Two weighted inequalities for maximal functions related to Cesàro convergence

Abstract

We characterize the pairs of weights (u, v) for which the maximal operator Mα- f (x) = supR>0 R-1-α ∫x-Rx-2R |f (s)|(x - R - s)α ds, -1 <α < 0, is of weak and restricted weak type (p, p) with respect to u(x) dx and v(x) dx. As a consequence we obtain analogous results for Mαf (x) = supR>0 R-1-α ∫R<|x-y|<2R |f(y)|(|x - y| -R)α dy. We apply the results to the study of the Cesàro-α convergence of singular integrals.