# New Characterizations of $\mathcal{N}(p,q,s)$ spaces on the unit ball of
$\mathbb{C}^{n}$.

Athanasios Beslikas

###### Abstract

In this note 1112020 Mathematics Subject Classification: 32A37. Key words and
phrases. $N(p,q,s)$-type spaces.  we provide Holland-Walsh-type
characterizations for functions on the $\mathcal{N}(p,q,s)$ spaces on the unit
ball for specific values of $p\geq 1.$ Characterizations for the holomorphic
function spaces $\mathcal{N}(p,q,s)$ were studied extensively in [2] and [3].

##  1 Introduction and notation

We denote by $z=(z_{1},...,z_{n}),z\in\mathbb{C}^{n}$ and
$|z|^{2}=|z_{1}|^{2}+...+|z_{n}|^{2}=\langle z,z\rangle,$ where
$\langle.,.\rangle$ denotes the usual inner product that induces the Euclidean
norm in $\mathbb{C}^{n}.$ Let $\mathbb{B}$ denote the open unit ball in
$\mathbb{C}^{n}$, that is $\mathbb{B}=\\{z\in\mathbb{C}^{n}:|z|<1\\}$. The
class of holomorphic functions on the unit ball will be denoted by
$\mathrm{Hol}(\mathbb{B}).$ We denote by $dV(z)$ the usual volume measure
normalized over the unit ball, and with

$dV_{\alpha}(z)=\frac{\Gamma(n+\alpha+1)}{n!\Gamma(\alpha+1)}(1-|z|^{2})^{\alpha}dV(z),\alpha\geq-1$

where $\Gamma(\cdot)$ is the usual Gamma function. For the whole length of our
paper we denote by $\Phi_{a}(z)$ the involutive automorphisms of the unit
ball. We have an explicit representation of such functions for
$a\in\mathbb{B}\setminus\\{0\\},$ given by the following formula:

$\Phi_{a}(z)=\frac{a-P_{a}z-s_{a}Q_{a}z}{1-\langle
z,a\rangle},z\in\mathbb{B},$

where $s_{a}=\sqrt{1-|a|^{2}}P_{a}$, $P_{a}=\frac{\langle
z,a\rangle}{|a|^{2}}$ is the orthogonal projection of $z$ onto the space
spanned by $a,$ and $Q_{a}=I-P_{a}.$ For more details the reader can check
[8]. Now we define the Möbius invariant measure by

$d\lambda(z)=(1-|z|^{2})^{-n-1}dV(z).$

It is called Möbius invariant because of the following equality:

$\int_{\mathbb{B}}f(z)d\lambda(z)=\int_{\mathbb{B}}f\circ\Phi_{a}(z)d\lambda(z).$

Also, by $\nabla f$ we symbolise the complex gradient of a holomorphic
function $f$, that is

$\nabla f(z)=\left(\frac{\partial f}{\partial z_{1}},...,\frac{\partial
f}{\partial z_{n}}\right)$

and by $\widetilde{\nabla}f$ the Möbius invariant gradient, that is:

$\widetilde{\nabla}f(z)=\nabla(f\circ\Phi_{z})(0).$

Furthermore, by $\mathcal{R}f(z)$ we denote the radial derivative of $f,$ that
is:

$\mathcal{R}f(z)=\sum_{k=1}^{n}z_{k}\frac{\partial f}{\partial z_{k}}.$

Lastly, whenever we encounter the notation $a\asymp b,$ we simply mean that
there exist two positive constants $C_{1},C_{2}$ such that $C_{1}a\leq b\leq
C_{2}a.$

##  2 Definitions and tools

In this section we give the definition of the $\mathcal{N}(p,q,s)$ spaces, and
some known characterizations. These will constitute our main tools for proving
our results.

