1 Introduction
Let \({\mathbb{C}}^{n}=\{z=(z_{1},\ldots,z_{n}): z_{1},\ldots,z_{n}\in{\mathbb{C}}\}\) denote the n dimensional complex vector space. For \(a=(a_{1},\ldots,a_{n})\in{\mathbb {C}}^{n}\), we define the Euclidean inner product \(\langle\cdot,\cdot\rangle\) by where \(\bar{a}_{k}\) (\(k\in\{1,\ldots,n\}\)) denotes the complex conjugate of \(a_{k}\). Then the Euclidean length of z is defined by Denote a ball in \({\mathbb{C}}^{n}\) with center a and radius \(r > 0\) by In particular, we let \(\mathbb{B}^{n}\) denote the unit ball \(\mathbb{B}^{n}(0,1)\) and let \({\mathbb{D}}\) be the unit disk in \({\mathbb{C}}\).
$$\langle z,a\rangle=z_{1}\bar{a}_{1}+\cdots+z_{n} \bar{a}_{n}, $$
$$|z|=\langle z,z\rangle^{\frac{1}{2}}=\bigl(|z_{1}|^{2}+ \cdots+|z_{n}|^{2}\bigr)^{\frac{1}{2}}. $$
$$\mathbb{B}^{n}(a,r)=\bigl\{ z\in{\mathbb{C}}^{n} : |z-a|< r \bigr\} . $$
A complex-valued function f of \(\mathbb{B}^{n}\) into \({\mathbb{C}}\) is called pluriharmonic if there are two holomorphic functions h and g, such that \(f=h+\bar{g}\). We denote by \(\mathcal{P}(\mathbb{B}^{n})\) the class of all pluriharmonic mappings on the unit ball of \({\mathbb{C}}^{n}\).
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Let \(f=h+\bar{g}\in\mathcal{P}(\mathbb{B}^{n})\). For a multi-index \(m=(m_{1},\ldots,m_{n})\), we employ the notations where \(|m|=m_{1}+\cdots+m_{n}\). Obviously, if \(f\in\mathcal{P}(\mathbb{B}^{n})\), then so does \(\hat{D}^{(m)}f\).
$$\begin{aligned}& \nabla f(z)=\biggl(\frac{\partial f}{\partial z_{1}},\ldots,\frac{\partial f}{\partial z_{n}}\biggr), \qquad \overline{\nabla} f(z)=\biggl(\frac{\partial f}{\partial\bar{z}_{1}},\ldots,\frac{\partial f}{\partial\bar{z}_{n}}\biggr), \\& \partial^{m} f=\frac{\partial^{m} f}{\partial z^{m}}=\frac{\partial ^{|m|} f}{\partial z_{1}^{m_{1}}\, \cdots \, \partial z_{n}^{m_{n}}},\qquad \overline { \partial}^{m} f=\frac{\partial^{m} f}{\partial\bar{z}^{m}}=\frac {\partial^{|m|} f}{\partial\bar{z}_{1}^{m_{1}}\, \cdots\, \partial\bar {z}_{n}^{m_{n}}}, \\& \hat{D}^{(m)}f=\partial^{m} f+ \overline{\partial}^{m} f= \partial^{m} h+ \overline{\partial}^{m} g, \end{aligned}$$
Similar to the planar case, the Bloch space
\(\mathcal{PB}(\mathbb{B}^{n})\) of \(\mathcal{P}(\mathbb{B}^{n})\) consists of all mappings \(f\in\mathcal{P}(\mathbb{B}^{n})\) such that the little Bloch space
\(\mathcal{PB}_{0}(\mathbb{B}^{n})\) consists of all mappings \(f\in \mathcal{PB}(\mathbb{B}^{n})\) such that
$$\|f\|= \sup_{z\in\mathbb{B}^{n}}\bigl(1-|z|^{2}\bigr) \bigl(\bigl\vert \nabla f(z)\bigr\vert + \bigl\vert \overline{\nabla} f(z)\bigr\vert \bigr)< \infty; $$
$$\lim_{|z|\rightarrow1^{-}}\bigl(1-|z|^{2}\bigr) \bigl(\bigl\vert \nabla f(z)\bigr\vert + \bigl\vert \overline{\nabla} f(z)\bigr\vert \bigr)=0. $$
Let \(d\lambda(z)=(1-|z|^{2})^{-n-1}\, dv(z)\), where dv is the normalized Lebesgue measure of \(\mathbb{B}^{n}\). For \(1\leq p<\infty\), the Besov space
\({\mathcal{B}}_{p}\) of \(\mathcal{P}(\mathbb {B}^{n})\) consists of all mappings \(f\in\mathcal{P}(\mathbb{B}^{n})\) such that \((1-|z|^{2})(|\nabla f(z)|+ |\overline{\nabla} f(z)|)\in L^{p}(\mathbb{B}^{n}, d\lambda)\), i.e.
