Hadamard Gaps and Nk-type Spaces in the Unit Ball

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

KEYWORDS

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 $〈z,w〉=\left({z}_{1}\stackrel{¯}{{w}_{1}},{z}_{2}\stackrel{¯}{{w}_{2}},\cdots ,{z}_{n}\stackrel{¯}{{w}_{n}}\right)$ , and write $|z|=\sqrt{〈z,w〉}$ .

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 ${\mathbb{B}}_{r}=\left\{z\in \mathbb{B}:|z|\le r\right\}$ .

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

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

The identity

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

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

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

${\phi }_{a}\left(z\right)=\frac{a-{P}_{a}\left(z\right)-{S}_{a}{Q}_{a}\left(z\right)}{1-〈z,a〉},$ (3)

where ${S}_{a}=\sqrt{1-{|z|}^{2}},{P}_{a}\left(z\right)=\frac{a〈z,a〉}{{|a|}^{2}},{P}_{0}=0$ and ${Q}_{a}=I-{P}_{a}\left(z\right)$ (see  or  ).

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

$1-〈{\phi }_{a}\left(z\right),{\phi }_{a}\left(w\right)〉=\frac{\left(1-{|a|}^{2}\right)\left(1-〈z,w〉\right)}{\left(1-〈z,a〉\right)\left(1-〈a,w〉\right)},$

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

$1-{|{\phi }_{a}\left(z\right)|}^{2}=\frac{\left(1-{|a|}^{2}\right)\left(1-{|z|}^{2}\right)}{{|1-〈z,a〉|}^{2}},$ (4)

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

$g\left(z\right)=\frac{n+1}{2n}\underset{|z|}{\overset{1}{\int }}{\left(1-{t}^{2}\right)}^{n-1}{t}^{1-2n}\text{d}t.$ (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 ${‖f‖}_{\infty }={\mathrm{sup}}_{z\in \mathbb{B}}|f\left(z\right)|$ .

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

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

${‖f‖}_{K}^{2}=\underset{z\in \mathbb{B}}{\mathrm{sup}}\underset{\mathbb{B}}{\int }{|f\left(z\right)|}^{2}K\left(G\left(z,a\right)\right)\text{d}v\left(z\right)<\infty .$ (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  , 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  and  .

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. 𝓝K 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 $f\left(z\right)={\sum }_{k=1}^{\infty }{a}_{k}{z}^{k}$ 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

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

By Jensen’s inequality on convexity,

${|{F}_{k}\left(z\right)|}^{2}\le \frac{1}{{\left(2\text{π}\right)}^{2n}}\underset{0}{\overset{\text{2π}}{\int }}\cdots \underset{0}{\overset{\text{2π}}{\int }}{|{U}_{\theta }f\left(z\right)|}^{2}\text{d}{\theta }_{1}\cdots \text{d}{\theta }_{n}.$ (10)

Consequently,

$\begin{array}{l}\underset{\mathbb{B}}{\int }{|{F}_{k}\left(z\right)|}^{2}K\left(G\left(z,a\right)\right)\text{d}\lambda \left(z\right)\\ \le {‖{U}_{\theta }f‖}_{K}^{2}\frac{1}{{\left(2\text{π}\right)}^{2n}}\underset{0}{\overset{\text{2π}}{\int }}\cdots \underset{0}{\overset{\text{2π}}{\int }}\text{d}{\theta }_{1}\cdots \text{d}{\theta }_{n}\le {‖{U}_{\theta }f‖}_{K}^{2}.\end{array}$ (11)

Because ${U}_{\theta }\left(z\right)\in Aut\left(\mathbb{B}\right)$ we have ${‖{U}_{\theta }f‖}_{K}={‖f‖}_{K}$ . Therefore,

${‖{F}_{k}f‖}_{K}={‖{a}_{k}{z}^{k}‖}_{K}\le {‖f‖}_{K}$

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

$\underset{0}{\overset{1}{\int }}{r}^{2n-1}K\left(g\left(r\right)\right)\text{d}r<\infty .$ (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 ${|f\left(z\right)|}^{2}\le M,\forall z\in \stackrel{¯}{\mathbb{B}}=\mathbb{B}\cup \mathbb{S}$ ). Then,

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

Since a is arbitrary, it follows that

${‖f‖}_{K}^{2}\le 2nM\underset{0}{\overset{1}{\int }}{r}^{2n-1}K\left(g\left(r\right)\right)\text{d}r<\infty .$ (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 $|\alpha |={\alpha }_{1}+{\alpha }_{2}+\cdots +{\alpha }_{n}\ge 1$ , $f\left(z\right)={z}^{\alpha }$ .

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

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

Thus,

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

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

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

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

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 $|z|>\delta$ . Then, we have

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

where

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

and

$\begin{array}{l}\underset{|z|\ge \delta }{\int }{|f\left({\varphi }_{a}\left(z\right)\right)|}^{2}{K}_{2}\left(g\left(z\right)\right)\text{d}v\left(z\right)\\ \le \underset{|z|\ge \delta }{\int }{|f\left({\varphi }_{a}\left(z\right)\right)|}^{2}{K}_{1}\left(g\left(z\right)\right)\text{d}v\left(z\right)\le {‖f‖}_{{K}_{1}}^{2}<\infty .\end{array}$

This show that ${‖f‖}_{{K}_{2}}<\infty$ 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  .

Theorem 2.4

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

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

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

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

Thus, ${‖f‖}_{K}<\infty$ 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 𝓝K 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 $f\left(z\right)={\sum }_{k=0}^{\infty }{P}_{{n}_{k}}\left(z\right)$ 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.  )

$\frac{{n}_{k+1}}{{n}_{k}}\ge c,\text{\hspace{0.17em}}\forall k\ge 0.$ (21)

Let ${\text{Λ}}_{n}\subset \mathbb{S}$ for $n={n}_{0},{n}_{0}+1,\cdots .$ The sequence of homogeneous polynomials

${P}_{n}\left(z\right)=\underset{\zeta \in {\Lambda }_{n}}{\sum }{〈z,\zeta 〉}^{n},$ (22)

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

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

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

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

${L}_{j}={\underset{\mathbb{S}}{\int }|{P}_{{n}_{j}}\left(\zeta \right)|}^{2}\text{d}\sigma \left(\zeta \right).$ (23)

Theorem 3.1

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

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

$\underset{k=0}{\overset{\infty }{\sum }}\frac{{n}_{k}^{m}}{{2}^{k}}K\left({n}_{k}^{-m}\right)\underset{{n}_{j}\in {I}_{k}}{\sum }{L}_{j}<\infty .$ (24)

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

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

where

${|\underset{k=0}{\overset{\infty }{\sum }}{P}_{{n}_{k}}\left(z\right)|}^{2}=\underset{k=0}{\overset{\infty }{\sum }}\frac{1}{{2}^{k}}{\underset{{n}_{j}\in {I}_{k}}{\sum }|{P}_{{n}_{k}}\left(\zeta \right)|}^{2}.$ (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,

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

Let ${k}^{\prime }$ be sufficiently large such that ${n}_{{k}^{\prime }}\mathrm{log}2\ge {c}_{1}+1$ . Then, for $k\ge {k}^{\prime }$ ,

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

And

$\underset{\mathbb{B}}{\int }{|f\left(z\right)|}^{2}K\left(G\left(z,a\right)\right)\text{d}v\left(z\right)\ge C\underset{k={k}^{\prime }}{\overset{\infty }{\sum }}\frac{{n}_{k}^{m}}{{2}^{k}}K\left({n}_{k}^{-m}\right)\underset{{n}_{j}\in {I}_{k}}{\sum }{L}_{j}.$ (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

$f\left(z\right)=\underset{k={k}_{0}}{\overset{\infty }{\sum }}{P}_{{2}^{k}}\left(z\right),$ (31)

where ${P}_{{2}^{k}}={\sum }_{\zeta \in {\Lambda }_{n}}{〈z,\zeta 〉}^{{2}^{k}}$ and $|{P}_{{2}^{k}}|=C{|z|}^{{2}^{k}}$ for $k\ge {k}_{0},{2}^{{k}_{0}}\ge {n}_{0}$ and $z\in \mathbb{B}$ .

Thus

$|f\left(z\right)|{\left(1-{|z|}^{2}\right)}^{n+1}\le {\left(1-{|z|}^{2}\right)}^{n+1}\underset{k={k}_{0}}{\overset{\infty }{\sum }}|{P}_{{2}^{k}}\left(z\right)|\le C\underset{n=1}{\overset{\infty }{\sum }}{|z|}^{n}\le C.$ (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

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

By (6), we have

$\underset{1/2}{\overset{1}{\int }}\frac{{r}^{2m-1}}{{\left(1-{r}^{2}\right)}^{m+1}}K\left(g\left(r\right)\right)\text{d}r\le {\int }_{0}^{{c}^{1/m}\mathrm{log}2}{t}^{-m-1}K\left({t}^{m}\right)\text{d}t.$ (34)

On the other hand,

$\begin{array}{c}{\int }_{0}^{1/2}{t}^{-m-1}K\left({t}^{m}\right)\text{d}t=\underset{k=1}{\overset{\infty }{\sum }}{\int }_{{2}^{-k-1}}^{{2}^{-k}}{t}^{-m-1}K\left({t}^{m}\right)\text{d}t\\ =\underset{k=1}{\overset{\infty }{\sum }}{2}^{-\left(k+1\right)}{2}^{-m-1}K\left({2}^{-mk}\right),\end{array}$ (35)

since K is non-decreasing. Thus,

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

Cite this paper

Bakhit, M. and Shammaky, A. (2017) Hadamard Gaps and Nk-type Spaces in the Unit Ball. Advances in Pure Mathematics, 7, 306-313. doi: 10.4236/apm.2017.74017.

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