Tuomas Hytönen: Recent advances in weighted norm inequalities

It is a classical theme to study the action of basic operators of real and harmonic analysis (like maximal functions and singular integrals) on L^p spaces equipped with a weighted measure. While many of the basic questions in this area were solved already in the 1970s by Muckenhoupt and others, there has also been substantial recent activity. I will report on two different but connected themes:

1. Sharp quantitative estimates. Over the first decade of the century, the sharp estimates for weighted operator norms were found for several particular operator and conjectured for general singular integrals. I confirmed this so-called A_2 conjecture in 2010. As it turned out, this was not the end of the story but rather a boost for several further developments.

2. Two-weight inequalities. This refers to a class of estimates, where the "input" and the "output" of an operator are weighted in unrelated ways, which is not unreasonable in applications. The problem of characterizing (in real-variable terms) the admissible pairs of weights for a two-weight inequality for the Hilbert transform was already raised by Muckenhoupt in the 1970s, but it was only recently solved by Lacey, Sawyer, Shen and Uriarte-Tuero in 2012-13. I extended their solution to general measures in place of weights, dealing in particular with the possibility of point masses.