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

Abstract

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