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Characterizations of some function spaces associated with Bloch type spaces on the unit ball of \(C^{n}\)
Journal of Inequalities and Applications volume 2015, Article number: 330 (2015)
Abstract
Motivated by characterizations of Bloch type spaces, we define some related function spaces and characterize them in this paper. Our results reveal the reason in theory that some equivalent characterizations of the Bloch type space \(B^{\alpha}\) require extra conditions for α.
1 Introduction
Let \(B_{n}\) be the unit ball of the complex Euclidean space \(C^{n}\) and \(\partial B_{n}\) be the unit sphere. The class of all holomorphic functions on \(B_{n}\) is denoted by \(H(B_{n})\). For \(0\leq\alpha<\infty\), let \(H_{\alpha}^{\infty}\) be the space of holomorphic functions f on \(B_{n}\) satisfying \(\sup _{z\in B_{n}}(1-|z|^{2})^{\alpha}|f(z)|<\infty\). When \(\alpha=0\), we write \(H^{\infty}\) for \(H_{0}^{\infty}\).
For \(f\in H(B_{n})\), its complex gradient is defined by
As introduced by Timoney in [1], the Bloch space on the unit ball \(B_{n}\) is the space of all \(f\in H(B_{n})\) such that \(\sup _{z\in B_{n}}Q_{f}(z)<\infty\), where
Let \(0<\alpha<\infty\). The Bloch type space \(B^{\alpha}\) consists of functions \(f\in H(B_{n})\) satisfying
It is well known that \(B^{\alpha}=H_{\alpha-1}^{\infty}\) for \(\alpha>1\). Moreover, when \(\alpha>1/2\), a function \(f\in B^{\alpha}\) if and only if
But it is not true for the case \(0<\alpha\leq1/2\) in the setting of several complex variables. For example (see [2]), when \(0<\alpha<1/2\), \(f\in B^{\alpha}\) if and only if \(\sup _{z\in B_{n}}(1-|z|^{2})^{\alpha-1}G_{f}(z)<\infty\), where
Thus an interesting question arises naturally: what is the property for a holomorphic function f on \(B_{n}\) satisfying (1) when \(0<\alpha\leq1/2\). Below we define the space
In Section 2 of this paper, we prove that \(f\in T_{1/2}\) if and only if the directional derivatives of f in the directions perpendicular to the radial direction are uniformly bounded. In particular, \(T_{\alpha}\) is a trivial space consisting of constants for \(\alpha <1/2\).
In 1986 Holland and Walsh [3] gave another characterization for the Bloch space on the unit disc D, namely, f belongs to the Bloch space if and only if
Ren and Tu [4] extended the above form to the unit ball \(B_{n}\). Zhao [5] generalized these results as follows.
Theorem A
Let \(0<\alpha\leq2\). Let λ be any real number satisfying the following properties:
-
(1)
\(0\leq\lambda\leq\alpha\) if \(0<\alpha<1\);
-
(2)
\(0<\lambda<1\) if \(\alpha=1\);
-
(3)
\(\alpha-1\leq\lambda\leq1\) if \(1<\alpha\leq2\).
Then a holomorphic function f on \(B_{n}\) is in \(B^{\alpha}\) if and only if
Zhao gave some examples showing that the conditions on α and λ in Theorem A cannot be improved.
Motivated by Theorem A, we denote the following function space:
where α and λ are real numbers. Our purpose is to characterize the space \(S_{\alpha,\lambda}\) for α and λ without satisfying those conditions in Theorem A. If \(\lambda<0\) or \(\lambda>\alpha\), by the maximum modulus principle, it is easy to see that \(S_{\alpha,\lambda}\) consists of constants. So we always assume \(0\leq\lambda\leq\alpha\). In Section 3 we show \(S_{\alpha,\lambda}\subset B^{\alpha}\) when α and λ do not satisfy the conditions of Theorem A. More explicitly, \(S_{\alpha,\lambda}\) coincides with the bounded space \(H^{\infty}\), or the Bloch type space \(B^{\lambda+1}\) or \(B^{\alpha-\lambda+1}\) in terms of different numbers α and λ. Our results reveal in theory instead of examples that the conditions on α and λ in Theorem A cannot be improved.
Throughout this paper, constants are denoted by C, and they are positive finite quantities and not necessarily the same in each occurrence.
