Commutators and dyadic paraproducts on weighted Lebesgue spaces

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.

Name thesis:degree_name

Mathematics

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics & Statistics

Year

2010

Author dc:creator

• Chung, Dae-Won

Contributors dc:contributor

• Pereyra, Maria Cristina
• Maria Cristina Pereyra
• Pedro Embid
• Matthew D. Blair
• Carlos Pérez

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

Language dc:language

English

Identifier

https://digitalrepository.unm.edu/math_etds/12

OAI identifier oai:identifier

oai:digitalrepository.unm.edu:math_etds-1011