Harmonic projections on Besov spaces in the real ball

As is well known, Bergman projection operator continuously maps weighted Bergman and some more general spaces onto their holomorphic or harmonic subspaces. In this paper, we turn to Besov spaces and define three-parameter Besov spaces \(\Lambda _\alpha ^{p,q}\) ${\mathrm{\Lambda }}_{\alpha }^{p,q}$ of smooth functions over the unit ball in \({{\mathbb {R}}}^n\) ${\mathbb{R}}^n$ . A family of Bergman type operators is constructed whose members continuously project the Besov space \(\Lambda _\alpha ^{p,q}\) ${\mathrm{\Lambda }}_{\alpha }^{p,q}$ onto its harmonic subspace \(h\Lambda _\alpha ^{p,q}\) $h{\mathrm{\Lambda }}_{\alpha }^{p,q}$ . A generalization with more general indices is also given.