Let
μ
be a positive Borel measure on the interval 0,1). Suppose
H
μ
is the Hankel matrix
(
μ
n
,
k
)
n
,
k
≥
0
with entries
μ
n
,
k
=
μ
n
+
k
, where
μ
n
=
∫
0
,
1
)
t
n
d
μ
(
t
)
. The matrix ...formally induces the operator
H
μ
(
f
)
(
z
)
=
∑
n
=
0
∞
(
∑
k
=
0
∞
μ
n
,
k
a
k
)
z
n
,
which has been widely studied in Bao and Wulan (J Math Anal Appl 409:228–235, 2014), Chatzifountas et al. (J Math Anal Appl 413:154–168, 2014), Galanopoulos and Peláez (Stud Math 200:201–220, 2010) and Girela and Merchán (Banach J Math Anal 12:374-398, 2018). In this paper, we define the Derivative-Hilbert operator as
DH
μ
(
f
)
(
z
)
=
∑
n
=
0
∞
∑
k
=
0
∞
μ
n
,
k
a
k
(
n
+
1
)
z
n
.
We mainly characterize the measures
μ
for which
DH
μ
is a bounded (resp., compact) operator on the Bloch space
B
. We also characterize those measures
μ
for which
DH
μ
is a bounded (resp., compact) operator from the Bloch space
B
into the Bergman space
A
p
,
1
≤
p
<
∞
.
It is well known that the Hilbert matrix
H
is bounded from the logarithmically weighted Bergman space
A
log
α
2
into Bergman space
A
2
when
α
>
2
. In this paper, we calculate lower bound and upper ...bound for the norm of the Hilbert matrix operator
H
from the logarithmically weighted Bergman space
A
log
α
2
into Bergman space
A
2
when
α
>
2
. We also calculate lower bound and upper bound for the norm of the Hilbert matrix operator from
A
log
α
p
into
A
p
, for
2
<
p
<
∞
and
α
>
1
.
In this note we express the norm of composition followed by differentiation D C φ from the logarithmic Bloch and the little logarithmic Bloch spaces to the weighted space H μ ∞ on the unit disk and ...give an upper and a lower bound for the essential norm of this operator from the logarithmic Bloch space to H μ ∞ .
Let $ \mu $ be a positive Borel measure on the interval $ 0, 1) $. The Hankel matrix $ \mathcal{H}_{\mu} = (\mu_{n, k})_{n, k\geq 0} $ with entries $ \mu_{n, k} = \mu_{n+k} $, where $ \mu_{n} = ...\int_{0, 1)}t^nd\mu(t) $, formally induces the operator as follows:
<disp-formula> <tex-math id="FE1"> \begin{document}$ \mathcal{DH}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , \; z\in \mathbb{D}, $\end{document} </tex-math></disp-formula>
where $ f(z) = \sum_{n = 0}^\infty a_nz^n $ is an analytic function in $ \mathbb{D} $. In this article, we characterize those positive Borel measures on $ 0, 1) $ such that $ \mathcal{DH}_\mu $ is bounded (resp., compact) from Bergman spaces $ \mathcal{A}^p $ into Hardy spaces $ H^q $, where $ 0 < p, q < \infty $.
Let
be a positive Borel measure on the interval
. The Hankel matrix
with entries
, where
, induces formally the operator as follows:
where
is an analytic function in
. In this article, we ...characterize those positive Borel measures on
for which
is bounded (resp. compact) from Dirichlet spaces
into
Let Hμ be the Hankel matrix with entries μn,k=∫0,1)(n+1)tn+kdμ(t), where μ is a positive Borel measure on the interval 0,1). The matrix acts on the space of all analytic functions in the unit disk by ...multiplication on Taylor coefficients and induces formally the operatorDHμ(f)(z)=∑n=0∞(∑k=0∞μn,kak)zn, where f(z)=∑n=0∞anzn is an analytic function in D. In this paper, we characterize the measures μ for which DHμ is a bounded (resp., compact) operator from the Bergman space Ap (0<p<∞) into the space Aq (q≥p and q>1), or from Ap (0<p≤1) into A1.
We characterize the boundedness and compactness of the weighted composition operator on the logarithmic Bloch space LB ={f ∈ H(D) : supD(1 − |z|2) ln( 21−|z| )|f0(z)| < +∞} and the little logarithmic ...Bloch space LB0. The results generalize the known corresponding results on the composition operator and the pointwise multiplier on the logarithmic Bloch space LB and the little logarithmic Bloch space LB0. KCI Citation Count: 10
We characterize the boundedness and compactness of the
weighted composition operator uCp from the general function space F(p,
q, s) into the logarithmic Bloch space βL on the unit disk. Some ...necessary
and sufficient conditions are given for which uCp is a bounded or a
compact operator from F(p, q, s), F0(p, q, s) into βL, β0L respectively. KCI Citation Count: 22
Let D = {z : |z| < 1} be the unit disk in the complex plane C, Φ : D → C is a analytic map. We study the multiplication operator MΦ on the logarithmic weighted BMOA space ...BMOAlog={g∈H(D):supa∈D(log21−|a|2)2∫D|g′(z)|2(1−|φa(z)|2)dm(z)<∞}. We obtain that a sufficient condition for the operator MΦ to be a bounded operator on BMOAlog. We also get that another necessary condition for the operator MΦ to be bounded on BMOAlog.
Sampling measure on Bergman spaces Zhuo, Zhengyuan; Ye, Shanli
Proceedings of the American Mathematical Society,
11/2023, Letnik:
151, Številka:
11
Journal Article
Recenzirano
In this paper, we investigate the relationship between sampling measure
μ
\mu
, Berezin transform
μ
~
\tilde {\mu }
and
r
r
-averaging transform
μ
^
r
\widehat {\mu }_r
on Bergman spaces. Compared ...with some results of Luecking Amer. J. Math. 107 (1985), pp. 85–111, our results provide an equivalent description of sampling measures, which reveals the reason why
1
/
μ
^
r
∈
L
∞
1/\hat {\mu }_r\in L^\infty
or
1
/
μ
~
∈
L
∞
1/\tilde {\mu }\in L^\infty
does not make sure that
μ
\mu
is a sampling measure on Bergman spaces.
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