Abstract
Let \(\mu \) be a finite positive Borel measure on [0, 1) and let \(H(\mathbb {D})\) be the space of all analytic function in the unit disc \(\mathbb {D}\). The Cesàro-like operator \(\mathcal {C}_\mu \) is defined in \(H(\mathbb {D})\) as follows: If \(f \in H(\mathbb {D})\), \(f(z)=\sum _{n=0}^{\infty }a_{n}z^{n} (z\in \mathbb {D})\), then
where for \(n\ge 0\), \(\mu _n\) denotes the n-th moment of the measure \(\mu \), that is, \(\mu _n=\int _{0}^{1} t^{n}d\mu (t)\). For \(s > 1\), let X be a Banach subspace of \(H(\mathbb {D})\) lying between the mean Lipschtz space \(\Lambda ^{s}_{\frac{1}{s}}\) and the Bloch space \(\mathcal {B}\). In this paper we characterize the measures \(\mu \) as above for which \(\mathcal {C}_\mu \) is bounded (compact) from X into any of the Hardy spaces \(H^{p} ~ (1\le p\le \infty )\).
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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 \(\mathbb {D}\).
The Bloch space \(\mathcal {B}\) consists of those functions \(f\in H(\mathbb {D}) \) for which
Let \(0<p\le \infty \), the classical Hardy space \(H^p\) consists of those functions \(f\in H(\mathbb {D})\) for which
where
For \(0<p<\infty \), the Dirichlet-type space \(D^{p}_{p-1}\) is the space of \(h\in H(\mathbb {D})\) such that
When \(p=2\), the space \(D^{2}_{1}\) is just the Hardy space \(H^{2}\).
The Hardy–Littlewood space HL(p) consists of those function \(f\in H(\mathbb {D})\) for which
The space \(HL(\infty )\) consist of \(f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}\in H(\mathbb {D})\) such that
We also need the space \(H(\infty ,p)=\{h\in H(\mathbb {D}): \Vert h\Vert ^{p}_{H(\infty ,p)}=\int _{0}^{1}M_{\infty }^{p}(r,h)dr<\infty \}\).
It is well known that
and
The proofs of (1) and (2) can be found in [10, 11, 18], and the proof of (3) appears in [21, p. 127] and [14, Lemma 4].
Let \(1\le p<\infty \) and \(0<\alpha \le 1\), 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 \(t\rightarrow 0\). 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 [10]) that \(\Lambda ^p_\alpha \) is a subset of \(H^p\) and
The space \(\Lambda ^p_\alpha \) is a Banach space with the norm \(\Vert \cdot \Vert _{\Lambda ^p_\alpha }\) given by
It is known (see e.g. [5, Theorem 2.5]) that
The Cesàro operator \(\mathcal {C}\) is defined in \(H(\mathbb {D})\) as follows:
If \(f\in H(\mathbb {D})\), \(f(z)=\sum _{n=0}^\infty a_nz^n\), then
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, Bloch space, \(Q_{p}\) space, mixed norm space have been widely studied. See, e.g., [1, 2, 6, 9, 19, 23,24,25,26] and the references therein.
Recently, Galanopoulos et al. [12] introduced a Cesàro-like operator \(\mathcal {C}_\mu \) on \(H(\mathbb {D})\), which is a natural generalization of the classical Cesàro operator \(\mathcal {C}\). For a finite positive Borel measure \(\mu \) on the interval [0, 1), the Cesàro-like operator \(\mathcal {C}_\mu \) is defined on \(H(\mathbb {D})\) as follows:
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 space, Bergman space, Bloch space, etc.).
The Cesàro-like operator \(\mathcal {C}_\mu \) defined above has attracted the interest of many mathematicians. For instance, Jin and Tang [16] studied the boundedness (compactness) of \(\mathcal {C}_\mu \) from one Dirichlet-type space \(\mathcal {D}_{\alpha }\) into another one \(\mathcal {D}_{\beta }\). Bao, Sun and Wulan [3] studied the range of \(\mathcal {C}_\mu \) acting on \(H^{\infty }\). Blasco [4] investigated the operators \(\mathcal {C}_\mu \) induced by complex Borel measures on [0, 1). Galanopoulos et al. [13] studied the behaviour of the operators \(\mathcal {C}_\mu \) on the Dirichlet space and on the analytic Besov spaces. The operators \(\mathcal {C}_\mu \) associated to arbitrary complex Borel measures on \(\mathbb {D}\) the reader is referred to Galanopoulos et al. [15] and Zhou [27].
