Matrix-weighted Besov spaces

Nazarov, Treil and Volberg defined matrix A(p) weights and extended the theory of weighted norm inequalities on 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) over dot(p)(alphap)(W) and matrix-weighted sequence Besov spaces (b) over dot(p)(alphaq)(W), as well as (b) over dot(p)(alphaq) ({A(Q)}), where the A(Q) are reducing operators for W. Under any of three different conditions on the weight W, we prove the norm equivalences parallel to(f) over right arrow parallel to(B) over dot(p)(alphaq) (W) approximate to parallel to {(s) over right arrow (Q)}Qparallel to(b) over dot (alphaq)(p)(W) approximate to parallel to{(s) over right arrow (Q)}Qparallel to(b) over dot(p)(alphaq)({A(Q)}), where {(s) over right arrow (Q)} Q is the vector-valued sequence of phi- transform coefficients of (f) over right arrow. In the process, we note and use an alternate, more explicit characterization of the matrix A(p) class. Furthermore, we introduce weighted version of almost diagonality and prove that an almost diagonal matrix is bounded on (b) over dot(p)(alphaq)(W) if W is doubling. We also obtain the boundedness of almost diagonal operators on (B) over dot(p)(alphaq) (W) under any of the three conditions on W. This leads to the boundedness of convolution and non-convolution type Calderon-Zygmund operators( CZOs) on (B) over dot(p)(alphaq) (W), in particular, the Hilbert transform. We apply these results to wavelets to show that the above norm equivalence holds if the phi-transform coefficients are replaced by the wavelet coefficients. Finally, we construct inhomogeneous matrix-weighted Besov spaces B-p(alphaq) (W) and show that results corresponding to those above are true also for the inhomogeneous case.