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

$$\langle z,a\rangle=z_{1}\bar{a}_{1}+\cdots+z_{n} \bar{a}_{n}, $$

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

$$|z|=\langle z,z\rangle^{\frac{1}{2}}=\bigl(|z_{1}|^{2}+ \cdots+|z_{n}|^{2}\bigr)^{\frac{1}{2}}. $$

Denote a ball in \({\mathbb{C}}^{n}\) with center a and radius \(r > 0\) by

$$\mathbb{B}^{n}(a,r)=\bigl\{ z\in{\mathbb{C}}^{n} : |z-a|< r \bigr\} . $$

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 \({\mathbb{C}}\).

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 \({\mathbb{C}}^{n}\).

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

$$\begin{aligned}& \nabla f(z)=\biggl(\frac{\partial f}{\partial z_{1}},\ldots,\frac{\partial f}{\partial z_{n}}\biggr), \qquad \overline{\nabla} f(z)=\biggl(\frac{\partial f}{\partial\bar{z}_{1}},\ldots,\frac{\partial f}{\partial\bar{z}_{n}}\biggr), \\& \partial^{m} f=\frac{\partial^{m} f}{\partial z^{m}}=\frac{\partial ^{|m|} f}{\partial z_{1}^{m_{1}}\, \cdots \, \partial z_{n}^{m_{n}}},\qquad \overline { \partial}^{m} f=\frac{\partial^{m} f}{\partial\bar{z}^{m}}=\frac {\partial^{|m|} f}{\partial\bar{z}_{1}^{m_{1}}\, \cdots\, \partial\bar {z}_{n}^{m_{n}}}, \\& \hat{D}^{(m)}f=\partial^{m} f+ \overline{\partial}^{m} f= \partial^{m} h+ \overline{\partial}^{m} g, \end{aligned}$$

where \(|m|=m_{1}+\cdots+m_{n}\). Obviously, if \(f\in\mathcal{P}(\mathbb{B}^{n})\), then so does \(\hat{D}^{(m)}f\).

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

$$\|f\|= \sup_{z\in\mathbb{B}^{n}}\bigl(1-|z|^{2}\bigr) \bigl(\bigl\vert \nabla f(z)\bigr\vert + \bigl\vert \overline{\nabla} f(z)\bigr\vert \bigr)< \infty; $$

the little Bloch space \(\mathcal{PB}_{0}(\mathbb{B}^{n})\) consists of all mappings \(f\in \mathcal{PB}(\mathbb{B}^{n})\) such that

$$\lim_{|z|\rightarrow1^{-}}\bigl(1-|z|^{2}\bigr) \bigl(\bigl\vert \nabla f(z)\bigr\vert + \bigl\vert \overline{\nabla} f(z)\bigr\vert \bigr)=0. $$

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

$$\beta_{f}=\sup_{z,w\in{\mathbb{D}},z\neq w}\frac {|f(z)-f(w)|}{{\operatorname{arctanh}}|\frac {z-w}{1-\bar{z}w}|}< \infty. $$

Let

$$l=\sup_{w\in D(z,r),z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(n-1)}f(z)-\hat {D}^{(n-1)}f(w)|}{|z-w|}, $$

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.

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

$$\phi_{a}(z)=\frac{a-P_{a}z-(1-|a|^{2})^{\frac{1}{2}}Q_{a}z}{1-\langle z,a\rangle}, $$

where \(a,z\in\mathbb{B}^{n}\),

$$P_{a}z=\frac{a\langle z,a\rangle}{\langle a,a\rangle}, \qquad Q_{a}z=z-P_{a}z. $$

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

$$E(z,r)=\bigl\{ w\in\mathbb{B}: \bigl\vert \phi_{z}(w)\bigr\vert < r \bigr\} . $$

Clearly, \(E(z,r)=\phi_{z}({\mathbb{B}}(0,r))\).

Lemma 1

([12])

Let \(0< r<1\) and \(w\in E(z,r)\). Then

$$1-|z|^{2}\asymp1-|w|^{2}\asymp \bigl\vert 1-\langle z, w \rangle\bigr\vert \asymp\bigl\vert E(z,r)\bigr\vert ^{\frac{1}{n+1}}, $$

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 \(B/C \leq A \leq BC\).

