Isomorphic Copies of $\ell ^\infty$ in the Weighted Hardy Spaces on the Unit Disc

Abstract

It is still unclear whether the density of analytic polynomials in an H-admissible space is sufficient to the minimality of the space? This question has a purely foundational background, relating fundamental concepts from the theory of $H^p$ spaces. We hypothesize that there is no general relationship between the density of analytic polynomials and the R-admissibility of an H-admissible space. We solve this problem by finding suitable counterexamples of Hardy spaces built upon some weighted Lebesgue spaces. In particular, we provide a direct construction of weights from Szegő class, which guarantees the existence of isomorphic copies of the space of bounded sequences in weighted Hardy spaces on the unit disc.

1 Introduction

Let $H(\mathbb D)$ denote the space of all analytic functions on the unit disc $\mathbb D= \{z \in \mathbb C: |z| < 1\}$. Let f be a complex-valued function on $\mathbb {D}$ and $0\leqslant r<1$. We write $f_r$ for the function $f_r(z):=f(r z)$, $z\in \mathbb T=\partial \mathbb {D}$ and $f_*$ for its radial limit. The class $H(\overline{\mathbb {D}})$ consists of functions which are analytic in a neighbourhood of the closed disc $\overline{\mathbb {D}}$. Let $u_n:\mathbb {D}\rightarrow \mathbb {C}$ denote the monomials $u_n(z) = z^n$, $n\in \mathbb {N}_0:=\left\{ 0, 1, 2, \dots \right\}$. The set $\mathcal {P}$ of all analytic polynomials, is defined to be the linear span of $\{u_n\}_{n\in \mathbb N_0}$.

Following Pavlović [10, 2nd rev.] let us recall that a quasinormed vector space $X\subset H(\mathbb {D})$ (i.e., equipped with a q-norm for some $0<q\leqslant 1$), is called H-admissible if it is complete, $H(\overline{\mathbb {D}})\subset X$, and the inclusion $X\subset H(\mathbb {D})$ is continuous. If an H-admissible space X satisfies

$$\begin{aligned} \sup _{0\leqslant r<1}\left\| f_r\right\| _X\leqslant C_X\left\| f\right\| _X,\quad f\in X, \end{aligned}$$

then we call it an R-admissible space. An H-admissible space is said to be minimal if

$$\begin{aligned} \lim _{r\rightarrow 1^{-}}\left\| f_r-f\right\| _{X}=0. \end{aligned}$$

The analytic polynomials always form a dense subset in a minimal space. The R-admissibility of a minimal space follows from the Banach–Steinhaus principle (see, e.g., [10, 11] for its more general variants). Moreover, if X is an R-admissible space, then the closure $X_\mathcal {P}$ of $\mathcal {P}$ in X equals

$$\begin{aligned} \Bigl \{f\,:\,\lim _{r\rightarrow 1^{-}}\left\| f-f_r\right\| _X=0\Bigr \} \end{aligned}$$

(see, e.g., [10, Lemma 1.21]). However, it is still unclear whether the density of analytic polynomials in an H-admissible space is related to the R-admissibility of this space?

We hypothesize that there is no general relationship between the density of analytic polynomials and the R-admissibility of an H-admissible space. We shall prove that neither the R-admissibility nor the density of analytic polynomials alone is sufficient for minimality to hold. Let us note here, that in the second revision of his book [10, p. 15], Pavlović posed a closely related problem whether the density of harmonic polynomials in an h-admissible space is sufficient to the minimality of the space?

The purpose of this paper is to study the weighted Hardy spaces $H^p_{\phi }(\mathbb D)$ and $H^p_{\phi }(\mathbb T)$ for Szegő’s weights. Here $0<p<\infty$ and $\mathcal {B}$ is the $\sigma$-algebra consisting of Borel sets of $\mathbb T$ and $\ell$ is the normalized Haar measure on $\mathbb T$. By a weight function we shall always mean a non-negative function $\phi$ on the unit circle $\mathbb T$. The Szegő class of weights, denoted by W,  is the collection of all weights on $\mathbb {T}$ satisfying $\phi \in L^1(\mathbb T)$ and $\log \phi \in L^1(\mathbb T)$. We adopt the convention that (a class of $\ell$-a.e. equal measurable functions) f belongs to $L^p_\phi (\mathbb T)$, for $0<p<\infty$, exactly when

$$\begin{aligned} \left\| f\right\| _{L^p_{\phi }(\mathbb {T})}=\Bigl (\frac{1}{2\pi }\int _{\mathbb {T}}\left| f(\xi )\right| ^p\phi (\xi )\left| d\xi \right| \Bigr )^{{1}/{p}}<\infty , \end{aligned}$$

and $f\in L^{\infty }(\mathbb T)$ whenever its essential supremum satisfies

$$\begin{aligned} \left\| f\right\| _{L^\infty (\mathbb {T})}=\mathop {\mathrm {ess \, sup}}\limits _{(\mathbb {T},\mathcal {B},\ell )}\left| f\right| <\infty . \end{aligned}$$

Here, every contour integral along $\mathbb T$ will be taken counter-clockwise. For $L^p(\mathbb {T})$ we take $\phi \equiv 1$. Moreover, the following functionals from $H(\mathbb {D})$ give rise to the quasi-norms of the weighted Hardy spaces $H^p_{\phi }(\mathbb D)$, $H^\infty (\mathbb D)$ and $H^p_{\phi }(\mathbb T)$, namely

$$\begin{aligned} f\mapsto \sup _{0\leqslant r<1}\left\| f_r\right\| _{L^p_{\phi }(\mathbb {T})},\quad f\mapsto \sup _{0\leqslant r<1}\left\| f_r\right\| _{L^\infty (\mathbb {T})} \quad \text {and}\quad f\mapsto \left\| f_*\right\| _{L^p_{\phi }(\mathbb {T})}. \end{aligned}$$

If $\phi \equiv 1$, then we recover the classical case $H^p(\mathbb D)\cong H^p(\mathbb T)$.

Recall that by Beurling’s theorem (see, e.g., [5, Theorem 7.4]), the analytic polynomials $\mathcal {P}$ are always dense in $H^p_{\phi }(\mathbb {T})$ for every $\phi \in W.$ Specifically, we plan to solve the aforementioned problem by addressing somewhat simpler claims:

$*$:

There exists a weight $\phi \in W$ such that $H^p_{\phi }(\mathbb {D})\ne H^p_{\phi }(\mathbb {T})$ for every $0<p<\infty$.

