Hadamard Gaps and N_k-type Spaces in the Unit Ball

M. A. Bakhit, A. E. Shammaky

Department of Mathematics, Faculty of Science, Jazan University, Jazan, KSA.

DOI: 10.4236/apm.2017.74017                

Abstract

In this paper, we introduce a class of holomorphic Banach spaces N_K of functions on the unit ball B of Cⁿ. We develop the necessary and sufficient condition for N_K(B) spaces to be non-trivial and we discuss the nesting property of N_K(B) spaces. Also, we obtain some characterizations of functions with Hadamard gaps in N_K(B) spaces. As a consequence, we prove a necessary and sufficient condition for that N_K(B) spaces coincides with the Beurling-type space.

Keywords

N_k-type Spaces, Beurling-Type Space, Hadamard Gaps

1. Introduction

Through this paper, $\mathbb{B}$ is the unit ball of the n-dimensional complex Euclidean space ${ℂ}^n$ , $\mathbb{S}$ is the boundary of $\mathbb{B}$ . We denote the class of all holomorphic functions, with the compact-open topology on the unit ball $\mathbb{B}$ by $\mathcal{H}\left(\mathbb{B}\right)$ .

For any $z=\left(z_1,z_2,\cdots ,z_n\right)$ , $w=\left(w_1,w_2,\cdots ,w_n\right)\in {ℂ}^n$ , the inner product is defined by $\langle z,w\rangle =\left(z_1\overline{w_1},z_2\overline{w_2},\cdots ,z_n\overline{w_n}\right)$ , and write $\left|z\right|=\sqrt{\langle z,w\rangle }$ .

Let $\text{d}v$ be the Lebesgue volume measure on ${ℂ}^n$ , normalized so that $v\left(\mathbb{B}\right)\equiv 1$ and $\text{d}\sigma$ be the surface measure on $\mathbb{S}$ . Once again, we normalize $\sigma$ so that $\sigma \left(\mathbb{B}\right)\equiv 1$ . For $z\in \mathbb{B}$ and $r>0$ let ${\mathbb{B}}_r=\left\{z\in \mathbb{B}:\left|z\right|\le r\right\}$ .

For $\zeta \in \mathbb{B}$ the measures $v$ and $\sigma$ are related by the following formula:

${\displaystyle \underset{\mathbb{B}}{\int }f\text{d}v}=2n{\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2n-1}\text{d}r{\displaystyle \underset{\mathbb{S}}{\int }f\left(r\zeta \right)\text{d}\sigma \left(\zeta \right)}.}$ (1)

The identity

${\displaystyle \underset{\mathbb{S}}{\int }f\text{d}\sigma }={\displaystyle \underset{\mathbb{S}}{\int }\text{d}\sigma \left(\zeta \right)}\frac{1}{\text{2π}}{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}f\left({\text{e}}^{i\theta }\zeta \right)\text{d}\theta ,}$ (2)

is called integration by slices, for all $0\le \theta \le 2\text{π}$ (see [1] ).

For every point $a\in \mathbb{B}$ the Möbius transformation ${\phi }_a:\mathbb{B}\to \mathbb{B}$ is defined by

${\phi }_a\left(z\right)=\frac{a-P_a\left(z\right)-S_a{Q}_a\left(z\right)}{1-\langle z,a\rangle },$ (3)

where $S_a=\sqrt{1-{\left|z\right|}^2},P_a\left(z\right)=\frac{a\langle z,a\rangle }{{\left|a\right|}^2},P_0=0$ and $Q_a=I-P_a\left(z\right)$ (see [1] or [2] ).