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

$$\sup_{z\in\mathbb{B}^{n}}\bigl(1-|z|^{2}\bigr)\bigl\vert \nabla h(z)\bigr\vert < \infty; $$

similarly, \(h\in\mathcal{B}_{0}\) if \(h\in \mathcal{B}\) and

$$\lim_{|z|\rightarrow1^{-}}\bigl(1-|z|^{2}\bigr)\bigl\vert \nabla h(z)\bigr\vert =0. $$

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. \(\mathcal{B}_{0}\)).

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

$$\sup_{z\in \mathbb{B}^{n}}\bigl(1-|z|^{2}\bigr)^{N}\biggl\vert \frac{\partial^{m} h(z)}{\partial z^{m}}\biggr\vert < \infty \qquad \biggl(\textit{resp. }\lim _{|z|\rightarrow 1^{-}}\bigl(1-|z|^{2}\bigr)^{N}\biggl\vert \frac{\partial^{m} h(z)}{\partial z^{m}}\biggr\vert =0\biggr) $$

for all values of the multi-index m with \(|m| = N\).

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

$$\sup_{z\in\mathbb{B}^{n}}\bigl(1-\vert z\vert ^{2} \bigr)^{N}\bigl(\bigl\vert \partial^{m}f\bigr\vert +\bigl\vert \overline {\partial}^{m}f\bigr\vert \bigr)=\sup _{z\in\mathbb{B}^{n}}\bigl(1-\vert z\vert ^{2} \bigr)^{N} \bigl(\bigl\vert \partial^{m} h\bigr\vert + \bigl\vert \overline{\partial}^{m} g\bigr\vert \bigr)< \infty, $$

respectively,

$$\lim_{\vert z\vert \rightarrow 1^{-}}\bigl(1-\vert z\vert ^{2} \bigr)^{N}\bigl(\bigl\vert \partial^{m}f\bigr\vert +\bigl\vert \overline{\partial}^{m}f\bigr\vert \bigr)=\lim _{\vert z\vert \rightarrow 1^{-}}\bigl(1-\vert z\vert ^{2} \bigr)^{N} \bigl(\bigl\vert \partial^{m} h\bigr\vert + \bigl\vert \overline{\partial}^{m} g\bigr\vert \bigr)\rightarrow0 $$

for all values of the multi-index m with \(|m|=N\).

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

$$z=w+\xi\overline{\nabla \biggl(\frac{\partial h}{\partial z_{j}}\biggr) (w)}\rightarrow w $$

with \(\xi\in {\mathbb{C}}\), we have

$$\bigl(1-|w|^{2}\bigr)^{2}\biggl\vert \nabla \biggl( \frac{\partial h}{\partial z_{j}}\biggr) (w)\biggr\vert \leq L, $$

for each \(j\in \{1,\ldots,n\}\). By Lemma 3, we see that \(h\in\mathcal{B}\).

For the converse, we assume that \(h\in\mathcal{B}\). Let \(h_{j}(z)=\frac{\partial h(z)}{\partial z_{j}}\), then for each \(j\in \{1,\ldots,n\}\),

$$\begin{aligned} \bigl\vert h_{j}(z)-h_{j}(w)\bigr\vert =&\biggl\vert \int^{1}_{0}\frac{dh_{j}}{ds} \bigl(sz+(1-s)w\bigr)\, ds\biggr\vert \\ \leq&\sum_{k=1}^{n}\biggl\vert (z_{k}-w_{k})\int^{1}_{0} \frac{\partial h_{j}}{\partial z_{k}}\bigl(sz+(1-s)w\bigr)\, ds \biggr\vert \\ \leq& \sqrt{n}\vert z-w\vert \int^{1}_{0} \bigl\vert \nabla h_{j}\bigl(sz+(1-s)w\bigr)\bigr\vert \, ds \\ \leq& C\vert z-w\vert \int^{1}_{0} \frac{ds}{(1-\vert sz+(1-s)w\vert ^{2})^{2}}. \end{aligned}$$

It follows from [7] that there exists \(0< C_{1}<\infty\) such that

$$\int^{1}_{0}\frac{ds}{(1-|sz+(1-s)w|^{2})^{2}}\leq \frac{C_{1}}{(1-|z|^{2})(1-|w|^{2})}. $$

This implies that

$$\sup_{z,w\in\mathbb{B}^{n}, z\neq w}\frac {(1-|z|^{2})(1-|w|^{2})}{|z-w|}\bigl\vert h_{j}(z)-h_{j}(w) \bigr\vert < \infty. $$

So the result follows. □

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

$$L_{f}=\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r, z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}< \infty $$

for all values of the multi-index m with \(|m|=N\), where \(\alpha +\beta=N+1\).