$$\| f\|_{L^{p}(d\lambda(z))}=\int_{\mathbb{B}^{n}} \bigl(\bigl(1-|z|^{2} \bigr) \bigl(\bigl\vert \nabla f(z)\bigr\vert + \bigl\vert \overline{\nabla} f(z)\bigr\vert \bigr) \bigr)^{p}\, d\lambda (z)< \infty. $$
For a planar harmonic mapping f in \({\mathbb{D}}\), Colonna [1] proved that \(f\in\mathcal{PB}(\mathbb{D})\) if and only if the Lipschitz number
$$\beta_{f}=\sup_{z,w\in{\mathbb{D}},z\neq w}\frac {|f(z)-f(w)|}{{\operatorname{arctanh}}|\frac {z-w}{1-\bar{z}w}|}< \infty. $$
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Let where \(D(z,r)\) is the Bergman disc with center \(z\in\mathbb{D}\) and radius r, \(n\geq1\) an integer and \(\alpha+\beta=n\). By means of it, Yoneda [2] characterized the spaces \(\mathcal{PB}(\mathbb{D})\) and \({\mathcal{B}}_{p}\) as follows.
$$l=\sup_{w\in D(z,r),z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(n-1)}f(z)-\hat {D}^{(n-1)}f(w)|}{|z-w|}, $$
Theorem A
Let
\(n\geq1\)
be an integer and
\(f\in \mathcal{P}(\mathbb{D})\). Then
\(f\in\mathcal{PB}(\mathbb{D})\)
if and only if
l
is bounded.
Theorem B
Let
\(n\geq1\)
be an integer and
\(f\in \mathcal{P}(\mathbb{D})\). Then
\(f\in{\mathcal{B}}_{p}\)
if and only if
$$\int_{{\mathbb{D}}}l^{p}\, d\lambda(z)< \infty. $$
In this article, we consider the corresponding problems in higher dimensional setting. We refer to [3‐7] for the related topics for holomorphic or harmonic functions. See [8‐12] for various characterizations of the Bloch, little Bloch, and Besov spaces in the unit ball of \({\mathbb{C}}^{n}\). In Section 2, we recall some basic facts for pluriharmonic mappings. Our main results are Theorems 1-4, whose proofs will be presented in Sections 3 and 4.
2 Preliminaries
Let \(\operatorname{Aut}(\mathbb{B}^{n})\) denote the group of biholomorphic mappings of \(\mathbb{B}^{n}\) onto itself. It is well known that \(\operatorname{Aut}(\mathbb{B}^{n})\) is generated by the unitary operators on \(\mathbb{B}^{n}\) and the involutions \(\phi_{a}\) of the form where \(a,z\in\mathbb{B}^{n}\),
$$\phi_{a}(z)=\frac{a-P_{a}z-(1-|a|^{2})^{\frac{1}{2}}Q_{a}z}{1-\langle z,a\rangle}, $$
$$P_{a}z=\frac{a\langle z,a\rangle}{\langle a,a\rangle}, \qquad Q_{a}z=z-P_{a}z. $$
For \(z,w\in\mathbb{B}^{n}\), we define \(\rho(z,w)=|\phi_{z}(w)|\). It is known that ρ is a distance function on \(\mathbb{B}^{n}\), and we call it pseudo-hyperbolic metric (cf. [6, 12]). For \(r\in(0,1)\), the pseudo-hyperbolic ball with center z and radius r is given by Clearly, \(E(z,r)=\phi_{z}({\mathbb{B}}(0,r))\).
$$E(z,r)=\bigl\{ w\in\mathbb{B}: \bigl\vert \phi_{z}(w)\bigr\vert < r \bigr\} . $$
Lemma 1
([12])
Let
\(0< r<1\)
and
\(w\in E(z,r)\). Then
where
\(|E(z,r)|\)
is the normalized volume of
\(E(z,r)\), \(A\asymp B\)
means that there is a constant
\(C>0\)
such that
\(B/C \leq A \leq BC\).
$$1-|z|^{2}\asymp1-|w|^{2}\asymp \bigl\vert 1-\langle z, w \rangle\bigr\vert \asymp\bigl\vert E(z,r)\bigr\vert ^{\frac{1}{n+1}}, $$
The following lemma is crucial [13].
