Maximal functions and the control of weighted inequalities for the fractional integral operator Maria CarroC. PerezFernando SoriaJavier Soria 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. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2714 10.1512/iumj.2005.54.2714 en Indiana Univ. Math. J. 54 (2005) 627 - 644 state-of-the-art mathematics http://iumj.org/access/