Title:
Weighted inequalities and dyadic harmonic analysis Abstract: The Hilbert
transform is a prototypical example of a CalderónZygmund 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
L^{p}(w) if the weight w is in the Muckenhoupt A_{p} class. Only recently, the optimal rate of
growth of the operator norm of the Hilbert transform with respect to the A_{p}characteristic of the weight was
discovered by Stephanie
Petermichl. The proof reduces to obtaining the correct
rate of growth (linear) in L^{2}(w) for
Petermichl's Ш (Sha), a dyadic operator, and then a
"sharp extrapolation theorem" provides the correct bound in L^{p}(w) .
The conjecture is that the same rate of growth in L^{p}(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.
