On $A_2$ conjecture and corona decomposition of weights

By C. P'erez, Sergei Treil, Alexander Volberg. 2010

49 citations recorded by the source corpus.

We consider here a problem of finding the sharp estimate for the boundedness of an arbitrary Calder\'on-Zygmund operator in $L^2(w)$, $w\in A_2$. We first prove that for $A_2$ weight $w$ one has that the norm a Calderon--Zygmund operator $T$ in $L^2(w)$ is bounded by the sum of its weak norm, the weak norm of its adjoint, and the $A_2$ norm of the weight. From this result we derive that $\|T\|_{L^2(w)\rightarrow L^2(w)} \le C\,[w]_{A_2}\log (1+[w]_{A_2})$. We believe that the logarithmic factor is superflous. The approach is based on $2$-weight estimates technique and, hence, on non-homogeneous harmonic analysis.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0c9d224850536d5e197016a9b23ea7fd9ff2d346

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

The A_2 theorem: Remarks and complements (2012, methodology): “My original proof of the A2 theorem [6] proceeded via the “top right” route corona → testing Sk → testing CZO → A2 for CZO, where the last step was borrowed from Pérez, Treil and Volberg [24].”
One and two weight norm inequalities for Riesz potentials (2012, background): “For the complete history of the problem we also refer the reader to [2, 18, 22, 24, 32] and the references they contain.”
Representation of singular integrals by dyadic operators, and the A_2 theorem (2011, methodology): “In the original proof of this theorem, in Summer 2010, the two reductions were done in a different order: the (quite complicated) reduction to testing condition was obtained for general Calderón–Zygmund operators by Pérez–Treil–Volberg [18]; my completion of the proof [6] then consisted of reducing ”
Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II (2012, result): “Partial information about the two-weight problem for singular integrals in [21] was basic to the resolution of the A2 conjecture in [3], and several related results (see [4], [6], [21], [22]).”
SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES (2010, unknown): “Hyt¨onen’s proof is based on approximating T by generalized dyadic Haar shift operators with good bounds combined with the key fact that to prove (1.2) it is enough to prove the corresponding weak type (2 , 2) estimate with the same linear bound as proved in [29].”