######  Definition 1.

Let $f\in\mathrm{Hol}(\mathbb{B}).$ For $p\geq 1,q>0,s>0$ we define the
holomorphic function spaces $\mathcal{N}(p,q,s),$ or $\mathcal{N}_{q,s}^{p}$
by:

$\mathcal{N}_{q,s}^{p}=\Biggl{\\{}f\in\mathrm{Hol}(\mathbb{B}):\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}|f(z)|^{p}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)<+\infty\Biggr{\\}}.$

For different values of the parameters $p,q,s$ we obtain different function
spaces. The reader can check [2] for such examples and more details about the
$\mathcal{N}(p,q,s)$ spaces. For instance, if we could allow
$\mathcal{N}(2,0,s)$, $\frac{n-1}{n}<s<1$ we would receive the classical
$Q_{s}$ spaces, and if the parameter $s$ satisfied $1<s<\frac{n}{n-1},$ then
we would receive the Bloch space $\mathcal{B}$ on the unit ball. By
definition, we are not allowed to do so, and this shows that the theory of
$Q_{s}$ spaces is completely independent to the one of $\mathcal{N}(p,q,s)$
spaces. Some basic information about the mentioned function spaces is that we
already know that for $p\geq 1,q,s>0$ the set of polynomials is dense in
$\mathcal{N}(p,q,s),$ if and only if $ns+q>n.$ Also for $p\geq 1,q,s>0$ we
know that these spaces are functional Banach spaces. For the purpose of our
work, we will now mention some known characterizations.

######  Theorem 2.1.

Let $f\in\mathrm{Hol}(\mathbb{B})$ and $p\geq 1,q>0$ and
$s>\max\\{0,1-\frac{q}{n}\\}.$ Then $f\in\mathcal{N}(p,q,s)$ if and only if:

$\displaystyle I_{1}=$
$\displaystyle\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}|\nabla
f(z)|^{p}(1-|z|^{2})^{p+q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)<+\infty.$
(2.1) $\displaystyle I_{2}=$
$\displaystyle\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}|\widetilde{\nabla}f(z)|^{p}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)<+\infty.$
(2.2) $\displaystyle I_{3}=$
$\displaystyle\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}|\mathcal{R}f(z)|^{p}(1-|z|^{2})^{p+q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)<+\infty.$
(2.3)

###### Proof.

The proofs for all of the 3 characterizations can be found in [3]. ∎

The next Theorem that we will state is of particular interest for us.

######  Theorem 2.2.

Suppose $f\in\mathrm{Hol}(\mathbb{B}),p\geq 1,q>0$ and also
$s>\max\\{0,1-\frac{q}{n}\\},\alpha>q+ns-n-1.$ Then $f\in\mathcal{N}(p,q,s)$
if and only if:

$\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(z)dV_{\alpha}(w)<+\infty$

###### Proof.

See Theorem 5.12 of [3]. ∎

##  3 Motivation and statement of results

Historically, the problem started from [1] and the Holland-Walsh
characterization of the classical and well studied Bloch space $\mathcal{B}$
on the unit disc. Later, in [9], Stroethoff, in a much simpler and elegant
way, gave the Holland-Walsh characterization for the Bloch space and also
provided a similar characterization for the Besov spaces $B_{p}$ on the unit
disc $\mathbb{D},$ for $p\geq 2.$ After defining and studying extensively the
Hardy, Bergman, Besov and Dirichlet-type spaces on the unit ball of
$\mathbb{C}^{n}$, some similar double integral characterizations emerged for
the cases of Dirichlet, Bergman, Besov, and Bloch-type spaces, and this was
followed up by similar characterizations for the $Q_{s}$ spaces. Later, the
authors of [3] treated partially the case of $\mathcal{N}(p,q,s)$ spaces. In
the present note, we provide three characterizations that have not been found
in [3], that are similar to the ones presented in [4], [6], [7] for specific
values of the parameter $p.$ Let us now state the main results.

