1 Introduction
Let
$\mathbb {D}=\{z\in \mathbb {C}:\vert z\vert <1\}$
denote the open unit disk of the complex plane
$\mathbb {C}$
and
$H(\mathbb {D})$
denote the space of all analytic functions in
$\mathbb {D}$
.
$H^{\infty }$
denote the set of bounded analytical functions on
$\mathbb {D}$
.
The Bloch space
$\mathcal {B}$
consists of those functions
$f\in H(\mathbb {D}) $
for which
$$ \begin{align*}\vert \vert f\vert \vert _{\mathcal {B}}=\vert f(0)\vert +\sup_{z\in \mathbb{D}}(1-\vert z\vert ^{2})\vert f'(z)\vert <\infty. \end{align*} $$
For
$1<p<\infty $
, the analytic Besov space
$B_p$
consists of those functions
$f\in H(\mathbb {D}) $
such that
$$ \begin{align*}\|f\|_{B_p}=|f(0)|+\left(\int_{\mathbb{D}}|f'(z)|^p(1-|z|)^{p-2}\mathrm{d}A(z)\right)^{\frac{1}{p}}<\infty, \end{align*} $$
where
$dA(z)=\frac {dxdy}{\pi }$
is the normalized area measure on
$\mathbb {D}$
. When
$p=2$
, then
$B_2$
is just the classic Dirichlet space
$\mathcal {D}$
. If
$1<p_1<p_2<\infty $
, then
$B_{p_1}\subsetneq B_{p_2} \subsetneq \mathcal {B}$
. It is known that the analytic Besov spaces are Möbius invariant and the Bloch space
$\mathcal {B}$
is the largest Möbius invariant space.
The space
$B_{1}$
consists of
$f\in H(\mathbb {D})$
such that
$$ \begin{align*}||f||_{B_{1}}=|f(0)|+|f'(0)|+\int_{\mathbb{D}}|f"(z)|dA(z)<\infty.\end{align*} $$
The space
$B_{1}$
is the smallest Möbius invariant Banach spaces of analytic function in
$\mathbb {D}$
and
$B_{1}\subsetneq H^{\infty }$
. See [Reference Zhu32, Chapter 5] for the theory of these spaces.
Let
$0<p\leq \infty $
, the classical Hardy space
$H^p$
consists of those functions
$f\in H(\mathbb {D})$
for which
$$ \begin{align*}||f||_{p}=\sup_{0\leq r<1} M_p(r, f)<\infty,\end{align*} $$
where
$$ \begin{align*}M_p(r, f)= \left(\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^p d\theta \right)^{1/p}, \ 0<p<\infty,\end{align*} $$
$$ \begin{align*}M_{\infty}(r, f)=\sup_{|z|=r}|f(z)|.\end{align*} $$
Let
$1\leq p<\infty $
and
$0<\alpha \leq 1$
, the mean Lipschitz space
$\Lambda ^p_\alpha $
consists of those functions
$f\in H(\mathbb {D})$
having a non-tangential limit almost everywhere such that
$\omega _p(t, f)=O(t^\alpha )$
as
$t\to 0$
. Here
$\omega _p(\cdot , f)$
is the integral modulus of continuity of order p of the function
$f(e^{i\theta })$
. It is known (see [Reference Duren11]) that
$\Lambda ^p_\alpha $
is a subset of
$H^p$
and
$$ \begin{align*}\Lambda^p_\alpha=\left(f\in H(\mathbb{D}):M_p(r, f')=O\left(\frac{1}{(1-r)^{1-\alpha}}\right), \ \ \mbox{as}\ r\rightarrow 1\right).\end{align*} $$
The space
$\Lambda ^p_\alpha $
is a Banach space with the norm
$||\cdot ||_{\Lambda ^p_\alpha }$
given by
$$ \begin{align*}\|f\|_{\Lambda^p_\alpha}=|f(0)|+\sup_{0\leq r<1}(1-r)^{1-\alpha}M_p(r, f'). \end{align*} $$
It is known (see, e.g., [Reference Buckley, Koskela and Vukotić8]) that
$$ \begin{align*}\Lambda^{p}_{\frac{1}{p}}\subsetneq \mathcal {B}. \ \ 1<p<\infty.\end{align*} $$
For
$f(z)=\sum _{n=0}^\infty a_nz^n\in H(\mathbb {D})$
, the Cesàro operator
$\mathcal {C}$
is defined by
$$ \begin{align*}\mathcal {C}(f)(z)=\sum_{n=0}^\infty\left(\frac{1}{n+1}\sum_{k=0}^n a_k\right)z^n, \ z\in\mathbb{D}. \end{align*} $$
The boundedness and compactness of the Cesàro operator
$\mathcal {C}$
and its generalizations on various spaces of analytic functions such as Hardy spaces, Bergman spaces, Dirichlet spaces, the Bloch space,
$Q_{p}$
spaces, mixed norm spaces have been widely studied. See, e.g., [Reference Aleman and Cima1, Reference Aleman and Siskakis2, Reference Danikas and Siskakis10, Reference Miao19, Reference Siskakis24–Reference Siskakis26, Reference Xiao30]. We refer the reader to the recent survey [Reference Ross, Condori, Pozzi, Ross and Sola22] for more on Cesàro operator.
