On an estimate of Calder\'on-Zygmund operators by dyadic positive operators

TL;DR

This paper establishes a bound for Calderón-Zygmund operators using dyadic positive operators, leading to significant advances in weighted inequalities and resolving longstanding conjectures in harmonic analysis.

Contribution

The paper introduces a new estimate linking Calderón-Zygmund operators to dyadic positive operators, enabling simplified proofs and extensions of key inequalities.

Findings

Proves the two-weight conjecture in full generality.

Simplifies the proof of the A2 conjecture.

Extends sharp A1 estimates to the maximal Calderón-Zygmund operator.

Abstract

Given a general dyadic grid ${\mathscr{D}}$ and a sparse family of cubes ${\mathcal S}=\{Q_j^k\}\in {\mathscr{D}}$, define a dyadic positive operator ${\mathcal A}_{{\mathscr{D}},{\mathcal S}}$ by $${\mathcal A}_{{\mathscr{D}},{\mathcal S}}f(x)=\sum_{j,k}f_{Q_j^k}\chi_{Q_j^k}(x).$$ Given a Banach function space $X({\mathbb R}^n)$ and the maximal Calder\'on-Zygmund operator $T_{\natural}$, we show that $$\|T_{\natural}f\|_X\le c(n,T)\sup_{{\mathscr{D}},{\mathcal S}}\|{\mathcal A}_{{\mathscr{D}},{\mathcal S}}f\|_{X}.$$ This result is applied to weighted inequalities. In particular, it implies: (i) the "two-weight conjecture" by D. Cruz-Uribe and C. P\'erez in full generality; (ii) a simplification of the proof of the "$A_2$ conjecture"; (iii) an extension of certain mixed $A_p$-$A_r$ estimates to general Calder\'on-Zygmund operators; (iv) an extension of sharp $A_1$ estimates (known for $T$) to the maximal Calder\'on-Zygmund operator $T_{\natural}$.