DUALITY IN SOME SPACES OF FUNCTIONS HARMONIC IN THE UNIT BALL

Abstract

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 have $h^{1}(\eta)^*\sim h_{\infty}(\eta)$ 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.