Perfect dyadic operators: weighted T(1) theorem and two weight estimates

Perfect dyadic operators were first introduced in \cite{AHMTT}, where a local $T(b)$ theorem was proved for such operators. In \cite{AY} it was shown that for every singular integral operator $T$ with locally bounded kernel on $\mathbb{R}^n \times \mathbb{R}^n$ there exists a perfect dyadic operator $\mathbb{T}$ such that $T -\mathbb{T}$ is bounded on $L^p (dx)$ for all $1<p<\infty$. In this paper we show a decomposition of perfect dyadic operators on real line into four well known operators: two selfadjoint operators, paraproduct and its adjoint. Based on this decomposition we prove a sharp weighted version of the $T(1)$ theorem for such operators, which implies $A_2$ conjecture for such operators with constant which only depends on $\|T(1)\|_{BMO^d}$, $\|T^*(1)\|_{BMO^d}$ and the constant in testing conditions for $T$. Moreover, the constant depends on these parameters at most linearly. In this paper we also obtain sufficient conditions for the two weight boundedness for a perfect dyadic operator and simplify these conditions under additional assumptions that weights are in the Muckenhoupt class $A_\infty^d$.