Besov spaces induced by doubling weights

TL;DR

This paper characterizes weighted Besov spaces induced by doubling weights, providing a new factorization theorem and conditions for products with inner functions, with implications for zero sets.

Contribution

It offers a characterization of Besov spaces depending only on the modulus, and extends classical factorization results within this weighted setting.

Findings

Characterization of $_{ u}^{p,q}$ depending only on $|f|$, $p$, $q$, and $ u$

A modified factorization theorem for functions in $_{ u}^{p,q}$

Conditions for the product of $H^p$ functions and inner functions to belong to $_{ u}^{p,q}$

Abstract

Let $1\le p<\infty$, $0<q<\infty$ and $\nu$ be a two-sided doubling weight satisfying $$\sup_{0\le r<1}\frac{(1-r)^q}{\int_r^1\nu(t)\,dt}\int_0^r\frac{\nu(s)}{(1-s)^q}\,ds<\infty.$$ The weighted Besov space $\mathcal{B}_{\nu}^{p,q}$ consists of those $f\in H^p$ such that $$\int_0^1 \left(\int_{0}^{2\pi} |f'(re^{i\theta})|^p\,d\theta\right)^{q/p}\nu(r)\,dr<\infty.$$ Our main result gives a characterization for $f\in \mathcal{B}_{\nu}^{p,q}$ depending only on $|f|$, $p$, $q$ and $\nu$. As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. In particular, we show the following modification of a classical factorization by F. and R. Nevanlinna: If $f\in \mathcal{B}_{\nu}^{p,q}$, then there exist $f_1,f_2\in \mathcal{B}_{\nu}^{p,q} \cap H^\infty$ such that $f=f_1/f_2$. In addition, we give a sufficient and necessary condition guaranteeing that the product of $f\in H^p$ and an inner function belongs to $\mathcal{B}_{\nu}^{p,q}$. Applying this result, we make some observations on zero sets of $\mathcal{B}_{\nu}^{p,p}$.

Equations

$$\underset{0\le r<1}{\sup }\frac{(1-r{)}^q}{{\int }_r^1\nu (t)dt}{\int }_0^r\frac{\nu (s)}{(1-s{)}^q}ds<\infty .$$

$${\int }_0^1{\left({\int }_0^{2\pi }|f^{\prime }(r{e}^{i\theta }){|}^{p}d\theta \right)}^{q/p}\nu (r)dr<\infty .$$

$$\Vert f{\Vert }_{H^p}=\underset{0\le r<1}{\sup }{M}_p(r,f)<\infty ,$$

$$\begin{array}{r}{M}_p(r,f)={\left(\frac{1}{2\pi }{\int }_0^{2\pi }|f(r{e}^{i\theta }){|}^{p}d\theta \right)}^{1/p}.\end{array}$$

$$\Vert f{\Vert }_{L^p}^p=\frac{1}{2\pi }{\int }_0^{2\pi }|f(e^{i\theta }){|}^{p}d\theta <\infty .$$

$$\Vert f{\Vert }_{A_{\nu }^{p,q}}^q={\int }_0^1{M}_p^q(r,f)\nu (r)dr<\infty .$$

$$\hat{\nu }(z)=\hat{\nu }(|z|)={\int }_{|z|}^1\nu (s)ds,z\in \mathbb{D}.$$

$$\begin{array}{r}\hat{\nu }(r)\ge C\hat{\nu }\left(1-\frac{1-r}{K}\right),0\le r<1.\end{array}$$

$${\hat{\mathcal{D}}}_p(\nu )=\underset{0\le r<1}{\sup }\frac{(1-r{)}^p}{\hat{\nu }(r)}{\int }_0^r\frac{\nu (s)}{(1-s{)}^p}ds<\infty$$

$${\omega }_p(t,f)=\underset{0<h<t}{\sup }{\left({\int }_0^{2\pi }|f(e^{i(\theta +h)})-f(e^{i\theta }){|}^{p}d\theta \right)}^{1/p},0<t\le 2\pi .$$

$${\int }_0^{\infty }\frac{{\omega }_p(t,f{)}^q}{t^{q-\alpha }}dt<\infty .$$

$${\int }_{1/2}^1{\omega }_p(1-r,f{)}^q\frac{\nu (r)}{(1-r{)}^q}dr\le C\Vert f^{\prime }{\Vert }_{A_{\nu }^{p,q}}^q$$

$$d{\mu }_z(\theta )=\frac{1-|z{|}^2}{|e^{i\theta }-z{|}^2}\frac{d\theta }{2\pi },z\in \mathbb{D},0\le \theta <2\pi ,$$

$$\begin{array}{rl}\Vert f^{\prime }{\Vert }_{A_{\nu }^{p,q}}^q& \le {\int }_0^1{\left({\int }_0^{2\pi }{\left({\int }_0^{2\pi }|f(e^{i\theta })-f(r{e}^{it})|d{\mu }_{r{e}^{it}}(\theta )\right)}^{p}dt\right)}^{q/p}\frac{\nu (r)}{(1-r{)}^q}dr\\ & \le C_1\left({\int }_0^1{\omega }_p(1-s,f{)}^q\frac{\nu (s)}{(1-s{)}^q}ds+\Vert f{\Vert }_{H^p}^q\right)\\ & \le C_2\left(\Vert f^{\prime }{\Vert }_{A_{\nu }^{p,q}}^q+\Vert f{\Vert }_{H^p}^q\right)\end{array}$$

$$\begin{array}{rl}& \Vert f^{\prime }{\Vert }_{A_{\nu }^{p,q}}^q\le C_1\left(F_1(f)+F_2(f)\right)\le C_2\left(\Vert f^{\prime }{\Vert }_{A_{\nu }^{p,q}}^q+\Vert f{\Vert }_{H^p}^q\right),f\in H^p,\end{array}$$

$$\begin{array}{r}{F}_1(f)={\int }_0^1{\left({\int }_0^{2\pi }{\left({\int }_0^{2\pi }|f(e^{i\theta })|d{\mu }_{r{e}^{it}}(\theta )-|f(r{e}^{it})|\right)}^{p}dt\right)}^{q/p}\frac{\nu (r)}{(1-r{)}^q}dr\end{array}$$

$$\begin{array}{r}{F}_2(f)={\int }_0^1{\left({\int }_0^{2\pi }{\left({\int }_0^{2\pi }\left||f(e^{i\theta })|-{\int }_0^{2\pi }|f(e^{is})|d{\mu }_{r{e}^{it}}(s)\right|d{\mu }_{r{e}^{it}}(\theta )\right)}^{p}dt\right)}^{q/p}\frac{\nu (r)}{(1-r{)}^q}dr.\end{array}$$

