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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*.

Define $h^{\infty }\left(E\right)$ as the subspace of $C^{\infty }(B̅L,E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^{-\infty }(E*)$ consisting of all harmonic E*-valued functions g such that ${(1-|x\left|\right)}^{m}f$ is bounded for some m>0. Then the dual $h^{\infty }(E*)$ is represented by $h^{-\infty }(E*)$ through $⟨f,g{⟩}_0=li{m}_{r\to 1}{ʃ}_B⟨f\left(rx\right),g\left(x\right)⟩dx$, $f\in h^{-\infty }(E*),g\in h^{\infty }\left(E\right)$. This extends the results of S. Bell in the scalar case.

We find necessary and sufficient conditions on radial weights w on the unit disc so that the Bergman type projections of Forelli-Rudin are bounded on L¹(w) and in the Herz spaces $K_p^q\left(w\right)$.