$**$:

There exists a weight $\phi \in W$ such that the set of analytic polynomials $\mathcal {P}$ is not dense in $H^p_{\phi }(\mathbb {D})$ for every $0<p<\infty$.

In contrast to the case of the Hardy spaces $H^p_{\phi }(\mathbb {T})$, the subspace structure of the corresponding disc spaces $H^p_{\phi }(\mathbb {D})$ is still not satisfactorily understood. Nonetheless, we also show that for the weighted Hardy spaces on the disc:

${*{**}}$:

There exists a weight $\phi \in W$ such that $H^p_{\phi }(\mathbb {D})$ contains an isomorphic copy of $\ell ^\infty$.

Finally, let us recall that the question of how to construct copies of $\ell _p$ in $H^p$ for $1\leqslant p\leqslant \infty$ was already considered in the literature (see, e.g. the survey [1]).

2 Notation, Definitions and Auxiliary Results

Note that $H(\mathbb D)$ is a Fréchet space equipped with the F-norm generated by the family of seminorms $\{\Vert \cdot \Vert _n: n\in \mathbb N\}$ given by

$$\begin{aligned} \sup \Bigl \{|f(z)| : |z|\le 1-\frac{1}{n}\Bigr \},\quad f\in H(\mathbb D). \end{aligned}$$

In the case where $f\in H(\mathbb {D})$ has non-tangential limits $\ell$-a.e. on $\mathbb T$ (which gives rise to boundary function $\ell$-a.e. on $\mathbb T$), we write $f_{*}$ for the $\ell$-a.e. defined radial limit $f_{*}= \lim _{r \rightarrow 1^-} f_r$.

A function $f\in H(\mathbb D)$ is said to be of the Nevanlinna class N if the integrals

$$\begin{aligned} \int _{\mathbb T}\log ^+|f_r| \,d\ell \end{aligned}$$

are bounded for $0\leqslant r < 1$. Since $\log ^+f$ is subharmonic, it follows that these integrals increase with r. The theorem due to F. and R. Nevanlinna says that a function belongs to N if and only if it is the quotient of two bounded analytic functions (see, e.g., [5, Theorem 2.1]). Therefore, for each non-zero function $f\in N$, there exists non-tangential limit $f_{*}$ and $\log \left| f_{*}\right|$ is integrable (see, e.g., [5, Theorem 2.2]).

Let us recall that an outer function for the class N is a function F of the form $e^{ic}\,F_\phi$ for some real constant c, where

$$\begin{aligned} F_\phi (z)=\exp \biggl (\frac{1}{2\pi }\int _{\mathbb T}\frac{\xi +z}{\xi -z} \log \phi (\xi ) \left| d\xi \right| \biggr ),\quad z\in \mathbb D \end{aligned}$$

and the weight $\phi$ satisfies $\log \phi \in L^1(\mathbb T)$. If $\phi$ is additionally assumed to satisfy $\phi \in L^p(\mathbb T)$ then $F=e^{ic}F_\phi$, $c\in \mathbb R$ is an outer function for the class $H^p(\mathbb T)$. The outer function satisfies the equality $|F_*|=\phi$. A function $f\in H(\mathbb D)$ satisfying

• (i) $\left| f\right| \leqslant 1$ on $\mathbb D,$ and

• (ii) $\left| f_*\right| =1$ a.e. on $\mathbb T$,

is called an inner function. In the case where an inner function has no zeros, it is called a singular inner function. It is well known that $S\in H(\mathbb D)$ is a singular inner function if and only if there exists a non-negative singular measure $\sigma$ on $\mathbb T$ satisfying

$$\begin{aligned} S(z)=e^{ic}\exp \biggl (-\frac{1}{2\pi }\int _{\mathbb T}\frac{\xi +z}{\xi -z} \, d\sigma (\xi )\biggr ),\quad z\in \mathbb D, \end{aligned}$$

where c is a real constant. Let us remark that both functions S and F do not vanish in $\mathbb {D}$.

The zeros of a non-zero function f from $H(\mathbb D)$ cannot cluster inside its domain of analyticity $\mathbb D$. In the case where the set of zeros is empty, we have $f\in N$ by Jensen’s formula and we set $B\equiv 1$ (the auxiliary function B will be defined below for the other cases). If this is not the case, we arrange the zeros $\{a_n\}$ in an order of non-decreasing absolute values, where each zero is counted according to its algebraic multiplicity. If f has infinitely many zeros $\{a_n\}$, then $\lim _{n\rightarrow \infty }\left| a_n\right| =1$. In the case where the function satisfies a growth condition, the zeros must tend more rapidly to the boundary. Let us recall the theorem due to Blaschke (see, e.g., [5, Theorem 2.3 and 2.4] or [10]). The zeros $\{a_n\}$ of a non-zero function $f\in H(\mathbb D)$ satisfy the Blaschke condition

$$\begin{aligned} \sum _{n}\bigl (1-\left| a_n\right| \bigr )<\infty \end{aligned}$$

if and only if $f\in N$. If a sequence $\{a_n\}\subset \mathbb D$ satisfies the Blaschke condition, then the product

$$\begin{aligned} B(z)=\prod _{n}\frac{\left| a_n\right| }{a_n}\frac{a_n-z}{1-\overline{a}_nz} \end{aligned}$$

converges in $H(\mathbb D)$ (i.e., uniformly on each disc $\bigl \{z\in \mathbb {C}:\,\left| z\right| \leqslant R<1\bigr \}$). By convention, we set ${\left| a_n\right| }/{a_n}=-1$ if $a_n=0$.

• (i) Each $a_n$ is a zero of B, with multiplicity equal to the number of times it occurs in the sequence, and B has no other zeros in $\mathbb D$.

• (ii) $\left| B\right| \leqslant 1$ on $\mathbb D$.

• (iii) $\left| B_*\right| =1$ a.e. on $\mathbb T$.

The function B is called a Blaschke product. A function $f\in N$ is said to be of the Smirnov class $N^+$ if

$$\begin{aligned} \lim _{r\rightarrow 1^-}\int _{\mathbb T}\log ^+|f_r| \,d\ell =\int _{\mathbb T}\log ^+|f_{*}| \,d\ell . \end{aligned}$$

Every non-zero function of the class $N^{+}$ can be expressed in the form $f=BSF$, where B is a Blaschke product, S is a singular inner function, and F is an outer function for the class N (see, e.g., [5, Theorem 2.10]). The Smirnov class $N^+$ is the natural limit space of $H^p(\mathbb D)$ as $p\rightarrow 0$ and the inclusions $H^p(\mathbb D)\subset N^+\subset N$ are proper for every $p >0$.