The map ${\phi }_a$ has the following properties that ${\phi }_a\left(0\right)=a$ , ${\phi }_a\left(a\right)=0$ , ${\phi }_a={\phi }_a^{-1}$ and

$1-\langle {\phi }_a\left(z\right),{\phi }_a\left(w\right)\rangle =\frac{\left(1-{\left|a\right|}^2\right)\left(1-\langle z,w\rangle \right)}{\left(1-\langle z,a\rangle \right)\left(1-\langle a,w\rangle \right)},$

where z and w are arbitrary points in $\mathbb{B}$ . In particular,

$1-{\left|{\phi }_a\left(z\right)\right|}^2=\frac{\left(1-{\left|a\right|}^2\right)\left(1-{\left|z\right|}^2\right)}{{\left|1-\langle z,a\rangle \right|}^2},$ (4)

For $a\in \mathbb{B}$ the Möbius invariant Green function in the unit ball $\mathbb{B}$ denoted by $G\left(z,a\right)=g\left({\phi }_a(z)\right)$ where $g\left(z\right)$ is defined by:

$g\left(z\right)=\frac{n+1}{2n}{\displaystyle \underset{\left|z\right|}{\overset{1}{\int }}{\left(1-t^2\right)}^{n-1}{t}^{1-2n}\text{d}t.}$ (5)

For $n>1$ , we have

$\frac{1}{C_n}{\left(1-r^2\right)}^n{t}^{-2\left(n-1\right)}\le C_n{\left(1-r^2\right)}^n{t}^{-2\left(n-1\right)},$ (6)

where $C_n$ is a constant depending on n only.

Let $H^{\infty }\left(\mathbb{B}\right)$ denote the Banach space of bounded functions in $\mathcal{H}\left(\mathbb{B}\right)$ with the norm ${\Vert f\Vert }_{\infty }={\sup }_{z\in \mathbb{B}}\left|f\left(z\right)\right|$ .

For $\alpha >0$ , the Beurling-type space (sometimes also called the Bers-type space) $H_{\alpha }^{\infty }\left(\mathbb{B}\right)$ in the unit ball $\mathbb{B}$ consists of those functions $f\in \mathcal{H}\left(\mathbb{B}\right)$ for which

${\Vert f\Vert }_{H_{\alpha }^{\infty }\left(\mathbb{B}\right)}=\underset{z\in \mathbb{B}}{\sup }\left|f\left(z\right)\right|{\left(1-{\left|z\right|}^2\right)}^{\alpha }<\infty .$ (7)

Let $K:\left(0,\infty \right)\to \left[0,\infty \right)$ is a right-continuous, non-decreasing function and is not equal to zero identically. The ${\mathcal{N}}_K\left(\mathbb{B}\right)$ space consists of all functions $f\in \mathcal{H}\left(\mathbb{B}\right)$ such that

${\Vert f\Vert }_K^2=\underset{z\in \mathbb{B}}{\sup }{\displaystyle \underset{\mathbb{B}}{\int }{\left|f\left(z\right)\right|}^2}K\left(G\left(z,a\right)\right)\text{d}v\left(z\right)<\infty .$ (8)

Clearly, if $K\left(t\right)=t^p$ , then ${\mathcal{N}}_K\left(\mathbb{B}\right)={\mathcal{N}}_p\left(\mathbb{B}\right)$ . For $K\left(t\right)=1$ it gives the Bergman space ${\mathcal{A}}^2\left(\mathbb{B}\right)$ . If ${\mathcal{N}}_K\left(\mathbb{B}\right)$ consists of just the constant functions, we say that it is trivial.

We assume from now that all $K:\left(0,\infty \right)\to \left[0,\infty \right)$ to appear in this paper are right-continuous and nondecreasing function, which is not equal to 0 identically.

In [3] , several basic properties of ${\mathcal{N}}_K\left(\mathbb{B}\right)$ are proved, in connection with the Beurling-type space $H_{\alpha }^{\infty }\left(\mathbb{B}\right)$ . In particular, an embedding theorem for ${\mathcal{N}}_K\left(\mathbb{B}\right)$ and $H_{\alpha }^{\infty }\left(\mathbb{B}\right)$ is obtained, together with other useful properties. Hadamard gaps series and Hadamard product on ${\mathcal{N}}_K$ spaces of holomorphic function in the case of the unit disk has been studied quite well in [4] and [5] .

