On an estimate of Calderón-Zygmund operators by dyadic positive operators (1202.1860v3)

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 $T_{\natural}$.