Two weighted inequalities for maximal functions related to Ces\`aro convergence

J. Aust. Math. Soc. 74 (2003), 111-120

Two weighted inequalities for maximal functions related to Cesàro convergence

A. L. Bernardis
IMAL
Güemes 3450
(3000) Santa Fe
Argentina
bernard@ceride.gov.ar

and

F. J. Martín-Reyes
Departamento de Análisis Matemático
Facultad de Ciencias
Universidad de Málaga
29071 Málaga
Spain
martin@anamat.cie.uma.es

Abstract

We characterize the pairs of weights

for which the maximal operator

$M_{\alpha}^-f(x) = \sup_{R>0} R^{-1-\alpha} \int_{x-2R}^{x-R} |f(s)| (x-R - s)^{\alpha} \, ds, \quad -1<\alpha < 0,$

is of weak and restricted weak type

with respect to $u(x)\,dx$ and $v(x)\,dx$ . As a consequence we obtain analogous results for

$M_{\alpha}f(x) = \sup_{R>0} R^{-1-\alpha} \int_{R<|x-y|<2R} |f(y)| (|x-y|-R)^{\alpha} \, dy.$

We apply the results to the study of the Cesàro- $\alpha$ convergence of singular integrals.