A matrix weighted \(T1\) theorem for matrix kernelled CZOs and a matrix weighted John-Nirenberg theorem (Q1651393)

The paper contains new results related to the weighted \(L^p\) boundedness of Calderón-Zygmund singular integral operators (CZO) \(T\) with matrix valued kernels acting on \(\mathbb C^n\) valued functions on \(\mathbb R^d\). The corresponding classes of matrix \(A_p\) weights \(W\) and the relevant BMO type spaces \(\mathrm{BMO}_W^p\) are introduced. The main result of the paper is the matrix weighted \(T1\) theorem, according to which \(T\) extends to a bounded operator on \(L^p(W)\) if and only if \(T1 \in\mathrm{BMO}_W^p\) and \(T^*1 \in\mathrm{BMO}_{W^{1-p'}}^{p'}\). Here \(T1\) and \(T^*1\) are defined via the action on \(H^1\) atoms. Another new result is the matrix weighted analogue of the John-Nirenberg theorem associated to \(\mathrm{BMO}_W^p\).