######  Theorem 3.1.

Let $f\in\mathrm{Hol}(\mathbb{B}),q>0$. Suppose also that the parameters
$p,q,s$ satisfy: $\alpha>q+ns-n-1,$ $s>\max\\{0,1-\frac{q}{n}\\}$ and $p\geq
2(n+1+\alpha).$ Then, $f\in\mathcal{N}(p,q,s),$ if and only if:

$\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|z-w|^{2(n+1+\alpha)}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(w)dV_{\alpha}(z)<+\infty.$

######  Remark 1.

Actually, because of the fact that $\alpha>q+ns-n-1$ the Theorem can hold for
$p>2(q+ns).$

Our second Theorem is quite similar to the first one:

######  Theorem 3.2.

Let $f\in\mathrm{Hol}(\mathbb{B}),q>0$. Suppose also that the parameters
$p,q,s$ satisfy: $\alpha>q+ns-n-1,$ $s>\max\\{0,1-\frac{q}{n}\\}$ and $p\geq
2(n+1+\alpha).$ Then, $f\in\mathcal{N}(p,q,s),$ if and only if:

$\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|w-P_{w}z-s_{w}Q_{w}z|^{2(n+1+\alpha)}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(w)dV_{\alpha}(z)<+\infty.$
(3.1)

Our third result is straightforward and can be deduced very easily by Theorem
2.3. The motive behind it can be found in [4] again.

######  Definition 2.

Let $f\in\mathrm{Hol}(\mathbb{B}).$ We define the $p$-Mean Oscillation of $f$
as follows:

$MO_{p}(f)(z)=\left(\int_{\mathbb{B}}|f(z)-f(w)|^{p}\frac{(1-|z|^{2})^{n+1}}{|1-\langle
z,w\rangle|^{2(n+1)}}dV(w)\right)^{1/p}.$

By the previous definition, we get:

######  Theorem 3.3.

Let $f\in\mathrm{Hol}(\mathbb{B}).$ If $p\geq 1,q>0$ and
$s>\max\\{0,1-\frac{q}{n}\\},$ then $f\in N(p,q,s)$ if and only if:

$J(f)=\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}MO^{p}_{p}(f)(z)(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)<+\infty.$

##  4 Some Lemmata and proofs

Initially, we provide the reader with some lemmata that will constitute our
main tools for the proof of Theorem 3.1. Let
$\Phi_{z}(w),z,w\in\mathbb{B},z\neq w,$ be an involutive automorphism of the
unit ball on $\mathbb{C}^{n}.$ As in [3],[4] and [5], we consider the Bergman-
pseudometric on the unit ball as:

$d(z,w)=|\Phi_{z}(w)|.$

We denote by

$E(z,r)=\\{w\in\mathbb{B}:|\Phi_{z}(w)|<r\\}$

the Bergman-metric ball, centered at $z$ with radius $r>0.$ The following well
known Lemma holds:

######  Lemma 4.1.

For any $r>0$ and $z\in\mathbb{B}$, let $E(z,r)$ be the Bergman metric ball
centered at $z.$ Then:

$(1-|z|^{2})\asymp(1-|w|^{2})\asymp|1-\langle z,w\rangle|$

for all $z\in\mathbb{B},w\in E(z,r).$

###### Proof.

The proof can be found in [10]. ∎

######  Lemma 4.2.

Let $f\in\mathrm{Hol}(\mathbb{B}).$ If $f(0)=0$ then, for all $p\geq
2(n+1+\alpha)$, $\alpha\geq-1,$ there exists a positive constant $C>0$ such
that:

$\int_{\mathbb{B}}\frac{|f(w)|^{p}}{|w|^{2(n+1+a)}}dV_{a}(w)\leq
C\int_{\mathbb{B}}|f(w)|^{p}dV_{\alpha}(w).$

###### Proof.

The Lemma holds for $p=2(n+1+\alpha)$ (see lemma 2.2. of [4]). So for all
$w\in\mathbb{B},$ we have the trivial inequality:

$\frac{1}{|w|^{2(n+1+\alpha)}}\leq\frac{1}{|w|^{p}},$

for all $p\geq 2(n+1+\alpha).$ ∎

######  Lemma 4.3.

Let $z,w\in\mathbb{B},z\neq w.$ Then, the following inequalities are true:

$|z-\Phi_{z}(w)|\geq\frac{|w|(1-|z|^{2})}{|1-\langle z,w\rangle|}$ (4.1)
$|z-\Phi_{z}(w)|^{2}\leq\frac{2(1-|z|^{2})}{|1-\langle z,w\rangle|}$ (4.2)

###### Proof.

The proof can be found again in [10]. ∎

We are ready to proceed with the proof of Theorem 3.1.

