Meromorphic Bergman spaces
UDC 517.5We introduce new spaces of holomorphic functions on the pointed unit disc in C that generalize classical Bergman spaces. We prove some fundamental properties of these spaces and their dual spaces. Finally, we extend the Hardy – Littlewood  and Fejer – Riesz inequalities to thes...
Saved in:
| Date: | 2022 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2022
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/6163 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512280226037760 |
|---|---|
| author | Ghiloufi, N. Zaway , M. Ghiloufi, N. Zaway , M. |
| author_facet | Ghiloufi, N. Zaway , M. Ghiloufi, N. Zaway , M. |
| author_sort | Ghiloufi, N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-10-24T09:23:19Z |
| description | UDC 517.5We introduce new spaces of holomorphic functions on the pointed unit disc in C that generalize classical Bergman spaces. We prove some fundamental properties of these spaces and their dual spaces. Finally, we extend the Hardy – Littlewood  and Fejer – Riesz inequalities to these spaces with application of the Toeplitz operators. ´ |
| doi_str_mv | 10.37863/umzh.v74i8.6163 |
| first_indexed | 2026-03-24T03:26:16Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i8.6163
UDC 517.5
N. Ghiloufi1 (Univ. Gabes, Faculty of Sciences of Gabes, Tunisia),
M. Zaway (Shaqra Univ., Saudi Arabia and Irescomath Laboratory, Gabes Univ., Tunisia)
MEROMORPHIC BERGMAN SPACES
МЕРОМОРФНI ПРОСТОРИ БЕРГМАНА
We introduce new spaces of holomorphic functions on the pointed unit disc in \BbbC that generalize classical Bergman spaces.
We prove some fundamental properties of these spaces and their dual spaces. Finally, we extend the Hardy – Littlewood
and Fejér – Riesz inequalities to these spaces with application of the Toeplitz operators.
Введено новi простори голоморфних функцiй на загостреному одиничному диску в \BbbC , якi узагальнюють класичнi
простори Бергмана. Доведено деякi фундаментальнi властивостi цих просторiв та дуальних до них. Насамкiнець
поширено нерiвностi Гардi – Лiттльвуда та Феєра – Рiсса на цi простори за допомогою операторiв Теплiца.
1. Introduction and preliminary results. Since the seventeenth of the last century the notion of
Bergman spaces has known an increasing use in mathematics and essentially in complex analysis and
geometry. The fundamental concept of this notion is the Bergman kernel. This kernel was computed
firstly for the unit disc \BbbD in \BbbC and then it was determined for any simply connected domain by the
famous Riemann’s theorem. However the determination of the Bergman kernels of domains in \BbbC n
is more delicate and it is determined for some type of domains and still unknown up to our day in
general. In this paper we generalize most properties of Bergman spaces of the unit disk by introducing
new spaces of holomorphic functions on the pointed unit disc \BbbD \ast that are square integrable with
respect to a probability measure d\mu \alpha ,\beta for some \alpha , \beta > - 1. In fact the classical Bergman space is
reduced to the case \beta = 0 (see [3] for more details). We call these new spaces meromorphic Bergman
spaces; indeed any element of such a space is a meromorphic function which has 0 as a pole of order
controlled by the parameter \beta . The originality of our idea is that the Bergman kernels of these spaces
may have zeros in the unit disk essentially when \beta is not an integer. This problem will be discussed
in a separate paper as a continuity of the present paper. For this reason we will concentrate here on
the topological properties of these spaces and prove some well-known inequalities.
Throughout this paper, \BbbD (a, r) will be the disc of \BbbC with center a and radius r > 0. In case
a = 0, we use \BbbD (r) (resp., \BbbD ) in stead of \BbbD (0, r) (resp., \BbbD (0, 1)). We set \BbbS (r) := \partial \BbbD (r) the circle
and \BbbD \ast := \BbbD \setminus \{ 0\} . For every - 1 < \alpha , \beta < +\infty , we consider the positive measure \mu \alpha ,\beta on \BbbD
defined by
d\mu \alpha ,\beta (z) :=
1
B(\alpha + 1, \beta + 1)
| z| 2\beta (1 - | z| 2)\alpha dA(z),
where B is the beta-function defined by
B(s, t) =
1\int
0
xs - 1(1 - x)t - 1dx =
\Gamma (s)\Gamma (t)
\Gamma (s+ t)
\forall s, t > 0
and
1 Corresponding author, e-mail: noureddine.ghiloufi@fsg.rnu.tn.
c\bigcirc N. GHILOUFI, M. ZAWAY, 2022
1060 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
MEROMORPHIC BERGMAN SPACES 1061
dA(z) =
1
\pi
dxdy =
1
\pi
rdrd\theta , z = x+ iy = rei\theta ,
the normalized area measure on \BbbD .
The general aim of this paper is to study the properties of the Bergman type space \scrA p
\alpha ,\beta (\BbbD
\ast )
defined for 0 < p < +\infty as the set of holomorphic functions on \BbbD \ast that belongs to the space
Lp(\BbbD , d\mu \alpha ,\beta ) = \{ f : \BbbD - \rightarrow \BbbC ; measurable function such that \| f\| \alpha ,\beta ,p < +\infty \} ,
where
\| f\| p\alpha ,\beta ,p :=
\int
\BbbD
\bigm| \bigm| f(z)\bigm| \bigm| pd\mu \alpha ,\beta (z).
When 1 \leq p < +\infty , the space
\bigl(
Lp(\BbbD , d\mu \alpha ,\beta ), \| .\| \alpha ,\beta ,p
\bigr)
is a Banach space; however for 0 < p < 1,
the space Lp(\BbbD , d\mu \alpha ,\beta ) is a complete metric space where the metric is given by d(f, g) = \| f -
- g\| p\alpha ,\beta ,p. The following proposition will be useful in the hole of the paper.
Proposition 1. For every 0 < r < 1 and 0 < \varepsilon < 1 there exists c\varepsilon (r) = c\varepsilon ,\alpha ,\beta (r) > 0 such
that, for any 0 < p < +\infty and f \in \scrA p
\alpha ,\beta (\BbbD
\ast ), we have\bigm| \bigm| f(z)\bigm| \bigm| p \leq B(\alpha + 1, \beta + 1)
c\varepsilon (r)
\| f\| p\alpha ,\beta ,p \forall z \in \BbbS (r).