If $\phi$ is a weight and $0<p<\infty$, then the weighted Hardy space $H^p_\phi (\mathbb {D})$, consists of all functions f which are analytic in $\mathbb {D}$ and satisfy $f_r\in L^p_\phi (\mathbb {T})$ for all $0\leqslant r<1$, where the corresponding $L^p_\phi (\mathbb {T})$ bounds are uniform with

$$\begin{aligned} \left\| f\right\| _{H^p_\phi (\mathbb D)}= \sup _{0\leqslant r<1} \left\| f_r\right\| _{L^{p}_\phi (\mathbb {T})} < \infty . \end{aligned}$$

The classical Hardy space $H^p(\mathbb {D})$ is obtained by taking $\phi \equiv 1$. At the other endpoint, the Hardy space $H^\infty (\mathbb {D})$ is the collection of all analytic functions $f\in H(\mathbb D)$ that satisfy

$$\begin{aligned} \left\| f\right\| _{H^\infty (\mathbb D)}= \sup _{0\leqslant r<1} \left\| f_r\right\| _{L^{\infty }(\mathbb {T})} < \infty . \end{aligned}$$

By $\mathcal {A}(\mathbb D)$ we denote the disc algebra, that is the set of all functions from $H(\mathbb D)$ which are continuous on $\overline{\mathbb {D}}$. We also have $H(\overline{\mathbb {D}})\subset \mathcal {A}(\mathbb D)\subset H^{\infty }(\mathbb D)$. Moreover, the assumption that $\phi \in L^1(\mathbb T)$ is equivalent to the inclusion $\mathcal {A}(\mathbb D)\subset H^{p}_{\phi }(\mathbb D)$. The Poisson integral of a function $f\in L^1(\mathbb T)$ is the harmonic function $\mathcal {P}[f]$ defined by

where P is the Poisson kernel, given by

$$\begin{aligned} P(z)=\frac{1-\left| z\right| ^2}{\left| 1-z\right| ^2}={{\,\mathrm{Re{\,\!\!}}\,}}\frac{1+z}{1-z},\quad z\in \mathbb D. \end{aligned}$$

It is a well-known fact that if $f\in H^{1}(\mathbb D)$, then $f_{*}\in L^{1}(\mathbb T)$ and $f=\mathcal {P}[f_{*}]$. For more details concerning classical Hardy spaces on the unit disc we refer the reader to [5, 10].

Assuming $\log \phi \in L^1(\mathbb T)$ and $0< p < \infty$, let $H^p_{\phi }(\mathbb T)$ denote the space of all analytic functions $f\in H(\mathbb D)$ that satisfy $fF_{\phi }^{{1}/{p}}\in H^p(\mathbb D)$ with

$$\begin{aligned} \left\| f\right\| _{H^p_{\phi }(\mathbb T)}=\bigl \Vert fF_{\phi }^{{1}/{p}}\bigr \Vert _{H^p(\mathbb D)}. \end{aligned}$$

Let us remark here that we always have $H^p_{\phi }(\mathbb T)\subset N^{+}$. Indeed, taking any non-zero $f\in H^p_{\phi }(\mathbb T)$ we have $fF_{\phi }^{{1}/{p}}=BSF_{\psi }$ for some $\psi$ satisfying $\log \psi \in L^1(\mathbb T)$. Since both $\log \psi$ and $\log \phi$ are members of $L^1(\mathbb T)$, it follows that $\log \varphi \in L^1(\mathbb T)$ for $\varphi ={\psi }/{\phi ^{{1}/{p}}}$, and the function

$$\begin{aligned} f=\frac{BSF_{\psi }}{F_{\phi ^{{1}/{p}}}}=BSF_{\varphi } \end{aligned}$$

belongs to $N^{+}$. Therefore, each $f\in H^p_{\phi }(\mathbb T)$ has a boundary function $f_{*}$ ($\ell$-a.e. on $\mathbb T$) and

$$\begin{aligned} \left\| f\right\| _{H^p_{\phi }(\mathbb T)}=\bigl \Vert f_{*}\phi ^{{1}/{p}}\bigr \Vert _{L^p(\mathbb T)}=\left\| f_{*}\right\| _{L^p_\phi (\mathbb T)}. \end{aligned}$$

(2.1)

We make the following provisional definition. If for $\log \psi \in L^1(\mathbb T)$ we have that $\psi \in L^p_\phi (\mathbb T)$, then we say that $F=e^{ic}F_\psi$, $c\in \mathbb R$, is an outer function for the class $H^p_\phi (\mathbb T)$.

Thus, we have proved the canonical factorization theorem for weighted Hardy spaces on the torus.

Theorem 2.1

If $\log \phi \in L^1(\mathbb T)$ and $0< p < \infty$, then every non-zero function f from $H^p_{\phi }(\mathbb T)$ admits a unique factorization of the form $f=BSF$, where B is a Blaschke product, S is a singular inner function, and F is an outer function for the class $H^p_\phi (\mathbb T)$. Conversely, every product BSF for the outer function F for the class $H^p_\phi (\mathbb T)$ is a member of $H^p_\phi (\mathbb T)$.

In fact, (2.1) allows us to treat $H^p_{\phi }(\mathbb T)$ as a subspace of $L^p_\phi (\mathbb T)$. If also we have $\phi \in L^1(\mathbb {T})$, then $A(\mathbb D)\subset H^p_{\phi }(\mathbb T)$, and moreover, $H^p_{\phi }(\mathbb T)$ is isometrically isomorphic to the closure of $\mathcal {P}$ in $L^p_\phi (\mathbb {T})$ by Beurling’s theorem (see, e.g., [5, Theorem 7.4] or [4]). In the case where $\phi \equiv 1$, we have $H^p(\mathbb {D})\cong H^p(\mathbb {T})$.

