Introduction and preliminaries

The solid hulls and cores of spaces of analytic functions on the unit disc \(\mathbb {D}= \{ z \in \mathbb {C} : |z| < 1 \}\) or the entire plane \(\mathbb {C}\) have been investigated by many authors. We refer the reader to the recent books [12] and [16] and the many references therein. In the series of articles [2,3,4,5], the authors have presented the solid hulls and cores of the weighted \(H^\infty\)-spaces \(H^\infty _v\) on \(\mathbb {D}\) or \(\mathbb {C}\) for a large class of radial weights v as well as their Bergman space analogues \(A_\mu ^p\) for \(1< p < \infty\). Earlier, the cases of standard weights and \(d \mu (r) = (1-r)^{\alpha }dr\), \(\alpha > 0\), were considered in [1] and [12].

In this part, we want to extend the results of [5] to weighted Bergman spaces \(A_{\mu }^p\) for \(p=1\). The spaces are defined on the unit disc \(\mathbb {D}\) or on the entire plane. (Fock spaces are usually considered the Bergman space analogues of spaces of entire functions, but these are defined with Gaussian weight functions, which is not required here. Thus, here we keep the term Bergman space also for the entire functions.) Consider \(R =1\) or \(R = \infty\). We study holomorphic functions \(f: R \cdot \mathbb {D} \rightarrow \mathbb {C}\) where \(R \cdot \mathbb {D} =\mathbb {D}\) if \(R=1\) and \(R \cdot \mathbb {D}= \mathbb {C}\) if \(R = \infty\). Let \(\hat{f}(k)\) be the Taylor coefficients of f, i.e., \(f(z) = \sum _{k=0}^{\infty } \hat{f}(k) z^k\). We take a non-atomic positive bounded Borel measure \(\mu\) on [0, R[ such that \(\mu ([r, R[) > 0\) for every \(r > 0\) and \(\int _0^R r^n d \mu (r) < \infty\) for all \(n> 0\). Put, for \(1 \le p < \infty\),

$$\begin{aligned} \Vert f \Vert _p = \left( \frac{1}{2 \pi }\int _0^R \int _0^{2 \pi }|f(re^{i \varphi })|^p d \varphi d \mu (r) \right) ^{1/p} \end{aligned}$$

and let

$$\begin{aligned} A_{\mu }^p = \{ f: R \cdot \mathbb {D} \rightarrow \mathbb {C} : f \text { holomorphic with } \Vert f \Vert _p < \infty \}. \end{aligned}$$

We will also consider the weighted spaces

$$\begin{aligned} H_v^\infty = \{ f : \mathbb {D} \rightarrow \mathbb {C} \text { holomorphic} : \Vert f \Vert _v = \sup \limits _{z \in \mathbb {D}} v(z) |f(z)| < \infty \} , \end{aligned}$$

where the weight \(v: \mathbb {D} \rightarrow (0, \infty )\) is a continuous and radial (\(v(z) =v(|z|)\)) function which is decreasing with respect to \(r = |z|\), and \(\lim _{r \rightarrow 1^-} v(r) = 0\).

Let A be a vector space of holomorphic functions on \(R \cdot \mathbb {D}\) containing the polynomials. The solid core is defined as

$$\begin{aligned} s(A) = \{ f \in A : g \in A \text { for all holomorphic } g \text { with } |\hat{g}(k)| \le |\hat{f}(k) | \text { for all } k \} \end{aligned}$$

and the solid hull as

$$\begin{aligned} S(A) = \{ g: \mathbb {D} \rightarrow \mathbb {C} \text { holomorphic : there is } f \in A \text { with } |\hat{g}(k)| \le |\hat{f}(k) | \text { for all } k \}. \end{aligned}$$

The space A is called solid if \(A= S(A)\). The concept of a solid hull will also be discussed at the beginning of Section “On solid hulls.”

Here, in Theorem 2.4, we will transfer Theorem 4.1 of [5] to the case \(p=1\). This result concerns the characterization of solid Bergman spaces \(A_{\mu }^1\), and it is motivated by the fact that such spaces indeed exist in the case \(R= \infty\) only (see Example 2.7 and Corollary 2.6). In Theorem 2.8, we determine the solid cores for all Bergman spaces \(A_{\mu }^1\).