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 \(r\in(0,1)\),

$$ \bigl\vert \nabla \bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert \leq \frac{C}{(1-\vert z\vert ^{2})}\int_{\partial\mathbb{B}^{n}}\bigl\vert \bigl(\hat {D}^{(m)}f\bigr) (z+\varrho\zeta)-\bigl(\hat{D}^{(m)}f\bigr) (z) \bigr\vert \, d\sigma(\zeta), $$

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

$$ \bigl\vert \nabla\bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert \leq \frac{C}{(1-\vert z\vert ^{2})}\sup_{w\in E(z,r)}\bigl\vert \bigl( \hat{D}^{(m)}f\bigr) (z)-\bigl(\hat{D}^{(m)}f\bigr) (w)\bigr\vert . $$

Since for each \(w \in E(z, r)\), \(w\neq z \),

$$\frac{(1-|z|^{2})^{\frac{1}{2}}(1-|w|^{2})^{\frac{1}{2}}}{|z-w|}\geq \frac{(1-r^{2})^{\frac{1}{2}}}{r}, $$

by Lemma 1, we can deduce that

$$\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}}{|z-w|}\geq C_{1}\bigl(1-|z|^{2} \bigr)^{N}. $$

Therefore, there exists a positive constant \(C_{2}\) such that

$$\bigl(1-\vert z\vert ^{2}\bigr)^{N+1}\bigl(\bigl\vert \nabla\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{ \nabla}\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert \bigr)\leq C_{2}L_{f}, $$

from which we see that \(f\in\mathcal{PB}(\mathbb{B}^{n})\).

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

$$\begin{aligned} \bigl\vert \bigl(\hat{D}^{(m)}f\bigr) (z)-\bigl(\hat{D}^{(m)}f \bigr) (w)\bigr\vert =&\biggl\vert \int^{1}_{0} \frac{d(\hat{D}^{(m)}f)}{ds}\bigl(sz+(1-s)w\bigr)\,ds\biggr\vert \\ \leq&\sum_{k=1}^{n}\biggl\vert (z_{k}-w_{k})\int^{1}_{0} \frac{\partial (\hat{D}^{(m)}f)}{\partial z_{k}}\bigl(sz+(1-s)w\bigr)\,ds \biggr\vert \\ &{}+\sum_{k=1}^{n}\biggl\vert ( \bar{z}_{k}-\bar{w}_{k})\int^{1}_{0} \frac {\partial (\hat{D}^{(m)}f)}{\partial\bar{z}_{k}}\bigl(sz+(1-s)w\bigr)\,ds\biggr\vert \\ \leq& \sqrt{n}\vert z-w\vert \int^{1}_{0} \bigl(\bigl\vert \nabla \bigl(\hat{D}^{(m)}f\bigr) \bigl(sz+(1-s)w\bigr) \bigr\vert \\ &{}+\bigl\vert \overline{\nabla} \bigl(\hat{D}^{(m)}f\bigr) \bigl(sz+(1-s)w\bigr)\bigr\vert \bigr)\,ds \\ \leq& C\vert z-w\vert \int^{1}_{0} \frac{ds}{(1-\vert sz+(1-s)w\vert )^{N+1}}. \end{aligned}$$

By Lemma 1 we infer that there exists \(\iota>0\) such that \(1-|w|=\iota(1-|z|)\) and

$$\begin{aligned} \frac{|(\hat{D}^{(m)}f)(z)-(\hat {D}^{(m)}f)(w)|}{|z-w|} \leq& C\int^{1}_{0} \frac {ds}{(s(1-|z|)+(1-s)(1-|w|))^{N+1}} \\ \leq&\frac{C'}{(1-|z|^{2})^{N+1}}\int^{1}_{0} \frac{ds}{[s+\iota (1-s)]^{N+1}} \\ \leq&\frac{C''}{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}}. \end{aligned}$$

Thus,

$$L_{f}=\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}< \infty. $$

So the proof is complete. □

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

$$\sup_{z,w\in \mathbb{B}^{n}, z\neq w}\bigl(1-|z|^{2}\bigr)^{\frac{N}{2}}\bigl(1-|w|^{2}\bigr)^{\frac{N}{2}}\biggl\vert \frac{(\hat {D}^{(m)}f)(z)-(\hat{D}^{(m)}f)(w)}{z-w}\biggr\vert < \infty $$

for all multi-index with \(|m|=N-1\).

