On $A_2$ conjecture and corona decomposition of weights

TL;DR

This paper establishes a sharp estimate for Calderón-Zygmund operators in weighted L^2 spaces with A_2 weights, introducing a bound involving the A_2 norm and logarithmic factors, using non-homogeneous harmonic analysis techniques.

Contribution

It provides a new bound for Calderón-Zygmund operators in weighted spaces, connecting the operator norm to the A_2 weight norm with a potential removal of the logarithmic factor.

Findings

Bound: T_{L^2(w) ightarrow L^2(w)} \u2264 C[w]_{A_2}\log(1+[w]_{A_2})

Operator norm controlled by A_2 norm and weak norms of T and T*

Approach based on 2-weight estimates and non-homogeneous harmonic analysis

Abstract

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.