###### Proof.

(Proof of Theorem 3.1.): Sufficiency: Let $f\in\mathrm{Hol}(\mathbb{B}),$ and
$p,q,s,\alpha$ as in the statement of the Theorem. Initially, for the
convenience of the reader we set:

$I_{a}(f)=\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|z-w|^{2(n+1+\alpha)}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(w)dV_{\alpha}(z)$
(4.3)

We assume firstly that $\sup_{a\in\mathbb{B}}I_{a}(f)<\infty.$ We have to
prove that $f\in\mathcal{N}(p,q,s).$ For the upcoming calculations, let

$k_{z}(w)=\frac{(1-|z|^{2})^{n+1+\alpha}}{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}},z,w\in\mathbb{B}.$

We will estimate $I_{a}(f)$ from below. To do so, we apply firstly a change of
variables $w=\Phi_{z}(w):$

$I_{a}(f)=\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f\circ\Phi_{z}(w)|^{p}}{|z-\Phi_{z}(w)|^{2(n+1+\alpha)}}\times\\\
(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}k_{z}(w)dV_{\alpha}(z)dV_{\alpha}(w)$
(4.4)

For convenience in the upcoming calculations, set
$F_{z}(w)=f(z)-f\circ\Phi_{z}(w),$ and $n+1+\alpha=\gamma.$ By applying
Fubini’s Theorem, and Lemma 4.3 (inequality (4.2)), we get:

$I_{a}(f)\geq
C\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{a}(z)\times\\\
\left(\int_{\mathbb{B}}\frac{|F_{z}(w)|^{p}}{\frac{(1-|z|^{2})^{n+1+\alpha}}{|1-\langle
z,w\rangle|^{n+1+\alpha}}}\frac{(1-|z|^{2})^{n+1+\alpha}}{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}dV_{\alpha}(w)\right).$ (4.5)

Fix $0<r<1,$ and take $E(z,r)\subset{\mathbb{B}}$ be the domain of the second
integral:

$\displaystyle I_{a}(f)\geq$ $\displaystyle
C\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{a}(z)\left(\int_{\mathbb{B}}\frac{|F_{z}(w)|^{p}}{|1-\langle
z,w\rangle|^{\gamma}}dV_{\alpha}(w)\right)$ (4.6) $\displaystyle\geq$
$\displaystyle
C\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{a}(z)\left(\int_{E(z,r)}\frac{|F_{z}(w)|^{p}}{|1-\langle
z,w\rangle|^{\gamma}}dV_{\alpha}(w)\right)$ (4.7) $\displaystyle\geq$
$\displaystyle
C\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{a}(z)\left(\int_{E(z,r)}\frac{|F_{z}(w)|^{p}}{(1-|z|^{2})^{\gamma}}dV_{\alpha}(w)\right).$
(4.8)

At this point, we move $(1-|z|^{2})^{-\gamma}$ to the outer integral. It is
now obvious that we will replace $(1-|z|^{2})^{-n-1-a}dV_{a}(z)$ with
$d\lambda(z):$

$\displaystyle I_{a}(f)\geq
C\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)\left(\int_{E(z,r)}|F_{z}(w)|^{p}dV_{\alpha}(w)\right).$
(4.9)

By the topology induced by the Bergman pseudometric, we can find $\rho<r$ and
a Euclidean ball $B(z,\rho)\subset E(z,r)$. By applying the
plurisubharmonicity property for the function $|F_{z}(w)|^{p},p\geq 1$ we
deduce:

$\displaystyle I_{a}(f)$ $\displaystyle\geq
C\int_{\mathbb{B}}|F_{z}(z)|^{p}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)$
(4.10)
$\displaystyle=C\int_{\mathbb{B}}|f(z)-f(0)|^{p}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z).$
(4.11)

By taking the supremum for $a\in\mathbb{B}$ in (4.10) we proved:

$\displaystyle+\infty>\sup_{a\in\mathbb{B}}I_{a}(f)\geq$
$\displaystyle\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}|f(z)|^{p}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)=||f||_{\mathcal{N}},$
(4.12)

where $||f||_{\mathcal{N}}$ denotes the pseudo-norm induced by the definition
of the $\mathcal{N}(p,q,s)$ spaces.