The Szegő class of weights, denoted by W, is the collection of all weights on $\mathbb {T}$ satisfying $\phi \in L^1(\mathbb T)$ and $\log \phi \in L^1(\mathbb T)$.

$$\begin{aligned} W=\bigl \{ \phi \in L^1(\mathbb {T})\,\,:\,\,\log \phi \in L^1(\mathbb {T})\bigr \} \end{aligned}$$

Carthy showed in [9] (see also [8]) that $N^+$, can be realized as a union of weighted Hardy spaces, namely

$$\begin{aligned} N^+=\bigcup _{\phi \in W}H^2_{\phi }(\mathbb T). \end{aligned}$$

Let us note that since a non-zero function $f\in N^+$ has a factorization $f=BSF$ and $g=F^{{p}/{2}}\in N^+$ for all $p>0$, it follows that

$$\begin{aligned} N^+=\bigcup _{\phi \in W}H^p_{\phi }(\mathbb T). \end{aligned}$$

(2.2)

Let us first discuss the relation between the weighted Hardy spaces $H^p_{\phi }(\mathbb {D})$ and $H^p_{\phi }(\mathbb {T})$ for $\phi \in W$. We state the following technical result, which is a more general version of Stoll’s lemma given in [3]. The proof that $fF_{\phi }^{{1}/{p}}\in H^p(\mathbb D)$ whenever $f\in H^p_{\phi }(\mathbb D)$ appeared first in [3, Lemma 2.3] and [7, Theorem 3]. Here is a simple direct proof.

Lemma 2.2

Let $0<p<\infty$. Assume that $\phi \in W$ and $f\in H^p_\phi (\mathbb D)$. Then

$$\begin{aligned} \left\| f\right\| _{H^p_\phi (\mathbb T)}\leqslant \left\| f\right\| _{H^p_\phi (\mathbb D)}. \end{aligned}$$

(2.3)

Moreover if $f\in H^p_\phi (\mathbb T)$ with $\limsup _{r\rightarrow 1^{-}}\left\| f_r\right\| _{L^p_\phi (\mathbb T)}<\infty$, then $f\in H^p_\phi (\mathbb D)$.

Proof

In the proof below, we shall use the mean convergence theorem and Hardy’s convexity theorem (see, e.g., [5, Theorems 1.5 and 2.6]). Fix $f\in H^p_{\phi }(\mathbb D)$. Since

$$\begin{aligned} \sup _{0\leqslant r<1}\bigl \Vert \bigl (f_{\rho }F_{\phi }^{{1}/{p}}\bigr )_r \bigr \Vert _{L^{p}(\mathbb T)}= \bigl \Vert f_{\rho }{\phi }^{^{{1}/{p}}} \bigr \Vert _{L^{p}(\mathbb T)}<\infty ,\quad 0\leqslant \rho <1 \end{aligned}$$

where $f_{\rho } {\phi }^{^{{1}/{p}}}=(f_{\rho }F_{\phi }^{{1}/{p}} )_*$, it follows that

$$\begin{aligned} \bigl \Vert fF_{\phi }^{{1}/{p}} \bigr \Vert _{H^{p}(\mathbb D)}&=\sup _{0\leqslant r<1}\bigl \Vert \bigl (fF_{\phi }^{{1}/{p}}\bigr )_r\bigr \Vert _{L^{p}(\mathbb T)} =\sup _{0\leqslant r<1}\Bigl \Vert \liminf _{\rho \rightarrow 1^-}f_{r\rho }\bigl (F_{\phi }^{{1}/{p}}\bigr )_r \Bigr \Vert _{L^{p}(\mathbb T)} \\&\leqslant \sup _{0\leqslant r<1}\liminf _{\rho \rightarrow 1^-} \bigl \Vert f_{r\rho }\bigl (F_{\phi }^{{1}/{p}}\bigr )_r \bigr \Vert _{L^{p}(\mathbb T)} \leqslant \liminf _{\rho \rightarrow 1^-} \bigl \Vert f_{\rho }{\phi }^{^{{1}/{p}}} \bigr \Vert _{L^{p}(\mathbb T)}\\&\leqslant \Vert f \Vert _{H^{p}_{\phi }(\mathbb D)} \end{aligned}$$

by Fatou’s lemma.

We now prove the remaining inequality (cf. [3, Remark 2.4]). By assumption, $\phi \in L^1(\mathbb T)$ and $\log \phi \in L^1(\mathbb T)$, and moreover $F_{\phi }^{{1}/{p}}$ is an outer function for the class $H^p(\mathbb D)$. Let $f\in H^p_{\phi }(\mathbb T)$. Then $f\in H(\mathbb D)$ with $fF_{\phi }^{{1}/{p}}\in H^{p}(\mathbb D)$. Since $f_{*}{\phi }^{^{{1}/{p}}}\in L^p(\mathbb T)$ and

$$\begin{aligned} \limsup _{r\rightarrow 1^-}\Vert f_r{\phi }^{^{{1}/{p}}}\Vert _{L^p(\mathbb T)}<\infty , \end{aligned}$$

and since moreover

$$\begin{aligned} \sup _{0\le r \leqslant R}\Vert f_r\Vert _{L^p_{\phi }(\mathbb T)}\leqslant \Vert f_R\Vert _{H^{\infty }(\mathbb D)} \Vert \phi \Vert _{L^1(\mathbb T)}<\infty \end{aligned}$$

for every $0<R<1$, it follows that $f\in H^p_{\phi }(\mathbb D)$. $\square$

Lemma 2.3

If $0<p<\infty$ and $\phi \in W,$ then both $H^p_{\phi }(\mathbb {D})$ and $H^p_{\phi }(\mathbb {T})$ are H-admissible.

Proof

For the sake of rigor, we first show that both $H^p_{\phi }(\mathbb {D})$ and $H^p_{\phi }(\mathbb {D})$ are complete. Fix $0<p<\infty$. Let $\{f_n\}$ be a Cauchy sequence in $H^p_{\phi }(\mathbb {T})$. Since $\{f_nF_{\phi }^{{1}/{p}}\}$ is a Cauchy sequence in $H^p(\mathbb {T})$, it follows that there is $g:=\lim _{n\rightarrow \infty }f_nF_{\phi }^{{1}/{p}}$ in $H^p(\mathbb T)$. On the other hand, $\{(f_n)_{*}\phi ^{{1}/{p}}\}$ is a Cauchy sequence in $L^p(\mathbb {T})$ with $g_{*}=\lim _{n\rightarrow \infty }(f_n)_{*}\phi ^{{1}/{p}}$ in $L^p(\mathbb {T})$. By Theorem 2.1,

$$\begin{aligned} f=\frac{g}{F_{\phi ^{{1}/{p}}}}=\frac{BSF_{g_{*}}}{F_{\phi ^{{1}/{p}}}} \end{aligned}$$

is a limit $\lim _{n\rightarrow \infty }f_n$ in $H^p_{\phi }(\mathbb T)$ with $f_{*}\in L^p_{\phi }(\mathbb {T})$.