Lemma 2
Suppose that
\(f:\mathbb{\overline{B}}^{n}(a,r) \rightarrow{\mathbb {C}}\)
is continuous and pluriharmonic in
\(\mathbb{B}^{n}(a,r)\). Then there exists
\(C>0\)
such that
$$\bigl\vert \nabla f(a)\bigr\vert +\bigl\vert \overline{\nabla} f(a)\bigr\vert \leq\frac{C}{r}\int_{\partial\mathbb{B}^{n}}\bigl\vert f(a+r \zeta)-f(a)\bigr\vert \, d\sigma(\zeta). $$
Let h be a holomorphic function in \(\mathbb{B}^{n}\). We say that \(h\in\mathcal{B}\) if similarly, \(h\in\mathcal{B}_{0}\) if \(h\in \mathcal{B}\) and It is obvious that a pluriharmonic mapping \(f=h+\bar{g} \in\mathcal{P}(\mathbb{B}^{n})\) (resp. \(\mathcal{PB}_{0}(\mathbb{B}^{n})\)) if and only if both \(h,g \in \mathcal{B}\) (resp. \(\mathcal{B}_{0}\)).
$$\sup_{z\in\mathbb{B}^{n}}\bigl(1-|z|^{2}\bigr)\bigl\vert \nabla h(z)\bigr\vert < \infty; $$
$$\lim_{|z|\rightarrow1^{-}}\bigl(1-|z|^{2}\bigr)\bigl\vert \nabla h(z)\bigr\vert =0. $$
The following is a characterization of the space \(\mathcal{B}\) (resp. \(\mathcal{B}_{0}\)).
Lemma 3
([12])
Let
h
be holomorphic in
\(\mathbb{B}^{n}\)
and
N
a positive integer. Then
\(h\in\mathcal{B}\) (resp. \(\mathcal{B}_{0}\)) if and only if
for all values of the multi-index
m
with
\(|m| = N\).
$$\sup_{z\in \mathbb{B}^{n}}\bigl(1-|z|^{2}\bigr)^{N}\biggl\vert \frac{\partial^{m} h(z)}{\partial z^{m}}\biggr\vert < \infty \qquad \biggl(\textit{resp. }\lim _{|z|\rightarrow 1^{-}}\bigl(1-|z|^{2}\bigr)^{N}\biggl\vert \frac{\partial^{m} h(z)}{\partial z^{m}}\biggr\vert =0\biggr) $$
Corollary 1
Let
\(f=h+\bar{g}\)
be a pluriharmonic mapping in
\(\mathbb{B}^{n}\)
and
N
a positive integer. Then
\(f\in\mathcal{PB}(\mathbb{B}^{n})\) (resp. \(\mathcal{PB}_{0}(\mathbb{B}^{n})\)) if and only if
respectively,
for all values of the multi-index
m
with
\(|m|=N\).
$$\sup_{z\in\mathbb{B}^{n}}\bigl(1-\vert z\vert ^{2} \bigr)^{N}\bigl(\bigl\vert \partial^{m}f\bigr\vert +\bigl\vert \overline {\partial}^{m}f\bigr\vert \bigr)=\sup _{z\in\mathbb{B}^{n}}\bigl(1-\vert z\vert ^{2} \bigr)^{N} \bigl(\bigl\vert \partial^{m} h\bigr\vert + \bigl\vert \overline{\partial}^{m} g\bigr\vert \bigr)< \infty, $$
$$\lim_{\vert z\vert \rightarrow 1^{-}}\bigl(1-\vert z\vert ^{2} \bigr)^{N}\bigl(\bigl\vert \partial^{m}f\bigr\vert +\bigl\vert \overline{\partial}^{m}f\bigr\vert \bigr)=\lim _{\vert z\vert \rightarrow 1^{-}}\bigl(1-\vert z\vert ^{2} \bigr)^{N} \bigl(\bigl\vert \partial^{m} h\bigr\vert + \bigl\vert \overline{\partial}^{m} g\bigr\vert \bigr)\rightarrow0 $$
As an application of Lemma 3, we obtain the following.
