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Known results Formulation of the result Sharp A2 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 A2 inequality

Hilbert transform

Theorem (S. Petermichl, 2000)

The Hilbert transform: Hf(x) = B.V.

• f(y)

x − y dy is recovered by averaging over certain dyadic shift operators H(f)(x) =

• I∈D

< f, hI > gI(x) dP(D)

Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A2 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 A2 inequality

Formulation of the result

Theorem (A. V., 2009)

Let the kernel K : (−∞, 0) ∪ (0, ∞) → R be odd and lim

x→∞ K(x) = lim x→∞ K ′(x) = 0

(1) and x3 K ′′(x) ∈ L∞(R) (2) Then there exists a coefficient-function γ : (0, ∞) → R, so that γ∞ ≤ Cx3K ′′(x)∞ and K(x − y) =

I∈D

γ(|I|) hI(x)gI(y) dP(D), for x y

Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A2 inequality

A2 inequalities

Definition

ωA2 : = sup 1 |Q|2

• Q

ω(x) dx ·

• Q

ω−1(x)dx

• ver intervals Q

Sharp A2 inequality TfL2(ω) ≤ CωA2fL2(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 A2 inequality

A2 inequalities

Theorem (A. Lerner, S. Ombrosi, C. P´ erez, 2009)

Let T be a Calder´

• n-Zygmund operator then,

TfL2,∞(ω) ≤ CωA2

• 1 + log ωA2
• fL2(ω)

Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts

Known results Formulation of the result Sharp A2 inequality

Sharp A2 inequality

Corollary (M. Lacey, S. Petermichl, M. Reguera, 2009)

Let T(f)(x) = P.V.

• R

K(x − y)f(y)dy be a one dimensional Calder´

• n-Zygmund convolution operator

whose kernel K is odd and satisfies (1) and (2), then TfL2(ω) ≤ CωA2fL2(ω)

Armen Vagharshakyan Georgia, Institute of Technology Recovering Singular Integrals from Haar Shifts