One can choose c\varepsilon (r) = r2\varepsilon a\varepsilon (r)b\varepsilon (r) with r\varepsilon = \varepsilon \mathrm{m}\mathrm{i}\mathrm{n}(r, 1 - r),
a\varepsilon (r) :=
\left\{
\bigl[
1 - (r + r\varepsilon )
2
\bigr] \alpha
if \alpha \geq 0,\bigl[
1 - (r - r\varepsilon )
2
\bigr] \alpha
if - 1 < \alpha < 0,
and
b\varepsilon (r) :=
\left\{ (r - r\varepsilon )
2\beta if \beta \geq 0,
(r + r\varepsilon )
2\beta if - 1 < \beta < 0.
Proof. Let 0 < r < 1, 0 < \varepsilon < 1 and 0 < p < +\infty be fixed reals. Let f \in \scrA p
\alpha ,\beta (\BbbD
\ast )
and z \in \BbbS (r). We set r\varepsilon = \varepsilon \mathrm{m}\mathrm{i}\mathrm{n}(r, 1 - r). It is easy to see that \BbbD (z, r\varepsilon ) \subset \BbbD \ast ; so thanks to the
subharmonicity of | f | p, we obtain\bigm| \bigm| f(z)\bigm| \bigm| p \leq 1
r2\varepsilon
\int
\BbbD (z,r\varepsilon )
| f(w)| pdA(w) \leq
\leq B(\alpha + 1, \beta + 1)
r2\varepsilon
\int
\BbbD (z,r\varepsilon )
| f(w)| p
| w| 2\beta
\bigl(
1 - | w| 2
\bigr) \alpha d\mu \alpha ,\beta (w). (1.1)
If w \in \BbbD (z, r\varepsilon ) then r - r\varepsilon \leq | w| \leq r + r\varepsilon . Thus, we obtain
| w| 2\beta \geq b\varepsilon (r) :=
\left\{ (r - r\varepsilon )
2\beta if \beta \geq 0,
(r + r\varepsilon )
2\beta if - 1 < \beta < 0,
and \bigl(
1 - | w| 2
\bigr) \alpha \geq a\varepsilon (r) :=
\left\{
\bigl[
1 - (r + r\varepsilon )
2
\bigr] \alpha
if \alpha \geq 0,\bigl[
1 - (r - r\varepsilon )
2
\bigr] \alpha
if - 1 < \alpha < 0.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1062 N. GHILOUFI, M. ZAWAY
It follows that inequality (1.1) gives\bigm| \bigm| f(z)\bigm| \bigm| p \leq B(\alpha + 1, \beta + 1)
r2\varepsilon a\varepsilon (r)b\varepsilon (r)
\int
\BbbD (z,r\varepsilon )
\bigm| \bigm| f(w)\bigm| \bigm| pd\mu \alpha ,\beta (w) \leq
\leq B(\alpha + 1, \beta + 1)
r2\varepsilon a\varepsilon (r)b\varepsilon (r)
\| f\| p\alpha ,\beta ,p.
Proposition 1 is proved.
Using the previous proof, one can improve the previous proposition as follows.
Remark 1. For any n \in \BbbN and 0 < r < 1, there exists c = c(n, r, \alpha , \beta ) > 0 such that, for every
f \in \scrA p
\alpha ,\beta (\BbbD
\ast ), we have
| f (n)(z)| p \leq c\| f\| p\alpha ,\beta ,p \forall z \in \BbbS (r).
As a first consequence of Proposition 1, we have the following corollary.
Corollary 1. For every - 1 < \alpha , \beta < +\infty and 0 < p < +\infty , the space \scrA p
\alpha ,\beta (\BbbD
\ast ) is closed in
Lp(\BbbD , \mu \alpha ,\beta ) and, for any z \in \BbbD \ast , the linear form \delta z : \scrA p
\alpha ,\beta (\BbbD
\ast ) - \rightarrow \BbbC defined by \delta z(f) = f(z) is
bounded on \scrA p
\alpha ,\beta (\BbbD
\ast ).
Proof. As Lp(\BbbD , d\mu \alpha ,\beta ) is complete, it suffices to consider a sequence (fn)n \subset \scrA p
\alpha ,\beta (\BbbD
\ast )
that converges to f \in Lp(\BbbD , d\mu \alpha ,\beta ) and to prove that f \in \scrA p
\alpha ,\beta (\BbbD
\ast ). Thanks to Proposition 1, the
sequence (fn)n converges uniformly to f on every compact subset of \BbbD \ast . Hence, the function f is
holomorphic on \BbbD \ast and we conclude that f \in \scrA p
\alpha ,\beta (\BbbD
\ast ).
For the second statement, one can see that \delta z is a linear functional well defined on \scrA p
\alpha ,\beta (\BbbD
\ast ).
For the continuity of \delta z, thanks to Proposition 1, for every z \in \BbbD \ast , there exists c > 0 such that,
for every f \in \scrA p
\alpha ,\beta (\BbbD
\ast ), we have | \delta z(f)| =
\bigm| \bigm| f(z)\bigm| \bigm| \leq c\| f\| \alpha ,\beta ,p. Thus the linear functional \delta z is
continuous on \scrA p
\alpha ,\beta (\BbbD
\ast ).
The corollary is proved.
In the following we give some immediate properties:
If f \in \scrA p
\alpha ,\beta (\BbbD
\ast ) then 0 can’t be an essential singularity for f, hence either 0 is removable for f
(so f is holomorphic on \BbbD ) or 0 is a pole for f with order \nu f = \nu f (0) that satisfies
\nu f \leq mp,\beta =
\left\{
\biggl\lfloor
2(\beta + 1)
p
\biggr\rfloor
if
2(\beta + 1)
p
\not \in \BbbN ,
2(\beta + 1)
p
- 1 if
2(\beta + 1)
p
\in \BbbN ,
(1.2)
where \lfloor .\rfloor is the integer part.
