Resumen de Besov spaces and the boudedness of weighted Bergman projections over symmetric tube domains
Daniele Debertol
We extend the analysis of weighted Bergman spaces $\mathcal{A}^{p,q}_{\mathbf{s}}$ on symmetric tube domains, contained in [2], to the case where the weights are positive powers $\Delta_{\mathbf{s}}\doteq\Delta_1^{s_1-s_2} \cdot\dotsc\cdot \Delta_{r-1}^{s_{r-1}-s_r} \Delta_r^{s_r}$ of the principal minors $\Delta_1,\dotsc,\Delta_r$ 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 [2]. We use these results to discuss multipliers between Besov spaces and the boundedness of the weighted Bergman projection $P_{\mathbf{s}}\colon L^{p,q}_{\mathbf{s}}\rightarrow\mathcal{A}^{p,q}_{\mathbf{s}}$. The situation in the rank two case is specifically dealt with.
