One and two weight norm inequalities for Riesz potentials

We consider weighted norm inequalities for the Riesz potentials $I_{\alpha }$, also referred to as fractional integral operators. First we prove mixed $A_p$-$A_{\mathrm{\infty }}$ 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.