If we set \widetilde f(z) = z\nu f f(z) then \widetilde f is a holomorphic function on \BbbD and f \in \scrA p
\alpha ,\beta (\BbbD
\ast ) if and only
if \widetilde f \in \scrA p
\alpha ,\beta -
p\nu f
2
(\BbbD \ast ) and
\| f\| \alpha ,\beta ,p =
\Biggl(
B(\alpha + 1, \beta - p\nu f
2 + 1)
B(\alpha + 1, \beta + 1)
\Biggr) 1
p
\| \widetilde f\|
\alpha ,\beta -
p\nu f
2
,p
.
Using the two previous properties, if we replace f by zmp,\beta f in the proof of Proposition 1, we can
obtain a more sharp estimate in Proposition 1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
MEROMORPHIC BERGMAN SPACES 1063
If - 1 < \beta < \beta \prime and - 1 < \alpha < \alpha \prime then \scrA p
\alpha ,\beta (\BbbD
\ast ) \subseteq \scrA p
\alpha \prime ,\beta \prime (\BbbD \ast ) and the canonical injection is
continuous. This is a consequence of the fact that we have
B(\alpha \prime + 1, \beta \prime + 1)\| f\| \alpha \prime ,\beta \prime ,p \leq B(\alpha + 1, \beta + 1)\| f\| \alpha ,\beta ,p
for every f \in \scrA p
\alpha ,\beta (\BbbD
\ast ).
Claim that if we set \BbbD \zeta := \BbbD \setminus \{ \zeta \} for any \zeta \in \BbbD , then all results on \scrA p
\alpha ,\beta (\BbbD
\ast ) can be extended
to the space \scrA p
\alpha ,\beta (\BbbD \zeta ) of holomorphic functions on \BbbD \zeta that are p-integrable with respect to the
positive measure | z - \zeta | 2\beta
\bigl(
1 - | z| 2
\bigr) \alpha
dA(z). Indeed h \in \scrA p
\alpha ,\beta (\BbbD \zeta ) if and only if h\circ \varphi \zeta \in \scrA p
\alpha ,\beta (\BbbD
\ast ),
where \varphi \zeta (z) =
\zeta - z
1 - \zeta z
.
2. Meromorphic Bergman kernels. In the case p = 2 we have \scrA 2
\alpha ,\beta (\BbbD \ast ) is a Hilbert space
and \scrA 2
\alpha ,\beta (\BbbD \ast ) = \scrA 2
\alpha ,m(\BbbD \ast ) for every \beta \in ]m - 1,m] with m \in \BbbN . If we set
en(z) =
\sqrt{}
B(\alpha + 1, \beta + 1)
B(\alpha + 1, n+ \beta + 1)
zn (2.1)
for every n \geq - m, then the sequence (en)n\geq - m is a Hilbert basis of \scrA 2
\alpha ,\beta (\BbbD \ast ). Furthermore, if
f, g \in \scrA 2
\alpha ,\beta (\BbbD \ast ) with
f(z) =
+\infty \sum
n= - m
anz
n, g(z) =
+\infty \sum
n= - m
bnz
n,
then
\langle f, g\rangle \alpha ,\beta =
+\infty \sum
n= - m
anbn
B(\alpha + 1, n+ \beta + 1)
B(\alpha + 1, \beta + 1)
,
where \langle ., .\rangle \alpha ,\beta is the inner product in \scrA 2
\alpha ,\beta (\BbbD \ast ) inherited from L2(\BbbD , d\mu \alpha ,\beta ).
Lemma 1. Let - 1 < \alpha < +\infty and m \in \BbbN . Then the reproducing (Bergman) kernel \BbbK \alpha ,m of
\scrA 2
\alpha ,m(\BbbD \ast ) is given by
\BbbK \alpha ,m(w, z) =
(\alpha + 1)B(\alpha + 1,m+ 1)
(wz)m(1 - wz)2+\alpha
.
Proof. The sequence (en)n\geq - m given by (2.1) is a Hilbert basis of \scrA 2
\alpha ,\beta (\BbbD \ast ), hence the
reproducing kernel of \scrA 2
\alpha ,\beta (\BbbD \ast ) is given by
\BbbK \alpha ,\beta (w, z) =
+\infty \sum
n= - m
en(w)en(z) =
+\infty \sum
n= - m
B(\alpha + 1, \beta + 1)
B(\alpha + 1, n+ \beta + 1)
wnzn =
=
1
(wz)m
+\infty \sum
n=0
B(\alpha + 1, \beta + 1)
B(\alpha + 1, n+ \beta - m+ 1)
(wz)n.
The computation of this kernel in the general case is more complicated. However, in our case for
\beta = m, we obtain
\BbbK \alpha ,m(w, z) =
1
(wz)m
+\infty \sum
n=0
B(\alpha + 1,m+ 1)
B(\alpha + 1, n+ 1)
(wz)n =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1064 N. GHILOUFI, M. ZAWAY
=
(\alpha + 1)B(\alpha + 1,m+ 1)
(wz)m(1 - wz)2+\alpha
.
Lemma 1 is proved.
Here we give some fundamental properties of the Bergman kernel as consequences of Lemma 1.
Corollary 2. Let - 1 < \alpha < +\infty and m \in \BbbN . Let \BbbP \alpha ,m be the orthogonal projection from
L2(\BbbD , d\mu \alpha ,m) onto \scrA 2
\alpha ,m(\BbbD \ast ). Then, for every f \in L2(\BbbD , d\mu \alpha ,m), we have
\BbbP \alpha ,mf(z) = (\alpha + 1)B(\alpha + 1,m+ 1)
\int
\BbbD
f(w)
(zw)m(1 - zw)2+\alpha
d\mu \alpha ,m(w).
Proof. This is a simple consequence of Lemma 1 and the fact that for every f \in L2(\BbbD , d\mu \alpha ,m)
we have \BbbP \alpha ,mf(z) = \langle f,\BbbK \alpha ,m(., z)\rangle \alpha ,m.
Using the density of \scrA 2
\alpha ,m(\BbbD \ast ) in \scrA 1
\alpha ,m(\BbbD \ast ), one can prove the following corollary.