Recently, Galanopoulos et al. [Reference Galanopoulos, Girela and Merchán13] introduced a Cesàro-like operator
$\mathcal {C}_\mu $
on
$H(\mathbb {D})$
, which is a natural generalization of the classical Cesàro operator
$\mathcal {C}$
. They considered the following generalization: For a positive Borel measure
$\mu $
on the interval
$[0, 1)$
they define the operator
$$ \begin{align*}\mathcal{C}_\mu (f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\widehat{f}(k)\right)z^n=\int_{0}^{1}\frac{f(tz)}{1-tz}d\mu(t), \ z\in\mathbb{D} \end{align*} $$
where
$\mu _{n}$
stands for the moment of order n of
$\mu $
, that is,
$\mu _{n}=\int _{0}^{1} t^{n}d\mu (t)$
. They studied the operators
$\mathcal {C}_\mu $
acting on distinct spaces of analytic functions (e.g., Hardy spaces, Bergman spaces, the Bloch space, etc.).
The Cesàro-like operator
$\mathcal {C}_\mu $
defined above has attracted the interest of many mathematicians. For instance, Jin and Tang [Reference Jin and Tang18] studied the boundedness (resp. compactness) of
$\mathcal {C}_\mu $
from one Dirichlet-type space
$\mathcal {D}_{\alpha }$
into another one
$\mathcal {D}_{\beta }$
. Bao et al. [Reference Bao, Sun and Wulan4] studied the range of
$\mathcal {C}_\mu $
acting on
$H^{\infty }$
. Blasco [Reference Blasco6] investigated the operators
$\mathcal {C}_\mu $
induced by complex Borel measures on
$[0, 1)$
, and extended the results of [Reference Galanopoulos, Girela and Merchán13] to this more general case. Galanopoulos et al. [Reference Galanopoulos, Girela, Mas and Merchán12] studied the behavior of the operators
$\mathcal {C}_{\mu }$
on the Dirichlet space and on the analytic Besov spaces
$B_{p}$
. Recently, Bao et al. [Reference Bao, Guo, Sun and Wang3] have complectly characterized the measure
$\mu $
such that
$\mathcal {C}_\mu $
is bounded (resp. compact) on the Dirichlet space. The reader is referred to [Reference Beltrán-Meneu, Bonet and Jordá5, Reference Blasco7, Reference Galanopoulos, Girela and Merchán14, Reference Tang28, Reference Tang29, Reference Zhou31] for more on Cesàro-like operators on spaces of analytic functions.
In [Reference Bao, Sun and Wulan4], Bao et al. introduced a more general Cesàro-like operator. Suppose that
$0<\alpha <\infty $
and
$\mu $
is a finite positive Borel measure on
$[0, 1)$
. For
$f(z)=\sum _{n=0}^{\infty } a_{n}z^{n} \in H(\mathbb {D})$
, they defined
$$ \begin{align*}\mathcal {C}_{\mu,\alpha}(f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!}a_k\right)z^n, \ z\in \mathbb{D}. \end{align*} $$
A simple calculation with power series gives the integral form of
$\mathcal {C}_{\mu ,\alpha }$
as follows.
$$ \begin{align*}\mathcal{C}_{\mu,\alpha}(f)(z)=\int_{0}^{1}\frac{f(tz)}{(1-tz)^{\alpha}}d\mu(t).\end{align*} $$
For
$1<s<\infty $
, let X be a Banach subspace of
$H(\mathbb {D})$
with
$\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$
. There are many well known spaces located between the mean Lipschitz space
$\Lambda ^{s}_{\frac {1}{s}}$
and the Bloch space
$\mathcal {B}$
. In [Reference Bao, Sun and Wulan4], the authors investigated the range of
$\mathcal {C}_{\mu ,\alpha }$
acting on
$H^{\infty }$
. They proved that if
$\max \{1,\frac {1}{\alpha }\}<s<\infty $
, then
$\mathcal {C}_{\mu ,\alpha }(H^{\infty })\subset X$
if and only if
$\mu $
is an
$\alpha $
-Carleson measure. Zhou [Reference Zhou31] considered the same problem for the measure
$\mu $
supported on
$\mathbb {D}$
. Guo et al. [Reference Guo, Tang and Zhang17] investigated the boundedness (resp. compactness) of
$\mathcal {C}_{\mu ,\alpha }$
acting from X to weighted Bergman spaces
$A^{p}_{\beta }$
. Galanopoulos et al. [Reference Galanopoulos, Siskakis and Zhao15] characterized the measure
$\mu $
such that
$\mathcal {C}_{\mu ,\alpha }$
is bounded from weighted Bergman space
$A^{p}_{\beta }$
to
$A^{q}_{\beta }$
when
$1\leq p\leq q<\infty $
and
$\beta>-1$
. Sun et al. [Reference Sun, Ye and Zhou27] studied the boundedness (resp. compactness) of the operator
$\mathcal {C}_{\mu ,1}$
acting from
$B_{p}$
to X. It remains open to characterize the boundedness and the compactness of
$\mathcal {C}_{\mu ,\alpha }$
from
$B_{p}$
to
$B_{p}$
when
$ p>1$
and
$p \neq 2$
. The Besov spaces
$B_{p}$
and Bloch space
$\mathcal {B}$
are Möbius invariant and the Bloch space
$\mathcal {B}$
can be regarded as the limit case of
$B_{p}$
as
$p\rightarrow +\infty $
. The purpose of this article is describe the measure
$\mu $
such that the operator
$\mathcal {C}_{\mu ,\alpha }$
is bounded (resp. compact) from X to
$B_{p}$
for
$1\leq p<\infty $
.
