Duality of matrix-weighted Besov spaces

We determine the duals of the homogeneous matrix-weighted Besov spaces (B) over dot(p)(alphaq) (W) and (b) over dot (alphaq)(p) (W) which were previously defined in [5]. If W is a matrix A(p) weight, then the dual of (B) over dot (alphaq)(p) (W) can be identified with (B) over dot(p')(-alphaq)' (W-p'/p) and, similarly, [(b) over dot(p)(alphaq) (W))]* approximate to (b) over dot(p')(alphaq') (W-p'/p). Moreover, for certain W which may not be in the A(p) class, the duals of (B) over dot(p)(alphaq) (W) and (b) over dot(p)(alphaq) (W) are determined and expressed in terms of the Besov spaces (B) over dot(p')(-alphaq') ({A(Q)(-1)}) and (b) over dot(p')(-alphaq') ({A(Q)(-1)}), which we define in terms of reducing operators {A(Q)}(Q) associated with W. We also develop the basic theory of these reducing operator Besov spaces. Similar results are shown for inhomogeneous spaces.