One and two weight norm inequalities for Riesz potentials

Abstract

We consider weighted norm inequalities for the Riesz potentials $I_{\alpha}$, also referred to as fractional integral operators. First, we prove mixed $A_{p}\mbox{-}A_{\infty}$ type estimates in the spirit of (Indiana Univ. Math. J. 61 (2012) 2041–2052, Anal. PDE 6 (2013) 777–818, Houston J. Math. 38 (2012) 799–814). 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 Pérez (Indiana Univ. Math. J. 43 (1994) 663–683) and answer a conjecture from (Weights, extrapolation and the theory of Rubio de Francia (2011) Birkhäuser). For both sets of results, our main tool is a corona decomposition adapted to fractional averages.