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Analytic classes on subframe and expanded disk and the \(\mathcal R\)s differential operator in polydisk. (English) Zbl 1190.32007

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.

MSC:

32A30 Other generalizations of function theory of one complex variable
30H05 Spaces of bounded analytic functions of one complex variable
32A36 Bergman spaces of functions in several complex variables
46E15 Banach spaces of continuous, differentiable or analytic functions
47B38 Linear operators on function spaces (general)
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References:

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