Duality in some spaces of functions harmonic in the unit ball

We introduce the Banach spaces $h_{\infty}(\varphi), h_0(\varphi)$ and $h^1(\eta)$ of functions harmonic in the unit ball in $\mathbb{R}^ n $, depending on weight function $\varphi$ and weighting measure $\eta$. The paper studies the following question: for which $\varphi$ and $\eta$ we $h^1(\eta)^* \sim h_{\infty} (\eta)$ and $h_0(\varphi)^* \sim h^1 (\eta)$. We prove that the necessary and sufficient condition for this is that certain linear operator, which projects $L^{\infty}(d\eta\, d\sigma)$ onto the subspace $\varphi h_{\infty}(\varphi)$, is bounded.