Weighted norm inequalities for the Hardy maximal function

The principal problem considered is the determination of all nonnegative functions, U (x), for which there is a constant, C, such thatwhere lU (x) dx S cj |/ (x) |/ (x) dx was the well-known one of Hardy and Littlewood 3 showing that (1. 1) is true if t/ (x) =l and 1 <p<co. Stein in 10 showed that (1. 1) is true for J= (-oo, oo) if l<p<co, U (x) = and -<a<\ -. Fefferman and Stein in 1 showed that (1. 1) is true for7= (-oo, oo) if 1 <p<oo and U* (x) CU (x) for almost every x. Theorems of this sort are important in proving weighted mean convergence results for orthogonal series since the error terms can almost always be majorized by some version of/* (x) ; this was done, for example, in 6, 7 and 8. They can also be used to prove mean summability results ; several examples of this are given in this paper. It also turns out that the results here are needed to determine all the