Our main results are included in the following.
Theorem 1.1 Suppose
$0<\alpha <\infty $
,
$1<s<\infty $
and
$\max \{1,\frac {1}{\alpha }\}\leq p<\infty $
. Let
$\mu $
be a finite positive Borel measure on
$[0, 1)$
and X be a Banach subspace of
$H(\mathbb {D})$
with
$\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$
. Then the following statements are equivalent.
-
(1) The operator
$\mathcal {C}_{\mu ,\alpha }$
is bounded from X to
$B_{p}$
. -
(2) The operator
$\mathcal {C}_{\mu ,\alpha }$
is compact from X to
$B_{p}$
. -
(3) The measure
$\mu $
satisfies
$$ \begin{align*}\sum_{n=0}^{\infty} (n+1)^{p\alpha-1}\mu_{n}^{p}\log^{p}(n+2)<\infty.\end{align*} $$
For
$p = 1$
, we have the following corollary.
Corollary 1.2 Suppose
$1\leq \alpha <\infty $
and
$1<s<\infty $
. Let
$\mu $
be a finite positive Borel measure on
$[0, 1)$
and X be a Banach subspace of
$H(\mathbb {D})$
with
$\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$
. Then the following statements are equivalent.
-
(1) The operator
$\mathcal {C}_{\mu ,\alpha }$
is bounded from X to
$B_{1}$
. -
(2) The operator
$\mathcal {C}_{\mu ,\alpha }$
is compact from X to
$B_{1}$
. -
(3) The measure
$\mu $
satisfies
$$ \begin{align*}\sum_{n=0}^{\infty} (n+1)^{\alpha-1}\mu_{n}\log(n+2)<\infty.\end{align*} $$
-
(4) The measure
$\mu $
satisfies
$$ \begin{align*}\int_{0}^{1}\frac{\log\frac{e}{1-t}}{(1-t)^{\alpha}}d\mu(t)<\infty.\end{align*} $$
When
$p=2$
, the space
$B_{2}$
is the classic Dirichlet space
$\mathcal {D}$
, so we have the following corollary.
Corollary 1.3 Suppose
$1\leq \alpha <\infty $
and
$1<s<\infty $
. Let
$\mu $
be a finite positive Borel measure on
$[0, 1)$
and X be a Banach subspace of
$H(\mathbb {D})$
with
$\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$
. Then the following statements are equivalent.
-
(1) The operator
$\mathcal {C}_{\mu ,\alpha }$
is bounded from X to
$\mathcal {D}$
. -
(2) The operator
$\mathcal {C}_{\mu ,\alpha }$
is compact from X to
$\mathcal {D}$
. -
(3) The measure
$\mu $
satisfies
$$ \begin{align*}\sum_{n=0}^{\infty} (n+1)^{2\alpha-1}\mu_{n}^{2}\log^{2}(n+2)<\infty.\end{align*} $$
Throughout the article, the letter C will denote an absolute constant whose value depends on the parameters indicated in the parenthesis, and may change from one occurrence to another. We will use the notation
$"P\lesssim Q\text{"}$
if there exists a constant
$C=C(\cdot ) $
such that
$" P \leq CQ,\text{"}$
and
$" P \gtrsim Q\text{"}$
is understood in an analogous manner. In particular, if
$"P\lesssim Q\text{"}$
and
$ "P \gtrsim Q,\text{"}$
then we will write
$"P\asymp Q\text{"}$
.
2 Preliminaries
To prove our main results, we need some preliminary results which will be repeatedly used throughout the rest of the article. We begin with a characterization of the functions
$f\in H(\mathbb {D})$
whose sequence of Taylor coefficients is decreasing which belong to
$B^{p}$
. For a proof, see, e.g., [Reference Girela and Merchán16, Theorem 3.10] and [Reference Pavlović20].
Lemma 2.1 Let
$1<p<\infty $
and
$f(z)=\sum _{n=0}^{\infty } a_{n}z^{n}\in H(\mathbb {D})$
. Suppose that the sequence
$\{a_{n}\}_{n=0}^{\infty }$
is a decreasing sequence of nonnegative real numbers. Then
$f\in B_{p}$
if and only if
$$ \begin{align*}\sum_{n=1}^{\infty}n^{p-1}a_{n}^{p}<\infty.\end{align*} $$
The following lemma contains a characterization of
$L^{p}$
-integrability of power series with nonnegative coefficients. For a proof, see [Reference Pavlović and Mateljević21, Theorem 1].
