Martingale inequalities in rearrangement invariant function spaces (Q1120190)

It is well-known that the \(L_{\Phi}\)-norms of the square function and the maximal function of a martingale on (0,1) are equivalent when \(\Phi\) is a convex symmetric Orlicz function with a growth condition. This equivalence is extended here to the case of a rearrangement invariant normed function space X on (0,1); it is shown to be true if and only if the upper Boyd index of X is finite. The setting comes from the book by \textit{J. Lindenstrauss} and \textit{L. Tzafriri}, Classical Banach spaces II: Function spaces. (1979; Zbl 0403.46022), and the proofs are adapted from the usual arguments for martingale inequalities.