Weighted inequalities and dyadic harmonic analysis
The Hilbert transform is a prototypical example of a Calderón-Zygmund singular integral operator (CZ SIO), and it is an element of the convex hull of very simple "dyadic operators". In this talk we are interested on boundedness properties of the commutator of the Hilbert transform, with a BMO function, on weighted Lebesgue spaces. It is well known that these operators are bounded in Lp(w) if the weight w is in the Muckenhoupt Ap class. Only recently, the optimal rate of growth of the operator norm of the Hilbert transform with respect to the Ap-characteristic of the weight was discovered by Stephanie Petermichl. The proof reduces to obtaining the correct rate of growth (linear) in L2(w) for Petermichl's Ш (Sha), a dyadic operator, and then a "sharp extrapolation theorem" provides the correct bound in Lp(w) . The conjecture is that the same rate of growth in Lp(w) is shared by all CZ SIOs, however this is only known so far for dyadic Haar shift operators, and operators on their convex hull such us Hilbert, Riesz, and Beurling transforms. The commutator of the Hilbert transform and a BMO function is not a CZ SIO, and that is reflected in a different rate of growth (quadratic) when considering weighted inequalities for the commutator, as shown by my student Daewon Chung.