The sharp weighted bound for general Calderon-Zygmund operators

for all Muckenhoupt weights w ∈ A2. This optimal estimate was known as the A2 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 Calderón–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.

[1]  Two weight inequalities for individual Haar multipliers and other well localized operators , 2007, math/0702758.

[2]  S. Treil,et al.  On $A_2$ conjecture and corona decomposition of weights , 2010, 1006.2630.

[3]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[4]  Yang Xiang.  Fast Algorithms for Calderón–Zygmund Singular Integral Operators , 1996 .

[5]  M. Lacey,et al.  Sharp A2 inequality for Haar shift operators , 2009, 0906.1941.

[6]  A. Lerner.  A pointwise estimate for the local sharp maximal function with applications to singular integrals , 2010 .

[7]  L. Grafakos,et al.  Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces , 2005 .

[8]  J. Wilson,et al.  The intrinsic square function , 2007 .

[9]  S. Treil,et al.  Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures , 2010, 1003.1596.

[10]  S. Petermichl.  The sharp weighted bound for the Riesz transforms , 2007 .

[11]  S. Buckley.  Estimates for operator norms on weighted spaces and reverse Jensen inequalities , 1993 .

[12]  A. Volberg,et al.  Sharp estimate of the Ahlfors-Beurling operator via averaging martingale transforms , 2003 .

[13]  A. Volberg,et al.  The proof of $A_2$ conjecture in a geometrically doubling metric space , 2011, 1106.1342.

[14]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[15]  A. Lerner.  Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals , 2010, 1005.1422.

[16]  S. Petermichl.  The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic , 2007 .

[17]  M. Lacey,et al.  A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure , 2008, 0807.0246.

[18]  Michael Christ,et al.  A T(b) theorem with remarks on analytic capacity and the Cauchy integral , 1990 .

[19]  Carlos Perez,et al.  Sharp weighted estimates for approximating dyadic operators , 2010, 1001.4724.

[20]  A. Volberg,et al.  Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular , 2002 .

[21]  E. Saksman,et al.  Beltrami operators in the plane , 2001 .

[22]  Fedor Nazarov,et al.  TheTb-theorem on non-homogeneous spaces , 2003 .

[23]  W. Schachermayer,et al.  Singular integral operators: a martingale approach , 1991 .

[24]  Vector-valued non-homogeneous Tb theorem on metric measure spaces , 2010, 1004.3176.

[25]  A. Volberg,et al.  A simple sharp weighted estimate of the dyadic shifts on metric spaces with geometric doubling , 2011, 1104.4893.

[26]  A. Vagharshakyan.  RECOVERING SINGULAR INTEGRALS FROM HAAR SHIFTS , 2009, 0911.4968.

[27]  T. Hytönen,et al.  Non-homogeneous Tb Theorem and Random Dyadic Cubes on Metric Measure Spaces , 2009, 0911.4387.