In Section “On solid hulls,” we also present how the duality theory can be used for new results on certain solid hulls (see the beginning of Section “On solid hulls” for detailed definitions). In particular, we construct the solid hull \(S_{BK} (A_\mu ^1)\) of \(A_\mu ^1\) for \(R=1\) by using the known solid core of the space \(H^\infty _v\) in [4]. This result is more special than the previous one for solid cores since we need to restrict to the case \(\mu\) is the weighted Lebesgue measure \(d \mu = v dA = v \pi ^{-1} r dr d \varphi\), where the weight v needs to satisfy some special assumptions in addition to those mentioned above. Examples of such weights include important cases like exponentially decreasing weights.

For a holomorphic g and \(r > 0\), we define

$$\begin{aligned} M_p(g,r) = \left( \frac{1}{2 \pi } \int _0^{2 \pi } |g(r e^{i \varphi })|^pd\varphi \right) ^{1/p} \end{aligned}$$

and denote the Dirichlet projections by \(P_n g(z) = \sum _{k=0}^n \hat{g}(k) z^k\), \(n \in \mathbb {N}\). It is well known that, for \(1< p < \infty\), there are constants \(c_p>0\), not depending on g, n or r, such that \(M_p(P_ng,r) \le c_p M_p(g,r)\). Moreover, we have \(\lim _{n \rightarrow \infty } M_p(g - P_ng, r) =0\). Hence, we obtain

$$\begin{aligned} \Vert P_nf \Vert _p \le c_p \Vert f \Vert _p \text { for all } f \in A_{\mu }^p \text { and all } n \text { and } \lim \limits _{n \rightarrow \infty } \Vert f - P_nf \Vert _p =0. \end{aligned}$$

In particular, we see that the monomials \(z \mapsto z^n\), \(n \in \mathbb {N}_0 = \{ 0,1,2, \ldots \} = \mathbb {N} \cup \{0 \}\), form a (Schauder) basis of \(A_{\mu }^p\) if \(1< p < \infty\). On the other hand, denoting by \(H^1\) the Hardy space of all holomorphic functions on \(\mathbb {D}\) which are bounded under \(\sup _{0 \le r <1}M_1(\cdot , r)\), it is well known that the operator norm of \(P_n : H^1 \rightarrow H^1\) tends to infinity as \(n \rightarrow \infty\). (See details in [9] and [17].) For the terminology and definitions on bases in Banach spaces, see also [13].

In the remaining part of the article, [r] denotes the largest integer smaller or equal to \(r>0\). By \(c, c_1, c_2, C, C'\), etc., we denote generic positive constants, the actual value of which may vary depending on the place.

Solid core and examples of solid \(A_{\mu }^1\)-spaces

In this section, we extend Theorem 4.1 of [5] concerning the characterization of solid Bergman spaces to the case \(p=1\) and also determine the solid cores for all spaces \(A_{\mu }^1\). We consider both cases \(R=1\) and \(R= \infty\) unless otherwise specified. First, we recall a fundamental result from [10], which concerns equivalent representations of the norm of the space \(A_{\mu }^1\).

Theorem 2.1

There are sequences \(0< s_1< s_2< \ldots < R\) and \(0=m_0< m_1< m_2 < \ldots\), non-negative numbers \(d_n\), \(t_{n,k}\) (with \(n \in \mathbb {N}\) and \([m_{n-1}]< k \le [m_{n+1}]\) ) and constants \(c_1\), \(c_2 > 0\) such that for all \(g(z) = \sum _{k=0}^{\infty } \alpha _k z^k\) we have

$$\begin{aligned} c_1 \Vert g \Vert _1 \le \sum \limits _{n=0}^{\infty } M_1(T_ng,s_n)d_n \le c_2 \Vert g \Vert _1, \end{aligned}$$
(2.1)

where

$$\begin{aligned} T_n g = \sum \limits _{[m_{n-1}]+1}^{[m_{n+1}]}t_{n,k} \alpha _k z^k. \end{aligned}$$
(2.2)

We will need the following consequence of this result.