Necessity: We assume that $f\in\mathcal{N}(p,q,s).$ We have to show that
$\sup_{a\in\mathbb{B}}I_{a}(f)$ is bounded. We recall that

$F_{z}(w)=f(z)-f\circ\Phi_{z}(w),z,w\in\mathbb{B},$

and we observe that $F_{z}(0)=0.$ We begin by applying Fubini’s Theorem and a
change of variables $w=\Phi_{z}(w),$ as before:

$I_{a}(f)=\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{a}(z)\times\\\
\left(\int_{\mathbb{B}}\frac{|F_{z}(w)|^{p}}{|z-\Phi_{z}(w)|^{2(n+1+\alpha)}}\frac{(1-|z|^{2})^{n+1+\alpha}}{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}dV_{\alpha}(w)\right).$ (4.13)

Now we apply the inequality (4.1) and we get:

$I_{a}(f)\leq\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{a}(z)\times\\\
\left(\int_{\mathbb{B}}\frac{|F_{z}(w)|^{p}}{\frac{|w|^{2(n+1+\alpha)}(1-|z|^{2})^{2(n+1+\alpha)}}{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}}\frac{(1-|z|^{2})^{n+1+\alpha}}{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}dV_{\alpha}(w)\right).$ (4.14)

After we simplify the equal terms in the numerator and denominator, we move
the term $(1-|z|^{2})^{-(n+1+\alpha)}$ to the outer integral, and we obtain

$I_{a}(f)\leq\int_{\mathbb{B}}(1-|z|^{2})^{q-(n+1+\alpha)}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{a}(z)\int_{\mathbb{B}}\frac{|F_{z}(w)|^{p}}{|w|^{2(n+1+\alpha)}}dV_{\alpha}(w).$

Now we apply Lemma 4.2 for the function $F_{z}(w).$ After another change of
variables we deduce:

$\displaystyle I_{a}(f)\leq$ $\displaystyle
C\int_{\mathbb{B}}(1-|z|^{2})^{q-(n+1+\alpha)}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(z)\int_{\mathbb{B}}|F_{z}(w)|^{p}dV_{a}(w)$
(4.15) $\displaystyle=$ $\displaystyle
C\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(z)\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|1-\langle
z,w\rangle|^{2(n+1+a)}}dV_{a}(w)$ (4.16) $\displaystyle=$ $\displaystyle
C\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(z)dV_{\alpha}(w),$
(4.17)

hence $I_{a}(f)<+\infty,$ by Theorem 2.3. ∎

We proceed now with the proof of Theorem 3.2.

###### Proof.

(Proof of Theorem 3.2) (Sufficiency:) Suppose that:

$\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|w-P_{w}z-s_{w}Q_{w}z|^{2(n+1+\alpha)}}\times\\\
(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(w)dV_{\alpha}(z)<+\infty.$
(4.18)

Then, by the trivial inequality:

$\frac{1}{|1-\langle z,w\rangle|}\leq\frac{1}{|w-P_{w}z-s_{w}Q_{w}z|}$

and Theorem 2.3 we deduce that $f\in\mathcal{N}(p,q,s).$

(Necessity:) Let $f\in\mathcal{N}(p,q,s).$ We will estimate from above the
following integral:

$J_{a}(f)=\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|w-P_{w}z-s_{w}Q_{w}z|^{2(n+1+\alpha)}}\times\\\
(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(w)dV_{\alpha}(z).$ (4.19)

Initially, we observe:

$J_{a}(f)=\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|\Phi_{z}(w)|^{2(n+1+a)}|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}\times\\\
(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(w)dV_{\alpha}(z).$ (4.20)

We apply the change of variables $w=\Phi_{z}(w),$ and Fubini’s Theorem as in
the proof of the previous Theorem:

$I_{a}(f)=\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(z)\times\\\
\left(\int_{\mathbb{B}}\frac{|f\circ\Phi_{z}(w)-f\circ\Phi_{z}(0)|^{p}k_{z}(w)}{|w|^{2(n+1+\alpha)}|1-\langle\Phi_{z}(w),z\rangle|^{2(n+1+\alpha)}}dV_{\alpha}(w)\right)$
(4.21)

By the properties of the involutive automorphisms of the unit ball, we know
that the following identity holds (e.g. [8].):

$\frac{1}{|1-\langle\Phi_{z}(w),z\rangle|^{2(n+1+\alpha)}}=\frac{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}{(1-|z|^{2})^{2(n+1+\alpha)}}.$ (4.22)

By applying (4.22) to (4.21) we obtain:

$J_{a}(f)=\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)\times\\\
\left(\int_{\mathbb{B}}\frac{|f\circ\Phi_{z}(w)-f\circ\Phi_{z}(0)|^{p}}{|w|^{2(n+1+\alpha)}}dV_{\alpha}(w)\right).$
(4.23)

Recall, once again, the previous notation
$F_{z}(w)=f\circ\Phi_{z}(w)-f\circ\Phi_{z}(0).$ At this stage, we will apply
Lemma 4.2. By doing so, we obtain a positive constant $C>0$ such that:

$\displaystyle J_{a}(f)\leq
C\int_{\mathbb{B}}(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)\left(\int_{\mathbb{B}}|F_{z}(w)|^{p}dV_{\alpha}(w)\right).$
(4.24)

By repeating the change of variables and applying Theorem 2.3, we obtain:

$\sup_{a\in\mathbb{B}}J_{a}(f)\leq\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|1-\langle
z,w\rangle|^{2(n+1+\alpha)}}\times\\\
(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV_{\alpha}(z)dV_{\alpha}(w)<+\infty$
(4.25)

and the second implication follows. ∎

###### Proof.

(Proof of Theorem 3.3:) Observe that:

$MO_{p}^{p}(f)=\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|1-\langle
z,w\rangle|^{2(n+1)}}(1-|z|^{2})^{n+1}dV(w).$

From Theorem 2.3. and by choosing $\alpha=0,$ we know that $f\in N(p,q,s)$ if
and only if

$J(f)=\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|1-\langle
z,w\rangle|^{2(n+1)}}\times\\\
(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}dV(z)dV(w)<+\infty.$ (4.26)

We write the integral of $J(f)$ as an iterated one and we observe that $f\in
N(p,q,s)$ if and only if

$\sup_{a\in\mathbb{B}}\int_{\mathbb{B}}\left(\int_{\mathbb{B}}\frac{|f(z)-f(w)|^{p}}{|1-\langle
z,w\rangle|^{2(n+1)}}(1-|z|^{2})^{n+1}dV(w)\right)\times\\\
(1-|z|^{2})^{q}(1-|\Phi_{a}(z)|^{2})^{ns}d\lambda(z)<+\infty$ (4.27)

which gives us the desired result. ∎

##  5 Discussion and questions

In this final section we pose some comments and questions for the interested
reader. The motivation behind our question is clear. We proved Theorem 3.1 and
Theorem 3.2. for $p\geq 2(n+1+\alpha),\alpha\geq 1.$

Question 1: Can we obtain Theorem 3.1 for more general values of the parameter
$p\geq 1?$ What happens for $1\leq p<2(n+1+\alpha)?$

The answer to this question seems to be technical, as in this case we cannot
apply Lemma 4.2 as in our proofs.

## References

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  * [2] B.Hu, S.Li, $\mathcal{N}(p,q,s)$-Type spaces on the unit ball of $\mathbb{C}^{n}$, arXiv:1609.00957v2, 2017.
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