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 Mat. Zametki, 2016, Volume 100, Issue 1, Pages 47–58 (Mi mz11159)

Mixed Norm Bergman–Morrey-type Spaces on the Unit Disc

A. N. Karapetyantsab, S. G. Samkoc

a Southern Federal University, Rostov-on-Don
b Don State Technical University, Rostov-on-Don

Abstract: We introduce and study the mixed-norm Bergman–Morrey space $\mathscr A^{q;p,\lambda}(\mathbb D)$, mixed-norm Bergman–Morrey space of local type $\mathscr A_{\mathrm{loc}}^{q;p,\lambda}(\mathbb D)$, and mixed-norm Bergman–Morrey space of complementary type ${^{\complement}}\mathscr A^{q;p,\lambda}(\mathbb D)$ on the unit disk $\mathbb D$ in the complex plane $\mathbb C$. The mixed norm Lebesgue–Morrey space $\mathscr L^{q;p,\lambda}(\mathbb D)$ is defined by the requirement that the sequence of Morrey $L^{p,\lambda}(I)$-norms of the Fourier coefficients of a function $f$ belongs to $l^q$ ($I=(0,1)$). Then, $\mathscr A^{q;p,\lambda}(\mathbb D)$ is defined as the subspace of analytic functions in $\mathscr L^{q;p,\lambda}(\mathbb D)$. Two other spaces $\mathscr A_{\mathrm{loc}}^{q;p,\lambda}(\mathbb D)$ and ${^{\complement}}\mathscr A^{q;p,\lambda}(\mathbb D)$ are defined similarly by using the local Morrey $L_{\mathrm{loc}}^{p,\lambda}(I)$-norm and the complementary Morrey ${^{\complement}}L^{p,\lambda}(I)$-norm respectively. The introduced spaces inherit features of both Bergman and Morrey spaces and, therefore, we call them Bergman–Morrey-type spaces. We prove the boundedness of the Bergman projection and reveal some facts on equivalent description of these spaces.

Keywords: Bergman–Morrey-type space, mixed norm.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-02732 S. G. Samko's work was supported by the Russian Foundation for Basic Research under grant 15-01-02732.

DOI: https://doi.org/10.4213/mzm11159

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English version:
Mathematical Notes, 2016, 100:1, 38–48

Bibliographic databases:

Document Type: Article
UDC: 517.53

Citation: A. N. Karapetyants, S. G. Samko, “Mixed Norm Bergman–Morrey-type Spaces on the Unit Disc”, Mat. Zametki, 100:1 (2016), 47–58; Math. Notes, 100:1 (2016), 38–48

Citation in format AMSBIB
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