Through this, paper, given two quantities $A_f$ and $B_f$ both depending on a function $f\in \mathcal{H}\left(\mathbb{B}\right)$ , we are going to write $A_f\lesssim B_f$ if there exists a constant $C>0$ , independent of $f$ , such that $A_f\le C{B}_f$ for all $f$ . When $A_f\lesssim B_f\lesssim A_f$ , we write $A_f\approx B_f$ . If the quantities $A_f$ and $B_f$ are equivalent, then in particular we have $A_f<\infty$ if and only if $B_f<\infty$ . As usual, the letter C will denote a positive constant, possibly different on each occurrence.

In this paper, we introduce ${\mathcal{N}}_K\left(\mathbb{B}\right)$ spaces, in terms of the right continuous and non-decreasing function $K:\left(0,\infty \right)\to \left[0,\infty \right)$ on the unit ball $\mathbb{B}$ . We discuss the nesting property of ${\mathcal{N}}_K\left(\mathbb{B}\right)$ . We prove a sufficient condition for

${\mathcal{N}}_K\left(\mathbb{B}\right)=H_{\alpha }^{\infty }\left(\mathbb{B}\right)$ , $\alpha =\frac{n+1}{2}$ (the Beurling-type space). Also we generalize

the necessary condetion to ${\mathcal{N}}_K\left(\mathbb{B}\right)$ for a kind of lacunary series. As aplplication, we show that the sufficient condition is also a necessary to ${\mathcal{N}}_K\left(\mathbb{B}\right)=H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ .

2. 𝓝_K Spaces in the Unit Ball

In this section we prove some basic Banach space properties of ${\mathcal{N}}_K\left(\mathbb{B}\right)$ space. A sufficient and necessary condition for ${\mathcal{N}}_K\left(\mathbb{B}\right)$ to be non-trivial is given. We discuss the nesting property of ${\mathcal{N}}_K\left(\mathbb{B}\right)$ spaces and prove a sufficient condition for ${\mathcal{N}}_K\left(\mathbb{B}\right)=H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ .

Lemma 2.1

Let $f\left(z\right)={\displaystyle {\sum }_{k=1}^{\infty }{a}_k{z}^k}$ be a non-constant function, where $k=\left(k_1,k_2,\cdots ,k_n\right)$ is an n-tuple of non-negative integers and $z^k=\left(z_1^{k_1},z_2^{k_2},\cdots ,z_n^{k_n}\right)$ .

Then, $z^k\in {\mathcal{N}}_K\left(\mathbb{B}\right)$ if $a_k\ne 0$ .

Proof:

Let k be such that Let k be such that $a_k\ne 0$ and let $F_k\left(z\right)=a_k{z}^k$ . Suppose that

$U_{\theta }f\left(z\right)=f\left(z_1{\text{e}}^{i{\theta }_1},z_2{\text{e}}^{i{\theta }_2},\cdots ,z_n{\text{e}}^{i{\theta }_n}\right)=f\circ U_{\theta }\left(z\right),$

where $U_{\theta }\left(z\right)=\left(z_1{\text{e}}^{i{\theta }_1},z_2{\text{e}}^{i{\theta }_2},\cdots ,z_n{\text{e}}^{i{\theta }_n}\right)$ . Then, we have

$\begin{array}{c}{F}_k\left(z\right)=\frac{1}{{\left(2\text{π}\right)}^n}{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}\cdots }{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}f\left(z_1{\text{e}}^{i{\theta }_1},\cdots ,z_n{\text{e}}^{i{\theta }_n}\right){\text{e}}^{-i{k}_1{\theta }_1}\cdots {\text{e}}^{-i{k}_n{\theta }_n}\text{d}{\theta }_n}\\ =\frac{1}{{\left(2\text{π}\right)}^n}{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}\cdots }{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}\left(U_{\theta }f\right)\left(z\right){\text{e}}^{-i{k}_1{\theta }_1}\cdots {\text{e}}^{-i{k}_n{\theta }_n}\text{d}}{\theta }_n.\end{array}$ (9)