Corollary 2.2

Let \((n_j)_{j=1}^\infty\) be an increasing sequence of indices such that \(n_{j+1} - n_j \ge 2\) for all j and let \(h_j(z) = {\sum }_{k= [m_{n_j}]+1}^{[m_{n_j+1}]} \alpha _ k z^k\) be a polynomial. We have

$$\begin{aligned} \Vert h \Vert _1 \le \sum \limits _{j=0}^{\infty } \Vert h_j \Vert _1 \le C \Vert h \Vert _1 \text { for all } h = \sum \limits _{j=1}^\infty h_j \in A_\mu ^1 . \end{aligned}$$
(2.3)

Proof

Applying (2.1) to \(h_j\) yields that \(\Vert h_j \Vert _1\) and

$$\begin{aligned} M_1(T_{n_j} h_j,s_{n_j}) d_{n_j } + M_1(T_{n_j +1 } h_j,s_{n_j +1 })d_{n_j +1} \end{aligned}$$

are proportional quantities. Moreover, \(T_nh = 0\), if n is not equal to \(n_j\) or \(n_j+1\) for any j, and

$$\begin{aligned} T_{n_j} h = T_{n_j} h_j , \ T_{n_j +1 } h = T_{n_j +1 } h_j \ \text {for all } j. \end{aligned}$$
(2.4)

Hence, by another application of (2.1),

$$\begin{aligned}&\Vert h \Vert _1 \le \sum _{j= 0}^\infty \Vert h_j\Vert _1 \le C \sum _{j= 0}^\infty \big ( M_1(T_{n_j} h_j,s_{n_j}) d_{n_j } + M_1(T_{n_j +1 } h_j,s_{n_j +1 })d_{n_j +1} \big ) \nonumber \\= & {} C \sum _{n= 0}^\infty M_1(T_n h,s_n) d_n \le C' \Vert h \Vert _1 . \end{aligned}$$

Let us make a remark concerning the numbers and constants in the above results.

Remark 2.3

\(1^{\circ }.\) Theorem 2.1 is a reformulation of Theorem 1.3 of [10], where the sequences \((s_n)_{n=1}^\infty\) and \((m_n)_{n=0}^\infty\) were chosen, by using induction, such that, for all \(n \in \mathbb {N}\),

$$\begin{aligned} \int _0^{s_n} r^{m_n}d \mu = b \int _{s_n}^Rr^{m_n}d \mu \ \ \ \text { and } \ \ \ \int _0^{s_n} r^{m_{n+1}}d \mu = \int _{s_n}^Rr^{m_{n+1}}d \mu . \end{aligned}$$

where \(b >5\) is some constant. Then, the numbers \(d_n\) were set to be

$$\begin{aligned} d_n= \left( \int _0^{s_n}\left( \frac{ r}{s_n}\right) ^{m_{n}}d \mu + \int _{s_n}^R\left( \frac{r}{s_n}\right) ^{m_{n+1}}d \mu \right) . \end{aligned}$$
(2.5)

As proven in Section 5 of [10], it is always possible to find these sequences, although calculating them exactly for given concrete weights seems to be difficult in general.

\(2^{\circ }\). If \(R=1\) and \(d \mu = r v(r) dr d\theta\) with \(v(r) = \exp \big ( - \alpha (1-r^\ell )^{-\beta }\big )\) for some constants \(\alpha , \beta , \ell > 0\), then the numbers \(m_n\) and \(s_n\), \(n \in \mathbb {N}\), (\(m_0 = 0\)), were calculated by a different method than in \(1^\circ\) in Propositions 3.1 and 3.3.(ii) of [6]:

$$\begin{aligned} m_n = \ell \beta ^2 \Big (\frac{\beta }{\alpha } \Big )^{1/\beta } n^{2 + 2/\beta } - \ell \beta ^2 n^2 \ \ \text { and } \ \ s_n = \Big ( 1 - \Big ( \frac{\alpha }{\beta } \Big )^{1/\beta } n^{- 2/\beta } \Big )^{1/ \ell } . \end{aligned}$$
(2.6)

\(3^{\circ }\). In the citations mentioned in \(1^\circ\) and \(2^\circ\), the numbers \(t_{n,k}\) were chosen as the coefficients of certain de la Valleé Poussin operators, more precisely,

$$\begin{aligned} t_{n,k}= \left\{ \begin{array}{ll} { \frac{k - [m_n] }{[m_n]- [m_{n-1}]} } , &{} \ \text { if } \ m_{n-1}< |k| \le m_n , \\ &{} \\ { \frac{[m_{n+1}] - k }{[m_{n+1}]- [m_n]} }, &{} \ \text { if } \ m_{n} < |k| \le m_{n+1} . \end{array} \right. \end{aligned}$$
(2.7)

Next let us state our result on the characterization of solid \(A_\mu ^1\)-spaces.

