A Matrix Weighted T1 Theorem for Matrix Kernelled CZOs and a Matrix Weighted John–Nirenberg Theorem

In this paper, we will prove a matrix weighted T1 theorem regarding the boundedness of certain matrix kernelled CZOs on matrix weighted L^p(W) for matrix A_p weights W. Using some of the ideas from the proof, we will also establish a natural matrix weighted John–Nirenberg result that extends to the matrix setting (in the case when one of the weights is the identity) a very recent characterization of both S. Bloom’s BMO space and the two weight boundedness of commutators by I. Holmes, M. Lacey, and B. Wick.