$$O_{\varphi }(z)=\exp \left(\frac{1}{2\pi }{\int }_0^{2\pi }\frac{e^{i\theta }+z}{e^{i\theta }-z}\log \varphi (e^{i\theta })d\theta \right),z\in \mathbb{D},$$

$$\begin{array}{r}|f(\xi )|=|O_{\varphi }(\xi )|=\varphi (\xi )\end{array}$$

$$|O_{\varphi }(z)|=\exp \left({\int }_0^{2\pi }\log \varphi (e^{i\theta })d{\mu }_z(\theta )\right),z\in \mathbb{D}.$$

$$\begin{array}{r}{\int }_0^{2\pi }|f(e^{i\theta })-f(z){|}^{2}d{\mu }_z(\theta )={\int }_0^{2\pi }|f(e^{i\theta }){|}^{2}d{\mu }_z(\theta )-|f(z){|}^2,z\in \mathbb{D};\end{array}$$

$$\begin{array}{r}{\int }_0^1{\left({\int }_0^{2\pi }{\left({\int }_0^{2\pi }|f(e^{i\theta }){|}^{2}d{\mu }_{r{e}^{it}}(\theta )-|f(r{e}^{it}){|}^2\right)}^{p/2}dt\right)}^{q/p}\frac{\nu (r)}{(1-r{)}^q}dr;\end{array}$$

$${\int }_{\{z\in \mathbb{D}:|f(z)|<\epsilon \}}\frac{dA(z)}{1-|z|}=\infty$$

$$\begin{array}{r}{F}_1(f)+F_2(f)=F_1(f)\asymp {\int }_{\mathbb{D}}(1-|f(z)|{)}^p(1-|z|{)}^{-1}dA(z)=\infty ,\end{array}$$

$$\begin{array}{r}\Vert f^{\prime }{\Vert }_{A_{\nu }^{p,q}}^q+\Vert f{\Vert }_{H^p}^q=\Vert f^{\prime }{\Vert }_{A_{p-1}^{p,p}}^p+1<\infty \end{array}$$

$$H^p\subset \{g:g^{\prime }\in A_{p-1}^{p,p}\},2\le p<\infty ,$$

$$\hat{\nu }(r)\le C{\left(\frac{1-r}{1-s}\right)}^{\beta }\hat{\nu }(s),0\le r\le s<1.$$

$${\int }_0^r{\left(\frac{1-r}{1-s}\right)}^{\gamma }\nu (s)ds\le C\hat{\nu }(r),0\le r<1.$$

$${\int }_0^1{s}^x\nu (s)ds\asymp \hat{\nu }\left(1-\frac{1}{x}\right),1\le x<\infty ,$$

$$\hat{\nu }(s)\le C{\left(\frac{1-s}{1-r}\right)}^{\alpha }\hat{\nu }(r),0\le r\le s<1.$$

Full text

[TABLE]

The weighted Besov space $\mathcal{B}_{\nu}^{p,q}$ consists of those $f\in H^{p}$ such that

[TABLE]

Our main result gives a characterization for $f\in\mathcal{B}_{\nu}^{p,q}$ depending only on $|f|$, $p$, $q$ and $\nu$.

As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. For instance, we show the following modification of a classical factorization by F. and R. Nevanlinna: If $f\in\mathcal{B}_{\nu}^{p,q}$, then there exist $f_{1},f_{2}\in\mathcal{B}_{\nu}^{p,q}\cap H^{\infty}$ such that $f=f_{1}/f_{2}$. Moreover, we give a sufficient and necessary condition guaranteeing that the product of $f\in H^{p}$ and an inner function belongs to $\mathcal{B}_{\nu}^{p,q}$. Applying this result, we make some observations on zero sets of $\mathcal{B}_{\nu}^{p,p}$.

This research was supported in part by Academy of Finland project no. 286877 and Finnish Cultural Foundation.

1. Introduction and characterizations

Let $\mathbb{D}$ be the open unit disc of the complex plane $\mathbb{C}$ and $\mathbb{T}$ the boundary of $\mathbb{D}$. The set of all analytic functions in $\mathbb{D}$ is denoted by $\mathcal{H}(\mathbb{D})$. For $0<p<\infty$, the Hardy space $H^{p}$ consists of those $f\in\mathcal{H}(\mathbb{D})$ such that

[TABLE]

where

[TABLE]

The Hardy space $H^{\infty}$ is the set of all bounded functions in $\mathcal{H}(\mathbb{D})$. Moreover, we recall that a measurable function $f$ on $\mathbb{T}$ belongs to $L^{p}(\mathbb{T})$ for some $p\in(0,\infty)$ if

[TABLE]

Alternatively, the Hardy space $H^{p}$ for $0<p<\infty$ can be characterized as follows: $f\in H^{p}$ if and only if $f\in\mathcal{H}(\mathbb{D})$, non-tangential limit $f(e^{i\theta})$ exists almost everywhere on $\mathbb{T}$ and $f(e^{i\theta})\in L^{p}(\mathbb{T})$. In particular, $\|f\|_{H^{p}}=\|f\|_{L^{p}}$ for $0<p<\infty$ and $f\in H^{p}$. This is due to Hardy’s convexity and the mean convergence theorems. These results and much more can be found in classic book [8] by P. Duren.

A function $\nu:\mathbb{D}\rightarrow[0,\infty)$ is called a (radial) weight if it is integrable over $\mathbb{D}$ and $\nu(z)=\nu(|z|)$ for all $z\in\mathbb{D}$. For $0<p,q<\infty$ and a weight $\nu$, the weighted mixed norm space $A^{p,q}_{\nu}$ consists of those $f\in\mathcal{H}(\mathbb{D})$ such that

[TABLE]

If $\nu(z)=(1-|z|)^{\alpha}$ for $-1<\alpha<\infty$, then the notation $A_{\alpha}^{p,q}$ is used for $A^{p,q}_{\nu}$. In this note, we study class $\mathcal{D}$ of so-called two-sided doubling weights, which originates from the work of J. A.  Peláez and J. Rättyä [19, 20]. For the definition of $\mathcal{D}$ we have to define two wider classes. For a weight $\nu$, set

[TABLE]

