Analytic classes on subframe and expanded disk and the \(\mathcal R\)s differential operator in polydisk (Q962421)

Let \(U^n\) denote the unit polydisc in \(\mathbb C^n\) and \(T^n\) the distinguished boundary of \(U^n\). Let \(dm_{2n}\) and \(dm_n\) denote the volume measure on \(U^n\) and the normalized Lebesgue measure on \(T^n\). The expanded disc is defined as \[ U^n_*:=\{(r_1\xi,\dots,r_n\xi)\in U^n:\xi\in T,\;r_j\in(0,1),\;j=1,\dots,n\}, \] and the subframe is defined by \[ \widetilde{U}^n:=\{(z_1,\dots,z_n)\in U^n:\exists_{r\in(0,1]}\;|z_j|=r,\;j=1,\dots,n\}. \] Let \(H(U^n)\) be the space of all holomorphic functions on \(U^n\). The authors introduce and study new Bergman classes on the expanded disc defined as \[ \begin{aligned} & A_{\alpha}^p(U^n_*):=\Big\{f\in H(U^n):\|f\|^p_{A_{\alpha}^p(U^n_*)}:=\\ & \int_0^1\dots\int_0^1\int_T|f(|z_1|\xi,\dots,|z_n|\xi)|^p\prod_{j=1}^n(1-|z_j|^2)^{\alpha_j}\,d|z_j|\,dm(\xi)<\infty\Big\},\end{aligned} \] where \(\alpha=(\alpha_1,\dots,\alpha_n)\in(-1,\infty)^n,\;p\in(0,\infty]\), and Bergman classes on the subframe defined as \[ \begin{aligned} &A_{\alpha}^p(\widetilde{U}^n):=\Big\{f\in H(U^n):\|f\|^p_{A_{\alpha}^p(\widetilde{U}^n)}:=\\ &\int_{T^n}\int_0^1|f(|z|\xi_1,\dots,|z|\xi_n)|^p(1-|z|^2)^{\alpha}\,dm_n(\xi)\,d|z|<\infty\Big\},\end{aligned} \] where \(\alpha\in(-1,\infty),\;p\in(0,\infty)\). The authors also present new results connected with operator of diagonal map in polydisc. In particular, they completely describe traces of Bergman classes \(A_{\alpha}^p(U^n_*)\) and \(A_{\alpha}^p(\widetilde{U}^n)\) on the unit disc \(U\). Finally, the authors study \(\mathcal R^s\) differential operator in polydisc defined as follows \[ \mathcal R^sf:=\sum_{k_1,\dots,k_n\geqslant0}(k_1+\dots+k_n+1)^sa_{k_1,\dots,k_n}z_1^{k_1}\dots z_n^{k_n}, \] where \(s\in\mathbb R,\;f\in H(U^n)\), \(f(z_1,\dots,z_n)=\sum_{k_1,\dots,k_n\geqslant0}a_{k_1,\dots,k_n}z_1^{k_1}\dots z_n^{k_n}\). They present various generalizations of well-known one-dimensional results providing at the same time new connections between standard classes of analytic functions with quasinorms on polydisc and \(\mathcal R^s\) differential operator with corresponding classes on the expanded disc and the subframe.