Theorem 2.4

The following are equivalent:

  1. (i)

    \(A_{\mu }^1\) is solid,

  2. (ii)

    \(s(A_{\mu }^1) = A_{\mu }^1\),

  3. (iii)

    The monomials \((z^n)_{n=0}^{\infty }\) are an unconditional basis of \(A_{\mu }^1\),

  4. (iv)

    The normalized monomials \((z^n/ \Vert z^n \Vert _1)_{n=0}^{\infty }\) are equivalent to the unit vector basis of \(\ell ^1\),

  5. (v)

    \(\sup \limits _{n \in \mathbb {N}}(m_{n+1}-m_n) < \infty\) for the numbers \(m_n\) in Theorem 2.1.

In the following, we retain the numbers \(m_n\), \(s_n\) of Theorem 2.1 and consider the Dirichlet projections \(P_n\).

Lemma 2.5

Assume that \(\limsup _{n \rightarrow \infty }(m_{n+1}-m_n)= \infty\). Then, for every \(N> 0\), there exists an arbitrarily large \(n \in \mathbb {N}\), an index \(M < m_{n+1}\) and a polynomial \(f(z )= \sum _{k=[m_n]+1}^{[m_{n+1}]} \alpha _k z^k\) with \(\Vert f \Vert _1 \le 1\) but \(\Vert P_{M}f \Vert _1 = \Vert ( P_{M} - P_{[m_n]} ) f \Vert _1 \ge N\).

Proof

Due to the unboundedness of the operator norms of \(P_n\) on \(H^1\) (see Section “Introduction and preliminaries”), we find an index K and a polynomial \(g(z)= \sum _{j=0}^L \beta _j z^j\) with \(M_1(g,1) = 1\) but \(M_1(P_K g,1) > N\). By assumption, we find \(n \in \mathbb {N}\), as large as we wish, such that \(m_{n+1}-m_n > L+1\). Then put

$$\begin{aligned} f(z) = \sum _{k=[m_n]+1}^{[m_n]+L+1}\beta _{k-[m_n]-1} \frac{1}{s_n^k}z^k. \end{aligned}$$

We obtain

$$\begin{aligned} M_1(f,s_n) = M_1(g,1)=1 \ \ \ \text{ and } \ \ \ M_1(P_{m_n+K}f, s_n) = M_1(P_Kg,1) > N. \end{aligned}$$

Put \(M= K+ [m_n]+1\) and use Theorem 2.1 to complete the proof of the lemma. We have \(P_M f = (P_{M} - P_{[m_n]}) f\) just by the choice of f.

Proof

of Theorem 2.4.

\((i) \Leftrightarrow (ii)\): follows from the definition.

\((iv) \Rightarrow (iii) \Rightarrow (ii)\): these are obvious.

\((ii) \Rightarrow (v)\): Assume that \(\limsup _{n \rightarrow \infty } (m_{n+1}-m_n)= \infty\). For every \(j \in \mathbb {N}\) we find, by Lemma 2.5, a polynomial \(f_j \in\) span\(\{z^{[m_{n_j}]+1}, \ldots , z^{[m_{n_{j+1}]}}\}\) for some \(m_{n_j}\) with

$$\begin{aligned} \Vert f_j \Vert _1=2^{-j} \text { and } \big \Vert P_{k_j} f_j \big \Vert _1 \ge 1 \text { for some } k_j \in ( m_{n_j}, m_{n_j +1 } ). \end{aligned}$$
(2.8)

We may assume that \(n_{j+1}- n_j \ge 2\). Put \(f= \sum _j f_j\) and \(g = \sum _j P_{k_j}f_j = \sum _j ( P_{k_j} - P_{[m_{n_j}]} )f_j\). Then, \(f \in A_{\mu }^1\) but in view of (2.8), (2.3) we have \(g \not \in A_{\mu }^1\). Hence \(f \not \in s(A_{\mu }^1)\).

