Duality on vector-valued weighted harmonic Bergman spaces

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