Lemma 2.2 Let
$0<\beta ,p<\infty $
,
$\{\lambda _{n}\}_{n=0}^{\infty }$
be a sequence of nonnegative numbers. Then
$$ \begin{align*}\int_{0}^{1}(1-r)^{p\beta-1}\left(\sum_{n=0}^{\infty} \lambda_{n}r^{n}\right)^{p}dr\asymp \sum_{n=0}^{\infty} 2^{-np\beta}\left(\sum_{k\in I_{n}}\lambda_{k}\right)^{p},\end{align*} $$
where
$I_{0}=\{0\}$
,
$I_{n}=[2^{n-1},2^{n})\cap \mathbb {N}$
for
$n\in \mathbb {N}$
.
The following lemma is a consequence of Theorem 2.31 on page 192 of the classical monograph [Reference Zygmund33].
Lemma 2.3
-
(a) The Taylor coefficients
$a_{n}$
of the function have the property
$$ \begin{align*}f(z)=\frac{1}{(1-z)^{\beta}}\log^{\gamma}\frac{2}{1-z}, \ \ \beta>0,\gamma\in \mathbb{R}, \ z\in \mathbb{D}\end{align*} $$
$a_{n}\asymp n^{\beta -1}(\log (n+1))^{\gamma }$
.
-
(b) The Taylor coefficients
$a_{n}$
of the function have the property
$$ \begin{align*}f(z)=\log^{\gamma}\frac{2}{1-z}, \ \ \gamma>0, z\in \mathbb{D}\end{align*} $$
$a_{n}\asymp n^{-1}(\log (n+1))^{\gamma -1}$
.
We also need the following estimates (see, e.g., [Reference Rudin23, Proposition 1.4.10]).
Lemma 2.4 Let
$\alpha $
be any real number and
$z\in \mathbb {D}$
. Then
$$ \begin{align*}\int^{2\pi}_0\frac{d\theta}{|1-ze^{-i\theta}|^{\alpha}}\asymp \begin{cases}1, & \enspace \text{if} \ \ \alpha<1,\\ \log\frac{2}{1-|z|^2}, & \enspace \text{if} \ \ \alpha=1,\\ \frac{1}{(1-|z|^2)^{\alpha-1}} ,& \enspace \text{if}\ \ \alpha>1, \end{cases} \end{align*} $$
The following lemma is useful in dealing with the compactness. The proof is similar to that of [Reference Cowen and MacCluer9, Proposition 3.11]. The details are omitted.
Lemma 2.5 Let
$p\geq 1$
,
$s>1$
, X be a Banach subspace of
$H(\mathbb {D})$
with
$\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$
. Suppose that T is a bounded operator from X to
$B_{p}$
. Then T is compact if and only if for any bounded sequence
$\{f_{k}\}$
in X which converges to
$0$
uniformly on every compact subset of
$\mathbb {D}$
, we have
$\lim _{k\rightarrow \infty }||T(f_{k})||_{B_{p}}=0$
.
3 Proofs of the main results
We now present the proofs of Theorem 1.1.
Proof of the implication (1) ⇒ (3)
Since the definition of the space
$B_{1}$
is slightly different from
$B_{p}$
when
$p>1$
, we split the proof into
$p=1$
and
$p>1$
.
Case 1:
$p=1$
.
Assume that
$\mathcal {C}_{\mu ,\alpha }$
is bounded from X to
$B_{1}$
. Let
$g(z)=\log \frac {1}{1-z}=\sum ^{\infty }_{k=1}\frac {z^{k}}{k}$
. It is easy to check that
$g\in \Lambda ^{s}_{\frac {1}{s}}\subset X$
. This implies that
$\mathcal {C}_{\mu ,\alpha }(g)\in B_{1}$
. For
$z\in \mathbb {D}$
, by the definition of
$\mathcal {C}_{\mu ,\alpha }$
we get
$$ \begin{align*}\mathcal{C}_{\mu,\alpha}(g)"(z)=\sum_{n=0}^{\infty}\left((n+2)(n+1)\mu_{n+2}\sum_{k=1}^{n+2}\frac{\Gamma(n+2-k+\alpha)}{\Gamma(\alpha)(n+2-k)!k}\right)z^{n}.\end{align*} $$
For
$0<r<1$
, by Hardy’s inequality we have that
$$ \begin{align*}M_{1}(r,\mathcal{C}_{\mu,\alpha}(g)")\gtrsim \sum_{n=0}^{\infty} \left((n+2)\mu_{n+2}\sum_{k=1}^{n+2}\frac{\Gamma(n+2-k+\alpha)}{\Gamma(\alpha)(n+2-k)!k}\right)r^{n}.\end{align*} $$
It follows that
$$\begin{align*}\begin{aligned} 1&\gtrsim ||g||_{X}\gtrsim ||\mathcal{C}_{\mu,\alpha}(g)||_{B_{1}}=\int_{\mathbb{D}}|\mathcal{C}_{\mu,\alpha}(g)"(z)|dA(z)\\ & = 2\int_{0}^{1}M_{1}(r,\mathcal{C}_{\mu,\alpha}(g)")rdr\\ & \gtrsim \int_{0}^{1} \sum_{n=0}^{\infty} \left((n+2)\mu_{n+2}\sum_{k=1}^{n+2}\frac{\Gamma(n+2-k+\alpha)}{\Gamma(\alpha)(n+2-k)!k}\right)r^{n+1}dr\\ & \gtrsim \sum_{n=0}^{\infty} \mu_{n+2}\sum_{k=1}^{n+2}\frac{\Gamma(n+2-k+\alpha)}{\Gamma(\alpha)(n+2-k)!k}. \end{aligned} \end{align*}$$
