In this course we will survey recent work on two weight norm inequalities for the fractional integral operator
and its commutator with BMO functions. We are interested in finding sufficient (and necessary and sufficient) conditions on pairs of weights (u,σ) for the weak and strong-type inequalities.
Recently, using the machinery developed to prove the A2 conjecture, there has been a great deal of progress in this area. We will first survey the history of this problem, starting with the work of Sawyer on testing conditions for pairs of weights (u,σ). We will then discuss the so-called Ap,q bump conditions. These conditions, which generalize the Muckenhoupt Ap weights, were introduced by Pérez in the 1990's and are closely related to the recently disproved Muckenhoupt-Wheeden conjectures.
Throughout our talks we will discuss the parallels with recent work on singular integrals.
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