Corollary 3. Let - 1 < \alpha < +\infty and m \in \BbbN . Then, for every f \in \scrA 1
\alpha ,m(\BbbD \ast ), we have
f(z) = (\alpha + 1)B(\alpha + 1,m+ 1)
\int
\BbbD
f(w)
(zw)m(1 - zw)2+\alpha
d\mu \alpha ,m(w).
The following result is well-known in general, its proof is based essentially on the fact that
\BbbK \alpha ,\beta (z, z) \not = 0 for every z \in \BbbD \ast .
Proposition 2. Let - 1 < \alpha , \beta < +\infty and \BbbK \alpha ,\beta be the reproducing (Bergman) kernel of
\scrA 2
\alpha ,\beta (\BbbD \ast ). Then, for every z \in \BbbD \ast , we have \BbbK \alpha ,\beta (z, z) > 0 and satisfies
\BbbK \alpha ,\beta (z, z) = \mathrm{s}\mathrm{u}\mathrm{p}
\Bigl\{ \bigm| \bigm| f(z)\bigm| \bigm| 2; f \in \scrA 2
\alpha ,\beta (\BbbD \ast ), \| f\| \alpha ,\beta ,2 \leq 1
\Bigr\}
=
= \mathrm{s}\mathrm{u}\mathrm{p}
\biggl\{
1
\| f\| \alpha ,\beta ,2
; f \in \scrA 2
\alpha ,\beta (\BbbD \ast ), f(z) = 1
\biggr\}
. (2.2)
In particular, the norm of the Dirac form \delta z on \scrA 2
\alpha ,\beta (\BbbD \ast ) is given by
\| \delta z\| = \| \BbbK \alpha ,\beta (., z)\| \alpha ,\beta ,2 =
\sqrt{}
\BbbK \alpha ,\beta (z, z).
One can find the proof of the first equality in Krantz book [5], however the second one is due to
Kim [4]. For the completeness of our paper we give the proof.
Proof. Thanks to the proof of Lemma 1, we have \BbbK \alpha ,\beta (z, z) > 0 for every z \in \BbbD \ast . To prove
the first equality in (2.2), we fix z \in \BbbD \ast and we consider
Q(z) := \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| f(z)| 2; f \in \scrA 2
\alpha ,\beta (\BbbD \ast ), \| f\| \alpha ,\beta ,2 \leq 1
\bigr\}
.
Let f \in \scrA 2
\alpha ,\beta (\BbbD \ast ) such that \| f\| \alpha ,\beta ,2 \leq 1. Then, thanks to the Cauchy – Schwarz inequality,
| f(z)| 2 =
\bigm| \bigm| \langle f,\BbbK \alpha ,\beta (., z)\rangle \alpha ,\beta
\bigm| \bigm| 2 \leq \| f\| \alpha ,\beta ,2
\bigm\| \bigm\| \BbbK \alpha ,\beta (., z)
\bigm\| \bigm\| 2
\alpha ,\beta ,2
\leq \BbbK \alpha ,\beta (z, z).
It follows that Q(z) \leq \BbbK \alpha ,\beta (z, z). Conversely, we set
g(\xi ) =
\BbbK \alpha ,\beta (\xi , z)\sqrt{}
\BbbK \alpha ,\beta (z, z)
, \xi \in \BbbD \ast .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
MEROMORPHIC BERGMAN SPACES 1065
Hence we have g \in \scrA 2
\alpha ,\beta (\BbbD \ast ), \| g\| \alpha ,\beta ,2 = 1 and | g(z)| 2 = \BbbK \alpha ,\beta (z, z) and the converse inequality
Q(z) \geq \BbbK \alpha ,\beta (z, z) is proved.
Now to prove the second equality in (2.2), we let
M (z) := \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
\| f\| \alpha ,\beta ,2; f \in \scrA 2
\alpha ,\beta (\BbbD \ast ), f(z) = 1
\bigr\}
.
If we set
h(\xi ) =
\BbbK \alpha ,\beta (\xi , z)
\BbbK \alpha ,\beta (z, z)
, \xi \in \BbbD \ast ,
then h(z) = 1 and h \in \scrA 2
\alpha ,\beta (\BbbD \ast ). Indeed,
\| h\| 2\alpha ,\beta ,2 =
\int
\BbbD
| h(\xi )| 2d\mu \alpha ,\beta (\xi ) =
\int
\BbbD
\BbbK \alpha ,\beta (\xi , z)
\BbbK \alpha ,\beta (z, z)
\BbbK \alpha ,\beta (z, \xi )
\BbbK \alpha ,\beta (z, z)
d\mu \alpha ,\beta (\xi ) =
=
1
\BbbK \alpha ,\beta (z, z)2
\BbbK \alpha ,\beta (z, z) =
1
\BbbK \alpha ,\beta (z, z)
.
It follows that
M (z) \leq 1
\BbbK \alpha ,\beta (z, z)
.
Conversely, for every f \in \scrA 2
\alpha ,\beta (\BbbD \ast ) such that f(z) = 1, we have\bigm| \bigm| f(\zeta )\bigm| \bigm| 2 = \bigm| \bigm| \langle f,\BbbK \alpha ,\beta (., \zeta )\rangle \alpha ,\beta
\bigm| \bigm| 2 \leq \| f\| 2\alpha ,\beta ,2\BbbK \alpha ,\beta (\zeta , \zeta ).
Thus we obtain
| f(\zeta )| 2
\BbbK \alpha ,\beta (\zeta , \zeta )
\leq \| f\| 2\alpha ,\beta ,2 \forall \zeta \in \BbbD \ast .
In particular, for \zeta = z,
1
\BbbK \alpha ,\beta (z, z)
\leq \| f\| 2\alpha ,\beta ,2.
We conclude that
1
\BbbK \alpha ,\beta (z, z)
\leq M (z).
Proposition 2 is proved.
3. Duality of meromorphic Bergman spaces. The aim of this part is to prove that the dual of
\scrA p
\alpha ,\beta (\BbbD
\ast ) is related to \scrA q
\alpha ,\beta (\BbbD
\ast ) with
1
p
+
1
q
= 1. This will be a consequence of the main result
(Theorem 1). But to prove the main result we need the following lemma.