By Jensen’s inequality on convexity,

${\left|F_k\left(z\right)\right|}^2\le \frac{1}{{\left(2\text{π}\right)}^{2n}}{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}\cdots }{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}{\left|U_{\theta }f\left(z\right)\right|}^2\text{d}{\theta }_1\cdots \text{d}{\theta }_n.}$ (10)

Consequently,

$\begin{array}{l}{\displaystyle \underset{\mathbb{B}}{\int }{\left|F_k\left(z\right)\right|}^2}K\left(G\left(z,a\right)\right)\text{d}\lambda \left(z\right)\\ \le {\Vert U_{\theta }f\Vert }_K^2\frac{1}{{\left(2\text{π}\right)}^{2n}}{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}\cdots }{\displaystyle \underset{0}{\overset{\text{2π}}{\int }}\text{d}{\theta }_1\cdots \text{d}{\theta }_n}\le {\Vert U_{\theta }f\Vert }_K^2.\end{array}$ (11)

Because $U_{\theta }\left(z\right)\in Aut\left(\mathbb{B}\right)$ we have ${\Vert U_{\theta }f\Vert }_K={\Vert f\Vert }_K$ . Therefore,

${\Vert F_{k}f\Vert }_K={\Vert a_k{z}^k\Vert }_K\le {\Vert f\Vert }_K$

and $z^k\in {\mathcal{N}}_K\left(\mathbb{B}\right)$ . The lemma is proved.

Theorem 2.1 The Holomorphic function spaces ${\mathcal{N}}_K\left(\mathbb{B}\right)$ , contains all polynomials if

${\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2n-1}K\left(g\left(r\right)\right)\text{d}r}<\infty .$ (12)

Otherwise, ${\mathcal{N}}_K\left(\mathbb{B}\right)$ contains only constant functions.

Proof:

First assume that (12) holds. Let $f\left(z\right)$ be a polynomial i.e. (there exists a $M>0$ such that ${\left|f\left(z\right)\right|}^2\le M,\forall z\in \bar{\mathbb{B}}=\mathbb{B}\cup \mathbb{S}$ ). Then,

$\begin{array}{l}{\displaystyle \underset{\mathbb{B}}{\int }{\left|f\left(z\right)\right|}^2}K\left(G\left(z,a\right)\right)\text{d}v\left(z\right)\\ =2n{\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2n-1}K\left(g\left(r\right)\right)\text{d}r}{\displaystyle \underset{\mathbb{S}}{\int }{\left|f\left({\varphi }_a\left(r\zeta \right)\right)\right|}^2}\text{d}\sigma \left(\zeta \right)\\ \le 2nM{\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2n-1}K\left(g\left(r\right)\right)\text{d}r}.\end{array}$ (13)

Since a is arbitrary, it follows that

${\Vert f\Vert }_K^2\le 2nM{\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2n-1}K\left(g\left(r\right)\right)\text{d}r}<\infty .$ (14)

Thus, $f\in {\mathcal{N}}_K\left(\mathbb{B}\right)$ and the first half of the theorem is proved.

Now, we assume that the integral in (12) is divergent. Let $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)$ is an n-tuple of non-negative integers $\left|\alpha \right|={\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n\ge 1$ , $f\left(z\right)=z^{\alpha }$ .