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 \(j\in \{1,\ldots,n\}\),

$$\begin{aligned}& \sup_{z,w\in \mathbb{B}^{n}, z\neq w}\frac{(1-\vert z\vert ^{2})(1-\vert w\vert ^{2})}{\vert z-w\vert }\biggl\vert \frac{\partial f(z)}{\partial z_{j}}+ \frac{\partial f(z)}{\partial\bar{z}_{j}}-\frac{\partial f(w)}{\partial z_{j}}-\frac{\partial f(w)}{\partial \bar{z}_{j}}\biggr\vert \\& \quad \leq\sup_{z,w\in\mathbb{B}^{n},z\neq w}\frac{(1-\vert z\vert ^{2})(1-\vert w\vert ^{2})}{\vert z-w\vert }\biggl(\biggl\vert \frac{\partial h(z)}{\partial z_{j}}-\frac{\partial h(w)}{\partial z_{j}}\biggr\vert +\biggl\vert \frac{\partial g(z)}{\partial z_{j}}-\frac{\partial g(w)}{\partial z_{j}}\biggr\vert \biggr). \end{aligned}$$

Since \(f\in\mathcal{PB}(\mathbb{B}^{n})\), \(h,g \in\mathcal{B}\), by Lemma 4,

$$\begin{aligned}& \sup_{z,w\in \mathbb{B}^{n},z\neq w}\frac{(1-\vert z\vert ^{2})(1-\vert w\vert ^{2})}{\vert z-w\vert }\biggl\vert \frac{\partial h(z)}{\partial z_{j}}- \frac{\partial h(w)}{\partial z_{j}}\biggr\vert < \infty, \\& \quad \leq\sup_{z,w\in \mathbb{B}^{n},z\neq w}\frac{(1-\vert z\vert ^{2})(1-\vert w\vert ^{2})}{\vert z-w\vert }\biggl\vert \frac{\partial g(z)}{\partial z_{j}}-\frac{\partial g(w)}{\partial z_{j}}\biggr\vert < \infty. \end{aligned}$$

This completes the proof. □

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

$$ \lim_{|z|\rightarrow1^{-}}\sup_{z\in\mathbb {B}^{n},\rho(z,w)< r,z\neq w} \frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}=0 $$
(1)

for all values of the multi-index m with \(|m|=N\), where \(\alpha +\beta=N+1\).

Proof

Sufficiency. Assume that (1) holds. Then for any \(\epsilon>0\), there exists \(\delta\in(0,1)\) such that

$$\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}< \epsilon $$

whenever \(\delta<|z|<1\). It follows from an argument similar to the proof of Theorem 1, that we have

$$\begin{aligned}& \bigl(1-|z|^{2}\bigr)^{N+1}\bigl(\bigl\vert \nabla \bigl( \hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{\nabla}\bigl( \hat{D}^{(m)}f\bigr)\bigr\vert \bigr) \\& \quad \leq C\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|} < C\epsilon, \end{aligned}$$

whenever \(\delta<|z|<1\). Hence

$$\lim_{|z|\rightarrow 1^{-}}\bigl(1-|z|^{2}\bigr)^{N+1}\bigl( \bigl\vert \nabla\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert \bigr)=0, $$

from which we see that \(f\in\mathcal{PB}_{0}(\mathbb{B}^{n})\).