Take an absolutely convergent series $\sum _{n=1}^{\infty } f_n$ in $H^p_{\phi }(\mathbb {D})$. Lemma 2.2 shows that this series converges in $H^p_{\phi }(\mathbb {T})$ to some analytic function, say $f=\sum _{n=1}^{\infty } f_n$. Observe that the series $\sum _{n=1}^{\infty } (f_n)_r$ is absolutely convergent in $L^p_{\phi }(\mathbb {T})$ to $f_r$ for every $0\leqslant r<1$, where

$$\begin{aligned} \left\| f_r\right\| _{L^p_{\phi }(\mathbb {T})}\leqslant \sum _{n=1}^{\infty } \left\| (f_n)_r\right\| _{L^p_{\phi }(\mathbb {T})}\leqslant \sum _{n=1}^{\infty }\left\| f_n\right\| _{H^p_{\phi }(\mathbb {D})}<\infty . \end{aligned}$$

That

$$\begin{aligned} \mathcal {A}(\mathbb D)\subset H^p_{\phi }(\mathbb {D})\subset H^p_{\phi }(\mathbb {T}), \end{aligned}$$

(2.4)

where the inclusions are continuous, follows immediately from $\phi \in L^1(\mathbb {T})$ and Lemma 2.2.

We will make use of the classical inequality

$$\begin{aligned} \left| f(z)\right| ^p(1-\left| z\right| ^2)\leqslant \left\| f\right\| _{H^p(\mathbb T)}^p,\quad z\in \mathbb {D} \end{aligned}$$

(see, e.g., [10]) which we apply to the case of weighted Hardy spaces. Take $f\in H^p_{\phi }(\mathbb {T})$ and $0\leqslant r<1$. Then

$$\begin{aligned} \begin{aligned} c_{r,\phi }\left| f(z)\right| ^p(1-\left| z\right| ^2)&\leqslant \left| f(z)\right| ^p\left| F_{\phi }(z)\right| (1-\left| z\right| ^2)\\ {}&\leqslant \bigl \Vert fF_{\phi }^{{1}/{p}}\bigr \Vert _{H^p(\mathbb T)}^p=\left\| f\right\| _{H^p_{\phi }(\mathbb T)}^p \end{aligned} \end{aligned}$$

(2.5)

is valid for $\left| z\right| \leqslant r$, where $c_{r,\phi }=\min _{\left| z\right| \leqslant r}\left| F_{\phi }(z)\right|$ is positive because of the fact that an outer function cannot vanish in $\mathbb {D}$. By (2.4) and (2.5), $H^p_{\phi }(\mathbb {D})\subset H^p_{\phi }(\mathbb {T})\subset H(\mathbb {D})$, where the last inclusion is continuous, too. $\square$

3 Main Results

Let us point out that the equality $H^p_{\phi }(\mathbb {D})=H^p_{\phi }(\mathbb {T})$ as sets, first announced in [6] without proof, was discussed in a series of papers [2, 3, 7] in the context of Muckenhoupt’s weights.

While the Hardy spaces $H^p_{\phi }(\mathbb {D})$ and $H^p_{\phi }(\mathbb {T})$ resemble the classical Hardy spaces $H^p(\mathbb {D})$ and $H^p(\mathbb {T})$, they in general need not be equal for weights from W.

Theorem 3.1

If $0<p<\infty$, then there exists a weight $\phi \in W$ such that $H^p_{\phi }(\mathbb {D})\ne H^p_{\phi }(\mathbb {T})$.

Proof

Fix $0<p<\infty$. We give a direct construction of both the weight $\phi \in W$ and the function $f\in H^p_\phi (\mathbb T)$ that satisfy $\limsup _{r\rightarrow 1^{-}}\left\| f_r\right\| _{L^p_\phi (\mathbb T)}=\infty$. Let $\left\{ I_n\right\} _{n\in \mathbb {N}_0}$ be a partition of $\mathbb {T}$ given by

$$\begin{aligned} I_0&:=\left\{ \xi \in \mathbb {T}\,:\, 2<{{\,\textrm{arg}\,}}\xi \leqslant 2\pi \right\} \\ I_{2n+1}&:=\left\{ \xi \in \mathbb {T}\,:\, {3}/{2^{n+1}}<{{\,\textrm{arg}\,}}\xi \leqslant {2}/{2^{n}}\right\} \quad \text {and}\\ I_{2n+2}&:=\left\{ \xi \in \mathbb {T}\,:\, {1}/{2^{n}}<{{\,\textrm{arg}\,}}\xi \leqslant {3}/{2^{n+1}}\right\} . \end{aligned}$$

Consider $1<c<2$. Then the following weight

$$\begin{aligned} \phi :=\chi _{I_0}+\sum _{n=0}^{\infty }\bigl (e^{-c^n}\chi _{I_{2n+1}}+\chi _{I_{2n+2}}\bigr ) \in W. \end{aligned}$$

Indeed,

$$\begin{aligned}&2\pi \int _{\mathbb {T}}\phi \, d\ell =(2\pi -2)+\sum _{n=0}^{\infty }\frac{e^{-c^n}+1}{2^{n+1}}<\infty ,\quad \text{ and }\\ {}&2\pi \int _{\mathbb {T}}\left| \log \phi \right| d\ell =\sum _{n=0}^{\infty }\frac{c^n}{2^{n+1}}<\infty . \end{aligned}$$

By the above, we also have $\log \phi ^{-1}\in L^1(\mathbb T)$. Now f is defined by setting an appropriate outer function, where $\psi :=\phi ^{-1}$, namely

$$\begin{aligned} \psi =\chi _{I_0}+\sum _{n=0}^{\infty }\bigl (e^{c^n}\chi _{I_{2n+1}}+\chi _{I_{2n+2}}\bigr ) \end{aligned}$$

and

$$\begin{aligned} f(z):=F_{\psi }^{{1}/{p}}=\exp \biggl (\frac{1}{2p\pi }\int _{\mathbb T}\frac{\xi +z}{\xi -z} \log \psi (\xi ) \left| d\xi \right| \biggr ),\quad z\in \mathbb D. \end{aligned}$$

Since $\left| f_{*}\right| ^p\equiv \psi$ $\ell$-a.e., it follows that $f\in H^p_{\phi }(\mathbb {T})$. Let us also define a family $\left\{ J_n\right\} _{n\in \mathbb {N}_0}$ of subsets of $\mathbb {T}$ by

$$\begin{aligned} J_{n}&:=\left\{ \xi \in \mathbb {T}\,:\, -{1}/{2^{n+1}}\leqslant {{\,\textrm{arg}\,}}\xi \leqslant 0\right\} . \end{aligned}$$

