Dae-Won Chung

Publication Date



We prove that the operator norm on weighted Lebesgue space L^2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2 characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is now a general theorem for a class of Haar shift operators that can be used instead to deduce similar results. We invoke this general theorem to obtain the corresponding result for the Riesz transforms and the Beurling-Ahlfors operator. We can then extrapolate to L^p(w), and the results are sharp for 1 < p < 1. We extend the linear bounds for the dyadic paraproduct on L^2(w) into several variable setting using Bellman function arguments, that is, we prove that the norm of the dyadic paraproduct on the weighted Lebesgue space L^2_{R^n}(w) is bounded with a bound that depends on [w]_{A^d_2} and \\|b\\|_{BMO^d} at most linearly. With this result, we can extrapolate to L^p_{R^n}(w) for 1 < p < \\infty. Furthermore, Bellman function arguments allow us to present the dimensionless linear bound in terms of the anisotropic weight characteristic.

Degree Name


Level of Degree


Department Name

Mathematics & Statistics

First Advisor

Pereyra, Maria Cristina

First Committee Member (Chair)

Pereyra, Maria Cristina

Second Committee Member

Embid, Pedro

Third Committee Member

Blair, Mattew

Fourth Committee Member

Perez, Carlos




Commutators (Operator theory), Lebesgue integral, Shift operators (Operator theory)

Document Type