## Illinois Journal of Mathematics

### 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.

#### Article information

Source
Illinois J. Math., Volume 57, Number 1 (2013), 295-323.

Dates
First available in Project Euclid: 23 June 2014

https://projecteuclid.org/euclid.ijm/1403534497

Digital Object Identifier
doi:10.1215/ijm/1403534497

Mathematical Reviews number (MathSciNet)
MR3224572

Zentralblatt MATH identifier
1297.42022

#### Citation

Cruz-Uribe, SFO, David; Moen, Kabe. One and two weight norm inequalities for Riesz potentials. Illinois J. Math. 57 (2013), no. 1, 295--323. doi:10.1215/ijm/1403534497. https://projecteuclid.org/euclid.ijm/1403534497

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