It is routine to verify that for $r_n:=1-{1}/{2^{n+1}}$ we have $P(r_n\xi )\geqslant 2^n$ for all $\xi \in J_n$ for each $n\in \mathbb {N}$, where P is the Poisson kernel. Thus,

$$\begin{aligned} \begin{aligned} \left\| f_{r_n}\right\| ^p_{L^p_\phi (\mathbb T)}&= \int _{\mathbb {T}}\left| f_{r_n}\right| ^p\phi \, d\ell \\&=\frac{1}{2\pi }\int _{\mathbb {T}} \exp \biggl (\frac{1}{2\pi }\int _{\mathbb {T}}P(r_n\zeta \xi ^{-1})\log \psi (\xi ) \left| d\xi \right| \biggr ) \phi (\zeta ) \left| d\zeta \right| \\&\geqslant \frac{1}{2\pi }\int _{I_{2n+2}}\exp \biggl (\frac{1}{2\pi }\int _{\zeta J_{n}}2^n\log \psi (\xi )\left| d\xi \right| \biggr ) \phi (\zeta ) \left| d\zeta \right| \\&\geqslant \frac{1}{2\pi }\exp \biggl (\frac{1}{2\pi }\frac{c^{n}}{4}\biggr ) \frac{1}{2^{n+1}}\\&=\frac{1}{4\pi }\exp \biggl (\frac{c^{n}}{8\pi }-n\log 2\biggr )\rightarrow \infty \quad \text {as}\quad n \rightarrow \infty , \end{aligned} \end{aligned}$$

(3.1)

and so $H^p_{\phi }(\mathbb {D})\ne H^p_{\phi }(\mathbb {T}),$ as required. $\square$

Actually, based on the ideas presented in Theorem 3.1 one can also find a pair consisting of the weight and the function satisfying more refined estimates.

Theorem 3.2

If $0<p<\infty$, then there exists a weight $\phi \in W$ such that $H^p_{\phi }(\mathbb {D})$ contains an isomorphic copy of $\ell ^\infty$.

Proof

We use the notation from the proof of Theorem 3.1, and the definitions of the intervals and functions given there ($I_n, \phi ,\ldots$). The purpose of this proof is to give the construction of a weight $\phi$ and a linear embedding $T:\ell ^\infty \rightarrow H^p_{\phi }(\mathbb {D})$. This will give rise to the countable family $\mathcal {F}$ of analytic functions $h_{n}:=T(e_n)$, $n\in \mathbb {N}_0$ on the disc satisfying

$$\begin{aligned} \biggl \Vert \sum _{n\in I}h_{n}\biggr \Vert _{H^p_{\phi }(\mathbb {D})}\asymp 1\quad \text {and}\quad \inf _{\begin{array}{c} I\ne J\\ J\subset \mathbb {N}_0 \end{array}}\biggl \Vert \sum _{n\in I}h_{n}-\sum _{n\in J}h_{n}\biggr \Vert _{H^p_{\phi }(\mathbb {D})}\asymp 1 \end{aligned}$$

for all $\emptyset \ne I\subset \mathbb {N}_0$. Our proof breaks into three parts.

As a first step we put

$$\begin{aligned} f_{n}(z):=F_{\psi _n}^{{1}/{p}}=\exp \biggl (\frac{1}{2p\pi }\int _{\mathbb T}\frac{\xi +z}{\xi -z} \log \psi _n(\xi ) \left| d\xi \right| \biggr ),\quad z\in \mathbb D, \end{aligned}$$

where

$$\begin{aligned} \psi _n:=\chi _{\mathbb {T}\setminus I_{2n+1}}+e^{c^n}\chi _{I_{2n+1}}, \quad n\in \mathbb {N}_0. \end{aligned}$$

It is also clear that $\left| (f_{n})_{*}\right| ^p=\psi _n$ $\ell$-a.e. and $\log \psi _n\in L^1(\mathbb T)$. Since $\left\| f\right\| _{L^p_\phi (\mathbb T)}\leqslant \left\| f\right\| _{L^p(\mathbb T)}$ for all $f\in L^p(\mathbb {T})$ and $\left\| g\right\| _{H^p(\mathbb D)}\leqslant \left\| g\right\| _{H^\infty (\mathbb {D})}$ for all $g\in H^\infty (\mathbb {T})$, and since moreover $\psi _n\leqslant e^{c^n}$ $\ell$-a.e., it follows that $\left\| f_{n}\right\| _{H^p_\phi (\mathbb D)}<\infty$, $n\in \mathbb {N}$. On the other hand, the key estimate (3.1) in the proof of Theorem 3.1 also gives

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| f_{n}\right\| _{H^p_\phi (\mathbb D)}=\infty . \end{aligned}$$

(3.2)

Indeed, a close inspection of the above proof reveals that

$$\begin{aligned} \left\| f_{n}\right\| ^p_{H^p_\phi (\mathbb D)}\geqslant \frac{1}{4\pi }\exp \biggl (\frac{c^{n}}{8\pi }-n\log 2\biggr ), \quad n\in \mathbb {N}_0. \end{aligned}$$

(3.3)

By the mean convergence theorem for $H^p(\mathbb {D})$ (see, e.g., [5, Theorem 2.6]) one can quickly check that

$$\begin{aligned} \lim _{r\rightarrow 1^{-}}\left\| (f_{n})_r-(f_{n})_{*}\right\| _{L^p_\phi (\mathbb T)}=0,\quad n\in \mathbb {N}_0. \end{aligned}$$

(3.4)

A standard verification shows that for all $0\leqslant r<1,$

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| (f_{n})_r\right\| _{H^\infty (\mathbb {D})}=\lim _{n\rightarrow \infty }\sup _{\left| z\right| \leqslant r}\left| f_n(z)\right| =1,\quad \text {and}\quad \sup _{n\in \mathbb {N}}\left\| (f_n)_*\right\| ^p_{L^p_\phi (\mathbb T)}<1. \end{aligned}$$

(3.5)

Indeed, this follows immediately from

$$\begin{aligned} \begin{aligned} \left| f_{n}(rz)\right| ^{p}&= \exp \biggl (\frac{1}{2\pi }\int _{\mathbb {T}}P(rz\xi ^{-1})\log \psi _n(\xi ) \left| d\xi \right| \biggr )\\&\leqslant \exp \biggl (\frac{1}{2\pi }\frac{1}{1-r}\int _{\mathbb {T}}\log \psi _n(\xi ) \left| d\xi \right| \biggr )\\&\leqslant \exp \biggl (\frac{1}{2\pi }\frac{1}{1-r}\frac{c^n}{2^{n+1}}\biggr )\rightarrow 1\quad \text {as}\quad n \rightarrow \infty ,\quad z\in \mathbb D, \end{aligned} \end{aligned}$$

(3.6)

and

$$\begin{aligned} \left\| (f_n)_*\right\| ^p_{L^p_\phi (\mathbb T)}&= \int _{\mathbb {T}}\psi _n\phi \, d\ell =\frac{1}{2\pi }\biggl (2\pi -2+ \sum _{\begin{array}{c} k=0\\ k\ne n \end{array}}^{\infty }\frac{e^{-c^k}+1}{2^{k+1}}+\frac{2}{2^{n+1}}\biggr )\\&\leqslant \frac{1}{2\pi }\biggl (2\pi -2+ \sum _{k=1}^{\infty }\frac{e^{-c^k}+1}{2^{k+1}}+1\biggr )<1, \end{aligned}$$

respectively.