Then, we have ${\left|f\left(r\xi \right)\right|}^2=r^{2\left|\alpha \right|}{\left|{\xi }^{\alpha }\right|}^2$ and

${\displaystyle \underset{\mathbb{S}}{\int }{\left|{\left(r\zeta \right)}^{\alpha }\right|}^2}\text{d}\sigma \left(r\zeta \right)\ge \frac{r^{2\left|\alpha \right|}\left(n-1\right)!\alpha !}{\left(n-1+\left|\alpha \right|\right)!}\ge C{r}^{2\left|\alpha \right|}.$ (15)

Thus,

${\Vert f\Vert }_K\ge \frac{nC}{2^{2\left|\alpha \right|-1}}{\displaystyle \underset{1/2}{\overset{1}{\int }}{r}^{2n-1}K\left(g\left(r\right)\right)\text{d}r}.$ (16)

There exists $a\in \mathbb{B}$ such that $f\left(a\right)\ne 0$ , by the subharmonicity of $\left|f\circ {\phi }_a\left(r\xi \right)\right|$ ,

${\Vert f\Vert }_K\ge \frac{3n}{2}{\left|f\left(a\right)\right|}^2{\displaystyle \underset{0}{\overset{1/2}{\int }}\frac{r^{2n-1}}{{\left(1-r^2\right)}^{n+1}}K\left(g\left(r\right)\right)\text{d}r}.$ (17)

Combining (17) and (18), we see that (12) implies that ${\Vert f\Vert }_K=\infty$ .

It is proved that $f\notin {\mathcal{N}}_K\left(\mathbb{B}\right)$ and, since $\alpha$ is arbitrary, any non-constant polynomial is not contained in ${\mathcal{N}}_K\left(\mathbb{B}\right)$ . Using Lemma 2.1, we conclude that ${\mathcal{N}}_K\left(\mathbb{B}\right)$ contains only constant functions. The theorem is proved.

Theorem 2.2

Let $K_1$ and $K_2$ satisfy (12). If there exist a constant $t_0>0$ such that $K_2\left(t\right)\lesssim K_1\left(t\right)$ for $t\in \left(0,t_0\right)$ , then ${\mathcal{N}}_{K_1}\left(\mathbb{B}\right)\subseteq {\mathcal{N}}_{K_2}\left(\mathbb{B}\right)$ . As a consequence, ${\mathcal{N}}_{K_1}\left(\mathbb{B}\right)={\mathcal{N}}_{K_2}\left(\mathbb{B}\right)$ . if $K_2\left(t\right)\approx K_1\left(t\right)$ for $t\in \left(0,t_0\right)$ .

Proof: Let $f\in {\mathcal{N}}_{K_1}\left(\mathbb{B}\right)$ . We note that from the property of $g\left(z\right)$ , there exists a constant $\delta >0$ , such that $g\left(z\right)<t_0$ if $\left|z\right|>\delta$ . Then, we have

${\displaystyle \underset{\mathbb{B}}{\int }{\left|f\left(z\right)\right|}^2}{K}_2\left(G\left(z,a\right)\right)\text{d}v\left(z\right)={\displaystyle \underset{{\mathbb{B}}_{\delta }}{\int }+}{\displaystyle \underset{\left|z\right|\ge \delta }{\int }{\left|f\left({\varphi }_a\left(z\right)\right)\right|}^2}{K}_2\left(g\left(z\right)\right)\text{d}v\left(z\right)$ (18)

where

$\begin{array}{l}{\displaystyle \underset{{\mathbb{B}}_{\delta }}{\int }{\left|f\left({\varphi }_a\left(z\right)\right)\right|}^2}{K}_2\left(g\left(z\right)\right)\text{d}v\left(z\right)\le {\Vert f\Vert }_{\infty }^2{\displaystyle \underset{{\mathbb{B}}_{\delta }}{\int }{\left(1-{\left|z\right|}^2\right)}^{-n}}{K}_2\left(g\left(z\right)\right)\text{d}v\left(z\right)\\ \le 2n{\Vert f\Vert }_{\infty }^2{\displaystyle \underset{0}{\overset{\delta }{\int }}{r}^{2n-1}}{K}_2\left(g\left(r\right)\right)\text{d}r<\infty ,\end{array}$

and

$\begin{array}{l}{\displaystyle \underset{\left|z\right|\ge \delta }{\int }{\left|f\left({\varphi }_a\left(z\right)\right)\right|}^2}{K}_2\left(g\left(z\right)\right)\text{d}v\left(z\right)\\ \le {\displaystyle \underset{\left|z\right|\ge \delta }{\int }{\left|f\left({\varphi }_a\left(z\right)\right)\right|}^2}{K}_1\left(g\left(z\right)\right)\text{d}v\left(z\right)\le {\Vert f\Vert }_{K_1}^2<\infty .\end{array}$

This show that ${\Vert f\Vert }_{K_2}<\infty$ and, consequently, $f\in {\mathcal{N}}_{K_2}\left(\mathbb{B}\right)$ .

Theorem 2.3

Let $K:\left(0,\infty \right)\to \left[0,\infty \right)$ be nondecreasing function, then ${\mathcal{N}}_K\left(\mathbb{B}\right)\subset H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ .

Proof: The theorem proved in [3] .

Theorem 2.4

${\mathcal{N}}_K\left(\mathbb{B}\right)=H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ if

${\displaystyle \underset{0}{\overset{1}{\int }}\frac{r^{2n-1}}{{\left(1-r^2\right)}^{n+1}}K\left(g\left(r\right)\right)\text{d}r}<\infty .$ (19)

Proof: Let $f\in H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ . Then,

$\begin{array}{l}{\displaystyle \underset{\mathbb{B}}{\int }{\left|f\left(z\right)\right|}^2}K\left(G\left(z,a\right)\right)\text{d}v\left(z\right)\\ \le {\Vert f\Vert }_{H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)}^2{\displaystyle \underset{\mathbb{B}}{\int }{\left(1-{\left|z\right|}^2\right)}^{-n}}K\left(g\left(z\right)\right)\frac{\text{d}v\left(z\right)}{{\left(1-{\left|z\right|}^2\right)}^{n+1}}\\ \le 2n{\Vert f\Vert }_{H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)}^2{\displaystyle \underset{0}{\overset{1}{\int }}\frac{r^{2n-1}}{{\left(1-r^2\right)}^{n+1}}}K\left(g\left(r\right)\right)\text{d}r.\end{array}$ (20)

Thus, ${\Vert f\Vert }_K<\infty$ and $f\in {\mathcal{N}}_K\left(\mathbb{B}\right)$ . This shows that $H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)\subset {\mathcal{N}}_K\left(\mathbb{B}\right)$ . By Theorem 2.3, we have ${\mathcal{N}}_K\left(\mathbb{B}\right)\subset H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ . The proof of theorem is complete.

3. Hadamard Gaps in 𝓝_K Spaces in the Unit Ball

In this section we prove a necessary condition for a lacunary series defined by a normal sequence to belong to ${\mathcal{N}}_K\left(\mathbb{B}\right)$ space. As an implication of Theorem

2.4, we prove that (19) is also necessary for ${\mathcal{N}}_K\left(\mathbb{B}\right)=H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ .

Recall that an $f\in \mathcal{H}\left(\mathbb{B}\right)$ written in the form $f\left(z\right)={\displaystyle {\sum }_{k=0}^{\infty }{P}_{n_k}\left(z\right)}$ where

$P_{n_k}$ is a homogeneous polynomial of degree $n_k$ , is said to have Hadamard gaps (also known as lacunary series) if there exists a constant $c>1$ such that (see e.g. [6] )

$\frac{n_{k+1}}{n_k}\ge c,\forall k\ge 0.$ (21)

Let ${\text{Λ}}_n\subset \mathbb{S}$ for $n=n_0,n_0+1,\cdots .$ The sequence of homogeneous polynomials

$P_n\left(z\right)={\displaystyle \underset{\zeta \in {\Lambda }_n}{\sum }{\langle z,\zeta \rangle }^n},$ (22)

is called a normal sequence if it possesses the following property (see [7] ):

・ $\left|P_n\left(z\right)\right|\le C{\left|z\right|}^n$ for $z\in \mathbb{B}$ ;

・ ${\displaystyle {\sum }_{\xi ,\zeta \in {\text{Λ}}_n}\xi ,{\zeta }^n}\ge \frac{n^{k+1}}{C}$ .

In what following, we will consider all lacunary series defined by normal sequences of homogeneous polynomials. To formulate our main result, we denote

$L_j={{\displaystyle \underset{\mathbb{S}}{\int }\left|P_{n_j}\left(\zeta \right)\right|}}^2\text{d}\sigma \left(\zeta \right).$ (23)

Theorem 3.1

Let $P_n\left(z\right)$ be a normal sequence and let $I_K=\left\{n\in \mathbb{N}:2^k\le n\le 2^{k+1}\right\}$ . Then a

lacunary series $f\left(z\right)={\displaystyle {\sum }_{k=0}^{\infty }{P}_{n_k}\left(z\right)}$ , belongs to ${\mathcal{N}}_K\left(\mathbb{B}\right)$ if

${\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{n_k^m}{2^k}K\left(n_k^{-m}\right)}{\displaystyle \underset{n_j\in I_k}{\sum }{L}_j}<\infty .$ (24)

Proof: Let $f\in {\mathcal{N}}_K\left(\mathbb{B}\right)$ . Then, we have

$\begin{array}{c}{\displaystyle \underset{\mathbb{B}}{\int }{\left|f\left(z\right)\right|}^2}K\left(G\left(z,a\right)\right)\text{d}v\left(z\right)\ge {\displaystyle \underset{\mathbb{B}}{\int }{\left|{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{P}_{n_k}}\left(z\right)\right|}^2}K\left(g\left(\left|z\right|\right)\right)\text{d}v\left(z\right)\\ \ge {\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{n_k}{2^k}}{\displaystyle \underset{n_j\in I_k}{\sum }{L}_j}{\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2m-1}}K\left(g\left(r\right)\right)\text{d}r,\end{array}$ (25)

where

${\left|{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{P}_{n_k}}\left(z\right)\right|}^2={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{1}{2^k}}{{\displaystyle \underset{n_j\in I_k}{\sum }\left|P_{n_k}\left(\zeta \right)\right|}}^2.$ (26)

By (6) for $\frac{1}{2}\le r\le 1$ , we have

$K\left(g\left(r\right)\right)\ge K\left(c^{-1}{\left(1-r\right)}^m\right).$ (27)

Consequently,

$\begin{array}{c}{\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2m-1}}K\left(g\left(r\right)\right)\text{d}r\ge {\displaystyle \underset{\frac{1}{2}}{\overset{1}{\int }}{r}^{2m-1}}K\left(c^{-1}{\left(1-r\right)}^m\right)\text{d}r\ge {\displaystyle \underset{0}{\overset{\log 2}{\int }}{\text{e}}^{-2mt}}K\left(c_1^{-1}{t}^m\right)\text{d}t\\ \ge K\left(n_k^{-m}\right){\displaystyle \underset{c_1{n}_k^{-1}}{\overset{\log 2}{\int }}{\text{e}}^{-2mt}}\text{d}t\ge n_k^{m-1}K\left(n_k^{-m}\right){\displaystyle \underset{c_1}{\overset{n_k\log 2}{\int }}{\text{e}}^{-2t}}\text{d}t.\end{array}$ (28)

Let $k^{\prime }$ be sufficiently large such that $n_{k^{\prime }}\log 2\ge c_1+1$ . Then, for $k\ge k^{\prime }$ ,

