \(\Phi\)-equivalence between \(S(f)\) and \(f^*\) (Q1276105)

One of the results obtained by \textit{G. I. Gaudry}, \textit{R. Long} and \textit{T. Qian} [Ann. Mat. Pura Appl., IV. Ser. 165, 369-394 (1993; Zbl 0814.42009)] entails the \(L^2\)-equivalence of the maximal function \(f^*\) of a Clifford martingale \(f\) and its square function \(S(f)\). In the present paper the \(\Phi\)-equivalence between the square function \(S(f)\) and \(f^*\) is derived for those Clifford martingales \(f\) which are predictably dominated in the sense that \(|\Delta_nf|\leq D_{n-1}\) for all \(n\), where \((D_n)\) is a nonnegative adapted process. Here, \(\Phi\) is a nondecreasing and continuous function from \(R^+\) to \(R^+\) satisfying \(\Phi(0)=0\) and the moderate growth condition \(\Phi(2u)\leq C_1\Phi(u)\), \(u>0\). If \(\Phi\) is e.g. additionally convex, one can get rid of \(D_\infty\) (which occurs in the inequalities obtained so far).