If a weight $\nu$ satisfies the condition $\widehat{\nu}(r)\leq C\widehat{\nu}(\frac{1+r}{2})$ for all $0\leq r<1$ and some $C=C(\nu)>0$, then we write $\nu\in\widehat{\mathcal{D}}$. Correspondingly, $\nu\in\widecheck{\mathcal{D}}$ if there exist $K=K(\nu)>1$ and $C=C(\nu)>1$ such that

[TABLE]

Class $\mathcal{D}$ is the intersection of $\widehat{\mathcal{D}}$ and $\widecheck{\mathcal{D}}$. In addition, we define the following subclass of $\widehat{\mathcal{D}}$: $\nu\in\widehat{\mathcal{D}}_{p}$ for some $p\in(0,\infty)$ if the condition

[TABLE]

is satisfied. As a concrete example, we mention that $\nu_{1}(z)=(1-|z|)^{\alpha}$ and $\nu_{2}(z)=(1-|z|)^{\alpha}\left(\log\frac{e}{1-|z|}\right)^{\beta}$ for any $\beta\in\mathbb{R}$ belong to $\mathcal{D}\cap\widehat{\mathcal{D}}_{p}$ if and only if $-1<\alpha<p-1$. Additional information about weights can be found in [18, 19, 20]. Some basic properties are recalled also in Section 2.

Define the weighted Besov space $\mathcal{B}_{\nu}^{p,q}$ by $\mathcal{B}_{\nu}^{p,q}=\{f:f^{\prime}\in A_{\nu}^{p,q}\}\cap H^{p}$. For $-1<\alpha<\infty$ and $\nu(z)=(1-|z|)^{\alpha}$, the notation $\mathcal{B}_{\alpha}^{p,q}$ is used for $\mathcal{B}_{\nu}^{p,q}$. The space $\mathcal{B}_{\nu}^{p,q}$ is the main research objective of this note. Hence it is worth pointing out that the definition is rational, which means that $H^{p}$ is not a subset of $\{f:f^{\prime}\in A_{\nu}^{p,q}\}$ in general, or conversely. The family of Blaschke products offers examples for the case where $f\in H^{\infty}$ and $f^{\prime}\notin A_{\nu}^{p,q}$; see for instance [24]. Moreover, it would be natural that certain lacunary series $g$ lie out of $H^{p}$, while $g^{\prime}\in A_{\nu}^{p,q}$. Arguments for this kind of examples can be found in M. Pavlović’s book [16], which contains numerous important observations on the topic of this note. The existence of both examples, of course, depends on $p$, $q$ and $\nu$. In other words, under certain hypotheses for $p$, $q$ and $\nu$, an inclusion relation between $\{f:f^{\prime}\in A_{\nu}^{p,q}\}$ and $H^{p}$ might be valid. However, this is not the case in general.

For $0<p<\infty$ and $f\in L^{p}(\mathbb{T})$, the $L^{p}$ modulus of continuity $\omega_{p}(t,f)$ is defined by

[TABLE]

We interpret $\omega_{p}(t,f)=\omega_{p}(2\pi,f)$ for $t>2\pi$. It is a well-known fact that, for $0<p,q<\infty$, $-1<\alpha<q-1$ and $f\in H^{p}$, the derivative of $f$ belongs to $A_{\alpha}^{p,q}$ if and only if

[TABLE]

This result originates from E. M. Stein’s book [26, Chapter V, Section 5], and the complete version is a consequence of [17, Theorems 2.1 and 5.1] or [12, Theorem 1.2] by M. Pavlović and M. Jevtić. Our first theorem is a partial generalization of the result. Its proof uses some ideas from [8, 21, 22].

Theorem 1**.**

Let $1\leq p<\infty$, $0<q<\infty$ and $\nu\in\mathcal{D}$. Then $\nu\in\widehat{\mathcal{D}}_{q}$ if and only if there exists a constant $C=C(p,q,\nu)>0$ such that

[TABLE]

for all $f\in H^{p}$.

Note that (1.2) is valid also if $0<p,q<\infty$, $\nu$ is a weight and there exists $\beta=\beta(q,\nu)<q-1$ such that $\nu(r)/(1-r)^{\beta}$ is increasing for $0\leq r<1$. This is due to [17, Theorem 5.1] and its proof. Even though the result is valid also for $0<p<1$, Theorem 1 is more useful for our purposes. In particular, it is worth underlining that the hypothesis $1\leq p<\infty$ in Theorems 2 and 3 below is natural.

Theorem 2 gives a practical estimate for $\|f^{\prime}\|_{A_{\nu}^{p,q}}^{q}+\|f\|_{H^{p}}^{q}$ when $f\in H^{p}$. As a by-product of its argument, we deduce that also the converse inequality of (1.2) holds if the norm $\|f\|_{H^{p}}^{q}$ is added into the right-hand side. Setting

[TABLE]

Theorem 2 reads as follows.

Theorem 2**.**

Let $1\leq p<\infty$, $0<q<\infty$ and $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{q}$. Then there exist positive constants $C_{1}$ and $C_{2}$ depending only on $p$, $q$ and $\nu$ such that

[TABLE]

for all $f\in H^{p}$.

By studying the classical weight $\nu(z)=(1-|z|)^{\alpha}$, where $-1<\alpha<q-1$, we obtain K. M. Dyakonov’s [9, Proposition 2.2(a)] as a direct consequence of Theorem 2. Hence it does not come as a surprise that the proofs of [9, Theorem 2.1] and Theorem 2 have some similarities. Nonetheless, it is worth mentioning that the presence of general weights complicates the argument; and consequently, our proof is quite technical. Note also that Theorem 1 plays an essential role in the proof.

Our main result below gives a characterization for functions $f$ in $\mathcal{B}_{\nu}^{p,q}$ depending only $|f|$, $p$, $q$ and $\nu$. This result improves B. Bøe’s [6, Theorem 1.1], which concentrates only on the case where $1\leq p,q<\infty$, $-1<\alpha<q-1$ and $\nu(z)=(1-|z|)^{\alpha}$. It also generalizes the essential contents of [2, Proposition 2.4] and [9, Proposition 2.2(b)] made by A. Aleman and K. M. Dyakonov, respectively.

