Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

We determine the duals of the homogeneous matrix-weighted Besov spaces $\displaystyle \dot{B}^{\alpha q}_p(W)$ and $\displaystyle \dot{b}^{\alpha q}_p(W)$ which were previously defined in [5]. If $W$ is a matrix $A_p$ weight, then the dual of $\dot{B}^{\alpha q}_p(W)$ can be identified with $\displaystyle \dot{B}^{-\alpha q'}_{p'}(W^{-p'/p})$ and, similarly, $\displaystyle [\dot{b}^{\alpha q}_p(W)]^* \approx \dot{b}^{-\alpha q'}_{p'}(W^{-p'/p})$. Moreover, for certain $W$ which may not be in the $A_p$ class, the duals of $\dot{B}^{\alpha q}_p(W)$ and $\dot{b}^{\alpha q}_p(W)$ are determined and expressed in terms of the Besov spaces $\displaystyle \dot{B}^{-\alpha q'}_{p'}(\{A^{-1}_Q\})$ and $\displaystyle \dot{b}^{-\alpha q'}_{p'}(\{A_Q^{-1}\})$, 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.