\((v) \Rightarrow (iv):\) Let \(g(z) = \sum _{k=[m_{n-1}]+1}^{[m_{n+1}]} \alpha _kz^k\). By (v) we obtain a constant independent of n, r, and g with

$$\begin{aligned} M_1(g,r) \le \sum \limits _{k=[m_{n-1}]+1}^{[m_{n+1}]} |\alpha _k| r^k \le c M_1(g,r). \end{aligned}$$

Then, (2.1) yields numbers \(\delta _k = t_{n,k}s_n^k d_n\) such that for all functions

$$\begin{aligned} f(z) = \sum \limits _{k=0}^{\infty } \alpha _k z^k \in A_\mu ^1 \end{aligned}$$

we have, with the universal constants \(c_1, c_2\),

$$\begin{aligned} c_1 \Vert f \Vert _1 \le \sum _{k=0}^{\infty } \delta _k |\alpha _k| \le c_2 \Vert f \Vert _1. \end{aligned}$$

This proves (iv).

Corollary 2.6

If \(R < \infty\) then \(A_{\mu }^1\) is never solid.

Proof

It follows from Proposition 2.1. of [10] that in this case we always have \(\limsup _{n \rightarrow \infty }(m_{n+1}-m_n)= \infty\).

Example 2.7

There are indeed examples where \(A_{\mu }^1\) is solid. Let \(R= \infty\) and \(d\mu (r) = \exp (-\log ^2(r))dr\). It was illustrated in [10], Example 2a that here \(\sup _n (m_{n+1}-m_n) < \infty\).

Theorem 2.8

Let \(m_n\), \(s_n\) and \(d_n\) be the numbers of Theorem 2.1. The solid core of \(A_{\mu }^1\) equals

$$\begin{aligned} s(A_{\mu }^1) =\bigg \{&g: R \cdot \mathbb {D} \rightarrow \mathbb {C} : g(z) = \sum _{k=0}^{\infty } \hat{g}(k)z^k \nonumber \\&\text { with } \sum _{n=1}^{\infty } d_n \Big ( \sum _{k=[m_n]+1}^{[m_{n+1}]} |\hat{g}(k)|^2 s_n^{2k} \Big )^{1/2} < \infty \bigg \}. \end{aligned}$$
(2.9)

Proof

For a holomorphic function \(g(z) = \sum _{k=0}^{\infty } \hat{g}(k) z^k\) we write

$$\begin{aligned} g_n(z) = \sum \limits _{k=[m_n]+1}^{[m_{n+1}]} \hat{g}(k)z^k \ \ \text { and } \ \ g_I(z) = \sum \limits _{n=0}^{\infty }g_{2n}(z), \ \ g_{II}(z) =\sum \limits _{n=0}^{\infty } g_{2n+1}(z). \end{aligned}$$

Let us denote by V the function space on the right-hand side of (2.9). Moreover, for all n, let \(\Delta _n = \{ +1, -1 \}^{[m_{n+1}] -[m_n]}\), and for \(\Theta _n = (\theta _{[m_{n}]+1}, \ldots , \theta _{[m_{n+1}]})\in \Delta _n\) put

$$\begin{aligned} g_{\Theta _n}(z) = \sum _{k=[m_n]+1}^{[m_{n+1}]} \theta _k \hat{g}(k) z^k. \end{aligned}$$

First, assume that \(g \in V\). Then \(g_I, g_{II} \in V\). Let f be holomorphic with \(|\hat{f}(k)| \le |\hat{g_I}(k)|\) for all k. By (2.3) and Theorem 2.1

$$\begin{aligned} \Vert f \Vert _1\le & {} \sum \limits _{n=0}^{\infty } \Vert f_{2n} \Vert _1 \le c \sum \limits _{n=0}^{\infty } d_{2n} M_1(f_{2n}, s_{2n})\\\le & {} c \sum \limits _{n=0}^{\infty } d_{2n} M_2(f_{2n}, s_{2n}) \le c \sum \limits _{n=0}^{\infty } d_{2n} M_2(g_{2n}, s_{2n}) < \infty \end{aligned}$$

where \(c >0\) is a universal constant. We also used the definition of the space V in the last step. Hence \(f\in A_{\mu }^1\), in particular \(g_I \in A_{\mu }^1\). We conclude \(g_I \in s(A_{\mu }^1)\). The same proof shows that \(g_{II}\) and hence \(g \in s(A_{\mu }^1)\).