Lemma 4
Let
h
be holomorphic in
\(\mathbb{B}^{n}\). Then
\(h\in\mathcal{B}\)
if and only if for each
\(j\in\{1,\ldots,n\}\),
$$L=\sup_{z,w\in\mathbb{B}^{n}, z\neq w}\frac{(1-|z|^{2})(1-|w|^{2})}{|z-w|}\biggl\vert \frac{\partial h(z)}{\partial z_{j}}-\frac{\partial h(w)}{\partial z_{j}}\biggr\vert < \infty. $$
Proof
Fixing a point w and letting with \(\xi\in {\mathbb{C}}\), we have for each \(j\in \{1,\ldots,n\}\). By Lemma 3, we see that \(h\in\mathcal{B}\).
$$z=w+\xi\overline{\nabla \biggl(\frac{\partial h}{\partial z_{j}}\biggr) (w)}\rightarrow w $$
$$\bigl(1-|w|^{2}\bigr)^{2}\biggl\vert \nabla \biggl( \frac{\partial h}{\partial z_{j}}\biggr) (w)\biggr\vert \leq L, $$
For the converse, we assume that \(h\in\mathcal{B}\). Let \(h_{j}(z)=\frac{\partial h(z)}{\partial z_{j}}\), then for each \(j\in \{1,\ldots,n\}\), It follows from [7] that there exists \(0< C_{1}<\infty\) such that This implies that So the result follows. □
$$\begin{aligned} \bigl\vert h_{j}(z)-h_{j}(w)\bigr\vert =&\biggl\vert \int^{1}_{0}\frac{dh_{j}}{ds} \bigl(sz+(1-s)w\bigr)\, ds\biggr\vert \\ \leq&\sum_{k=1}^{n}\biggl\vert (z_{k}-w_{k})\int^{1}_{0} \frac{\partial h_{j}}{\partial z_{k}}\bigl(sz+(1-s)w\bigr)\, ds \biggr\vert \\ \leq& \sqrt{n}\vert z-w\vert \int^{1}_{0} \bigl\vert \nabla h_{j}\bigl(sz+(1-s)w\bigr)\bigr\vert \, ds \\ \leq& C\vert z-w\vert \int^{1}_{0} \frac{ds}{(1-\vert sz+(1-s)w\vert ^{2})^{2}}. \end{aligned}$$
$$\int^{1}_{0}\frac{ds}{(1-|sz+(1-s)w|^{2})^{2}}\leq \frac{C_{1}}{(1-|z|^{2})(1-|w|^{2})}. $$
$$\sup_{z,w\in\mathbb{B}^{n}, z\neq w}\frac {(1-|z|^{2})(1-|w|^{2})}{|z-w|}\bigl\vert h_{j}(z)-h_{j}(w) \bigr\vert < \infty. $$
3 The Bloch space for pluriharmonic mappings
In this section, we give some characterizations of the spaces \(\mathcal{PB}(\mathbb{B}^{n})\) and \(\mathcal{PB}_{0}(\mathbb{B}^{n})\) which can be viewed as the generalizations of Yoneda’s results in the higher dimensional case.
Theorem 1
Let
\(f\in\mathcal{P}(\mathbb{B}^{n})\), \(N\geq0\)
be an integer and
\(0< r<1\). Then
\(f\in\mathcal{PB}(\mathbb{B}^{n})\)
if and only if
for all values of the multi-index
m
with
\(|m|=N\), where
\(\alpha +\beta=N+1\).
$$L_{f}=\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r, z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}< \infty $$
Proof
First we prove the sufficiency. Let \(f(z)\in \mathcal{P}(\mathbb{B}^{n})\), then for each multi-index m with \(|m|=N\), \(\hat{D}^{(m)}f(z)\) is also pluriharmonic. According to Lemma 2, for \(z\in\mathbb{B}^{n}\) and \(r\in(0,1)\), where \(\varrho=\frac{r(1-|z|^{2})}{2}\). By a simple computation, we see that \(\mathbb{B}^{n} (z, \varrho)\subset E(z, r)\), so Since for each \(w \in E(z, r)\), \(w\neq z \), by Lemma 1, we can deduce that Therefore, there exists a positive constant \(C_{2}\) such that from which we see that \(f\in\mathcal{PB}(\mathbb{B}^{n})\).