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 \(|m|=N\),

$$\begin{aligned}& \frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat {D}^{(m)}(f-f_{\lambda})(z)-\hat{D}^{(m)}(f-f_{\lambda})(w)|}{|z-w|} \\& \quad \leq C_{1}\bigl(1-|\xi|^{2}\bigr)^{N+1}\bigl( \bigl\vert \nabla\hat{D}^{(m)}(f-f_{\lambda}) (\xi)\bigr\vert + \bigl\vert \overline{\nabla}\hat{D}^{(m)}(f-f_{\lambda}) (\xi)\bigr\vert \bigr) \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} &\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat {D}^{(m)}f_{\lambda}(z)-\hat{D}^{(m)}f_{\lambda}(w)|}{|z-w|} \\ &\quad \leq \frac{C_{2}\lambda}{(1-|\lambda|^{2})^{N+1}}\bigl(1-|\eta|^{2}\bigr)^{N+1} \bigl(\bigl\vert \nabla\bigl( \hat{D}^{(m)}f_{\lambda}\bigr) (\eta) \bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f_{\lambda}\bigr) (\eta)\bigr\vert \bigr) \end{aligned} \end{aligned}$$

for all \(z,w\in\mathbb{B}^{n}\), \(\rho(z,w)< r\) and \(\xi, \eta\in E(z,r)\). So

$$\begin{aligned} L_{f} \leq&C_{1}\bigl(1-\vert \xi \vert ^{2} \bigr)^{N+1}\bigl(\bigl\vert \nabla\hat {D}^{(m)}(f-f_{\lambda}) (\xi)\bigr\vert + \bigl\vert \overline{\nabla}\hat{D}^{(m)}(f-f_{\lambda}) (\xi)\bigr\vert \bigr) \\ &{}+ \frac{C_{2}\lambda}{(1-\vert \lambda \vert ^{2})^{N+1}}\bigl(1-\vert \eta \vert ^{2} \bigr)^{N+1}\bigl(\bigl\vert \nabla\bigl( \hat{D}^{(m)}f_{\lambda}\bigr) (\eta)\bigr\vert + \bigl\vert \overline{\nabla}\bigl( \hat{D}^{(m)}f_{\lambda}\bigr) (\eta)\bigr\vert \bigr). \end{aligned}$$

First letting \(|z|\rightarrow1^{-}\) and then letting \(\lambda \rightarrow1^{-}\), we obtain the desired result. □

From Theorem 2 and the proof of Theorem 3, we have the following.

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

$$\lim_{|z|\rightarrow 1^{-}}\sup_{z,w\in\mathbb{B}^{n}, z\neq w}\bigl(1-|z|^{2} \bigr)^{\frac{N}{2}}\bigl(1-|w|^{2}\bigr)^{\frac{N}{2}} \biggl\vert \frac{(\hat {D}^{(m)}f)(z)-(\hat{D}^{(m)}f)(w)}{z-w}\biggr\vert =0 $$

for all multi-index with \(|m|=N-1\).

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

$$\sup_{z\in\mathbb{B}^{n}}\bigl(1-|z|^{2}\bigr)^{N+1} \bigl(\bigl\vert \nabla\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{\nabla}\bigl(\hat{D}^{(m)}f\bigr)\bigr\vert \bigr)\in L^{p}\bigl(\mathbb{B}^{n}, d\lambda\bigr) $$

for all values of the multi-index m with \(|m| = N\), and \(p(N+1)\geq n\).

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

$$\begin{aligned} \sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\biggl\vert \frac{h(z)-h(w)}{z-w}\biggr\vert \leq&C\sup_{\zeta\in E(z,r)}\bigl\vert \nabla h(\zeta)\bigr\vert \\ \leq& \frac{C}{|E(z,r')|}\int_{E(z,r')}\bigl\vert \nabla h( \zeta)\bigr\vert \, dv(\zeta) \\ \leq&K\int_{E(z,r')}\bigl\vert \nabla h(\zeta)\bigr\vert \, d\lambda(\zeta) \end{aligned}$$

for some \(r'>r\). □

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

$$K_{f}=\int_{\mathbb{B}^{n}} \biggl(\sup_{z\in\mathbb{B}^{n},\rho (z,w)< r,z\neq w} \frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|} \biggr)^{p}\, d\lambda(z)< \infty $$

for all values of the multi-index m with \(|m|=N\), where \(\alpha +\beta=N+1\), and \(p(N+1)\geq n\).

