On the \(T(1)\) theorem on product domains (Q1326660)

The paper gives an \(\mathbb R^ n \times \mathbb R^ m\)-version of the Coifman-Meyer's theorem: Let \(T\) be a singular integral operator with its kernel \(K(x,y)\) satisfying a \(\delta\)-smoothness condition for \(y\), such that \(T(1) \in \text{BMO} (\mathbb R^ n)\), and \(T\) has the weak boundedness property. Then the operator \(S = \int^ \infty_ 0 q_ t Q_ t TP^ 2_ t {dt \over t}\) is \(L^ 2\)-bounded, where \(q_ t\), \(Q_ t\), and \(P_ t\) are the convolution operators with the nice functions \(\widetilde \psi_ t (x)\), \(\psi_ t (x)\), and \(\varphi_ t (x)\) \((\int_{\mathbb R^ n} \widetilde \psi (x)\, dx = 0 = \int_{\mathbb R^ n} \psi (x)\, dx\), \(\int_{\mathbb R^ n} \varphi (x)\, dx = 1)\), respectively. The result was the main step of the simpler proof of David-Journé's \(T(1)\) theorem. The product-version of this result asserts the \(L^ 2 (\mathbb R^ n) \times L^ 2 (\mathbb R^ m)\)-boundedness of the \(S'\) product-version. As its consequence, the paper gets some kind of \(T(1)\) theorem, which gives sufficient conditions for the boundedness of \(\delta\)-singular integral operator on \(\mathbb R^ n \times \mathbb R^ m\).