$$ \bigl\vert \nabla \bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert \leq \frac{C}{(1-\vert z\vert ^{2})}\int_{\partial\mathbb{B}^{n}}\bigl\vert \bigl(\hat {D}^{(m)}f\bigr) (z+\varrho\zeta)-\bigl(\hat{D}^{(m)}f\bigr) (z) \bigr\vert \, d\sigma(\zeta), $$
$$ \bigl\vert \nabla\bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert \leq \frac{C}{(1-\vert z\vert ^{2})}\sup_{w\in E(z,r)}\bigl\vert \bigl( \hat{D}^{(m)}f\bigr) (z)-\bigl(\hat{D}^{(m)}f\bigr) (w)\bigr\vert . $$
$$\frac{(1-|z|^{2})^{\frac{1}{2}}(1-|w|^{2})^{\frac{1}{2}}}{|z-w|}\geq \frac{(1-r^{2})^{\frac{1}{2}}}{r}, $$
$$\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}}{|z-w|}\geq C_{1}\bigl(1-|z|^{2} \bigr)^{N}. $$
$$\bigl(1-\vert z\vert ^{2}\bigr)^{N+1}\bigl(\bigl\vert \nabla\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{ \nabla}\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert \bigr)\leq C_{2}L_{f}, $$
Now we prove the necessity. Let \(w\in E(z,r)\), \(w\neq z\). Then for each multi-index m with \(|m|=N\), we have By Lemma 1 we infer that there exists \(\iota>0\) such that \(1-|w|=\iota(1-|z|)\) and Thus, So the proof is complete. □
$$\begin{aligned} \bigl\vert \bigl(\hat{D}^{(m)}f\bigr) (z)-\bigl(\hat{D}^{(m)}f \bigr) (w)\bigr\vert =&\biggl\vert \int^{1}_{0} \frac{d(\hat{D}^{(m)}f)}{ds}\bigl(sz+(1-s)w\bigr)\,ds\biggr\vert \\ \leq&\sum_{k=1}^{n}\biggl\vert (z_{k}-w_{k})\int^{1}_{0} \frac{\partial (\hat{D}^{(m)}f)}{\partial z_{k}}\bigl(sz+(1-s)w\bigr)\,ds \biggr\vert \\ &{}+\sum_{k=1}^{n}\biggl\vert ( \bar{z}_{k}-\bar{w}_{k})\int^{1}_{0} \frac {\partial (\hat{D}^{(m)}f)}{\partial\bar{z}_{k}}\bigl(sz+(1-s)w\bigr)\,ds\biggr\vert \\ \leq& \sqrt{n}\vert z-w\vert \int^{1}_{0} \bigl(\bigl\vert \nabla \bigl(\hat{D}^{(m)}f\bigr) \bigl(sz+(1-s)w\bigr) \bigr\vert \\ &{}+\bigl\vert \overline{\nabla} \bigl(\hat{D}^{(m)}f\bigr) \bigl(sz+(1-s)w\bigr)\bigr\vert \bigr)\,ds \\ \leq& C\vert z-w\vert \int^{1}_{0} \frac{ds}{(1-\vert sz+(1-s)w\vert )^{N+1}}. \end{aligned}$$
$$\begin{aligned} \frac{|(\hat{D}^{(m)}f)(z)-(\hat {D}^{(m)}f)(w)|}{|z-w|} \leq& C\int^{1}_{0} \frac {ds}{(s(1-|z|)+(1-s)(1-|w|))^{N+1}} \\ \leq&\frac{C'}{(1-|z|^{2})^{N+1}}\int^{1}_{0} \frac{ds}{[s+\iota (1-s)]^{N+1}} \\ \leq&\frac{C''}{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}}. \end{aligned}$$
$$L_{f}=\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}< \infty. $$
Theorem 2
Let
\(f\in\mathcal{P}(\mathbb{B}^{n})\)
and
\(N=1,2\). Then
\(f\in \mathcal{PB}(\mathbb{B}^{n})\)
if and only if
for all multi-index with
\(|m|=N-1\).