Lemma 2. For every - 1 < \sigma , \gamma < +\infty , we set
I\omega (z) =
\int
\BbbD
(1 - | w| 2)\sigma | w| 2\gamma
| 1 - zw| 2+\sigma +\omega
dA(w).
Then I\omega is continuous on \BbbD and
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1066 N. GHILOUFI, M. ZAWAY
I\omega (z) \sim
\left\{
1 if \omega < 0,
\mathrm{l}\mathrm{o}\mathrm{g}
1
1 - | z| 2
if \omega = 0,
1
(1 - | z| 2)\omega
if \omega > 0,
when | z| - \rightarrow 1 - where \varphi \sim \psi means that there exist 0 < c1 < c2 such that we have c1\varphi (z) \leq
\leq \psi (z) \leq c2\varphi (z).
Proof. The proof is similar to [3] (Theorem 1.7).
Theorem 1. For every - 1 < \alpha , a, b < +\infty and m \in \BbbN , we consider the two integral operators
T and S defined by
Tf(z) =
1
zm
\int
\BbbD
f(w)(1 - | w| 2)\alpha - awm
| w| 2b(1 - zw)2+\alpha
d\mu a,b(w),
Sf(z) =
1
| z| m
\int
\BbbD
f(w)(1 - | w| 2)\alpha - a| w| m - 2b
| 1 - zw| 2+\alpha
d\mu a,b(w).
Then, for every 1 \leq p < +\infty , the following assertions are equivalent:
(1) T is bounded on Lp(\BbbD , d\mu a,b),
(2) S is bounded on Lp(\BbbD , d\mu a,b),
(3) p(\alpha + 1) > a+ 1 and
\Biggl\{
m - 2 < 2b \leq m if p = 1,
mp - 2 < 2b < mp - 2 + 2p if p > 1.
Proof. (2) =\Rightarrow (1) is obvious.
(1) =\Rightarrow (2) can be deduced using the transformation
\Omega zf(w) =
(1 - zw)2+\alpha | w| m
| 1 - zw| 2+\alpha wm
f(w).
(2) =\Rightarrow (3). Now assume that S is bounded on Lp(\BbbD , d\mu a,b). If we apply S to fN (z) = (1 -
- | z| 2)N for N large enough, we get
\| SfN\| pa,b,p =
\int
\BbbD
(1 - | z| 2)a| z| 2b - mp
\left( \int
\BbbD
(1 - | w| 2)\alpha +N | w| m
| 1 - zw| 2+\alpha
dA(w)
\right) p
dA(z)
is finite. Thanks to Lemma 2, we obtain b >
mp
2
- 1.
To prove the other inequalities, we need S \star the adjoint operator of S with respect to the inner
product \langle ., .\rangle a,b. It is given by
S \star g(w) = (1 - | w| 2)\alpha - a| w| m - 2b
\int
\BbbD
g(z)
| z| m| 1 - zw| 2+\alpha
d\mu a,b(z) =
= (1 - | w| 2)\alpha - a| w| m - 2b
\int
\BbbD
g(z)| z| 2b - m(1 - | z| 2)a
| 1 - zw| 2+\alpha
dA(z).
We distinguish two cases:
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
MEROMORPHIC BERGMAN SPACES 1067
1. Case p = 1: S is bounded on L1(\BbbD , d\mu a,b) gives S \star is bounded on L\infty (\BbbD , d\mu a,b). By
applying S \star on the constant function g \equiv 1, we have
\mathrm{s}\mathrm{u}\mathrm{p}
w\in \BbbD \ast
\bigl(
1 - | w| 2
\bigr) \alpha - a| w| m - 2b
\int
\BbbD
| z| 2b - m(1 - | z| 2)a
| 1 - zw| 2+\alpha
dA(z) < +\infty .
Thanks to Lemma 2, we get m - 2b \geq 0 and \alpha - a > 0. The desired inequalities are proved.
2. Case p > 1: Let q > 1 such that
1
p
+
1
q
= 1. Again by applying S \star on the function fN for
N large enough, we obtain
\| S \star fN\| qa,b,q =
\int
\BbbD
(1 - | w| 2)a+q(\alpha - a)| w| 2b+(m - 2b)q
\left( \int
\BbbD
| z| 2b - m(1 - | z| 2)a+N
| 1 - zw| 2+\alpha
dA(z)
\right) q
dA(w)
is finite and hence all inequalities
mp
2
- 1 < b <
mp
2
- 1 + p
and p(\alpha + 1) > a+ 1 hold.
(3) =\Rightarrow (2). We start by the case p = 1. We assume that m - 2 < 2b \leq m and \alpha > a. Using
Lemma 2, one can prove easily the boundedness of S on L1(\BbbD , d\mu a,b).
Now for p > 1, to prove the boundedness of S on Lp(\BbbD , d\mu a,b) we will use the Schur test. We
set
h(z) =
1
| z| t(1 - | z| 2)s
and \kappa (z, w) =
(1 - | w| 2)\alpha - a| w| m - 2b
| z| m| 1 - zw| 2+\alpha
.
Thanks to Lemma 2, if
m
q
\leq t <
m+ 2
q
, 0 < s <
\alpha + 1
q
, (3.1)
then \int
\BbbD
\kappa (z, w)h(w)qd\mu a,b(w) =
1
| z| m
\int
\BbbD
(1 - | w| 2)\alpha - sq| w| m - tq
| 1 - zw| 2+\alpha
dA(w) \leq
\leq c1
| z| m(1 - | z| 2)sq
= c1| z| tq - mh(z)q \leq c1h(z)
q
for some positive constant c1 > 0.