The following result was proved by Blasco [4, Theorem 3.17].
Theorem A
Let \(\mu \) be a finite positive Borel measure on [0, 1). Then \(\mathcal {C}_\mu : \mathcal {B}\rightarrow H^{2}\) is bounded if and only if
It is natural to seek conditions on the measure \(\mu \) which makes the operator \(\mathcal {C}_\mu \) bounded (compact) from \(\mathcal {B}\) to \(H^{p}\) when \(p\ne 2\). The result for \(p\ne 2\) has not been given in the existing literature. In this paper, we extend the Bloch space \(\mathcal {B}\) to a class of spaces X and consider the target space \(H^{p}\) with any \(1\le p\le \infty \), where X is a Banach subspace of \(H(\mathbb {D})\) with \(\Lambda ^{s}_{\frac{1}{s}}\subset X\subset \mathcal {B}\) (\(1<s<\infty )\). We complete characterize the measure \(\mu \) for which \(\mathcal {C}_\mu \) is bounded (compact) from X to \(H^{p}\) \((1\le p\le \infty )\). It turns out that the boundedness and compactness from X to \(H^{p}\) are equivalent. This conclusion appears to be new even for \(p = 2\). At the end of the article, we also consider the case \(0<p<1\).
Our main results are included in the following.
Theorem 1.1
Suppose \(1\le p<\infty \), \(1<s<\infty \), and let \(\mu \) be a finite positive Borel measure on [0, 1). Let \(Y_{p}\in \{D^{p}_{p-1}, H^{p}, HL(p), H(\infty ,p)\}\) and let 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 \(\mathcal {C}_\mu \) is bounded from X to \(Y_{p}\).
-
(2)
The operator \(\mathcal {C}_\mu \) is compact from X to \(Y_{p}\).
-
(3)
The measure \(\mu \) satisfies
Theorem 1.2
Let \(1<s<\infty \) and \(\mu \) be a finite positive Borel measure on [0, 1). Let X be a Banach subspace of \(H(\mathbb {D})\) with \(\Lambda ^{s}_{\frac{1}{s}}\subset X\subset \mathcal {B}\). Then the operator \(\mathcal {C}_\mu \) is bounded from X in \(H^{\infty }\) if and only if
Throughout the paper, the measure \(\mu \) will be a positive finite Borel measure on the radius [0, 1), 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 \(``P\lesssim Q"\) if there exists a constant \(C=C(\cdot ) \) such that \(`` P \le CQ"\), and \(`` P \gtrsim Q"\) is understood in an analogous manner. In particular, if \(``P\lesssim Q"\) and \( ``P \gtrsim Q"\), then we will write \(``P\asymp Q"\).
2 Proofs and some related results
We begin with some preliminary results which will be repeatedly used throughout the rest of the paper. The first lemma contains a characterization of \(L^{p}\)-integrability of power series with nonnegative coefficients. For a proof, see [20, Theorem 1].
Lemma 2.1
Let \(0<\beta ,p<\infty \), \(\{\lambda _{n}\}_{n=0}^{\infty }\) be a sequence of non-negative numbers. Then
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 [28].
Lemma 2.2
-
(a)
The Taylor coefficients \(a_{n}\) of the function
$$\begin{aligned} f(z)=\frac{1}{(1-z)^{\beta }}\log ^{\gamma }\frac{2}{1-z}, \quad \beta >0,\gamma \in \mathbb {R}, \quad z\in \mathbb {D} \end{aligned}$$have the property \(a_{n}\asymp n^{\beta -1}(\log (n+1))^{\gamma }\).
-
(b)
The Taylor coefficients \(a_{n}\) of the function
$$\begin{aligned} f(z)=\log ^{\gamma }\frac{2}{1-z}, \quad \gamma >0,\quad z\in \mathbb {D} \end{aligned}$$have the property \(a_{n}\asymp n^{-1}(\log (n+1))^{\gamma -1}\).
We also need the following estimates (see, e.g. Proposition 1.4.10 in [22]).
Lemma 2.3
Let \(\alpha \) be any real number and \(z\in \mathbb {D}\). Then
Proof of the implication \((1)\Rightarrow (3)\) Assume \(\mathcal {C}_\mu \) is bounded from X to \(Y_{p}\). Let \(f(z)=\log \frac{1}{1-z}\), then it is easy to check that \(f\in \Lambda ^{s}_{\frac{1}{s}}\subset X\). This implies that \(\mathcal {C}_\mu (f)\in Y_{p}\).