$$\sup_{z,w\in \mathbb{B}^{n}, z\neq w}\bigl(1-|z|^{2}\bigr)^{\frac{N}{2}}\bigl(1-|w|^{2}\bigr)^{\frac{N}{2}}\biggl\vert \frac{(\hat {D}^{(m)}f)(z)-(\hat{D}^{(m)}f)(w)}{z-w}\biggr\vert < \infty $$
Proof
The sufficiency follows from Theorem 1. We only need to prove the necessity. When \(N=1\), we refer to [8, 11]. Now we prove \(N=2\). Let \(f=h+\bar{g}\). Then for each \(j\in \{1,\ldots,n\}\), Since \(f\in\mathcal{PB}(\mathbb{B}^{n})\), \(h,g \in\mathcal{B}\), by Lemma 4, This completes the proof. □
$$\begin{aligned}& \sup_{z,w\in \mathbb{B}^{n}, z\neq w}\frac{(1-\vert z\vert ^{2})(1-\vert w\vert ^{2})}{\vert z-w\vert }\biggl\vert \frac{\partial f(z)}{\partial z_{j}}+ \frac{\partial f(z)}{\partial\bar{z}_{j}}-\frac{\partial f(w)}{\partial z_{j}}-\frac{\partial f(w)}{\partial \bar{z}_{j}}\biggr\vert \\& \quad \leq\sup_{z,w\in\mathbb{B}^{n},z\neq w}\frac{(1-\vert z\vert ^{2})(1-\vert w\vert ^{2})}{\vert z-w\vert }\biggl(\biggl\vert \frac{\partial h(z)}{\partial z_{j}}-\frac{\partial h(w)}{\partial z_{j}}\biggr\vert +\biggl\vert \frac{\partial g(z)}{\partial z_{j}}-\frac{\partial g(w)}{\partial z_{j}}\biggr\vert \biggr). \end{aligned}$$
$$\begin{aligned}& \sup_{z,w\in \mathbb{B}^{n},z\neq w}\frac{(1-\vert z\vert ^{2})(1-\vert w\vert ^{2})}{\vert z-w\vert }\biggl\vert \frac{\partial h(z)}{\partial z_{j}}- \frac{\partial h(w)}{\partial z_{j}}\biggr\vert < \infty, \\& \quad \leq\sup_{z,w\in \mathbb{B}^{n},z\neq w}\frac{(1-\vert z\vert ^{2})(1-\vert w\vert ^{2})}{\vert z-w\vert }\biggl\vert \frac{\partial g(z)}{\partial z_{j}}-\frac{\partial g(w)}{\partial z_{j}}\biggr\vert < \infty. \end{aligned}$$
Theorem 3
Let
\(f\in\mathcal{PB}(\mathbb{B}^{n})\), \(N\geq0\)
be an integer and
\(0< r<1\). Then
\(f\in\mathcal{PB}_{0}(\mathbb{B}^{n})\)
if and only if
for all values of the multi-index
m
with
\(|m|=N\), where
\(\alpha +\beta=N+1\).
$$ \lim_{|z|\rightarrow1^{-}}\sup_{z\in\mathbb {B}^{n},\rho(z,w)< r,z\neq w} \frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}=0 $$
(1)
Proof
Sufficiency. Assume that (1) holds. Then for any \(\epsilon>0\), there exists \(\delta\in(0,1)\) such that whenever \(\delta<|z|<1\). It follows from an argument similar to the proof of Theorem 1, that we have whenever \(\delta<|z|<1\). Hence from which we see that \(f\in\mathcal{PB}_{0}(\mathbb{B}^{n})\).
$$\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}< \epsilon $$
$$\begin{aligned}& \bigl(1-|z|^{2}\bigr)^{N+1}\bigl(\bigl\vert \nabla \bigl( \hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{\nabla}\bigl( \hat{D}^{(m)}f\bigr)\bigr\vert \bigr) \\& \quad \leq C\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|} < C\epsilon, \end{aligned}$$
$$\lim_{|z|\rightarrow 1^{-}}\bigl(1-|z|^{2}\bigr)^{N+1}\bigl( \bigl\vert \nabla\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert \bigr)=0, $$
Necessity. For \(\lambda\in(0,1)\), let \(f_{\lambda}(z)=f(\lambda z)\). By Lemma 1 and the proof of Theorem 1, we see that for each multi-index m with \(|m|=N\), and for all \(z,w\in\mathbb{B}^{n}\), \(\rho(z,w)< r\) and \(\xi, \eta\in E(z,r)\). So First letting \(|z|\rightarrow1^{-}\) and then letting \(\lambda \rightarrow1^{-}\), we obtain the desired result. □
$$\begin{aligned}& \frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat {D}^{(m)}(f-f_{\lambda})(z)-\hat{D}^{(m)}(f-f_{\lambda})(w)|}{|z-w|} \\& \quad \leq C_{1}\bigl(1-|\xi|^{2}\bigr)^{N+1}\bigl( \bigl\vert \nabla\hat{D}^{(m)}(f-f_{\lambda}) (\xi)\bigr\vert + \bigl\vert \overline{\nabla}\hat{D}^{(m)}(f-f_{\lambda}) (\xi)\bigr\vert \bigr) \end{aligned}$$
