Besov spaces and the boundedness of weighted Bergman projections over symmetric tube domains (Q558342)

We extend the analysis of weighted Bergman spaces \(\mathcal{A}^{p,q}_{\mathbf s}\) on symmetric tube domains, contained in [\textit{D. Békollé, A. Bonami, G. Garrigós} and \textit{F. Ricci}, Proc. Lond. Math. Soc. (3) Ser. 89, No. 2, 317--360 (2004; Zbl 1079.42015)], to the case where the weights are positive powers \(\Delta_{\mathbf s}\doteq\Delta_1^{s_1-s_2} \dots \Delta_{r-1}^{s_{r-1}-s_r} \Delta_r^{s_r}\) of the principal minors \(\Delta_1,\dots,\Delta_r\) on the symmetric cone \(\Omega\). We discuss the realization of the boundary distributions of functions in \(\mathcal{A}^{p,q}_{\mathbf s}\) in terms of Besov-type spaces \(B^{p,q}_{\mathbf s}\) adapted to the structure of the cone. We give a necessary and a sufficient condition on the values of \(p\), \(q\) and \(\mathbf s\) for which this identification between \(\mathcal{A}^{p,q}_{\mathbf s}\) and \(B^{p,q}_{\mathbf s}\) holds. We also present a continuous version of these latter spaces which is new even for the case \(s_1 =\dots =s_r\) considered in the cited paper. We use these results to discuss multipliers between Besov spaces and the boundedness of the weighted Bergman projection \(P_{\mathbf s}:L^{p,q}_{\mathbf s}\rightarrow\mathcal{A}^{p,q}_{\mathbf s}\). The situation in the rank two case is specifically dealt with.