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 l<p<oo, J is a fixed interval, C is independent of f, and/* is the Hardy maximal function,The main result is that U(x) is such a function if and only ifwhere I is any subinterval of J, \I\ denotes the length of / and AT is a constant independent of /.Various related problems are also considered.These include weak type results, the problem when there are different weight functions on the two sides of the inequality, the case whenp = l orp = oo, a weighted definition of the maximal function, and the result in higher dimensions.Applications of the results to mean summability of Fourier and Gegenbauer series are also given.1. Introduction.The original inequality of the type (1.1) $ [f*(x)]>U(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)=\x\a and -\lp<a<\ -\jp.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