Matrix-weighted Besov spaces

Nazarov, Treil and Volberg defined matrix A p A_p weights and extended the theory of weighted norm inequalities on L p L^p to the case of vector-valued functions. We develop some aspects of Littlewood-Paley function space theory in the matrix weight setting . In particular, we introduce matrix- weighted homogeneous Besov spaces B ˙ p α q ( W ) Ḃα q_p(W) and matrix-weighted sequence Besov spaces b ˙ p α q ( W ) ḃα q_p(W) , as well as b ˙ p α q ( { A Q } ) ḃα q_p(\{A_Q\}) , where the A Q A_Q are reducing operators for W W . Under any of three different conditions on the weight W W , we prove the norm equivalences ‖ <mml:mrow class="MJX-TeXAtom-ORD"