2010/01/05 by Daewon Chung, Chung, Daewon
Mathematics · #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.1001.0755
We prove that the operator norm on weighted Lebesgue space L2(w) of the\ncommutators of the Hilbert, Riesz and Beurling transforms with a BMO function b\ndepends quadratically on the A2-characteristic of the weight, as opposed to the\nlinear dependence known to hold for the operators themselves. It is known that\nthe operator norms of these commutators can be controlled by the norm of the\ncommutator with appropriate Haar shift operators, and we prove the estimate for\nthese commutators. For the shift operator corresponding to the Hilbert\ntransform we use Bellman function methods, however there is now a general\ntheorem for a class of Haar shift operators that can be used instead to deduce\nsimilar results. We invoke this general theorem to obtain the corresponding\nresult for the Riesz transforms and the Beurling-Ahlfors operator. We can then\nextrapolate to Lp(w), and the results are sharp for 1 < p < 1.\n