2 Characterizations of \(T_{\alpha}\)
Theorem 2.1
The following statements are equivalent:
-
(i)
\(f\in T_{1/2}\);
-
(ii)
There exists a constant \(C>0\) such that
$$\begin{aligned} \bigl|\bigl\langle \nabla f(z),\overline{\zeta}\bigr\rangle \bigr|\leq C \end{aligned}$$(3)for all \(z\in B_{n}\) and \(\zeta\in\partial B_{n}\) with \(\langle z,\zeta\rangle=0\);
-
(iii)
For all \(1\leq i,j\leq n\),
$$\begin{aligned} \sup_{z\in B_{n}}\biggl\vert \overline{z_{i}} \frac{\partial f}{\partial z_{j}}(z)-\overline{z_{j}}\frac{\partial f}{\partial z_{i}}(z)\biggr\vert < \infty. \end{aligned}$$(4)
Proof
(i) ⇔ (ii): For \(z\in B_{n}\) and \(\zeta\in\partial B_{n}\) with \(\langle z,\zeta\rangle=0\), we have
This shows that (i) ⇒ (ii).
For the converse, if (ii) holds, we have \((1-|z|^{2})^{1/2}|\langle\nabla f(z),\overline{z}\rangle|\leq C\) for all \(z\in B_{n}\) (see [6]). When \(1/2\leq|z|<1\) and \(0\neq w\in C^{n}\), by the projection theorem, there exists \(\zeta\in\partial B^{n}\) such that \(\langle\zeta,z\rangle=0\) and
where \(w_{1}=\langle w,z\rangle/|z|^{2}\) and \(|w|^{2}=|w_{1}|^{2}|z|^{2}+|w_{2}|^{2}\). Thus,
It follows that
for all \(1/2\leq|z|<1\) and \(0\neq w\in C^{n}\). On the other hand, for \(0\leq|z|<1/2\), we have
This proves (i).
(ii) ⇔ (iii): Without loss of generality, we only need to show that
If \(z_{1}=z_{2}=0\), there is nothing to prove. If \(|z_{1}|^{2}+|z_{2}|^{2}\neq0\), put
Obviously, \(\zeta\in\partial B_{n}\) and \(\langle z,\zeta\rangle=0\). Therefore,
Conversely, suppose (iii) holds. When \(|z|\leq1/2\), it is clear that (3) holds. For \(|z|= \sqrt{|z_{1}|^{2}+\cdots+|z_{n}|^{2}}>1/2\), there exists \(z_{i}\) (\(1\leq i\leq n\)) such that \(|z_{i}|>1/(2\sqrt{n})\). We may assume that \(|z_{1}|>1/(2\sqrt{n})\). Let \(V=\{w\in C^{n}:\langle z,w\rangle=0\}\). Then V is a subspace of \(C^{n}\) and a basis of V is \(\{v_{1},\ldots,v_{n-1}\}\), where
Therefore, for \(\zeta=(\zeta_{1},\ldots,\zeta_{n})\in\partial B_{n}\) with \(\langle z,\zeta\rangle=0\), there exist scalars \(k_{1},\ldots,k_{n-1}\) such that ζ is expressed as a linear combination of \(v_{1},\ldots,v_{n-1}\) in only one way. That is,
and
Note that \(|\zeta|=1\) and \(|z_{1}|>1/(2\sqrt{n})\). Hence, it follows from (8) that
Thus we have
The desired result follows from (4) and (9). This finishes the proof of Theorem 2.1. □
From Theorem 2.1 and the result of [6], we see that \(T_{1/2} \subset B^{1/2}\). Meanwhile, it is evident that \(B^{\alpha}\subset T_{1/2} \) for \(0<\alpha<1/2\). Below we give two examples to show that these inclusions are strict.
Example 1
Let
Note that \(|z_{1}|^{2}+|z_{2}|^{2}<1\) for \(z=(z_{1},z_{2})\in B_{2}\). Then
On the other hand, when \(0<\alpha<1/2\) and \(z=(r,0)\rightarrow(1,0)\) (\(0< r<1\)),
Therefore \(f\in T_{1/2}\) by Theorem 2.1, but \(f\,\overline{\in}\, B^{\alpha}\) (\(0<\alpha<1/2\)).
Example 2
Let
Then, for \(z=(z_{1},z_{2})\in B_{2}\),
Meanwhile, let \(z=(r,0)\rightarrow(1,0)\) (\(0< r<1\)), we have
Thus \(g\,\overline{\in}\, T_{1/2}\) but \(g\in B^{1/2}\).
Lemma 2.1
Let \(n>1\), \(1\leq i,j\leq n\) and \(i\neq j\). Suppose that \(f\in H(B_{n})\), \(g\in H(B_{n})\) and
Then \(f\equiv0\) and \(g\equiv0\).