Using the Stirling’s formula, we get
$$ \begin{align*}\sum_{k=1}^{n+2}\frac{\Gamma(n+2-k+\alpha)}{\Gamma(\alpha)(n+2-k)!k}\asymp \sum_{k=1}^{n+2}\frac{(n+3-k)^{\alpha-1}}{k}.\end{align*} $$
For
$n\geq 1 $
, simple estimations lead us to the following
$$\begin{align*}\begin{aligned} \sum_{k=1}^{n+2}\frac{(n+3-k)^{\alpha-1}}{k}&=\left(\sum_{k=1}^{[\frac{n+2}{2}]}+\sum_{k=[\frac{n+2}{2}]+1}^{n+2}\right)\frac{(n+3-k)^{\alpha-1}}{k}\\ & \asymp (n+1)^{\alpha-1}\sum_{k=1}^{[\frac{n+2}{2}]}\frac{1}{k} +\frac{1}{n+1} \sum_{k=[\frac{n+2}{2}]+1}^{n+2}(n+3-k)^{\alpha-1}\\ & \asymp (n+1)^{\alpha-1}\log(n+2)+(n+1)^{\alpha-1}\\ & \asymp (n+1)^{\alpha-1}\log(n+2). \end{aligned} \end{align*}$$
Therefore,
$$\begin{align*}\begin{aligned} 1&\gtrsim \sum_{n=0}^{\infty} \mu_{n+2}\sum_{k=1}^{n+2}\frac{\Gamma(n+2-k+\alpha)}{\Gamma(\alpha)(n+2-k)!k}\\ & \gtrsim \sum_{n=0}^{\infty} (n+1)^{\alpha-1}\mu_{n}\log(n+2). \end{aligned} \end{align*}$$
Case 2:
$p>1$
.
Let q be the conjugate index of p, that is,
$\frac {1}{p}+\frac {1}{q}=1$
. It is known that
$(B_{q})^{\ast }\cong B_{p}$
(see [Reference Zhu32, Theorem 5.24]) under the paring
$$ \begin{align*}\langle F,G\rangle=\int_{\mathbb{D}}F'(z)\overline{G'(z)}dA(z), \ \ F\in B_{p},G\in B_{q}.\end{align*} $$
This means that
$\mathcal {C}_{\mu ,\alpha }$
is bounded from X to
$B_{p}$
if and only if
$$ \begin{align*}|\langle \mathcal{C}_{\mu,\alpha}(F),G\rangle|\lesssim ||F||_{X}||G||_{B_{q}}\ \mbox{for all}\ F\in X, G\in B_{q}.\end{align*} $$
Now, suppose that
$\mathcal {C}_{\mu ,\alpha }$
is bounded from X to
$B_{p}$
. Take
$g(z)=\sum _{n=0}^{\infty }\widehat { g}(n)z^{n}\in B_{q}$
and the sequence of its Taylor coefficients is a decreasing sequence of the nonnegative real numbers. Let
$f(z)=\log \frac {1}{1-z}=\sum _{n=1}\frac {z^{n}}{n}\in X$
, we have that
$$ \begin{align*}|\langle \mathcal{C}_{\mu,\alpha}(f),g\rangle|\lesssim ||f||_{X}||g||_{B_{q}}.\end{align*} $$
A simple calculation shows that
$$ \begin{align*}|\langle \mathcal{C}_{\mu,\alpha}(f),g\rangle|=\sum^{\infty}_{n=1}n\mu_{n}\left(\sum_{k=1}^{n}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!k}\right)\widehat{g}(n).\end{align*} $$
This implies that
$$ \begin{align*}|\langle \mathcal{C}_{\mu,\alpha}(f),g\rangle|=\sum^{\infty}_{n=1}n^{\frac{1}{q}}\mu_{n}\left(\sum_{k=1}^{n}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!k}\right)\widehat{g}(n)n^{\frac{q-1}{q}}<\infty.\end{align*} $$
By Lemma 2.1, the sequence
$\{\widehat {g}(n)n^{\frac {q-1}{q}}\}_{n=1}^{\infty }\in l^{q}$
. The well known duality
$(l^{q})^{\ast }=l^{p}$
yields that
$$ \begin{align*}\left\{n^{\frac{1}{q}}\mu_{n}\left(\sum_{k=1}^{n}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!k}\right)\right\}^{\infty}_{n=1}\in l^{p}.\end{align*} $$
Using the estimate
$\sum _{k=1}^{n}\frac {\Gamma (n-k+\alpha )}{\Gamma (\alpha )(n-k)!k}\asymp (n+1)^{\alpha -1}\log (n+2)$
we deduce that
$$ \begin{align*}\sum_{n=0}^{\infty} (n+1)^{p\alpha-1}\mu_{n}^{p}\log^{p}(n+2)<\infty.\end{align*} $$
Proof of the implication (3) ⇒ (2)
Let
$\{f_{k}\}_{k=1}^{\infty }$
be a bounded sequence in X which converges to
$0$
uniformly on every compact subset of
$\mathbb {D}$
. Without loss of generality, we may assume that
$f_{k}(0)=0$
and
$\sup _{k\geq 1}||f||_{X}\leq 1$
. It suffices to prove that
$\lim _{k\rightarrow \infty }||\mathcal {C}_{\mu ,\alpha }(f_{k})||_{B_{p}}=0$
by using Lemma 2.5. As before, we divide the proof into
$p = 1$
and
$p>1$
.
