IUMJ

Title: Maximal functions and the control of weighted inequalities for the fractional integral operator

Authors: Carlos Perez, Maria J. Carro, Fernando Soria and Javier Soria

Issue: Volume 54 (2005), Issue 3, 627-644

Abstract:

We study weak-type $(1,1)$ weighted inequalities for the fractional integral operator $I_{\alpha}$. We show that the fractional maximal operator $M_{\alpha}$ controls these inequalities when the weight is radially decreasing. However, we exhibit some counterexamples which show that $M_{\alpha}$ is not appropriate for this control on general weights. We do provide, nevertheless, some positive results related to this problem by considering other suitable maximal functions.