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

We study the duals of the spaces $A^{p\alpha }\left(X\right)$ of harmonic functions in the unit ball of ${ℝ}^n$ with values in a Banach space X, belonging to the Bochner $L^p$ space with weight ${(1-|x\left|\right)}^{\alpha }$, denoted by $L^{p\alpha }\left(X\right)$. For 0 < α < p-1 we construct continuous projections onto $A^{p\alpha }\left(X\right)$ providing a decomposition $L^{p\alpha }\left(X\right)=A^{p\alpha }\left(X\right)+M^{p\alpha }\left(X\right)$. We discuss the conditions on p, α and X for which $A^{p\alpha }\left(X\right)*=A^{q\alpha }(X*)$ and $M^{p\alpha }\left(X\right)*=M^{q\alpha }(X*)$, 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

This paper is an extension of the work done in [Morsli M., Bedouhene F., Boulahia F., Duality properties and Riesz representation theorem in the Besicovitch-Orlicz space of almost periodic functions, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 103--117] to the Besicovitch-Musielak-Orlicz space of almost periodic functions. Necessary and sufficient conditions for the reflexivity of this space are given. A Riesz type ``duality representation theorem'' is also stated.

In [6], the classical Riesz representation theorem is extended to the class of Besicovitch space of almost periodic functions $B^q$ a.p., $q\in ]1,+\infty [$ . It is also shown that this space is reflexive. We shall consider here such results in the context of Orlicz spaces, namely in the class of Besicovitch-Orlicz space of almost periodic functions $B^{\phi }$ a.p., where $\phi$ is an Orlicz function.

We study atomic decompositions and their relationship with duality and reflexivity of Banach spaces. To this end, we extend the concepts of "shrinking" and "boundedly complete" Schauder basis to the atomic decomposition framework. This allows us to answer a basic duality question: when an atomic decomposition for a Banach space generates, by duality, an atomic decomposition for its dual space. We also characterize the reflexivity of a Banach space in terms of properties of its atomic decompositions....

Let $X$ be a completely regular Hausdorff space, $E$ a real normed space, and let $C_b(X,E)$ be the space of all bounded continuous $E$-valued functions on $X$. We develop the general duality theory of the space $C_b(X,E)$ endowed with locally solid topologies; in particular with the strict topologies ${\beta }_z(X,E)$ for $z=\sigma ,\tau ,t$. As an application, we consider criteria for relative weak-star compactness in the spaces of vector measures $M_z(X,E^{\text{'}})$ for $z=\sigma ,\tau ,t$. It is shown that if a subset $H$ of $M_z(X,E^{\text{'}})$ is relatively $\sigma (M_z(X,E^{\text{'}}),C_b(X,E))$-compact, then the set $conv\left(S\right(H\left)\right)$ is still relatively $\sigma (M_z(X,E^{\text{'}}),C_b(X,E))$-compact...