Case 1:
${p>1}$
.
Assume that
$\sum _{n=1}^{\infty }(n+1)^{p\alpha -1}\mu _{n}^{p}\log ^{p}(n+1)<\infty $
, then
$$\begin{align*}\begin{aligned} \sum_{n=1}^{\infty}(n+1)^{p\alpha-1}\mu_{n}^{p}\log^{p}(n+1)&= \sum_{n=1}^{\infty}\left(\sum_{k=2^{n-1}}^{2^{n}-1}(k+1)^{p\alpha-1}\mu_{k}^{p}\log^{p}(k+1)\right)\\ & \gtrsim \sum_{n=1}^{\infty}2^{np\alpha}\mu_{2^{n}}^{p}\log^{p}(2^{n}+1)\\ & \gtrsim \sum_{n=1}^{\infty}2^{-n(p-1)}\left(\sum_{k=2^{n}}^{2^{n+1}-1}(k+1)^{\alpha-\frac{1}{p}}\mu_{k}\log(k+1)\right)^{p}. \end{aligned} \end{align*}$$
This shows that
$$ \begin{align*}\sum_{n=1}^{\infty}2^{-n(p-1)}\left(\sum_{k=2^{n}}^{2^{n+1}-1}(k+1)^{\alpha-\frac{1}{p}}\mu_{k}\log(k+1)\right)^{p}<\infty.\end{align*} $$
By Lemma 2.2 we have that
$$\begin{align*}\begin{aligned} & \ \ \ \ \int_{0}^{1}(1-r)^{p-2}\left(\sum_{n=0}^{\infty} (n+1)^{\alpha-\frac{1}{p}}\mu_{n}\log(n+1)r^{n}\right)^{p}dr \\ &\asymp \sum_{n=0}^{\infty}2^{-n(p-1)}\left(\sum_{k=2^{n}}^{2^{n+1}-1}(k+1)^{\alpha-\frac{1}{p}}\mu_{k}\log(k+1)\right)^{p}<\infty. \end{aligned} \end{align*}$$
Therefore, for any
$\varepsilon>0$
there exists a
$0<r_{0}<1$
such that
$$ \begin{align} \int_{r_{0}}^{1}(1-r)^{p-2}\left(\sum_{n=0}^{\infty} (n+1)^{\alpha-\frac{1}{p}}\mu_{k}\log(k+1)r^{n}\right)^{p}dr<\varepsilon. \end{align} $$
It is clear that
$$\begin{align*}\begin{aligned} ||\mathcal{C}_{\mu,\alpha}(f_{k})||^{p}_{B_{p}}&=\left(\int_{|z|\leq r_{0}}+\int_{r_{0}<|z|<1}\right)|\mathcal{C}_{\mu,\alpha}(f_{k})'(z)|^{p}(1-|z|)^{p-2}dA(z)\\ & := J_{1,k}+J_{2,k}. \end{aligned} \end{align*}$$
By the integral representation of
$\mathcal {C}_{\mu ,\alpha }$
we get
$$ \begin{align} \mathcal{C}_\mu(f_{k})'(z)=\int_{0}^{1}\frac{tf^{\prime}_{k}(tz)}{(1-tz)^{\alpha}}d\mu(t)+\int_{0}^{1}\frac{\alpha tf_{k}(tz)}{(1-tz)^{\alpha+1}}d\mu(t).\end{align} $$
Cauchy integral theorem implies that the sequence
$\{f^{\prime }_{k}\}_{k=1}^{\infty }$
is also converge to
$0$
uniformly on every compact subset of
$\mathbb {D}$
. Thus, for
$|z|\leq r_{0}$
we have that
$$\begin{align*}\begin{aligned} |\mathcal{C}_{\mu,\alpha}(f_{k})'(z)|& \lesssim \int_{0}^{1}\frac{|f_{k}'(tz)|}{|1-tz|^{\alpha}}+ \frac{|f_{k}(tz)|}{|1-tz|^{\alpha+1}}d\mu(t)\\ & \lesssim \sup_{|w|<r_{0}}\left(|f_{k}(w)|+|f_{k}'(w)|\right)\int_{0}^{1}\frac{1}{(1-tr_{0})^{\alpha+1}}d\mu(t)\\ & \lesssim \sup_{|w|<r_{0}}\left(|f_{k}(w)|+|f_{k}'(w)|\right). \end{aligned} \end{align*}$$
It follows that
$$ \begin{align*}J_{1,k} \rightarrow 0, \ (k\rightarrow \infty).\end{align*} $$
Next, we estimate
$J_{2,k}$
.
Since
$X\subset \mathcal {B}$
, we have
$$ \begin{align} |f_{k}(z)|\lesssim \log\frac{e}{1-|z|}\ \ \mbox{and}\ \ |f_{k}'(z)|\lesssim \frac{1}{1-|z|}\ \ \mbox{for all}\ \ k\geq 1, z\in \mathbb{D} .\end{align} $$
By (2) and (3), Minkowski inequity, Lemma 2.4 we get
$$ \begin{align*}M_{p}(r,\mathcal{C}_{\mu,\alpha}(f_{k})') &= \left\{\int_{0}^{2\pi}\left|\int_{0}^{1}\frac{tf_{k}'(tre^{i\theta})}{(1-tre^{i\theta})^{\alpha}}+\frac{tf_{k}(tre^{i\theta})}{(1-tre^{i\theta})^{\alpha+1}}d\mu(t)\right|{}^{p}d\theta\right\}^{\frac{1}{p}} \\[3pt]&\lesssim \left\{\int_{0}^{2\pi}\left(\int_{0}^{1}\frac{1}{(1-tr)|1-tre^{i\theta}|^{\alpha}}d\mu(t)\right)^{p}d\theta\right\}^{\frac{1}{p}}\\[3pt]& \quad +\left\{\int_{0}^{2\pi}\left(\int_{0}^{1}\frac{\log\frac{e}{1-tr}}{|1-tre^{i\theta}|^{\alpha+1}}d\mu(t)\right)^{p}d\theta\right\}^{\frac{1}{p}}\\[3pt]&\lesssim \int_{0}^{1}\frac{1}{1-tr}\left(\int_{0}^{2\pi}\frac{d\theta}{|1-tre^{i\theta}|^{p\alpha}}\right)^{\frac{1}{p}}d\mu(t)\\[3pt]&\quad + \int_{0}^{1}\log\frac{e}{1-tr}\left(\int_{0}^{2\pi}\frac{d\theta}{|1-tre^{i\theta}|^{p(\alpha+1)}}\right)^{\frac{1}{p}}d\mu(t)\\[3pt]&\lesssim\int_{0}^{1}H(t,r)d\mu(t), \end{align*} $$
where
$$\begin{align*}H(t,r)=\left\{\!\!\! \begin{array}{cc} \displaystyle{\frac{\log\frac{e}{1-tr}}{(1-tr)^{\alpha+1-\frac{1}{p}}},} & \text{if} \ \ \displaystyle{p>\frac{1}{\alpha}},\\ \displaystyle{\frac{\log\frac{e}{1-tr}}{1-tr}, } & \text{if} \ \ \displaystyle{p=\frac{1}{\alpha}}. \end{array}\right.\end{align*}$$
Lemma 2.3 yields that
$$ \begin{align*}M_{p}(r,\mathcal{C}_{\mu,\alpha}(f_{k})') \lesssim\int_{0}^{1}H(t,r)d\mu(t) \asymp \sum_{n=0}^{\infty} (n+1)^{\alpha-\frac{1}{p}}\mu_{n}\log(n+1)r^{n}.\end{align*} $$
This together with (1) implies
$$\begin{align*}\begin{aligned} J_{2,k}& =\int_{r_{0}<|z|<1}|\mathcal{C}_{\mu,\alpha}(f_{k})'(z)|^{p}(1-|z|)^{p-2}dA(z)\\ & \lesssim \int_{r_{0}}^{1}(1-r)^{p-2}M^{p}_{p}(r,\mathcal{C}_{\mu,\alpha}(f_{k})')dr\\ & \lesssim \int_{r_{0}}^{1}(1-r)^{p-2}\left(\sum_{n=0}^{\infty} (n+1)^{\alpha-\frac{1}{p}}\mu_{n}\log(n+1)r^{n}\right)^{p}dr\\ & \lesssim \varepsilon. \end{aligned} \end{align*}$$
Consequently,
$$ \begin{align*}\lim_{k\rightarrow \infty}||\mathcal{C}_{\mu,\alpha}(f_{k})||_{B_{p}}=0.\end{align*} $$
Case 2:
$ {p=1}$
.
When
$p=1$
, Lemma 2.3 shows that the condition
$\sum _{n=0}^{\infty } (n+1)^{\alpha -1}\mu _{n}\log (n+2)<\infty $
is equivalent to
$\int _{0}^{1}\frac {\log \frac {e}{1-t}}{(1-t)^{\alpha }}d\mu (t)<\infty .$
Hence, for any
$\varepsilon>0$
there exists a
$0<t_{0}<1$
such that
$$ \begin{align} \int_{t_{0}}^{1}\frac{\log\frac{e}{1-t}}{(1-t)^{\alpha}}d\mu(t)<\varepsilon. \end{align} $$
By the integral representation of
$\mathcal {C}_{\mu ,\alpha }$
we have
$$ \begin{align} \mathcal{C}_{\mu,\alpha}(f)"(z)=\int_{0}^{1}\left(\frac{t^{2}f"(tz)}{(1-tz)^{\alpha}} +\frac{2\alpha t^{2}f'(tz)}{(1-tz)^{\alpha+1}}+ \frac{\alpha(\alpha+1)t^{2}f(tz)}{(1-tz)^{\alpha+2}}\right)d\mu(t).\end{align} $$
For
$0<r<1$
, we have
$$\begin{align*}\begin{aligned} M_{1}(r, \mathcal{C}_{\mu,\alpha}(f_{k})")&\lesssim \sup_{|w|\leq t_{0}}\left(|f_{k}"(w)|+|f_{k}'(w)|+|f_{k}(w)|\right)\int_{0}^{t_{0}}\frac{d\mu(t)}{(1-t_{0}r)^{\alpha+2}}\\ & \ \ \ \ +\int_{0}^{2\pi}\int_{t_{0}}^{1}\frac{|f_{k}"(tz)|}{|1-tre^{i\theta}|^{\alpha}}+\frac{|f_{k}'(tz)|}{|1-tre^{i\theta}|^{\alpha+1}} +\frac{|f_{k}(tz)|}{|1-tre^{i\theta}|^{\alpha+2}}d\mu(t)d\theta. \end{aligned} \end{align*}$$
Since
$\{f_{k}\}\subset X\subset \mathcal {B}$
, we see that
$$ \begin{align} |f_{k}"(z)|\lesssim \frac{1}{(1-|z|)^{2}}\ \mbox{ for all }\ k \geq 1.\end{align} $$
The assumption of p means that
$\alpha \geq 1$
. By Fubini’s theorem, (3), (6), and Lemma 2.4, we have
$$\begin{align*}\begin{aligned} &\ \ \ \ \int_{0}^{2\pi}\int_{t_{0}}^{1}\frac{|f_{k}"(tre^{i\theta})|}{|1-tre^{i\theta}|^{\alpha}}+\frac{|f_{k}'(tre^{i\theta})|}{|1-tre^{i\theta}|^{\alpha+1}} +\frac{|f_{k}(tre^{i\theta})|}{|1-tre^{i\theta}|^{\alpha+2}}d\mu(t)d\theta\\ &\lesssim \int_{t_{0}}^{1}\int_{0}^{2\pi}\left(\frac{1}{(1-tr)^{2}|1-tre^{i\theta}|^{\alpha}}+\frac{1}{(1-tr)|1-tre^{i\theta}|^{\alpha+1}} +\frac{\log\frac{e}{1-tr}}{|1-tre^{i\theta}|^{\alpha+2}}\right)d\theta d\mu(t)\\ &\lesssim \int_{t_{0}}^{1} \frac{\log\frac{e}{1-tr}}{(1-tr)^{\alpha+1}}d\mu(t). \end{aligned} \end{align*}$$
Hence,
$$\begin{align*}\begin{aligned} & \ \ \ \ \int_{t_{0}<|z|<1}|\mathcal{C}_{\mu,\alpha}(f_{k})"(z)|dA(z)\\ &= 2\int_{t_{0}}^{1}M_{1}(r, \mathcal{C}_{\mu,\alpha}(f_{k})")rdr\\ & \lesssim \sup_{|w|\leq t_{0}}\left(|f_{k}"(w)|+|f_{k}'(w)|+|f_{k}(w)|\right) +\int_{t_{0}}^{1} \int_{t_{0}}^{1} \frac{\log\frac{e}{1-tr}}{(1-tr)^{\alpha+1}}d\mu(t)dr\\ & \lesssim \sup_{|w|\leq t_{0}}\left(|f_{k}"(w)|+|f_{k}'(w)|+|f_{k}(w)|\right) +\int_{t_{0}}^{1}\log\frac{e}{1-t}\int_{0}^{1}\frac{dr}{(1-tr)^{\alpha+1}}d\mu(t)\\ & \lesssim \sup_{|w|\leq t_{0}}\left(|f_{k}"(w)|+|f_{k}'(w)|+|f_{k}(w)|\right)+\int_{t_{0}}^{1}\frac{\log\frac{e}{1-t}}{(1-t)^{\alpha}}d\mu(t)\\ &\lesssim \sup_{|w|\leq t_{0}}\left(|f_{k}"(w)|+|f_{k}'(w)|+|f_{k}(w)|\right)+\varepsilon. \end{aligned} \end{align*}$$
The uniform convergence of
$\{f_{k}\}$
on compact subsets of
$\mathbb {D}$
implies that
$$ \begin{align*}\int_{|z|\leq t_{0}}|\mathcal{C}_{\mu,\alpha}(f_{k})"(z)|dA(z)\lesssim \sup_{|w|\leq t_{0}}\left(|f_{k}"(w)|+|f_{k}'(w)|+|f_{k}(w)|\right)\rightarrow 0, \ \ \mbox{as}\ k\rightarrow0.\end{align*} $$
Since
$f_{k}(0)=0$
, so we have that
$|\mathcal {C}_{\mu ,\alpha }(f_{k})(0)|+|\mathcal {C}_{\mu ,\alpha }(f_{k})'(0)|\asymp |f_{k}'(0)|\rightarrow 0$
as
$k\rightarrow \infty $
.
Therefore, we deduce that
$$ \begin{align*}\lim_{k\rightarrow \infty}||\mathcal{C}_{\mu,\alpha}(f_{k})||_{B_{1}}=0.\end{align*} $$
Thus, the operator
$\mathcal {C}_{\mu ,\alpha }$
is compact from X to
$B_{1}$
.
Conflicts of interest
The authors declare that there is no conflict of interest.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study: the article describes entirely theoretical research.