Similarly, if
2b - m
p
\leq t <
2b - m+ 2
p
,
a - \alpha
p
< s <
a+ 1
p
(3.2)
then \int
\BbbD
\kappa (z, w)h(z)pd\mu a,b(z) = (1 - | w| 2)\alpha - a| w| m - 2b
\int
\BbbD
| z| 2b - m - tp(1 - | z| 2)a - sp
| 1 - zw| 2+\alpha
dA(z) \leq
\leq c2
| w| m - 2b
(1 - | w| 2)sp
= c2| w| m - 2b+tph(w)p \leq c2h(w)
p
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1068 N. GHILOUFI, M. ZAWAY
with c2 > 0. Thanks to the hypothesis given in assertion (3), we have\biggr]
m
q
,
m+ 2
q
\biggl[
\cap
\biggr]
2b - m
p
,
2b - m+ 2
p
\biggl[
\not = \varnothing ,
\biggr]
0,
\alpha + 1
q
\biggl[
\cap
\biggr]
a - \alpha
p
,
a+ 1
p
\biggl[
\not = \varnothing .
This proves the existence of t and s satisfying (3.1) and (3.2). Thanks to Schur’s test, S is bounded
on Lp(\BbbD , d\mu a,b).
Theorem 1 is proved.
Theorem 2. For every 1 < p < +\infty and - 1 < a, b < +\infty , the topological dual of \scrA p
a,b(\BbbD
\ast )
is the space \scrA q
a,b(\BbbD
\ast ) under the integral pairing
\langle f, g\rangle a,b =
\int
\BbbD
f(z)g(z)d\mu a,b(z) \forall f \in \scrA p
a,b(\BbbD
\ast ), g \in \scrA q
a,b(\BbbD
\ast ),
where q is the conjugate exponent of p.
Proof. Thanks to Hölder inequality, every function g \in \scrA q
a,b(\BbbD
\ast ) defines a bounded linear form
on \scrA p
a,b(\BbbD
\ast ) via the above integral pairing. Conversely, let G be a bounded linear form on \scrA p
a,b(\BbbD
\ast ).
Then thanks to Hahn – Banach extension theorem, one can extend G to a bounded linear form on
Lp(\BbbD , d\mu a,b) (still denoted by G) with the same norm. By duality, there exists \psi \in Lq(\BbbD , d\mu a,b)
such that
G(f) = \langle f, \psi \rangle a,b \forall f \in \scrA p
a,b(\BbbD
\ast ).
Claim that if m = mp,b given in (1.2), then, thanks to Theorem 1, \BbbP a,m maps continuously
Lp(\BbbD , d\mu a,b) onto \scrA p
a,b(\BbbD
\ast ) and \BbbP a,mf = f for every f \in \scrA p
a,b(\BbbD
\ast ). It follows that
G(f) = \langle f, \psi \rangle a,b = \langle \BbbP a,mf, \psi \rangle a,b = \langle f,\BbbP \star
a,m\psi \rangle a,b \forall f \in \scrA p
a,b(\BbbD
\ast ).
If we set g = \BbbP \star
a,m\psi then g \in \scrA q
a,b(\BbbD
\ast ) and G(f) = \langle f, g\rangle for every f \in \scrA p
a,b(\BbbD
\ast ).
Theorem 2 is proved.
4. Inequalities on \bfscrA \bfitp
\bfitalpha ,\bfitbeta (\BbbD
\ast ). The aim here is to extend the two famous Hardy – Littlewood and
Fejér – Riesz inequalities to our new spaces, these inequalities were proved firstly on Hardy spaces,
then on Bergman spaces with some applications. In our case we give only one application on Toeplitz
operators. To reach this aim, for a holomorphic function f on \BbbD \ast and 0 < r < 1, we consider the
main value on the circle:
Mp(r, f) :=
\left( 1
2\pi
2\pi \int
0
| f(rei\theta )| pd\theta
\right)
1
p
,
M\infty (r, f) := \mathrm{s}\mathrm{u}\mathrm{p}
\theta \in [0,2\pi ]
\bigm| \bigm| f(rei\theta )\bigm| \bigm| .
We set
J (r) = J\alpha ,\beta ,p(r) :=
2rpmp,\beta
B(\alpha + 1, \beta + 1)
1\int
r
t2\beta - pmp,\beta +1(1 - t2)\alpha dt.
4.1. Hardy – Littlewood inequality. To prove the Hardy – Littlewood inequality on \scrA p
\alpha ,\beta (\BbbD
\ast ),
we need to prove firstly the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
MEROMORPHIC BERGMAN SPACES 1069
Lemma 3. For every p > 1 and f \in \scrA p
\alpha ,\beta (\BbbD
\ast ), we have
Mp(r, f) \leq
\| f\| \alpha ,\beta ,p
J (r)
1
p
.
In particular,
Mp(r, f) \leq
\kappa 1\| f\| \alpha ,\beta ,p
r
\mathrm{m}\mathrm{a}\mathrm{x}( 2\beta
p
,\nu f )(1 - r2)
\alpha +1
p
,
where \kappa 1 =
\bigl(
(\alpha + 1)B(\alpha + 1, \beta + 1)
\bigr) 1
p .
Proof. Let f \in \scrA p
\alpha ,\beta (\BbbD
\ast ) and 0 < r < 1. We set F (z) = zmf(z) with m = mp,\beta . As F is
holomorphic on \BbbD , then we obtain
\| f\| p\alpha ,\beta ,p =
1
\pi B(\alpha + 1, \beta + 1)
1\int
0
2\pi \int
0
| f(tei\theta )| p(1 - t2)\alpha t2\beta +1dtd\theta =
=
2
B(\alpha + 1, \beta + 1)
1\int
0
Mp
p (t, f)(1 - t2)\alpha t2\beta +1dt =
=
2
B(\alpha + 1, \beta + 1)
1\int
0
Mp
p (t, F )(1 - t2)\alpha t2\beta - pm+1dt \geq
\geq 2
B(\alpha + 1, \beta + 1)
Mp
p (r, F )
1\int
r
(1 - t2)\alpha t2\beta - pm+1dt =Mp
p (r, f)J (r)
and the first inequality is proved. The particular case can be deduced from the following inequality:
J (r) =
rpm
B(\alpha + 1, \beta + 1)
1\int
r2
(1 - t)\alpha t\beta - pm/2dt \geq
\geq
\left\{
r2\beta (1 - r2)\alpha +1
(\alpha + 1)B(\alpha + 1, \beta + 1)
if 2\beta \geq pm,
rpm(1 - r2)\alpha +1
(\alpha + 1)B(\alpha + 1, \beta + 1)
if 2\beta < pm
\geq
\geq r\mathrm{m}\mathrm{a}\mathrm{x}(2\beta ,pm)(1 - r2)\alpha +1
(\alpha + 1)B(\alpha + 1, \beta + 1)
.