Case \(Y_{p}=H(\infty ,p)\). For \(0<r<1\), we have
Hence, by Lemma 2.1 we have
Case \(Y_{p}\ne H(\infty ,p)\). If \(p\ge 2\), then (2) and (3) show that \(Y_{p}\subset H(\infty ,p)\). This implies that
The desired result follows from the previous case.
If \(1\le p\le 2\), then (1) shows that \(Y_{p}\subset HL(p)\). Hence,
\(\square \)
Proof of the implication \((3)\Rightarrow (2)\) 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 \(f_{k}(0)=0\) for all \(k\ge 1\) and \(\sup _{k\ge 1}\Vert f\Vert _{X}\le 1\).
Case \(1\le p\le 2\). Assume \(\sum _{n=1}^{\infty }(n+1)^{p-2}\mu _{n}^{p}\log ^{p}(n+1)<\infty \), then
It follows that
By Lemma 2.1 we have that
Therefore, for any \(\varepsilon >0\) there exists a \(0<r_{0}<1\) such that
By (1) and (3), we have that \(D_{p-1}^{p}\subset Y_{p}\). Hence, it is suffices to prove that
It is clear that
By the integral representation of \(\mathcal {C}_\mu \) we get
Since \(\{f_{k}\}_{k=1}^{\infty }\) is converge to 0 uniformly on every compact subset of \(\mathbb {D}\), the Cauchy’s integral theorem implies that \(\{f'_{k}\}_{k=1}^{\infty }\) converges to 0 uniformly on every compact subset of \(\mathbb {D}\). Thus, for \(|z|\le r_{0}\), we have that
It follows that
Next, we estimate \(J_{2,k}\).
Since \(X\subset \mathcal {B}\), for each \(k\ge 1\), we have
By (5) and (6), Minkowski’s inequity, Lemma 2.3 we have that
where
Lemma 2.2 yields that
This together with (4) imply that
Consequently,
Case \(p>2\).
By (2) and (3) we see that \(HL(p)\subset Y_{p}\). To complete the proof, we have to prove that \(\lim _{k\rightarrow \infty }\Vert \mathcal {C}_\mu (f_{k})\Vert _{HL(p)}=0.\)
Since \(\sum _{n=1}^{\infty }(n+1)^{p-2}\mu _{n}^{p}\log ^{p}(n+1)<\infty \), for any \(\varepsilon >0\) there exists a positive integer N such that
Set
Since \(\{f_{k}\}_{k=1}^{\infty }\) converges to 0 uniformly on every compact subsets of \(\mathbb {D}\), it follows that \(\widehat{f_{k}}(j)\longrightarrow 0\) as \(k\rightarrow \infty \) for every j. This yields that there exists \(K_{0}\in \mathbb {N}\) such that
Note that \(\{f_{k}\}_{k=1}^{\infty }\subset X\subset \mathcal {B}\), then it follows from Corollary D in [17] we have that
Hence, for \(k> K_{0}\), by (7)–(9) we have that
Therefore,
The proof is complete. \(\square \)
Taking \(p = 1\) in Theorem 1.1, we can obtain the following corollary.
Corollary 2.4
Let \(1<s<\infty \), \(\mu \) be a finite positive Borel measure on [0, 1). Let X be a Banach subspace of \(H(\mathbb {D})\) with \(\Lambda ^{s}_{\frac{1}{s}}\subset X\subset \mathcal {B}\) and \(Y_{1}\in \{D, H^{1}, HL(1), H(\infty ,1)\}\). Then the following statements are equivalent.
-
(1)
The operator \(\mathcal {C}_\mu \) is bounded from X into \(Y_{1}\).
-
(2)
The operator \(\mathcal {C}_\mu \) is compact from X in to \(Y_{1}\).
-
(3)
The measure \(\mu \) satisfies
Carleson measures play a key role when we study the Cesàro-like operators. Recall that if \(\mu \) is a positive Borel measure on [0, 1), \(0\le \gamma <\infty \) and \(0< s < \infty \), then \(\mu \) is a \(\gamma \)-logarithmic s-Carleson measure if there exists a positive constant C such that
In particular, \(\mu \) is an s-Carleson measure if \(\gamma =0\).
In the following, we provide a sufficient condition and a necessary condition in term of Carleson-type measure.
