Recovering Singular Integrals from Haar Shifts Armen Vagharshakyan - PowerPoint PPT Presentation

Known results Formulation of the result Sharp A 2 inequality Recovering Singular Integrals from Haar Shifts Armen Vagharshakyan Georgia, Institute of Technology November 19, 2009 Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A 2 inequality Hilbert transform Theorem (S. Petermichl, 2000) The Hilbert transform: � f ( y ) Hf ( x ) = B . V . x − y dy is recovered by averaging over certain dyadic shift operators � � H ( f )( x ) = < f , h I > g I ( x ) d P ( D ) I ∈ D Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A 2 inequality Beurling, Riesz transforms Similar representations were obtained for: Riesz transform, S. Petermichl, S. Treil, A. Volberg (2000). Beurling transform, O. Dragiˇ cevi´ c, A. Volberg (2003). Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A 2 inequality Formulation of the result Theorem (A. V., 2009) Let the kernel K : ( −∞ , 0 ) ∪ ( 0 , ∞ ) → R be odd and x →∞ K ′ ( x ) = 0 x →∞ K ( x ) = lim lim ( 1 ) and x 3 K ′′ ( x ) ∈ L ∞ ( R ) ( 2 ) Then there exists a coefficient-function γ : ( 0 , ∞ ) → R , so that � γ � ∞ ≤ C � x 3 K ′′ ( x ) � ∞ and � � K ( x − y ) = γ ( | I | ) h I ( x ) g I ( y ) d P ( D ) , for x � y I ∈ D Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A 2 inequality A 2 inequalities Definition � � � ω � A 2 : = sup 1 ω − 1 ( x ) dx ω ( x ) dx · over intervals Q | Q | 2 Q Q Sharp A 2 inequality � Tf � L 2 ( ω ) ≤ C � ω � A 2 � f � L 2 ( ω ) For the case of: Beurling transform, S. Petermichl, A. Volberg (2002). Hilbert transform, S. Petermichl (2007). Riesz transform, S. Petermichl (2008). Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A 2 inequality A 2 inequalities Theorem (A. Lerner, S. Ombrosi, C. P´ erez, 2009) Let T be a Calder´ on-Zygmund operator then, � � � Tf � L 2 , ∞ ( ω ) ≤ C � ω � A 2 1 + log � ω � A 2 � f � L 2 ( ω ) Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A 2 inequality Sharp A 2 inequality Corollary (M. Lacey, S. Petermichl, M. Reguera, 2009) Let � T ( f )( x ) = P . V . K ( x − y ) f ( y ) dy R be a one dimensional Calder´ on-Zygmund convolution operator whose kernel K is odd and satisfies (1) and (2), then � Tf � L 2 ( ω ) ≤ C � ω � A 2 � f � L 2 ( ω ) Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts