Duality of matrix-weighted Besov spaces

Svetlana Roudenko¹

¹ Mathematics Department Duke University Durham, NC 27708, U.S.A.

Studia Mathematica, Tome 160 (2004) no. 2, pp. 129-156

Résumé

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.

Détail

DOI : 10.4064/sm160-2-3

Keywords: determine duals homogeneous matrix weighted besov spaces displaystyle dot alpha displaystyle dot alpha which previously defined matrix weight dual dot alpha identified displaystyle dot alpha p similarly displaystyle dot alpha * approx dot alpha p moreover certain which may class duals dot alpha dot alpha determined expressed terms besov spaces displaystyle dot alpha displaystyle dot alpha which define terms reducing operators associated develop basic theory these reducing operator besov spaces similar results shown inhomogeneous spaces