A Fefferman-Stein inequality for the martingale square and maximal functions

Suppose that f is a martingale and let |f|∗, S(f) denote the associated maximal and square functions. We prove that for any weight w we have |||f|∗||L≤C||S(f)||Lwith C=16(2+1)=38.62742…. The proof rests on the construction of an appropriate special function, enjoying certain size and concavity conditions. Furthermore, we show that the term w∗ on the right cannot be replaced by the r-maximal function of w for any 0