Conversely, let \(g \in s(A_{\mu }^1)\). Then \(g_I, g_{II} \in s(A_{\mu }^1)\). Let \({\tilde{\Theta }}_n \in \Delta _n\) be such that

$$\begin{aligned} a_1\left( \sum \limits _{k=[m_n]+1}^{[m_{n+1}]}|\hat{g}(k)|^2\right) ^{1/2} \le \frac{1}{2^{[m_{n+1}] -[m_n]}} \sum \limits _{\Theta _n \in \Delta _n}M_1(g_{\Theta _n}, s_n) \le M_1(g_{{\tilde{\Theta }}_n} ,s_n). \end{aligned}$$

Here we used the Khintchine inequality (see [18], Ch. V, Thm. 8.4) with the Khintchine constant \(a_1\). Put \(h_I = \sum _{n=0}^{\infty } g_{ {\tilde{\Theta }}_{2n}}\). Then we obtain \(|\hat{h_I}(k)| = |\hat{g_I}(k)|\) for all k. Hence \(h_I \in A_{\mu }^1\). The choice of \({\tilde{\Theta }}_n\) and Theorem 2.1 applied to \(h_I\) yield

$$\begin{aligned}&\sum \limits _{n=0}^{\infty } d_{2n} \Big ( \sum \limits _{k=[m_{2n}]+1}^{[m_{2n+1}]} | \hat{g}(k)|^2 s_{2n}^{2k} \Big )^{1/2} = \sum \limits _{n=0}^{\infty } d_{2n} \Big ( \sum \limits _{k=[m_{2n}]+1}^{[m_{2n+1}]} |\hat{h_I}(k)|^2 s_{2n}^{2k} \Big )^{1/2} \nonumber \\\le & {} \frac{ 1}{a_1}\sum \limits _{n=0}^{\infty } d_{2n} M_1(g_{{\tilde{\Theta }}_{2n}} ,s_{2n}) \le \frac{c_2}{a_1} \Vert h_I \Vert _1 < \infty . \end{aligned}$$

Here, \(c_2\) is the constant of Theorem 2.1. We conclude \(g_I \in V\), and similarly we see that \(g_{II} \in V\). Hence \(g \in V\), which implies \(V = s(A_{\mu }^1)\).

On solid hulls

In this section, we assume \(R=1\). We start off with the remark that in addition to the definition of a solid hull (see Section “Introduction and preliminaries”), there exist two other priori different definitions in the literature: in [1], the solid hull \(S_{\text {vect}}(X)\) of a space X of analytic functions on \(\mathbb {D}\) is defined as the intersection of all solid vector spaces of analytic functions on \(\mathbb {D}\). Obviously, S(X) is a vector space if and only if for every \(f,g \in X\) there is \(h \in X\) such that the Taylor coefficients satisfy \(|\hat{f}(k)|+|\hat{g}(k)| \le |\hat{h}(k)|\) for all k.

Another variant appears in the theory of the so-called BK-spaces. By definition, a BK-space is a vector space of complex sequences \(f= (f_k)_{k=0}^\infty\) endowed with a norm which makes it into a Banach space, such that the coordinate functionals become bounded operators. In the theory of BK-spaces (see [8]), the solid hull \(S_{BK}(X)\) of a BK-space X is defined as the intersection of all solid BK-spaces containing X. By using the Taylor coefficients, we consider Banach spaces of analytic functions on \(\mathbb {D}\) as BK-spaces, and, in particular, we will characterize in the sequel the solid hull \(S_{BK}(A_\mu ^1)\) although we will avoid using the terminology of BK-spaces, except for the proof of Proposition 3.1. It is obvious that

$$\begin{aligned} S(X) \subset S_{\text {vect}}(X) \subset S_{BK} (X) \end{aligned}$$
(3.1)

for a BK-space X as above. All results on solid hulls S(X) in the literature known to the authors are vector spaces which can be endowed with norms making them into solid BK-spaces. Thus, in all of these cases, one actually has \(S(X) = S_{BK}(X)\).

