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(infinity) 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 Perez (Indiana Univ. Math. J. 43 (1994) 663-683) and answer a conjecture from (Weights, extrapolation and the theory of Rubio de Francia (2011) Birkhauser). For both sets of results, our main tool is a corona decomposition adapted to fractional averages.