${\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2m-1}}K\left(g\left(r\right)\right)\text{d}r\ge n_k^{m-1}K\left(n_k^{-m}\right).$ (29)

And

${\displaystyle \underset{\mathbb{B}}{\int }{\left|f\left(z\right)\right|}^2}K\left(G\left(z,a\right)\right)\text{d}v\left(z\right)\ge C{\displaystyle \underset{k=k^{\prime }}{\overset{\infty }{\sum }}\frac{n_k^m}{2^k}}K\left(n_k^{-m}\right){\displaystyle \underset{n_j\in I_k}{\sum }{L}_j}.$ (30)

This shows (24) and the theorem is proved.

Theorem 3.2

${\mathcal{N}}_K\left(\mathbb{B}\right)=H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ if and only if (18) holds.

Proof: The sufficient condition was proved by Theorem 2.4. Now we prove the necessary condition, assume that ${\mathcal{N}}_K\left(\mathbb{B}\right)=H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ . Among lacunary series defined by normal sequences, we consider

$f\left(z\right)={\displaystyle \underset{k=k_0}{\overset{\infty }{\sum }}{P}_{2^k}\left(z\right),}$ (31)

where $P_{2^k}={\displaystyle {\sum }_{\zeta \in {\Lambda }_n}{\langle z,\zeta \rangle }^{2^k}}$ and $\left|P_{2^k}\right|=C{\left|z\right|}^{2^k}$ for $k\ge k_0,2^{k_0}\ge n_0$ and $z\in \mathbb{B}$ .

Thus

$\left|f\left(z\right)\right|{\left(1-{\left|z\right|}^2\right)}^{n+1}\le {\left(1-{\left|z\right|}^2\right)}^{n+1}{\displaystyle \underset{k=k_0}{\overset{\infty }{\sum }}\left|P_{2^k}\left(z\right)\right|\le C{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{\left|z\right|}^n\le C.}}$ (32)

This shows that $f\in H_{\frac{n+1}{2}}^{\infty }\left(\mathbb{B}\right)$ and, consequently, $f\in {\mathcal{N}}_K\left(\mathbb{B}\right)$ . By Theorem

3.1, we have

${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{2}^{k\left(m-1\right)}K\left(2^{-mk}\right)}<\infty .$ (33)

By (6), we have

${\displaystyle \underset{1/2}{\overset{1}{\int }}\frac{r^{2m-1}}{{\left(1-r^2\right)}^{m+1}}}K\left(g\left(r\right)\right)\text{d}r\le {\displaystyle {\int }_0^{c^{1/m}\log 2}{t}^{-m-1}}K\left(t^m\right)\text{d}t.$ (34)

On the other hand,

$\begin{array}{c}{\displaystyle {\int }_0^{1/2}{t}^{-m-1}}K\left(t^m\right)\text{d}t={\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{\displaystyle {\int }_{2^{-k-1}}^{2^{-k}}{t}^{-m-1}}K\left(t^m\right)\text{d}t}\\ ={\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{2}^{-\left(k+1\right)}{2}^{-m-1}K\left(2^{-mk}\right),}\end{array}$ (35)

since K is non-decreasing. Thus,

${\displaystyle \underset{1/2}{\overset{1}{\int }}\frac{r^{2m-1}}{{\left(1-r^2\right)}^{m+1}}}K\left(g\left(r\right)\right)\text{d}r<\infty .$ (36)

Combining this, we obtain (18). The theorem is proved.

4. Conclusion

Our aim of the present paper is to characterize the holomorphic functions with Hadamard gaps in ${\mathcal{N}}_K$ -type spaces on the unit ball, where K is the right continuous and non-decreasing function. Our main results will be of important uses in the study of operator theory of holomorphic function spaces.

Acknowledgements

The authors are thankful to the referee for his/her valuable comments and very useful suggestions.

Conflicts of Interest

The authors declare no conflicts of interest.

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