Two weight inequalities for fractional integral operators and commutators

摘要

1. Introduction In these lecture notes we describe some recent work on two weight norm inequalities for fractional integral operators, also known as Riesz potentials, and for commutators of fractional integrals. Our point of view is strongly influenced by the groundbreaking work on dyadic operators that led to the proof of the A2 conjecture by Hyt?nen [42] and the simplification of that proof by Lerner [58, 59], (See also [43] for a more detailed history and bibliography of this problem.) Fractional integrals are of interest in their own right and have important applications in the study of Sobolev spaces and PDEs. They are positive operators and in many instances proofs are much easier for fractional integrals than they are for Calderón-Zygmund singular integrals. But as we will see, in some cases they are more difficult to work with, and we will give several examples of results which are known to hold for singular integrals but remain conjectures for fractional integrals.