Our aim is to use the known duality relations between the weighted \(A^1\) and \(H^\infty\)-spaces and existing results of the solid core of \(H_v^\infty\) in order to find the solid hull \(S_{BK} ( A_\mu ^1)\). We focus on the case where the measure \(\mu\) is the weighted Lebesgue measure vdA with a radial weight v making the Bergman space into a “large” one: the admissible weights include the exponentially decreasing weights (see Example 3.3 below).

We start with some general considerations.

Given a sequence \(\theta = (\theta _k)_{k=0}^\infty\) with \(|\theta _k|\le 1\) for all k, we denote by \(M_\theta\) the operator \(M_\theta \sum _{k=0}^\infty \hat{f}(k) z^k = \sum _{k=0}^\infty \theta _ k \hat{f}(k) z^k\). We will need to consider analytic function spaces on \(\mathbb {D}\) such that the norm of the space satisfies

$$\begin{aligned} \Vert M_\theta f \Vert \le \Vert f \Vert \end{aligned}$$
(3.2)

for all \(f = \sum _{k=0}^\infty \hat{f}(k) z^k\in X\) and all sequences \(\theta = (\theta _k)_{k=0}^\infty\) with \(|\theta _k|\le 1\) for all k.

The following result is essentially known.

Proposition 3.1

If \(( X, \Vert \cdot \Vert _X )\) is a Banach space of analytic functions on the unit disc \(\mathbb {D}\) such that all coordinate functionals \(f \mapsto \hat{f}(k)\) are bounded operators, then its solid hull \(S_{BK}(X)\) can be endowed with a norm \(\Vert \cdot \Vert _S\) such that

  1. (i)

    the embedding \(X \hookrightarrow S_{BK}(X)\) is continuous,

  2. (ii)

    the norm \(\Vert \cdot \Vert _S\) satisfies (3.2),

  3. (iii)

    if \(p : S_{BK}(X) \rightarrow \mathbb {R}_0^+\) is any norm with (3.2) such that \(p(f) \le \Vert f \Vert _X\) for all \(f \in X\), then \(p(f) \le C \Vert f \Vert _S\) for a constant \(C > 0\) and all \(f \in S_{BK}(X)\),

  4. (iv)

    the normed space \(\big ( S_{BK}(X), \Vert \cdot \Vert _S \big )\) is complete, and

  5. (v)

    if the subspace of polynomials \(\mathcal {P}\) is dense in X, then it is dense in \(\big ( S_{BK}(X), \Vert \cdot \Vert _S \big )\), too.

Proof

Let us explain the way the claims follow from the theory of BK-spaces; see [7, 8]. For the sake of the simplicity of the notation, let us consider X as a BK-sequence space in the following, that we can do by assumption. We denote by \(y \cdot f\) the coordinatewise product of two complex sequences y and f. The space \(\ell ^\infty \widehat{\otimes }X\) is defined in [7] to consist of sequences \(g= (g_k)_{k=0}^\infty\) having a coordinatewise convergent representation

$$\begin{aligned} g = \sum \limits _{j=1}^\infty y^{(j)} \cdot f^{(j)} \ \ \ \text { with } y^{(j)} = \big ( y^{(j)}_k \big )_{k=0}^\infty \in \ell ^\infty , f^{(j)} = \big ( f^{(j)}_k \big )_{k=0}^\infty \in X \ \forall \,j \end{aligned}$$
(3.3)

such that

$$\begin{aligned} \sum \limits _{j=1}^\infty \Vert y^{(j)} \Vert _{\ell ^\infty } \Vert f^{(j)} \Vert _X < \infty . \end{aligned}$$
(3.4)

The norm \(\Vert \cdot \Vert _S\) of \(g \in \ell ^\infty \widehat{\otimes }X\) is defined by taking the infimum of the quantity (3.4) over all possible representations (3.3) of g. Theorem 3 of [7] yields that the resulting space is complete, and Theorem 8 of [8] says that \(\ell ^\infty \widehat{\otimes }X\) equals the solid hull \(S_{BK}(X)\). The completeness of the space is included in the same reference; hence, property (iv) holds.

If \(f \in X\), then we have \(\mathrm{e} \cdot f = f\), where \(\mathrm{e} = (1,1,1,\dots ) \in \ell ^\infty\), and in view of the previous definition of the norm \(\Vert \cdot \Vert _S\), this implies that \(\Vert f \Vert _S \le \Vert f \Vert _X\) for all \(f \in X\) so that the embedding of X into \(\big ( S_{BK}(X), \Vert \cdot \Vert _S \big )\) is continuous.

