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This article is cited in 13 scientific papers (total in 13 papers)
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
c Universidade do Algarve, Portugal
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.
Received: 17.02.2016
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
Linking options:
https://www.mathnet.ru/eng/mzm11159https://doi.org/10.4213/mzm11159 https://www.mathnet.ru/eng/mzm/v100/i1/p47
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