Corollary 2.5
Suppose \(1<p<\infty \), \(1<s<\infty \) and \(\gamma >1+\frac{1}{p}\), \(\mu \) is a positive Borel measure on [0, 1). Let \(Y_{p}\in \{D^{p}_{p-1}, H^{p}, HL(p), H(\infty ,p)\}\) and let X be a Banach subspace of \(H(\mathbb {D})\) with \(\Lambda ^{s}_{\frac{1}{s}}\subset X\subset \mathcal {B}\), then the following statements hold.
-
(1)
If \(\mu \) is a \(\gamma \)-logarithmic \(1-\frac{1}{p}\)-Carleson measure, then \(\mathcal {C}_\mu \) is bounded (equivalent to compact) from X to \( Y_{p}\).
-
(2)
If \(\mathcal {C}_\mu \) is bounded from X to \( Y_{p}\), then \(\mu \) is a 1-logarithmic \(1-\frac{1}{p}\)-Carleson measure.
Proof
-
(1)
Suppose \(\mu \) is a \(\gamma \)-logarithmic \(1-\frac{1}{p}\)-Carleson measure. Integrating by parts we have
$$\begin{aligned} \begin{aligned} \mu _{n}&=\int _{0}^{1}t^{n}d\mu (t)=n \int _{0}^{1}t^{n-1}\mu ([t,1))dt\\&\lesssim n \int _{0}^{1}t^{n-1}(1-t)^{1-\frac{1}{p}}\log ^{-\gamma }\frac{e}{1-t}dt.\\ \end{aligned} \end{aligned}$$It is easy to estimate that
$$\begin{aligned} n \int _{0}^{1}t^{n-1}(1-t)^{1-\frac{1}{p}}\log ^{-\gamma }\frac{e}{1-t}dt \asymp \frac{\log ^{-\gamma }(n+1)}{(n+1)^{1-\frac{1}{p}}}. \end{aligned}$$This shows that
$$\begin{aligned} \mu _{n}= O\left( \frac{\log ^{-\gamma }(n+1)}{(n+1)^{1-\frac{1}{p}}}\right) . \end{aligned}$$Hence,
$$\begin{aligned} \sum _{n=0}^{\infty }(n+1)^{p-2}\mu _{n}^{p}\log ^{p}(n+1)\lesssim \sum _{n=0}^{\infty }\frac{1}{(n+1)\log ^{p(\gamma -1)}(n+1)}\lesssim 1. \end{aligned}$$Now, Theorem 1.1 implies that \(\mathcal {C}_\mu \) is bounded (equivalent to compact) from X to \( Y_{p}\).
-
(2)
If \(\mathcal {C}_\mu \) is bounded from X to \( Y_{p}\), then for any \(N\ge 2\) we have
$$\begin{aligned} \begin{aligned} 1&\gtrsim \sum _{n=0}^{\infty }(n+1)^{p-2}\mu _{n}^{p}\log ^{p}(n+1)\\&\gtrsim \sum _{n=0}^{N}(n+1)^{p-2}\mu _{n}^{p}\log ^{p}(n+1)\\&\gtrsim \mu ^{p}_{N} \sum _{n=0}^{N}(n+1)^{p-2}\log ^{p}(n+1)\\&\asymp \mu ^{p}_{N}N^{p-1} \log ^{p}(N+1). \end{aligned} \end{aligned}$$This yields that \(\mu \) is a 1-logarithmic \(1-\frac{1}{p}\)-Carleson measure.
\(\square \)
Proof of Theorem 1.2 Suppose \(\int _{0}^{1}\frac{\log \frac{e}{1-t}}{1-t}d\mu (t)<\infty \), then for any \(f\in X\) we have
This shows that \(\mathcal {C}_\mu \) is bounded from X to \(H^{\infty }\).
Conversely, assume that \(\mathcal {C}_\mu \) is bounded from X to \(H^{\infty }\). Take \(g(z)=\log \frac{e}{1-z}\in X\), then
By the Lebesgue control convergence theorem we have \(\int _{0}^{1}\frac{\log \frac{e}{1-t}}{1-t}d\mu (t)<\infty \). \(\square \)
Remark 2.6
In contrast with what happens in the case \(1\le p<\infty \), Theorem 1.2 is no longer valid for \(HL(\infty )\). In fact, \(\mathcal {C}_\mu \) is bounded from X to \(HL(\infty )\) if and only if \(\mu \) is a 1-logarithmic 1-Carleson measure. The proof is not difficult, we leave it to the interested reader. However, the condition \(\int _{0}^{1}\frac{\log \frac{e}{1-t}}{1-t}d\mu (t)<\infty \) implies that \(\mu \) is a 1-logarithmic 1-Carleson measure. But the reverse is not true, as the measure \(d\mu (t)=\log ^{-1}\frac{e}{1-t}dt\) shows. It is easy to check that \(\mu \) is a 1-logarithmic 1-Carleson measure but \(\int _{0}^{1}\frac{\log \frac{e}{1-t}}{1-t}d\mu (t)=\infty \).
