A WEIGHTED INEQUALITY FOR A DYADIC-LIKE MAXIMAL OPERATOR

Assume that Omega = [0; 1](d) is the unit cube in R-d, the functions f, w is an element of L-1(Omega) are nonnegative, M is the dyadic maximal operator and 0 < p < 1. We prove the inequality parallel to(Mf)(p)w parallel to(1) <= 1/1-p parallel to f parallel to(p)(1)parallel to w parallel to(1) + p(2)/1-pE(M) (f, w), where E-M(f, w) is an appropriate error term. Both constants 1/(1-p) and p(2)/(1-p) are optimal. Actually, the assertion is established in the more general context of probability spaces equipped with a dyadic-like structure.