Lemma 3 is proved.
Now we can prove the Hardy – Littlewood inequality on meromorphic Bergman spaces.
Theorem 3. For every 1 < p \leq \tau \leq \infty , there exists a positive constant \kappa such that, for every
f \in \scrA p
\alpha ,\beta (\BbbD
\ast ), we have
M\tau (r, f) \leq
\kappa \| f\| \alpha ,\beta ,p
r
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl(
2\beta
p
,m
\Bigr)
(1 - r2)
\alpha +2
p
- 1
\tau
,
where m = mp,\beta .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1070 N. GHILOUFI, M. ZAWAY
The Hardy – Littlewood inequality is proved in [6] for classical Bergman spaces (\beta = 0).
Proof. The case \tau = p is simply the previous lemma. Let we start by the case \tau = \infty . Let
f \in \scrA p
\alpha ,\beta (\BbbD
\ast ) and 0 < r < 1. We set F (z) = zmf(z). Again as F is holomorphic on \BbbD , then,
thanks to the Cauchy formula, we have
F (rei\theta ) =
s
2\pi
2\pi \int
0
smeimtf(seit)
seit - rei\theta
eitdt,
where s =
1 + r
2
. Applying Hölder’s inequality
\biggl(
1
p
+
1
q
= 1
\biggr)
and Lemma 3, we obtain
rm| f(rei\theta )| \leq
\left( 1
2\pi
2\pi \int
0
spm| f(seit)| pdt
\right)
1
p
\left( 1
2\pi
2\pi \int
0
sq
| seit - rei\theta | q
dt
\right)
1
q
\leq
\leq smMp(s, f)
\left( \kappa 2\biggl(
1 - r
1 + r
\biggr) q - 1
\right)
1
q
\leq
\leq sm
\kappa 1\| f\| \alpha ,\beta ,p
s
\mathrm{m}\mathrm{a}\mathrm{x}( 2\beta
p
,m)
(1 - s2)
\alpha +1
p
\kappa 3
(1 - r2)
1 - 1
q
\leq
\leq
\kappa 4\| f\| \alpha ,\beta ,p
r
\mathrm{m}\mathrm{a}\mathrm{x}( 2\beta
p
- m,0)
(1 - r2)
\alpha +2
p
.
It follows that
M\infty (r, f) \leq
\kappa 4\| f\| \alpha ,\beta ,p
r
\mathrm{m}\mathrm{a}\mathrm{x}( 2\beta
p
,m)
(1 - r2)
\alpha +2
p
.
Let now p < \tau <\infty . We have
M\tau (r, f) =
\left( 1
2\pi
2\pi \int
0
| f(reit)| p| f(reit)| \tau - pdt
\right)
1
\tau
\leq
\leq M
1 - p
\tau \infty (r, f)M
p
\tau
p (r, f) \leq
\leq
\Biggl(
\kappa 4\| f\| \alpha ,\beta ,p
r
\mathrm{m}\mathrm{a}\mathrm{x}( 2\beta
p
,m)
(1 - r2)
\alpha +2
p
\Biggr) 1 - p
\tau
\Biggl(
\kappa 1\| f\| \alpha ,\beta ,p
r
\mathrm{m}\mathrm{a}\mathrm{x}( 2\beta
p
,m)
(1 - r2)
\alpha +1
p
\Biggr) p
\tau
=
=
\kappa \| f\| \alpha ,\beta ,p
r
\mathrm{m}\mathrm{a}\mathrm{x}( 2\beta
p
,m)
(1 - r2)
\alpha +2
p
- 1
\tau
.
Theorem 3 is proved.
4.2. Fejér – Riesz inequality. The aim here is to prove a generalization of the following lemma
to meromorphic Bergman spaces.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
MEROMORPHIC BERGMAN SPACES 1071
Lemma 4 (see [2]). Let g be a holomorphic function in the Hardy space Hp(\BbbD ). Then, for any
\xi \in \BbbC with | \xi | = 1, we have
1\int
- 1
| g(t\xi )| pdt \leq 1
2
\| g\| pHp :=
1
2
2\pi \int
0
| g(ei\theta )| pd\theta .
Theorem 4 (Fejér – Riesz inequality). For every f \in \scrA p
\alpha ,\beta (\BbbD
\ast ) and \xi \in \BbbC with | \xi | = 1, we
have
1\int
- 1
\bigm| \bigm| f(t\xi )\bigm| \bigm| pJ \bigl(
| t|
\bigr)
dt \leq \pi \| f\| p\alpha ,\beta ,p.
Claim that if \beta = 0 then we find the Zhu result [7].
Proof. Let f \in \scrA p
\alpha ,\beta (\BbbD
\ast ) and F (z) = zmf(z) where m = mp,\beta . If we set Fr(z) = F (rz) for
0 < r < 1, Then Fr \in Hp(\BbbD ) and, thanks to Lemma 4, for every \xi \in \BbbC , | \xi | = 1,
1\int
- 1
| Fr(t\xi )| pdt \leq
1
2
2\pi \int
0
| Fr(e
i\theta )| pd\theta ,
that is,
1\int
- 1
| F (rt\xi )| pdt \leq 1
2
2\pi \int
0
| F (rei\theta )| pd\theta . (4.1)
Thanks to inequality (4.1) and Fubini theorem, we have
\| f\| p\alpha ,\beta ,p =
1
\pi B(\alpha + 1, \beta + 1)
1\int
0
2\pi \int
0
| f(rei\theta )| pr2\beta +1(1 - r2)\alpha drd\theta =
=
1
\pi B(\alpha + 1, \beta + 1)
1\int
0
\left( 2\pi \int
0
| F (rei\theta )| pd\theta
\right) r2\beta - pm+1(1 - r2)\alpha dr \geq
\geq 2
\pi B(\alpha + 1, \beta + 1)
1\int
0
\left( 1\int
- 1
| F (rt\xi )| pdt
\right) r2\beta - pm+1(1 - r2)\alpha dr =
=
2
\pi B(\alpha + 1, \beta + 1)
1\int
0
\left( r\int
- r
| F (s\xi )| pds
\right) r2\beta - pm(1 - r2)\alpha dr =
=
2
\pi B(\alpha + 1, \beta + 1)
1\int
- 1
| F (s\xi )| p
\left( 1\int
| s|
r2\beta - pm(1 - r2)\alpha dr
\right) ds =
=
2
\pi B(\alpha + 1, \beta + 1)
1\int
- 1
| f(s\xi )| p
\left( | s| pm
1\int
| s|
r2\beta - pm(1 - r2)\alpha dr
\right) ds \geq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1072 N. GHILOUFI, M. ZAWAY
\geq 1
\pi
1\int
- 1
| f(s\xi )| pJ
\bigl(
| s|
\bigr)
ds.