Theorem 3**.**

Let $1\leq p<\infty$, $0<q<\infty$ and $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{q}$. Then there exist positive constants $C_{1}$ and $C_{2}$ depending only on $p$, $q$ and $\nu$ such that

[TABLE]

where

[TABLE]

and

[TABLE]

Before we talk about the argument of Theorem 3, recall the inner-outer factorization. An inner function is a member of $H^{\infty}$ having unimodular radial limits almost everywhere on $\mathbb{T}$. For $0<p\leq\infty$, an outer function for $H^{p}$ takes the form

[TABLE]

where $\phi$ is a non-negative function in $L^{p}(\mathbb{T})$ and $\log\phi\in L^{1}(\mathbb{T})$. The inner-outer factorization asserts that $f\in H^{p}$ can be represented as the product of an inner and outer function; see for instance [8, Theorem 2.8]. It is worth noting that the factorization is unique, and

[TABLE]

for almost every $\xi\in\mathbb{T}$ if $O_{\phi}$ is the outer function from the factorization of $f$. Equation (1.5) is due to the definition of inner functions, Poisson integral formula, harmonicity of $\log|O_{\phi}(z)|$ and fact that

[TABLE]

The last inequality in (1.4) can be proved by applying Theorem 2. In the argument of the first inequality, the inner-outer factorization, Schwarz-Pick lemma and an upper estimate for $|O_{\phi}^{\prime}|$ from [6] are the main tools. It is worth underlining that this Bøe’s idea to make an upper estimate for $|f^{\prime}|$ by using the factorization seems to be quite effective. Another way to prove results like Theorem 3 is to use a modification of Theorem 2 together with the well-known equation

[TABLE]

but this Dyakonov’s method has the obvious defect that it works only when $f\in H^{2}$. The advantage of this method in the case where $2\leq p<\infty$, $0<q<\infty$, $q/2-1<\alpha<q-1$ and $\nu(z)=(1-|z|)^{\alpha}$ is that $F_{1}(f)+F_{2}(f)$ in Theorem 3 can be replaced by

[TABLE]

see [9, Proposition 2.2(b)]. It is an open problem to prove a corresponding estimate for general weights.

Next we give an example which shows that the hypothesis $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{q}$ in Theorem 3 for $p\geq 2$ is sharp in a certain sense. Note that the example is a modification of [22, Example 8]. Before the statement we fix some notation. Write $f\lesssim g$ if there exists a constant $C>0$ such that $f\leq Cg$, while $f\gtrsim g$ is understood analogously. If $f\lesssim g$ and $f\gtrsim g$, then we write $f\asymp g$.

Example 4**.**

Let $2\leq p<\infty$, $q=p$, $\nu(z)=(1-|z|)^{p-1}$ and $dA(z)$ be the two-dimensional Lebesgue measure $dxdy$. Let $f$ be an inner function such that

[TABLE]

for some $\varepsilon\in(0,1)$. The existence of such $f$ is guaranteed by [7, Theorem 5]. Then

[TABLE]

while

[TABLE]

by the well-known inclusion

[TABLE]

which originates from [13].

We close the section by explaining how the remainder of this note is organized. Auxiliary results on weights are recalled in the next section. The utility of Theorem 3 is demonstrated in Sections 3 and 4. More precisely, in Section 3, we prove the factorization which states that, for any $f\in\mathcal{B}_{\nu}^{p,q}$, there exist $f_{1},f_{2}\in\mathcal{B}_{\nu}^{p,q}\cap H^{\infty}$ such that $f=f_{1}/f_{2}$. Section 4 begins with a result giving a sufficient and necessary condition guaranteeing that the product of $f\in H^{p}$ and an inner function belongs to $\mathcal{B}_{\nu}^{p,q}$. As a consequence of this theorem, we obtain some results on zero sets of $\mathcal{B}_{\nu}^{p,p}$. Sections 5, 6 and 7 consist of the proofs of Theorems 1, 2 and 3, respectively.

2. Auxiliary results on weights

In this section, we recall some basic properties of weights in $\widehat{\mathcal{D}}$ and $\widecheck{\mathcal{D}}$. These properties are needed in next sections. Another reason for these results is to help the reader to understand the nature of weights in $\mathcal{D}$. We begin with a result which is essentially [19, Lemma 3]; see also [18, Lemma 2.1].

Lemma A**.**

Let $\nu$ be a weight. Then the following statements are equivalent:

• (i)

$\nu\in\widehat{\mathcal{D}}$.

• (ii)

There exist $C=C(\nu)>0$ and $\beta=\beta(\nu)>0$ such that

[TABLE]

• (iii)

There exist $C=C(\nu)>0$ and $\gamma=\gamma(\nu)>0$ such that

[TABLE]

• (iv)

The estimate

[TABLE]

is satisfied.

For the point view of our main results Lemma A(iii) is interesting because it states that $\nu\in\widehat{\mathcal{D}}$ if and only if $\nu\in\widehat{\mathcal{D}}_{p}$ for some $p>0$. This means that $\widehat{\mathcal{D}}=\bigcup_{p>0}\widehat{\mathcal{D}}_{p}$. Nevertheless, Lemma A(ii) gives maybe the most interesting description for $\widehat{\mathcal{D}}$. Together with its $\widecheck{\mathcal{D}}$ counterpart below it offers a very practical characterization for weights in $\mathcal{D}$. Essentially this characterization says that $\widehat{\nu}$ is normal in the sense of A. L. Shields and D. L. Williams [25].

Lemma B**.**

Let $\nu$ be a weight. Then $\nu\in\widecheck{\mathcal{D}}$ if and only if there exist $C=C(\nu)>0$ and $\alpha=\alpha(\nu)>0$ such that

[TABLE]

Lemma B originates from [20], and it can be proved in a corresponding manner as Lemma A(ii). See in particular the proof of [18, Lemma 2.1].

By the definition of class $\widehat{\mathcal{D}}_{p}$, it is clear that $\widehat{\mathcal{D}}_{p}\subset\widehat{\mathcal{D}}_{p+\varepsilon}$ for any $\varepsilon>0$. Next we state [21, Lemma 3], which shows that also the converse inclusion is true for sufficiently small $\varepsilon=\varepsilon(\nu,p)>0$. The proof of this result is based on integration by parts. Note that $\widehat{\mathcal{D}}_{p}(\nu)$ in the statement is defined by (1.1).

Lemma C**.**

If $0<p<\infty$ and $\nu\in\widehat{\mathcal{D}}_{p}$, then $\nu\in\widehat{\mathcal{D}}_{p-\varepsilon}$ for any $\varepsilon\in\left(0,\frac{p}{\widehat{\mathcal{D}}_{p}(\nu)+1}\right)$.

The last result of this section is [22, Lemma 5], which shows that $\nu\in\mathcal{D}$ in the norm $\|f\|_{A_{\nu}^{p,q}}$ can be replaced by $\widehat{\nu}(z)/(1-|z|)$ without losing any essential information.

Lemma D**.**

Let $0<p,q<\infty$ and $\nu$ be a weight.

• (i)

If $\nu\in\widecheck{\mathcal{D}}$, then there exists $C=C(\nu)>0$ such that

[TABLE]

• (ii)

If $\nu\in\widehat{\mathcal{D}}$, then there exists $C=C(\nu)>0$ such that

[TABLE]

For $\nu\in\mathcal{D}$, Lemmas A(ii) and B yield

[TABLE]

In [22], Lemma D is proved by applying this fact together with partial integrations. An alternative way to prove results like Lemma D is to split the integral with respect to $dr$ to infinitely many parts by using a dyadic partition, and then apply (2.1) together with the monotonicity of $M_{p}^{q}(r,f)$. An advantage of the last method is that $M_{p}^{q}(r,f)$ can be easily replaced by a certain monotonic function $g(r)$. This observation will be utilized several times in the argument of Theorem 2.

3. Quotient factorization

Recall that if $f\in H^{p}$ for some $p>0$, then there exist $f_{1},f_{2}\in H^{\infty}$ such that $f=f_{1}/f_{2}$. This is an important consequence of classical factorization [8, Theorem 2.1] by F. and R. Nevanlinna. The main purpose of this section is to give the following $\mathcal{B}_{\nu}^{p,q}$ counterpart for the above-mentioned result.

Theorem 5**.**

Let $1\leq p<\infty$, $0<q<\infty$ and $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{q}$. If $f\in\mathcal{B}_{\nu}^{p,q}$, then there exist $f_{1},f_{2}\in\mathcal{B}_{\nu}^{p,q}\cap H^{\infty}$ such that $f=f_{1}/f_{2}$ and $f_{2}$ is an outer function.

It is worth mentioning that [16, Theorem 9.19] is a similar type of result as Theorem 5 with a different hypothesis for $\nu$. Moreover, we note that Theorem 5 generalizes [2, Corollary 2.7], [6, Theorem 3.4] and [9, Corollary 3.4]. For its argument we need an extension of [6, Theorem 3.3]. Note that a part of the next pages is really inspired by [6].

Proposition 6**.**

Let $1\leq p<\infty$, $0<q<\infty$, $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{q}$ and $f\in H^{p}$ be the product of an inner function $I$ and an outer function $O_{\phi}$. Then there exists a constant $C=C(p,q,\nu)>0$ such that

[TABLE]

Before the proof of Proposition 6, we note that the quantities $F_{1}(f)$ and $F_{2}(f)$ in Theorem 3 are used repeatedly in the future.

Proof.

Let us begin by noting that $|O_{\phi}(e^{i\theta})|=\phi(e^{i\theta})$, $|O_{\max\{\phi,1\}}(e^{i\theta})|=\max\{\phi(e^{i\theta}),1\}$ and

[TABLE]

for all $z\in\mathbb{D}$ and almost every $\theta\in[0,2\pi)$. Using these facts together with Jensen’s inequality [10, Chapter I, Lemma 6.1] and the definition of outer functions, we obtain

[TABLE]

Consequently, the obvious inequality $|f(z)|\leq|O_{\phi}(z)|$ yields

[TABLE]

Write $z=re^{it}$. Raising both sides of (3.1) to power $p$, integrating from [math] to $2\pi$ with respect to $dt$, then raising both sides to power $q/p$ and finally integrating from [math] to $1$ with respect to $\nu(r)dr/(1-r)^{q}$, we obtain $F_{1}(O_{\max\{\phi,1\}})\leq F_{1}(f)$.

Next we show $F_{2}(O_{\max\{\phi,1\}})\leq F_{2}(f)$. Set

[TABLE]

and

[TABLE]

Then elementary calculations yield

[TABLE]

Consequently, we obtain $F_{2}(O_{\max\{\phi,1\}})\leq F_{2}(O_{\phi})=F_{2}(f)$ by doing a similar integral procedure as above. Now Theorem 3 together with the inequalities for $F_{1}(f)$ and $F_{2}(f)$ gives

[TABLE]

By (3.3) it suffices to show

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

by arguing as above using Jensen’s inequality. It follows that

[TABLE]

Hence it is easy to deduce $F_{1}(IO_{\min\{\phi,1\}})\leq F_{1}(f)$. Since

[TABLE]

can be shown by using a modification of (3.2), the desired estimate (3.4) follows from Theorem 3. This completes the proof. ∎

Now we can easily prove Theorem 5 by using Proposition 6.

Proof of Theorem 5. By the inner-outer factorization, there exist an inner function $I$ and an outer function $O_{\phi}$ such that $f=IO_{\phi}$. Since $O_{\phi}=O_{\min\{\phi,1\}}O_{\max\{\phi,1\}}$, we have $f=f_{1}/f_{2}$, where $f_{1}=IO_{\min\{\phi,1\}}$ and $f_{2}=1/O_{\max\{\phi,1\}}$. Applying Proposition 6 together with the inequalities

[TABLE]

and

[TABLE]

we can check that $f_{1}$ and $f_{2}$ belong to $\mathcal{B}_{\nu}^{p,q}\cap H^{\infty}$. Moreover, it is obvious that $f_{2}$ is an outer function. Hence the proof is complete. $\Box$

4. Product of $f\in H^{p}$ and an inner function in $\mathcal{B}_{\nu}^{p,q}$

Theorem 7 below gives a sufficient and necessary condition guaranteeing that the product of $f\in H^{p}$ and an inner function belongs to $\mathcal{B}_{\nu}^{p,q}$. This result generalizes [6, Corollary 3.2], the essential contents of [6, Corollary 3.1] and [9, Theorem 3.2].

Theorem 7**.**

Let $1\leq p<\infty$, $0<q<\infty$, $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{q}$, $f\in H^{p}$ and $I$ be an inner function. Then $fI\in\mathcal{B}_{\nu}^{p,q}$ if and only if $f\in\mathcal{B}_{\nu}^{p,q}$ and

[TABLE]

Proof.

We have

[TABLE]

for all $z\in\mathbb{D}$. Write $z=re^{it}$. Raising both sides of (4.1) to power $p$, integrating from [math] to $2\pi$ with respect to $dt$, then raising both sides to power $q/p$, integrating from [math] to $1$ with respect to $\nu(r)dr/(1-r)^{q}$ and finally splitting the right-hand side into two parts by using well-known inequalities, we obtain

[TABLE]

Since

[TABLE]

the assertion follows from Theorem 3. ∎

Recall that a subspace $X$ of $H^{p}$ satisfies the $F$-property if the hypothesis $fI\in X$, where $f\in H^{p}$ and $I$ is an inner function, implies $f\in X$. The $F$-property for $\mathcal{B}_{\nu}^{p,q}$ is a direct consequence of Theorem 7. However, it is worth mentioning that if one just aims to prove the $F$-property for $\mathcal{B}_{\nu}^{p,q}$, our argument is not maybe the simplest one, taking into account the length of proofs of Theorem 3 and its auxiliary results. Ideas for an alternative proof can be found, for instance, in [16, Section 5.8.3].

A sequence $\{z_{n}\}\subset\mathbb{D}$ is said to be a zero set of $\mathcal{B}_{\nu}^{p,q}$ if there exists $f\in\mathcal{B}_{\nu}^{p,q}$ such that $\{z:f(z)=0\}=\{z_{n}\}$. Here each zero $z_{n}$ is repeated according to its multiplicity and function $f$ is not identically zero. Applying Theorem 7, we make some observations on zero sets of $\mathcal{B}_{\nu}^{p,p}$. More precisely, we concentrate on the case where $\{z_{n}\}$ is separated, which means that there exists $\delta=\delta(\{z_{n}\})>0$ such that $d(z_{n},z_{k})>\delta$ for all $n\neq k$, where

[TABLE]

is the pseudo-hyperbolic distance between points $z$ and $w$. Before these results some basic properties of Hardy spaces are recalled.

For $\{z_{n}\}\subset\mathbb{D}$ satisfying the Blaschke condition $\sum_{n}(1-|z_{n}|)<\infty$ and a point $\theta\in[0,2\pi)$, the Blaschke product with zeros $\{z_{n}\}$ is defined by

[TABLE]

For $z_{n}=0$, the interpretation $|z_{n}|/z_{n}=-1$ is used. By factorization [8, Theorem 2.5] made by F. Riesz, we know that any $f\in H^{p}$ for some fixed $p\in(0,\infty]$ can be represented in the form $f=Bg$, where $B$ is a Blaschke product and $g\in H^{p}$ does not vanish in $\mathbb{D}$. More precisely, Beurling factorization [8, Theorem 2.8] asserts that $g$ is the product of an outer function and a singular inner function

[TABLE]

where $\theta\in[0,2\pi)$ is a constant and $\sigma$ a positive measure on $\mathbb{T}$, singular with respect to the Lebesgue measure. Consequently, every zero set of $\mathcal{B}_{\nu}^{p,q}$ satisfies the Blaschke condition. With these preparations we are ready to state and prove the following result.

Corollary 8**.**

Let $1\leq p<\infty$, $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{p}$, and assume that $\{z_{n}\}$ is a finite union of separated sequences and zero set of $\mathcal{B}_{\nu}^{p,p}$. Then there exists an outer function $O_{\phi}\in\mathcal{B}_{\nu}^{p,p}$ such that

[TABLE]

Proof.

Let $\{z_{n}\}=\bigcup_{j=1}^{M}\{z_{n}^{j}\}$, where $M\in\mathbb{N}$ and each $\{z_{n}^{j}\}$ is separated. Let $B$ be the Blaschke product with zeros $\{z_{n}\}$, $S$ a singular inner function and $O_{\phi}$ an outer function such that $BSO_{\phi}\in\mathcal{B}_{\nu}^{p,q}$. By Theorem 7, we know that $O_{\phi}$ and $BO_{\phi}$ belong to $\mathcal{B}_{\nu}^{p,p}$. For $w\in\mathbb{D}$ and $0<r<1$, set

[TABLE]

Since each $\{z_{n}^{j}\}$ is separated, we find $R_{j},\delta_{j}\in(0,1)$ such that, for a fixed $j$, discs $\Lambda(z_{n}^{j},R_{j})$ are pairwise disjoint and the inclusion $\Delta(z_{n}^{j},\delta_{j})\subset\Lambda(z_{n}^{j},R_{j})$ is valid for every $n$. Hence $\widehat{\nu}$ is essentially constant in each disc $\Delta(z_{n}^{j},\delta_{j})$ by Lemma A(ii). Moreover,

[TABLE]

Using these facts together with the subharmonicity of $|O_{\phi}|^{p}$, we obtain

[TABLE]

where $dA(z)$ is the two-dimensional Lebesgue measure. Now it suffices to show that the last integral in (4.2) is finite.

Set $\psi(z)=\widehat{\nu}(z)/(1-|z|)$ for $z\in\mathbb{D}$. Note that $\widehat{\nu}(r)\asymp\widehat{\psi}(r)$ for $0\leq r<1$ by Lemmas A(ii) and B. Moreover, integrating by parts, one can show that $\nu\in\widehat{\mathcal{D}}_{p}$ if and only if

[TABLE]

In particular, $\psi\in\mathcal{D}\cap\widehat{\mathcal{D}}_{p}$ by the hypotheses of $\nu$. Since Lemma D implies $BO_{\phi}$ in $\mathcal{B}_{\psi}^{p,p}$, Theorem 7 gives

[TABLE]

This completes the proof. ∎

Recall that a sequence $\{z_{n}\}\subset\mathbb{D}$ is said to be uniformly separated if

[TABLE]

and a finite union of uniformly separated sequences is called a Carleson-Newman sequence. It is worth mentioning that any Carleson-Newman sequence is a finite union of separated sequences satisfying the Blaschke condition, but the converse statement is not true. For $1<p<\infty$, $p-2<\alpha<p-1$ and a Carleson-Newman sequence $\{z_{n}\}$, we can give a sufficient and necessary condition for $\{z_{n}\}$ to be a zero set of $\mathcal{B}_{\alpha}^{p,p}$. This is a straightforward consequence of Theorem 7, Corollary 8 and the reasoning made in paper [4] by N. Arcozzi, D. Blasi and J. Pau.

Corollary 9**.**

Let $1<p<\infty$, $p-2<\alpha<p-1$ and $\{z_{n}\}$ be a Carleson-Newman sequence. Then $\{z_{n}\}$ is a zero set of $\mathcal{B}_{\alpha}^{p,p}$ if and only if there exists an outer function $O_{\phi}\in\mathcal{B}_{\alpha}^{p,p}$ such that

[TABLE]

Proof.

Let $B$ be the Blaschke product with zeros $\{z_{n}\}$ and $O_{\phi}\in\mathcal{B}_{\alpha}^{p,p}$ an outer function satisfying (4.4). Then [14, Theorem 3.5] together with some elementary calculations gives

[TABLE]

Following the reasoning in the proof of [4, Proposition 3.2], it is easy to check that $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are finite. More precisely, estimating in a natural manner, one can show

[TABLE]

In the argument of $\mathcal{I}_{2}\lesssim\|O_{\phi}^{\prime}\|_{A_{\alpha}^{p,p}}^{p}<\infty$, [5, Lemma 2.1] and the hypothesis that $\{z_{n}\}$ is a Carleson-Newman sequence play key roles.

Since $O_{\phi}\in\mathcal{B}_{\alpha}^{p,p}$ and the first integral in (4.5) is finite, $BO_{\phi}$ belongs to $\mathcal{B}_{\alpha}^{p,p}$ by Theorem 7. Consequently, the implication $\Leftarrow$ is valid. The converse implication is a direct consequence of Corollary 8. Hence the proof is complete. ∎

It is an open problem to prove a $\mathcal{B}_{\nu}^{p,p}$ counterpart of Corollary 9. One could try prove such result, for instance, assuming $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{p}$ and

[TABLE]

In this case, the implication $\Leftarrow$ is the problematic part. An idea to approach this problem is to follow the argument of [4, Proposition 3.2] and aim to apply therein [3, Theorem 3.1] instead of [5, Lemma 2.1]. The down side of this method is that it leads to laborious computations of Bekollé-Bonami weights.

Corollaries 8 and 9 are related to some main results in [15] by J. Pau and J. A.  Peláez. In particular, the equivalence $\rm(i)\Leftrightarrow(ii)$ in [15, Theorem 1] follows from Corollary 9 by setting $p=2$. Moreover, Corollary 8 shows that the implication $\rm(i)\Rightarrow(ii)$ in [15, Theorem 1] is valid also if $\{z_{n}\}$ in the statement is a finite union of separated sequences. Applying the last observation, we can also replace a Carleson-Newman sequence in [15, Corollary 1] by a finite union of separated sequences: If $0<\alpha<1$, $\{z_{n}\}$ is a finite union separated sequences and zero set of $\mathcal{B}_{\alpha}^{2,2}$, then

[TABLE]

This result offers a practical way to construct Blaschke sequences which are not zero sets of $\mathcal{B}_{\alpha}^{2,2}$; see [15, Theorem 2] and its proof.

Note that (4.2) and (4.5) together with the estimates for $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are valid also if outer function $O_{\phi}$ is replaced by an arbitrary $f\in H^{p}$. Using this observation and Theorem 7, we can rewrite Corollary 9 in the following form.

Corollary 10**.**

Let $1<p<\infty$, $p-2<\alpha<p-1$, $f\in H^{p}$ and $B$ be a Blaschke product associated with a Carleson-Newman sequence $\{z_{n}\}$. Then $fB\in\mathcal{B}_{\alpha}^{p,p}$ if and only if $f\in\mathcal{B}_{\alpha}^{p,p}$ and

[TABLE]

Corollary 10 is a partial improvement of the main result in M. Jevtić’s paper [11]. More precisely, this paper contains an extended counterpart of Corollary 10 (in the sense of $p$ and $q$) with the defect $f\equiv 1$. It is also worth mentioning that Corollary 10 is not valid if the Carleson-Newman sequence $\{z_{n}\}$ is replaced by an arbitrary Blaschke sequence. This can be shown by studying the case where $f\equiv 1$ and $B$ is a Blaschke product with zeros on the positive real axis. More precisely, the counter example follows from [23, Theorem 1], which asserts that all such Blaschke products belong to $\mathcal{B}_{\alpha}^{p,p}$ for $1/2<p<\infty$ and $p-3/2<\alpha<\infty$.

Theorem 7 for $f\equiv 1$ (or Theorem 3 for inner functions) has also extended counterpart [22, Theorem 1].

Theorem E**.**

Let $0<p,q<\infty$ and $\nu\in\mathcal{D}$. Then $\nu\in\widehat{\mathcal{D}}_{q}$ if and only if

[TABLE]

for all inner functions $I$. Here the comparison constants may depend only on $p$, $q$ and $\nu$.

Theorem E confirms that the hypothesis $\nu\in\widehat{\mathcal{D}}_{q}$ in Theorems 3 and 7 is sharp in a certain sense. Studying the argument of this result in [22], we can also deduce that the proof of Theorem 3 is more straightforward when $f$ is an inner function, and the statement is valid for all $0<p<\infty$. It is also worth mentioning that results like Theorem E have turned out to be useful in the theory of inner functions. Several by-products of Theorem E can be found in [22, 24].

5. Proof of Theorem 1

Before the proof of Theorem 1 we recall [22, Lemma 6], which is a modification of [1, Lemma 5].

Lemma F**.**

If $0<p\leq 1$ and $g:[0,1)\rightarrow[0,\infty)$ is measurable, then

[TABLE]

for $0\leq r<1$.

Proof of Theorem 1. Let $\frac{4}{5}\leq s<1$ and choose $n=n(s)\in\mathbb{N}\setminus\{1,2,3,4\}$ such that $1-\frac{1}{n}\leq s<1-\frac{1}{n+1}$. Set $f_{n}(z)=z^{n}$ for $z\in\mathbb{D}$. Since

[TABLE]

we have

[TABLE]

Using the hypothesis $\nu\in\widehat{\mathcal{D}}$ together with Lemma A(iv)(ii) in a similar manner as in the proof of [21, Theorem 1], we obtain

[TABLE]

Finally combining the estimates above and using the inequality

[TABLE]

we deduce that if $\nu\in\widehat{\mathcal{D}}$ and (1.2) is satisfied for all $f\in H^{p}$, then $\nu\in\widehat{\mathcal{D}}_{p}$. Hence it suffices to prove the converse statement.

Let $f\in H^{p}$, $0\leq\theta<2\pi$, $\frac{1}{2}<r<1$ and $0<h<\frac{1}{2}$. Set $\rho=r-h$ and $\Gamma$ be the contour which goes first rapidly from $re^{i\theta}$ to $\rho e^{i\theta}$, then along the circle $\{z:|z|=\rho\}$ to $\rho e^{i(\theta+h)}$ and finally rapidly to $re^{i(\theta+h)}$. Since

[TABLE]

we have

[TABLE]

Consequently, the discrete and continuous forms of Minkowski’s inequality, a change of variable and Hardy’s convexity theorem yield

[TABLE]

Note that the deduction above can be found, for instance, in the proof of [8, Theorem 5.4].

By raising both sides of (5.1) to power $q$, adding $\sup_{0<h<1-t}$ and then integrating from $1/2$ to $r$ with respect to $\nu(t)\,dt/(1-t)^{q}$, we obtain

[TABLE]

Letting $r\rightarrow 1^{-}$, using the monotone and mean convergence theorems together with the hypothesis $f\in H^{p}$, we deduce

[TABLE]

Hence it suffices to show $\mathcal{I}\lesssim\|f^{\prime}\|_{A_{\nu}^{p,q}}^{q}$. Note that the argument of this estimate uses ideas from [22].

If $q\leq 1$, then Lemma F with the choice $g(s)=M_{p}(s,f^{\prime})$, Hardy’s convexity theorem, Fubini’s theorem, the hypothesis $\nu\in\widehat{\mathcal{D}}_{q}$ and Lemma D give

[TABLE]

for all $f\in\mathcal{H}(\mathbb{D})$. Hence the assertion for $q\leq 1$ is proved. If $q>1$, $0<\varepsilon<q/(\widehat{\mathcal{D}}_{q}(\nu)+1)$ and $h(s)=(1-s)^{\frac{q-1-\varepsilon}{q}}$, then Hölder’s inequality and Fubini’s theorem yield

[TABLE]

for all $f\in\mathcal{H}(\mathbb{D})$. Since $\nu\in\widehat{\mathcal{D}}_{q-\varepsilon}$ by Lemma C, the assertion for $q>1$ follows from Lemma D. This completes the proof. $\Box$

Since

[TABLE]

by Minkowski’s inequality, Theorem 1 has the following consequence.

Corollary 11**.**

Let $1\leq p<\infty$, $0<q<\infty$ and $\nu\in\mathcal{D}\cap\widehat{\mathcal{D}}_{q}$. Then there exists a constant $C=C(p,q,\nu)>0$ such that

[TABLE]

for all $f\in H^{p}$.

Note that Corollary 11 is a part of Theorem 2. We state it here as an independent result because it is needed for the proof of Theorem 2.

6. Proof of Theorem 2

We go directly to the proof of Theorem 2.

Proof of Theorem 2. Let $0\leq r<1$ and $0\leq t<2\pi$. Since

[TABLE]

Cauchy’s integral formula gives

[TABLE]

Raising both sides to power $p$, integrating from [math] to $2\pi$ with respect to $dt$, then raising both sides to power $q/p$ and finally integrating from [math] to 1 with respect to $\nu(r)\,dr$, we obtain

[TABLE]

for all $f\in H^{p}$.

Let $f\in H^{p}$, $0<q\leq 1$, and set

[TABLE]

By the proof of [9, Theorem 2.1], we know that

[TABLE]

Hence the sub-additivity of $g(x)=x^{q}$ for $x\geq 0$ and Fubini’s theorem give

[TABLE]

Next we show that the weight $\nu(r)$ in the right-hand side can be replaced by $\frac{\widehat{\nu}(r)}{1-r}$ without losing any essential information.

Set $\psi(z)=\widehat{\nu}(z)/(1-|z|)$ for $z\in\mathbb{D}$, and remind that $\widehat{\nu}(r)\asymp\widehat{\psi}(r)$ for $0\leq r<1$ by Lemmas A(ii) and B. In particular, $\psi$ belongs to class $\mathcal{D}$; and thus, there exist $K=K(\psi)>1$ and $C=C(\psi)>1$ such that

[TABLE]

Let $k\in\mathbb{N}\cup\{0\}$ and $r_{n}=1-K^{-n}$ for $n\in\mathbb{N}\cup\{0\}$. Using (6.4) together with Lemma A(ii), we obtain

[TABLE]

Now Minkowski’s inequality, the monotonicity of $\omega_{p}(s,f)$ with $s$ and (6.5) yield

[TABLE]

It is worth noting that a similar deduction works also in the opposite direction.

Using (6.3) and (6.6), we obtain

[TABLE]

Minkowski’s inequality, (4.3) with $p$ being replaced by $q$ and Lemma B yield

[TABLE]

for some $\alpha=\alpha(\nu)>0$. The continuity of $\widehat{\nu}$, changes of variables, Fubini’s theorem and the hypothesis $\nu\in\mathcal{D}$ give

[TABLE]

Summarizing, we have shown

[TABLE]

Applying a similar argument as in (6.6), we can replace $\widehat{\nu}(s)$ in the right-hand side of (6.7) by $\nu(s)(1-s)$. Consequently, (6.1) and Corollary 11 imply (1.3) for all $f\in H^{p}$. Hence the assertion for $q\leq 1$ is proved.

Let $1<q<\infty$. Then (6.2), the continuous form of Minkowski’s inequality, (6.6) and well-known inequalities give

[TABLE]

Minkowski’s inequality, (4.3) with $p$ being replaced by $q$ and Lemma B yield

[TABLE]

By Lemma B, there exists a constant $\alpha=\alpha(\nu)>0$ such that

[TABLE]

Using this together with a change of variable and modification of (6.6), we get

[TABLE]

Finally (6.1), (6.8) and Corollary 11 imply (1.3) for all $f\in H^{p}$. This completes the proof. $\Box$

7. Proof of Theorem 3

Before the proof of Theorem 3 we recall the following result, which is a part of the argument of [6, Theorem 1.1].

Lemma G**.**

If $O_{\phi}$ is an outer function, then

[TABLE]

for all $z\in\mathbb{D}$.

Proof of Theorem 3. Let $f\in H^{p}$. Then there exist an inner function $I$ and an outer function $O_{\phi}$ such that $f=IO_{\phi}$. Hence the Schwarz-Pick lemma, Lemma G and the fact that $\phi(\xi)=|f(\xi)|$ for almost every $\xi\in\mathbb{T}$ yield

[TABLE]

for all $z\in\mathbb{D}$. Write $z=re^{it}$. Raising both sides of (7.1) to power $p$, integrating from [math] to $2\pi$ with respect to $dt$, then raising both sides to power $q/p$, integrating from [math] to $1$ with respect to $\nu(r)dr/(1-r)^{q}$ and finally splitting the right-hand side into two parts, we obtain

[TABLE]

which is the first inequality in (1.4).

Set

[TABLE]

Then elementary calculations together with the subharmonicity of $|f|$ yield

[TABLE]

It follows that

[TABLE]

Doing a corresponding integration procedure for this estimate as above and applying Theorem 2, we obtain

[TABLE]

which is the last inequality in (1.4). This completes the proof. $\Box$

Acknowledgements. The author thanks Antti Perälä and Toshiyuki Sugawa for valuable comments, Tohoku University for hospitality during his visit there, and the referees for careful reading of the manuscript.