The sharp weighted bound for general Calderon-Zygmund operators

Abstract

For a general Calderon-Zygmund operator $T$ on $R^N$, it is shown that $\Vert Tf{\Vert }_{L^2(w)}\le C(T)\Vert w{\Vert }_{A_2}\Vert f{\Vert }_{L^2(w)}$ for all Muckenhoupt weights $w\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Perez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the Nazarov-Treil-Volberg method of random dyadic systems with just one random system and completely without bad parts; (ii) a resulting representation of a general Calderon-Zygmund operator as an average of dyadic shifts; and (iii) improvements of the Lacey-Petermichl-Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

28 pages

References in corpus (3)

• Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures
• On $A_2$ conjecture and corona decomposition of weights
• Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals

Cited by in corpus (8)

• Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture
• Sharp $A_2$ estimates of Haar shifts via Bellman function
• Sharp weighted bounds involving A_\infty
• A sharp estimate of weighted dyadic shifts of complexity 0 and 1
• Random "dyadic" lattice in geometrically doubling metric space and $A_2$ conjecture
• A simple proof of the sharp weighted estimate for Calderon-Zygmund operators on homogeneous spaces
• Some remarks on extrapolation with "flat" weights
• A dyadic analysis approach to the problem of continuity of weighted estimates with respect to the $A_p$ characteristic