In the second part, we define a new sequence of functions by setting

$$\begin{aligned} g_n:=\frac{f_n}{\left\| f_{n}\right\| _{H^p_\phi (\mathbb D)}}, \quad n\in \mathbb {N}_0. \end{aligned}$$

By (3.2), (3.4) and (3.5), $\left\| g_{n}\right\| _{H^p_\phi (\mathbb D)}=1$ and

$$\begin{aligned} \lim _{r\rightarrow 1^{-}}\left\| (g_{n})_r-(g_{n})_{*}\right\| _{L^p_\phi (\mathbb T)}=0,\quad \lim _{n\rightarrow \infty }\left\| (g_{n})_{*}\right\| _{L^p_\phi (\mathbb T)}=0,\quad \lim _{n\rightarrow \infty }\left\| (g_{n})_r\right\| _{H^p_\phi (\mathbb D)}=0 \end{aligned}$$

for all $n\in \mathbb {N}_0$ and $0\leqslant r<1$. Fix $0<\varepsilon <1$ and $0<\varepsilon _n$, $n\in \mathbb {N}_0$ satisfying

$$\begin{aligned} \sum _{n=0}^{\infty }\varepsilon _n<\varepsilon \quad \text{ and }\quad \sum _{n=0}^{\infty }\varepsilon ^p_n<\varepsilon .\end{aligned}$$

Now that we have the above claim, we can select a subsequence of $\left\{ g_n\right\} _{n\in \mathbb {N}_0}$ of appropriately separated functions (in the sense of $H^p_\phi (\mathbb D)$ norm). For the base step of the induction, we take $r_0:=0$ and choose $n_0\in \mathbb {N}_0$ to satisfy

$$\begin{aligned} \sup _{0\leqslant r\leqslant r_0}\left\| (g_{n_0})_{r}\right\| _{L^p_\phi (\mathbb D)}<\varepsilon _0\quad \text {and}\quad \left\| (g_{n_0})_{*}\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _0. \end{aligned}$$

There exists $r_0<r_1<1$ such that

$$\begin{aligned} \sup _{r_1\leqslant r< 1}\left\| (g_{n_0})_{r}\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _0. \end{aligned}$$

The next step is to find $n_0<n_1\in \mathbb {N}$ satisfying

$$\begin{aligned} \sup _{0\leqslant r \leqslant r_1}\left\| (g_{n_1})_r\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _1\quad \text {and}\quad \left\| (g_{n_1})_{*}\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _1. \end{aligned}$$

There is $r_1<r_2<1$ that satisfy

$$\begin{aligned} \sup _{r_2\leqslant r< 1}\left\| (g_{n_1})_{r}\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _1. \end{aligned}$$

Let us prove the induction step. Suppose that for all $0\leqslant k \leqslant m-1$ the following three statements hold

$$\begin{aligned} \begin{aligned} \sup _{0\leqslant r \leqslant r_k}&\left\| (g_{n_k})_r\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _k, \quad \left\| (g_{n_k})_{*}\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _k\quad \text {and}\\ \sup _{r_{k+1}\leqslant r< 1}&\left\| (g_{n_k})_{r}\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _k, \end{aligned} \end{aligned}$$

(3.7)

where $0=r_0<r_1<\ldots<r_{m-1}<r_m<1$ and $0\leqslant n_0<n_1<\ldots<n_{m-2}<n_{m-1}$. Now $n_{m-1}<n_m\in \mathbb {N}$ is chosen to satisfy

$$\begin{aligned} \sup _{0\leqslant r \leqslant r_m}\left\| (g_{n_m})_r\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _m\quad \text {and}\quad \left\| (g_{n_m})_{*}\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _m. \end{aligned}$$

There exists $r_m<r_{m+1}<1$ that satisfy

$$\begin{aligned} \sup _{r_{m+1}\leqslant r< 1}\left\| (g_{n_m})_{r}\right\| _{L^p_\phi (\mathbb T)}<\varepsilon _m. \end{aligned}$$

By the induction hypothesis, it follows that (3.7) holds for all $0\leqslant k \leqslant m$. We proceed by induction.

Let us move to the final step of the proof. We check first that

$$\begin{aligned} g_\lambda :=\sum _{k=0}^{\infty }\lambda _k g_{n_k} \end{aligned}$$

is a member of $H^p_\phi (\mathbb D)$ for every sequence $\lambda :=\left\{ \lambda _k\right\} _{k\in \mathbb {N}_0}\in \ell ^{\infty }$.

That $g_\lambda \in H(\mathbb D)$ follows from (3.1) and (3.6). Indeed, if $0\leqslant r<1$, then by (3.3)

$$\begin{aligned} \sum _{k=0}^{\infty }\left\| (g_{n_k})_r\right\| _{\infty }\leqslant \sum _{n=0}^{\infty }\frac{\left\| (f_{n})_r\right\| _{\infty }}{\left\| f_{n}\right\| _{H^p_\phi (\mathbb D)}}< \sum _{n=0}^{N_r}\frac{\left\| (f_{n})_r\right\| _{\infty }}{\left\| f_{n}\right\| _{H^p_\phi (\mathbb D)}}+ \sum _{n=N_r+1}^{\infty }\frac{2}{e^{n}}<\infty \end{aligned}$$

for some $N_r\in \mathbb {N}_0$. We also have

$$\begin{aligned} \begin{aligned} \left\| g_{\lambda }\right\| ^q_{H^p_\phi (\mathbb D)}&=\sup _{m\in \mathbb {N}_0}\sup _{r_{m}\leqslant r\leqslant r_{m+1}}\left\| (g_{\lambda })_{r}\right\| ^q_{L^p_\phi (\mathbb T)}\\&\leqslant \sup _{m\in \mathbb {N}_0}\biggl (\left| \lambda _m\right| ^q+\sum _{m\ne k\in \mathbb {N}_0}^{\infty }\varepsilon _k^q\left| \lambda _k\right| ^q\biggr )\leqslant (1+\varepsilon )(\left\| \lambda \right\| _{\infty })^q, \end{aligned} \end{aligned}$$

