A \(T(1)\) theorem for non-doubling measures with atoms. (Q2766390)

Let \(\mu\) be a measure on \textbf{C}, which may have atoms. Let \(K(x,\, y)\) be a Calderón--Zygmund kernel on \textbf{C} that satisfies the standard dimension \(1\) estimates (a prototypic example is the Cauchy kernel \((x-y)^{-1}\)). It is proved that the operator \(Tf(x)=\int K(x,\, y)f(y)d\mu (y)\) is bounded on \(L^2 (\mu)\) if and only if \(T\) and \(T'\) are bounded on the normalized characteristic functions of squares \textit{and} the folowing two conditions are satisfied: (1) \(\mu (B(x,\, r)\setminus u)\leq Cr\) for all \(x\in {\mathbf C},\; r>0\), where \(u\) is an atom of maximal \(\mu\)-measure included in \(B(x,\, r)\), and (2) \(\int_{y\neq x} | x-y| ^2 d\mu (y) \leq C\mu (x)\) for every \(\mu\)-atom \(x\in {\mathbf C}\).