Weighted square function inequalities

For an integrable function f on [0, 1)d, let S(f) and M f denote the corresponding dyadic square function and the dyadic maximal function of f, respectively. The paper contains the proofs of the following statements. (i) If w is a dyadic A1 weight on [0, 1)d, then ||S(f)||L1(w) ≤√ 5[w] 1/2 A1 ||M f||L1(w). The exponent 1/2 is shown to be the best possible. (ii) For any p > 1, there are no constants cp, αp  epending only on p such that for all dyadic Ap weights w on [0, 1)d, ||S(f)||L1(w) ≤ cp[w] αp Ap ||M f||L1(w).