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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 $1<p<\\infty$.\n  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. 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Beznosova","submitted_at":"2016-02-07T02:13:24Z","abstract_excerpt":"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 $1<p<\\infty$.\n  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. 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