$$\begin{aligned} \begin{aligned} &\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat {D}^{(m)}f_{\lambda}(z)-\hat{D}^{(m)}f_{\lambda}(w)|}{|z-w|} \\ &\quad \leq \frac{C_{2}\lambda}{(1-|\lambda|^{2})^{N+1}}\bigl(1-|\eta|^{2}\bigr)^{N+1} \bigl(\bigl\vert \nabla\bigl( \hat{D}^{(m)}f_{\lambda}\bigr) (\eta) \bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f_{\lambda}\bigr) (\eta)\bigr\vert \bigr) \end{aligned} \end{aligned}$$
$$\begin{aligned} L_{f} \leq&C_{1}\bigl(1-\vert \xi \vert ^{2} \bigr)^{N+1}\bigl(\bigl\vert \nabla\hat {D}^{(m)}(f-f_{\lambda}) (\xi)\bigr\vert + \bigl\vert \overline{\nabla}\hat{D}^{(m)}(f-f_{\lambda}) (\xi)\bigr\vert \bigr) \\ &{}+ \frac{C_{2}\lambda}{(1-\vert \lambda \vert ^{2})^{N+1}}\bigl(1-\vert \eta \vert ^{2} \bigr)^{N+1}\bigl(\bigl\vert \nabla\bigl( \hat{D}^{(m)}f_{\lambda}\bigr) (\eta)\bigr\vert + \bigl\vert \overline{\nabla}\bigl( \hat{D}^{(m)}f_{\lambda}\bigr) (\eta)\bigr\vert \bigr). \end{aligned}$$
Corollary 2
Let
\(f\in\mathcal{PB}(\mathbb{B}^{n})\)
and
\(N=1,2\). Then
\(f\in \mathcal{PB}_{0}(\mathbb{B}^{n})\)
if and only if
for all multi-index with
\(|m|=N-1\).
$$\lim_{|z|\rightarrow 1^{-}}\sup_{z,w\in\mathbb{B}^{n}, z\neq w}\bigl(1-|z|^{2} \bigr)^{\frac{N}{2}}\bigl(1-|w|^{2}\bigr)^{\frac{N}{2}} \biggl\vert \frac{(\hat {D}^{(m)}f)(z)-(\hat{D}^{(m)}f)(w)}{z-w}\biggr\vert =0 $$
4 The Besov space for pluriharmonic mappings
In order to state and prove our next result, we need the following lemmas.
Lemma 5
Let
\(f\in\mathcal{P}(\mathbb{B}^{n})\). Then
\(f\in{\mathcal{B}}_{p}\)
if and only if
for all values of the multi-index
m
with
\(|m| = N\), and
\(p(N+1)\geq n\).
$$\sup_{z\in\mathbb{B}^{n}}\bigl(1-|z|^{2}\bigr)^{N+1} \bigl(\bigl\vert \nabla\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert \bigr)\in L^{p}\bigl(\mathbb{B}^{n}, d\lambda\bigr) $$
Proof
This follows from [12], Theorem 6.1. □
Lemma 6
Let
h
be holomorphic in
\(\mathbb{B}^{n}\)
and
\(0< r<1\). Then there exist constants
\(K>0\), \(r< r'<1\)
such that
$$\sup_{z\in\mathbb{B}^{n}, \rho(z,w)< r,z\neq w}\biggl\vert \frac{h(z)-h(w)}{z-w}\biggr\vert \leq K \int_{E(z,r')}\bigl\vert \nabla h(u)\bigr\vert \, d\lambda(u). $$
Proof
By the subharmonicity and Lemma 1, for each \(w\in \mathbb{B}^{n}\), we have for some \(r'>r\). □
$$\begin{aligned} \sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\biggl\vert \frac{h(z)-h(w)}{z-w}\biggr\vert \leq&C\sup_{\zeta\in E(z,r)}\bigl\vert \nabla h(\zeta)\bigr\vert \\ \leq& \frac{C}{|E(z,r')|}\int_{E(z,r')}\bigl\vert \nabla h( \zeta)\bigr\vert \, dv(\zeta) \\ \leq&K\int_{E(z,r')}\bigl\vert \nabla h(\zeta)\bigr\vert \, d\lambda(\zeta) \end{aligned}$$
Theorem 4
Let
\(f\in\mathcal{P}(\mathbb{B}^{n})\), \(N\geq0\)
be an integer and
\(0< r<1\). Then
\(f\in{\mathcal{B}}_{p}\)
if and only if
for all values of the multi-index
m
with
\(|m|=N\), where
\(\alpha +\beta=N+1\), and
\(p(N+1)\geq n\).
$$K_{f}=\int_{\mathbb{B}^{n}} \biggl(\sup_{z\in\mathbb{B}^{n},\rho (z,w)< r,z\neq w} \frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|} \biggr)^{p}\, d\lambda(z)< \infty $$
Proof
Let \(f=h+\bar{g}\in\mathcal{P}(\mathbb{B}^{n})\). Suppose that Let It follows from the proof of Theorem 1 that we have Since \(L_{f}(z)\leq L_{f}\), we see that which yields \(f\in{\mathcal{B}}_{p}\).