Theorem 4 is proved.
As an application of the Fejér – Riesz inequality on the Toeplitz operators, we have the following
result.
Theorem 5. For every \xi \in \BbbD \ast , if we consider the Toeplitz operator \scrT defined by
\scrT f(z) =
1\int
- 1
f(\xi x)\BbbK \alpha ,\beta (z, \xi x)J\alpha ,\beta ,2(| x| )dx,
then \scrT is a positive bounded linear operator on \scrA \alpha ,\beta (\BbbD \ast ).
When \beta = 0, this result is due to Andreev [1] proved in a restricted case.
Proof. Thanks to Fubini theorem, for every f \in \scrA \alpha ,\beta (\BbbD \ast ) one has
\langle \scrT f, f\rangle \alpha ,\beta =
\int
\BbbD
\scrT f(z)f(z)d\mu \alpha ,\beta (z) =
=
\int
\BbbD
\left( 1\int
- 1
f(\xi x)\BbbK \alpha ,\beta (z, \xi x)J\alpha ,\beta ,2(| x| )dx
\right) f(z)d\mu \alpha ,\beta (z) =
=
1\int
- 1
f(\xi x)
\int
\BbbD
\BbbK \alpha ,\beta (\xi x, z)f(z)d\mu \alpha ,\beta (z)J\alpha ,\beta ,2
\bigl(
| x|
\bigr)
dx =
=
1\int
- 1
f(\xi x)f(\xi x)J\alpha ,\beta ,2(| x| )dx \leq \pi \| f\| 2\alpha ,\beta ,2.
The last inequality is the Fejér – Riesz one in the particular case p = 2.
This proves that the operator \scrT is positive and thus it is self-adjoint and bounded with norm
\| \scrT \| \leq \pi . Indeed,
\| \scrT \| = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| \langle \scrT f, f\rangle \alpha ,\beta | ; \| f\| \alpha ,\beta ,2 = 1
\bigr\}
\leq \pi .
Theorem 5 is proved.
References
1. V. V. Andreev, Fejér – Riesz type inequalities for Bergman spaces, Rend. Circ. Mat. Palermo, 61, 385 – 392 (2012).
2. P. L. Duren, Theory of Hp spaces, Acad. Press (1970).
3. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces, Grad. Texts Math., 199 (2000).
4. H. Kim, On the localization of the minimum integral related to the weighted Bergman kernel and its application,
C. R. Acad. Sci. Paris, Ser. I, 355, 420 – 425 (2017).
5. S. G. Krantz, Geometric analysis of the Bergman kernel and metric, Grad. Texts Math., 268 (2013).
6. P. Sobolewski, łInequalities on Bergman spaces, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 61, 137 – 143 (2007).
7. K. Zhu, Translating inequalities between Hardy and Bergman spaces, Amer. Math. Monthly, 111, 520 – 525 (2004).
Received 15.06.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
|
| id | umjimathkievua-article-6163 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:16Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/af/1c4e695a3373a8de3ec92297bad078af.pdf |
| spelling | umjimathkievua-article-61632022-10-24T09:23:19Z Meromorphic Bergman spaces Meromorphic Bergman spaces Ghiloufi, N. Zaway , M. Ghiloufi, N. Zaway , M. Bergman spaces, Bergman Kernels, Hardy-Littlewood and Riesz-Fejer inequalities 30H20, 30A10 UDC 517.5We introduce new spaces of holomorphic functions on the pointed unit disc in C that generalize classical Bergman spaces. We prove some fundamental properties of these spaces and their dual spaces. Finally, we extend the Hardy – Littlewood&nbsp; and Fejer – Riesz inequalities to these spaces with application of the Toeplitz operators. ´ УДК 517.5 Мероморфнi простори БергманаВведено новi простори голоморфних функцiй на загостреному одиничному диску в C, якi узагальнюють класичнi простори Бергмана. Доведено деякi фундаментальнi властивостi цих просторiв та дуальних до них. Насамкiнець поширено нерiвностi Гардi – Лiттльвуда та Феєра – Рiсса на цi простори за допомогою операторiв Теплiца. Institute of Mathematics, NAS of Ukraine 2022-10-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6163 10.37863/umzh.v74i8.6163 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 8 (2022); 1060 - 1072 Український математичний журнал; Том 74 № 8 (2022); 1060 - 1072 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6163/9286 Copyright (c) 2022 mohamed zaway |
| spellingShingle | Ghiloufi, N. Zaway , M. Ghiloufi, N. Zaway , M. Meromorphic Bergman spaces |
| title | Meromorphic Bergman spaces |
| title_alt | Meromorphic Bergman spaces |
| title_full | Meromorphic Bergman spaces |
| title_fullStr | Meromorphic Bergman spaces |
| title_full_unstemmed | Meromorphic Bergman spaces |
| title_short | Meromorphic Bergman spaces |
| title_sort | meromorphic bergman spaces |
| topic_facet | Bergman spaces Bergman Kernels Hardy-Littlewood and Riesz-Fejer inequalities 30H20 30A10 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6163 |
| work_keys_str_mv | AT ghiloufin meromorphicbergmanspaces AT zawaym meromorphicbergmanspaces AT ghiloufin meromorphicbergmanspaces AT zawaym meromorphicbergmanspaces |