TheTb-theorem on non-homogeneous spaces

0 Introduction: main objects and results 3 0.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.2 An application of T1-heorem: electric intensity capacity . . . . . . . . . . . . 7 0.3 How to interpret Calderon–Zygmund operator T? . . . . . . . . . . . . . . . 9 0.3.1 Bilinear form is defined on Lipschitz functions . . . . . . . . . . . . . 10 0.3.2 Bilinear form is defined for smooth functions . . . . . . . . . . . . . . 11 0.3.3 Apriori boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.4 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

[1]  E. Sawyer.  A characterization of a two-weight norm inequality for maximal operators , 1982 .

[2]  M. Christ.  Lectures on singular integral operators , 1991 .

[3]  Nguyen Xuan Uy.  Removable sets of analytic functions satisfying a Lipschitz condition , 1979 .

[4]  J. Verdera.  On theT(1)-theorem for the Cauchy integral , 2000 .

[5]  Joan Verdera,et al.  The Cauchy integral, analytic capacity, and uniform rectifiability , 1996 .

[6]  Guy David.  Singular integrals on curves and surfaces , 1991 .

[7]  G. David.  Wavelets and Singular Integrals on Curves and Surfaces , 1991 .

[8]  X. Tolsa.  $L^2$-boundedness of the Cauchy integral operator for continuous measures , 1999 .

[9]  Sergei Treil,et al.  Wavelets and the Angle between Past and Future , 1997 .

[10]  S. Treil,et al.  Accretive system Tb-theorems on nonhomogeneous spaces , 2002 .

[11]  G. David.  Analytic capacity, Calderón-Zygmund operators, and rectifiability , 1999 .

[12]  Littlewood–Paley Theory and the T(1) Theorem with Non-doubling Measures , 2000, math/0006039.

[13]  B. Øksendal,et al.  Analytic capacity and differentiability properties of finely harmonic functions , 1982 .

[14]  G. David,et al.  Removable sets for Lipschitz harmonic functions in the plane , 2000 .

[15]  G. David.  Unrectictifiable 1-sets have vanishing analytic capacity , 1998 .

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

[17]  Ronald R. Coifman,et al.  Analyse harmonique non-commutative sur certains espaces homogènes : étude de certaines intégrales singulières , 1971 .

[18]  G. David,et al.  Opérateurs intégraux singuliers sur certaines courbes du plan complexe , 1984 .

[19]  G. David,et al.  Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation , 1985 .

[20]  A Real Variable Method for the Cauchy Transform, and Analytic Capacity , 1988 .

[21]  P. V. Paramonov,et al.  On geometric properties of harmonic ${\rm Lip}_1$-capacity. , 1995 .

[22]  X. Tolsa.  COTLAR'S INEQUALITY WITHOUT THE DOUBLING CONDITION AND EXISTENCE OF PRINCIPAL VALUES FOR THE CAUCHY INTEGRAL OF MEASURES , 1998 .

[23]  S. Treil,et al.  Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces , 1997, math/9711210.

[24]  Peter W. Jones,et al.  BMO from dyadic BMO , 1982 .

[25]  R. Coifman,et al.  Two elementary proofs of the ² boundedness of Cauchy integrals on Lipschitz curves , 1989 .

[26]  G. David,et al.  A boundedness criterion for generalized Caldéron-Zygmund operators , 1984 .