A matrix weighted T1 theorem for matrix kernelled CZOs and a matrix weighted John-Nirenberg theorem

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In this paper, we will prove a matrix weighted T1 theorem regarding the boundedness of certain matrix kernelled CZOs on matrix weighted Lᵖ(W) for matrix Aₚ 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.