Proof

Let \(f=h+\bar{g}\in\mathcal{P}(\mathbb{B}^{n})\). Suppose that

$$\int_{\mathbb{B}^{n}} \biggl(\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w} \frac{(1-|z^{2}|)^{\alpha}(1-|w^{2}|)^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|} \biggr)^{p}\, d\lambda(z)< \infty. $$

Let

$$L_{f}(z)=\lim_{z\rightarrow w}\sup\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat {D}^{(m)}f(z)-\hat{D}^{(m)}f(w)|}{|z-w|}. $$

It follows from the proof of Theorem 1 that we have

$$\bigl(1-|z|^{2}\bigr)^{N+1}\bigl(\bigl\vert \nabla \bigl( \hat{D}^{(m)}f\bigr) (z)\bigr\vert + \bigl\vert \overline{\nabla} \bigl(\hat{D}^{(m)}f\bigr) (z)\bigr\vert \bigr)\leq CL_{f}(z). $$

Since \(L_{f}(z)\leq L_{f}\), we see that

$$\begin{aligned}& \int\bigl(1-|z|^{2}\bigr)^{(N+1)p} \bigl(\bigl\vert \nabla \bigl(\hat{D}^{(m)}f\bigr)\bigr\vert + \bigl\vert \overline{\nabla} \bigl(\hat{D}^{(m)}f\bigr)\bigr\vert \bigr)^{p}\, d\lambda(z) \\& \quad \leq C\int_{\mathbb{B}^{n}} \biggl(\sup_{z\in\mathbb{B}^{n},\rho (z,w)< r,z\neq w} \frac{(1-|z^{2}|)^{\alpha}(1-|w^{2}|)^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|} \biggr)^{p}\, d\lambda(z), \end{aligned}$$

which yields \(f\in{\mathcal{B}}_{p}\).

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,

$$\begin{aligned} L_{f} \leq&\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}(|\partial^{m}h(z)-\partial ^{m}h(w)|+|\partial^{m}g(z)-\partial^{m}g(w)|)}{|z-w|} \\ \leq& C\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{N+1}|\partial^{m}h(z)-\partial^{m}h(w)|}{|z-w|} \\ &{}+C\sup_{z\in\mathbb{B}^{n},\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{N+1}|\partial^{m}g(z)-\partial^{m}g(w)|}{|z-w|} \\ \leq& C_{1}\int_{E(z,r')}\bigl(1-|u|^{2} \bigr)^{N+1}\bigl(\bigl\vert \nabla\bigl(\partial^{m}h\bigr) (u)\bigr\vert +\bigl\vert \nabla \bigl(\partial^{m}g\bigr) (u)\bigr\vert \bigr)\, d\lambda(u). \end{aligned}$$

Since

$$\int_{E(z,r')}\, d\lambda(u)< \infty, $$

by Hölder’s inequality and Fubini’s theorem, we can obtain

$$\begin{aligned} K_{f} \leq& C\int_{\mathbb{B}^{n}} \biggl(\int _{E(z,r')}\bigl(1-|u|^{2}\bigr)^{N+1}\bigl(\bigl\vert \nabla \bigl(\partial^{m}h\bigr) (u)\bigr\vert +\bigl\vert \nabla\bigl(\partial^{m}g\bigr) (u)\bigr\vert \bigr)\, d\lambda(u) \biggr)^{p}\, d\lambda(z) \\ \leq&C\int_{\mathbb{B}^{n}} \biggl(\int_{E(z,r')} \bigl(1-|u|^{2}\bigr)^{(N+1)p}\bigl(\bigl\vert \nabla\bigl( \partial^{m}h\bigr) (u)\bigr\vert +\bigl\vert \nabla \bigl( \partial^{m}g\bigr) (u)\bigr\vert \bigr)^{p}\, d\lambda(u) \biggr)\, d\lambda(z) \\ \leq&C'\int_{\mathbb{B}^{n}}\bigl(1-|u|^{2} \bigr)^{(N+1)p}\bigl(\bigl\vert \nabla\bigl(\partial ^{m}h\bigr) (u)\bigr\vert +\bigl\vert \nabla\bigl(\partial^{m}g\bigr) (u)\bigr\vert \bigr)^{p}\, d\lambda(u). \end{aligned}$$

It follows from Lemma 5 that \(K_{f}\) is bounded. This completes the proof. □