Proof
We may assume \(i=1\), \(j=2\) without loss of generality. Let \(h(z)=\overline{z_{1}}f(z)-\overline{z_{2}}g(z)\). For each fixed \(\zeta=(\zeta_{1},\ldots,\zeta_{n})\in \partial B_{n}\), define the slice function \(h_{\zeta}(\lambda)=h(\lambda \zeta)\) on the unit disk \(D=\{\lambda:|\lambda|<1\}\). Then
That is,
Since the function \(\overline{ \zeta_{1}}f(\lambda\zeta_{1},\ldots,\lambda\zeta_{n})- \overline{ \zeta_{2}}g(\lambda\zeta_{1},\ldots,\lambda\zeta_{n})\) is holomorphic on the unit disk \(|\lambda|<1\), by the maximum modulus principle of one complex variable, it follows that
for all \(\lambda\in D\). Therefore, for any \(\zeta\in \partial B_{n}\) and \(\lambda\in D\),
When \(0\neq z\in B_{n}\), let \(\lambda=|z|\) and \(\zeta=z/|z|\). Then we get
Hence, for all \(z\in B_{n}\),
Since f and g are holomorphic on \(B_{n}\), we can conclude \(f(z)=g(z)\equiv0\). The proof is finished. □
An immediate consequence of Lemma 2.1 is the following theorem.
Theorem 2.2
Let \(n>1\) and \(\alpha<1/2\). If \(f\in T_{\alpha}\), then f is constant.
Proof
By (5) it follows that \(|\langle\nabla f(z),\overline{\zeta }\rangle| \leq C(1-|z|)^{1/2-\alpha}\) for \(z\in B_{n}\) and \(\zeta\in\partial B_{n}\) with \(\langle z,\zeta\rangle=0\). Since \(\alpha<1/2\), we get \(\lim _{|z|\rightarrow1}|\langle\nabla f(z),\overline{\zeta}\rangle|=0\). Using a similar argument as in the proof of Theorem 2.1, we have \(\lim _{|z|\rightarrow1}\vert \overline{z_{j}}\frac{\partial f}{\partial z_{i}}(z)-\overline{z_{i}}\frac{\partial f}{\partial z_{j}}(z)\vert =0\) for all \(1\leq i,j\leq n\). Thus the desired result follows from Lemma 2.1. □
3 Characterizations of \(S_{\alpha,\lambda}\)
In the section we will characterize \(S_{\alpha,\lambda}\) explicitly for real numbers α and λ in several cases. For this, we need the following lemma which plays an important role in the proof of Theorem 3.1.
Lemma 3.1
Let \(\alpha>1\). Let λ be any real number satisfying the following properties:
-
(1)
\(0<\lambda<\alpha-1\) if \(1<\alpha\leq2\);
-
(2)
\(0<\lambda\leq{\alpha}/{2}\) if \(\alpha>2\).
Let
Then there exists a constant \(C>0\) such that \(H(x,y)\leq C\) for any x and y satisfying \(0< x,y\leq1\) and \(x\neq y\).
Proof
Let \(t=x/y\). Then \(t\in(0,1)\cup(1,\infty)\), and
Let \(s={\tau}/{y}\), we have
where
It is evident that \(G(t)\) is continuous on \((0,1)\cup(1,\infty)\), and
For the case (1), since \(0<\lambda<1\), we get
Noticing that \(y^{\alpha-(\lambda+1)}\leq1\) for \(y\in(0,1]\), we conclude that \(H(x,y)\) is bounded in this case. For the case (2), we easily see that \(0<\lambda<\alpha-1\). Write
It is clear that
The above limit is \(1/\lambda\) for \(\lambda={\alpha}/{2}\) and 0 for \(0<\lambda<{\alpha}/{2}\). Therefore \(H(x,y)\) is also bounded in the case (2) since \(x^{\alpha-(\lambda+1)}\leq1\) for \(x\in(0,1]\). The proof is complete. □
Theorem 3.1
Let \(\alpha>1\). Let λ be any real number satisfying the following properties:
-
(1)
\(0<\lambda<\alpha-1\) if \(1<\alpha\leq2\);
-
(2)
\(0<\lambda\leq{\alpha}/{2}\) if \(\alpha>2\).
Then \(S_{\alpha,\lambda}=B^{\lambda+1}\).
Proof
Let \(f\in B^{\lambda+1}\). For any \(z,w\in B_{n}\), since
we get
If \(|z|=|w|\), noting that \(\lambda+1<\alpha\), we get
If \(|z|\neq|w|\), let \(\tau=t(1-|z|)+(1-t)(1-|w|)\). By Lemma 3.1, we have
Therefore,
which shows that \(f\in S_{\alpha,\lambda}\).
