Hardy-Littlewood Maximal Function and class (Ap , Ap)

Abstract

In this article, we begin with class (A_p , A_p) and Maximal function. We then establish the behavior of Maximal function on the functions of class (A_p , A_p). Precisely, we show if the pair of weights (u, w) is of class (A_p , A_p) for some 1< p <∞, then the Hardy Littlewood function M may not map L^p(w) to L^p(u). But in contrast if (u, w) ∈ (A_p , A_p), the Maximal function M maps L^p(w) to L^p,∞(u) with norm at most C(n, p)[u, w]^1/p_(Ap,Ap).