  We consider weighted norm inequalities for the Riesz potentials $I_\alpha$,
also referred to as fractional integral operators. First we prove mixed
$A_p$-$A_\infty$ type estimates in the spirit of [13, 15, 17]. Then we prove
strong and weak type inequalities in the case $p<q$ using the so-called log
bump conditions. These results complement the strong type inequalities of
P\'erez [30] and answer a conjecture from [3]. For both sets of results our
main tool is a corona decomposition adapted to fractional averages.