Conversely, if \(f\in S_{\alpha,\lambda}\), it follows that
Thus we get
namely, \(f\in H_{\lambda}^{\infty}\) and so \(f\in B^{\lambda+1}\). This finishes the proof of the theorem. □
Theorem 3.2
Let \(\alpha>1\). Let λ be any real number satisfying the following properties:
-
(1)
\(1<\lambda<\alpha\) if \(1<\alpha\leq2\);
-
(2)
\(\alpha/2<\lambda<\alpha\) if \(\alpha>2\).
Then \(S_{\alpha,\lambda}=B^{\alpha-\lambda+1}\).
Proof
If \(1<\lambda<\alpha\) for \(1<\alpha\leq2\), then \(0<\alpha-\lambda<\alpha-1\). If \(\alpha/2<\lambda<\alpha\) for \(\alpha>2\), then \(0<\alpha-\lambda <\alpha/2\). Applying Theorem 3.1, we immediately conclude that \(S_{\alpha,\lambda}= B^{\alpha-\lambda+1}\). □
For any point \(w\in B_{n}-\{0\}\), we recall that the bi-holomorphic mapping \(\varphi_{w}\) of \(B_{n}\), which interchanges the points 0 and w, is defined by
where \(s_{w}=\sqrt{1-|w|^{2}}\), \(P_{w}(z)=\frac{\langle z,w\rangle}{|w|^{2}}w\) and \(Q_{w}(z)=z-P_{w}(z)\). When \(w=0\), let \(\varphi_{w}(z)=-z\). The pseudo-hyperbolic distance between w and z is denoted by \(\rho(w,z)=|\varphi_{w}(z)|\).
Theorem 3.3
Let \(\alpha\geq1\). Then \(S_{\alpha,\lambda}=H^{\infty}\) for \(\lambda=0\) or \(\lambda=\alpha\).
Proof
It suffices to prove for \(\alpha\geq1\) and \(\lambda=0\). If \(f\in S_{\alpha,0}\), that is,
Then
which implies that \(f\in H^{\infty}\).
Conversely, assume \(f\in H^{\infty}\). Then we have (see Lemma 1 in [7])
for all \(z,w\in B_{n}\). On the other hand,
and so
By (11) and (12), we get (10) since \(\alpha\geq1\). Thus \(f\in S_{\alpha,0}\). The proof is complete. □
Remark 3.1
We easily see that \(\lambda+1<\alpha\) in Theorem 3.1, and \(\lambda>1\) in Theorem 3.2. Combining with Theorem 3.3, we conclude that \(S_{\alpha,\lambda}\subset B^{\alpha}\) and the inclusion is strict for real numbers α and λ which do not satisfy the conditions of Theorem A.
References
Timoney, R: Bloch functions in several complex variables I. Bull. Lond. Math. Soc. 12, 241-267 (1980)
Chen, H, Gauthier, P: Composition operators on μ-Bloch spaces. Can. J. Math. 61, 50-75 (2009)
Holland, F, Walsh, D: Criteria for membership of Bloch spaces and its subspace, BMOA. Math. Ann. 273, 317-335 (1986)
Ren, G, Tu, C: Bloch spaces in the unit ball of \(C^{n}\). Proc. Am. Math. Soc. 133, 719-726 (2005)
Zhao, R: A characterization of Bloch-type spaces on the unit ball of \(C^{n}\). J. Math. Anal. Appl. 330, 291-297 (2007)
Zhuo, W: Derivatives of Bloch functions and α-Carleson measure on the unit ball. Acta Math. Sci. 14, 351-360 (1994)
Toews, C: Topological components of the set of composition operators on \(H^{\infty}(B_{N})\). Integral Equ. Oper. Theory 48, 265-280 (2004)
Acknowledgements
JD was supported by the Natural Science Foundation of China under 11301404 and the Fundamental Research Funds for the Central Universities (WUT: 2015IVA069). BW was supported by the Natural Science Foundation of China under 11201348, the Fund of China Scholarship Council and the special Fund of Basic Scientific Research of Central Colleges of South Central University for Nationalities under CZQ13015.
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BW proposed the problem. JD and BW together finished the proof. All authors read and approved the final manuscript.
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Dai, J., Wang, B. Characterizations of some function spaces associated with Bloch type spaces on the unit ball of \(C^{n}\) . J Inequal Appl 2015, 330 (2015). https://doi.org/10.1186/s13660-015-0846-6
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DOI: https://doi.org/10.1186/s13660-015-0846-6