(3.8)

where $q=\min \left\{ 1,p\right\}$. We define a mapping $T:\ell ^{\infty }\rightarrow H^p_\phi (\mathbb D)$ by

$$\begin{aligned} T:\lambda \mapsto g_\lambda . \end{aligned}$$

(3.9)

By the above, T is well defined, linear and bounded with $\left\| T\right\| \leqslant (1+\varepsilon )^{{1}/{q}}$. Moreover, T is bounded below. To see this, we shall bound $\left\| g_\lambda \right\| _{H^p_\phi (\mathbb D)}$ from below by $\left\| \lambda \right\| _{\infty }$ up to a constant. Indeed,

$$\begin{aligned} \left| \lambda _m\right| ^q&= \left\| \lambda _m g_{n_m}\right\| ^q_{H^p_\phi (\mathbb D)}\leqslant \left\| g_\lambda \right\| ^q_{H^p_\phi (\mathbb D)}+\sum _{m\ne k\in \mathbb {N}_0}^{\infty }\varepsilon _k^q\left| \lambda _k\right| ^q,\quad m\in \mathbb {N}_0 \end{aligned}$$

and therefore

$$\begin{aligned} (1-\varepsilon )(\left\| \lambda \right\| _{\infty })^q\leqslant \left\| g_\lambda \right\| ^q_{H^p_\phi (\mathbb D)}. \end{aligned}$$

$\square$

A close inspection of the proof reveals that with the new weight

$$\begin{aligned} \varphi :=\chi _{K}+\sum _{k=0}^{\infty }e^{-c^{n_k}}\chi _{I_{2n_k+1}}+\chi _{I_{2n_k+2}} \quad \text {for}\quad K:=\mathbb {T}\setminus \bigcup _{k=0}^{\infty }(I_{2n_k+1}\cup I_{2n_k+2}) \end{aligned}$$

this procedure can be repeated with previously chosen family of functions $\left\{ g_{n_k}\right\}$.

Since minimal spaces are separable, we actually proved that $H^p_{\phi }(\mathbb {D})$ is not minimal.

Corollary 3.3

If $0<p<\infty$, then the set of analytic polynomials $\mathcal {P}$ is not dense in $H^p_{\phi }(\mathbb {D})$.

Proof

This is a consequence of Theorem 3.2. $\square$

We finish with the following result which follows directly from the proof of Theorem 3.2.

Proposition 3.4

If $0<p<\infty$, then there exists a weight $\phi \in W$ such that the closure of the set $\mathcal {P}$ of all analytic polynomials in $H^p_{\phi }(\mathbb {D})$ contains an isomorphic copy of $c_0$.

Proof

The proof proceeds along the same lines as the proof of Theorem 3.2 and will only be indicated briefly. Fix $0<p<\infty$ and $0<\eta <{1}/{2}$. There is the weight $\phi \in W$ and the mapping $T:\ell ^{\infty }\rightarrow H^p_\phi (\mathbb D)$ satisfying

$$\begin{aligned} (1-\eta )(\left\| \lambda \right\| _{\infty })^q \leqslant \left\| T(\lambda )\right\| ^q_{H^p_\phi (\mathbb D)} \leqslant (1+\eta )(\left\| \lambda \right\| _{\infty })^q \end{aligned}$$

(3.10)

where $q=\min \left\{ 1,p\right\}$ and $\lambda :=\left\{ \lambda _k\right\} _{k\in \mathbb {N}_0}\in \ell ^{\infty }$. Since $T(e_n)\in H^{\infty }(\mathbb {D})$ for each $n\in \mathbb {N}_0$, it follows that there are analytic polynomials $p_n\in \mathcal {P}$, $n\in \mathbb {N}_0$ satisfying

$$\begin{aligned} \left\| T(e_n)-p_n\right\| ^q_{H^p_{\phi }(\mathbb D)}<{\eta }/{2^{n+1}},\quad n\in \mathbb {N}_0. \end{aligned}$$

(3.11)

By the above,

$$\begin{aligned} p_\lambda :=\sum _{n=0}^{\infty }\lambda _n p_{n}\in H^p_\phi (\mathbb D) \end{aligned}$$

for every $\lambda \in c_0$. In a similar fashion we define a mapping $S:c_0\rightarrow H^p_\phi (\mathbb D)$ by

$$\begin{aligned} S:\lambda \mapsto p_\lambda . \end{aligned}$$

(3.12)

Then (3.10) combined with (3.11) gives

$$\begin{aligned} (1-2\eta )(\left\| \lambda \right\| _{\infty })^q \leqslant \left\| S(\lambda )\right\| ^q_{H^p_\phi (\mathbb D)} \leqslant (1+2\eta )(\left\| \lambda \right\| _{\infty })^q, \end{aligned}$$

which completes the proof. $\square$

4 Conclusions

Recall that by Beurling’s theorem (see, e.g., [5, Theorem 7.4]), the analytic polynomials $\mathcal {P}$ are always dense in $H^p_{\phi }(\mathbb {T})$ for every $\phi \in W$. With Lemma 2.3 at hand, Theorem 3.1 gives an example of an H-admissible space which is not R-admissible but yet analytic polynomials are dense in it.

That the weighted Hardy space $H^p_{\phi }(\mathbb {D})$ on the disc is R-admissible follows from the definition and Lemma 2.3. Indeed,

$$\begin{aligned} \left\| f_{\rho }\right\| _{H^p_\phi (\mathbb D)}=\sup _{0\leqslant r< 1}\left\| f_{r\rho }\right\| _{L^p_\phi (\mathbb D)} \leqslant \sup _{0\leqslant r< 1}\left\| f_{r}\right\| _{L^p_\phi (\mathbb D)}=\left\| f\right\| _{H^p_\phi (\mathbb D)},\quad 0\leqslant \rho <1. \end{aligned}$$

On the other hand, Corollary 3.3 shows that there exists an H-admissible space which is R-admissible but not minimal. In conclusion, there is no general relationship between the density of analytic polynomials and the R-admissibility of an H-admissible space.

We would like to thank both reviewers for their comments.