Recall that the mixed norm space \(H(p,q,\alpha )\), \(0<p,q<\infty \), \(0<\alpha <\infty \), consists of those analytic functions on \(\mathbb {D}\) such that
Hardy and Littlewood proved (see [10, Theorem 5.11]) that
In particular, for \(0<p<1\), \(H^{p}\subset H(1,p,\frac{1}{p}-1)\). In the following, we show that the range of \(\mathcal {C}_\mu \) acting on X is contained in \(H(1,p,\frac{1}{p}-1)\) for \(0<p<1\).
Theorem 2.7
Suppose \(1<s<\infty \), \(0<p<1\) and \(\mu \) is a finite positive Borel measure on [0, 1). Let X be a Banach subspace of \(H(\mathbb {D})\) with \(\Lambda ^{s}_{\frac{1}{s}}\subset X\subset \mathcal {B}\). Then \(\mathcal {C}_\mu (X)\subset H(1,p,\frac{1}{p}-1)\).
Proof
Let \(f\in X\subset \mathcal {B}\), then it is known that
By Fubini theorem and Lemma 2.1–2.3 we have
This yields that \(\mathcal {C}_\mu (X)\subset H\Bigg (1,p,\frac{1}{p}-1\Bigg )\). \(\square \)
We recall that \(f\in H(\mathbb {D})\) is a Cauchy transform if it admits a representation
where \(\lambda \) is a finite complex Borel measure on \(\partial \mathbb {D}\). The space of all Cauchy transforms is denoted by \(\mathcal {K}\). We let \(\mathcal {A}\) denote the disc algebra, that is, the space of analytic functions in \(\mathbb {D}\) with a continuous extension to the closed unit disc, endowed with the \(\Vert \cdot \Vert _{\infty }\)-norm. It turns out [8] that
It is known (see [7]) that \( H^{1}\subsetneq \mathcal {K}\subsetneq \bigcap _{0<p<1}H^{p}\). In view of Theorem 2.7, the following question arises naturally.
Question
Suppose \(1<s<\infty \), \(\mu \) is a finite positive Borel measure on [0, 1) and X is a Banach subspace of \(H(\mathbb {D})\) with \(\Lambda ^{s}_{\frac{1}{s}}\subset X\subset \mathcal {B}\). Is the operator \(\mathcal {C}_\mu \) bounded from X to \(\bigcap _{0<p<1}H^{p}\)?
We do not know the answer to this question. However, there exists a finite positive measure \(\mu \) on [0, 1) such that \(\mathcal {C}_\mu (X)\nsubseteq \mathcal {K}\). Actually, we have the following result.
Proposition 2.8
Suppose \(1<s<\infty \) and X is a Banach subspace of \(H(\mathbb {D})\) with \(\Lambda ^{s}_{\frac{1}{s}}\subset X\subset \mathcal {B}\). Then there exist \(f\in X\) and a finite positive measure \(\mu \) on [0, 1) such that \(\mathcal {C}_\mu (f)\notin \mathcal {K}\).
Proof
For \(1<\gamma <2\), let \(d\mu (t)=\left( (1-t)\left( \log \frac{e}{1-t}\right) ^{3-\gamma }\right) ^{-1}dt\). Then it is clear that \(\mu \) is a finite positive measure \(\mu \) on [0, 1). By Lemma 2.2 we have
For \(n\ge 2\), we have
It is easy to see that
Also, for every \(n\ge 2\) we have that
Hence, we have that
Now, let \(f(z)=\log \frac{1}{1-z}\in X\) and \(g(z)=\sum _{n=0}^{\infty }\frac{z^{n}}{(n+1)(\log (n+2))^{\gamma }}\in \mathcal {A}\). To complete the proof, it is suffices to show that
In fact, it follows from (10) that
The proof is complete. \(\square \)
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Tang, P. Cesàro-like operators acting on a class of analytic function spaces. Anal.Math.Phys. 13, 96 (2023). https://doi.org/10.1007/s13324-023-00858-y
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DOI: https://doi.org/10.1007/s13324-023-00858-y