$$\int_{\mathbb{B}^{n}} \biggl(\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w} \frac{(1-|z^{2}|)^{\alpha}(1-|w^{2}|)^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|} \biggr)^{p}\, d\lambda(z)< \infty. $$
$$L_{f}(z)=\lim_{z\rightarrow w}\sup\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat {D}^{(m)}f(z)-\hat{D}^{(m)}f(w)|}{|z-w|}. $$
$$\bigl(1-|z|^{2}\bigr)^{N+1}\bigl(\bigl\vert \nabla \bigl( \hat{D}^{(m)}f\bigr) (z)\bigr\vert + \bigl\vert \overline{\nabla} \bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert \bigr)\leq CL_{f}(z). $$
$$\begin{aligned}& \int\bigl(1-|z|^{2}\bigr)^{(N+1)p} \bigl(\bigl\vert \nabla \bigl(\hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{\nabla} \bigl(\hat{D}^{(m)}f\bigr)\bigr\vert \bigr)^{p}\, d\lambda(z) \\& \quad \leq C\int_{\mathbb{B}^{n}} \biggl(\sup_{z\in\mathbb{B}^{n},\rho (z,w)< r,z\neq w} \frac{(1-|z^{2}|)^{\alpha}(1-|w^{2}|)^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|} \biggr)^{p}\, d\lambda(z), \end{aligned}$$
To prove the necessity, we suppose that \(f=h+\bar{g}\in{\mathcal{B}}_{p}\). By Lemmas 1 and 6, for each multi-index m, Since by Hölder’s inequality and Fubini’s theorem, we can obtain It follows from Lemma 5 that \(K_{f}\) is bounded. This completes the proof. □
$$\begin{aligned} L_{f} \leq&\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}(|\partial^{m}h(z)-\partial ^{m}h(w)|+|\partial^{m}g(z)-\partial^{m}g(w)|)}{|z-w|} \\ \leq& C\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{N+1}|\partial^{m}h(z)-\partial^{m}h(w)|}{|z-w|} \\ &{}+C\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{N+1}|\partial^{m}g(z)-\partial^{m}g(w)|}{|z-w|} \\ \leq& C_{1}\int_{E(z,r')}\bigl(1-|u|^{2} \bigr)^{N+1}\bigl(\bigl\vert \nabla\bigl(\partial^{m}h\bigr) (u)\bigr\vert +\bigl\vert \nabla \bigl(\partial^{m}g\bigr) (u)\bigr\vert \bigr)\, d\lambda(u). \end{aligned}$$
$$\int_{E(z,r')}\, d\lambda(u)< \infty, $$
$$\begin{aligned} K_{f} \leq& C\int_{\mathbb{B}^{n}} \biggl(\int _{E(z,r')}\bigl(1-|u|^{2}\bigr)^{N+1}\bigl(\bigl\vert \nabla \bigl(\partial^{m}h\bigr) (u)\bigr\vert +\bigl\vert \nabla\bigl(\partial^{m}g\bigr) (u)\bigr\vert \bigr)\, d\lambda(u) \biggr)^{p}\, d\lambda(z) \\ \leq&C\int_{\mathbb{B}^{n}} \biggl(\int_{E(z,r')} \bigl(1-|u|^{2}\bigr)^{(N+1)p}\bigl(\bigl\vert \nabla\bigl( \partial^{m}h\bigr) (u)\bigr\vert +\bigl\vert \nabla \bigl( \partial^{m}g\bigr) (u)\bigr\vert \bigr)^{p}\, d\lambda(u) \biggr)\, d\lambda(z) \\ \leq&C'\int_{\mathbb{B}^{n}}\bigl(1-|u|^{2} \bigr)^{(N+1)p}\bigl(\bigl\vert \nabla\bigl(\partial ^{m}h\bigr) (u)\bigr\vert +\bigl\vert \nabla\bigl(\partial^{m}g\bigr) (u)\bigr\vert \bigr)^{p}\, d\lambda(u). \end{aligned}$$
Acknowledgements
The authors heartily thank the referee for a careful reading of this paper as well as for many helpful comments and suggestions. The research was partly supported by program for NSF of China (Nos. 11501374, 11501284), NSFs of Zhejiang (No. LQ14A010006) and Hunan (No. 2015JJ6095).
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.