Also, if \(g \in S_{BK}(X)\) has a representation (3.3) and \(\theta\) is given as in (3.2), then \(M_\theta g\) has a coordinatewise convergent representation

$$\begin{aligned} M_\theta g = \sum \limits _{j=1}^\infty ( M_\theta y^{(j)} ) \cdot f^{(j)} , \end{aligned}$$
(3.5)

and property (ii) follows from the definition of \(\Vert \cdot \Vert _{BK}\).

In the proof of Theorem 3 of [7], it is shown that if p is the norm of any BK-space containing \(S_{BK}(X)\), then there exists \(C> 0\) such that

$$\begin{aligned} p(y \cdot f) \le C \Vert y \Vert _\infty \, \Vert f \Vert _X \end{aligned}$$

for all \(y \in \ell ^\infty\), \(f \in X\). This implies

$$\begin{aligned} p\Big ( \sum _{j=1}^\infty y^{(j)} \cdot f^{(j)} \Big ) \le C \sum _{j=1}^\infty \Vert y^{(j)} \Vert _\infty \, \Vert f^{(j)} \Vert _X \ \ \text {for all} \ g = \sum _{j=1}^\infty y^{(j)} \cdot f^{(j)} \in \ell ^\infty \widehat{\otimes }X, \end{aligned}$$

and property (iii) follows from the definition of \(\Vert \cdot \Vert _S\). Finally, as for property (v), it follows from Theorem 2 of [7] that finite linear combinations of functions \(y \cdot f\), \(y \in \ell ^\infty\), \(f \in X\), form a dense subspace of \(\ell ^\infty \widehat{\otimes }X = S(X)\). If y, f, and \(\varepsilon > 0\) are given, we use the assumption in (v) to find a polynomial h such that \(\Vert f- h\Vert _X < \varepsilon / (1 + \Vert y \Vert _\infty )\). Then, \(y \cdot h\) is a polynomial, which satisfies

$$\begin{aligned} p( y \cdot f - y \cdot h) = p( y \cdot ( f-h)) \le \Vert y \Vert _\infty \Vert f-h \Vert _X \le \varepsilon . \end{aligned}$$

Property (v) follows from these arguments.

Lemma 3.2

Let X be a Banach space of analytic functions on the unit disc \(\mathbb {D}\) such that the subspace \(\mathcal {P}\) of polynomials is dense in X, and let w be a radial weight function on \(\mathbb {D}\). Let Y be the space of all analytic functions g on the disc such that

$$\begin{aligned} \sup \limits _{f \in B_X} |\langle f , g \rangle | < \infty , \ \ \ \text { where } \ \ \ \ \langle f , g \rangle = \int \limits _\mathbb {D} f \overline{g} w dA \end{aligned}$$
(3.6)

and \(B_X\) denotes the unit ball of X. If X is solid and there exists a constant \(C > 0\) such that

$$\begin{aligned} \Vert M_\theta f \Vert _X \le C \Vert f \Vert _X \end{aligned}$$
(3.7)

for all numerical sequences \(\theta = (\theta _k)_{k=0}^\infty\) with \(|\theta _k| \le 1\), then Y is solid, too.

We point out that given the Banach space X as in the assumption, it is not in general known whether its dual space has a representation as a space of analytic functions with dual norm coming from (3.6).

Proof

If \(g = \sum _{k=0}^\infty \hat{g}(k) z^k \in Y\) and \(\theta\) is as above, then for \(M_ \theta g\) we have by (3.7)

$$\begin{aligned}&\sup \limits _{f \in B_X} |\langle f , M_\theta g \rangle | = \sup \limits _{f \in B_X} \sum \limits _{k=0}^\infty \bar{\theta }_k \hat{f}(k) \overline{ \hat{g}(k)} \int \limits _0^1 r^{2k+1} w(r)dr \nonumber \\= & {} \sup \limits _{f \in B_X} |\langle M_{\bar{\theta }} f , g \rangle | \le \sup \limits _{{\mathop {\Vert f \Vert _X \le C }\limits ^{ f \in X }} } |\langle f , g \rangle | < \infty . \end{aligned}$$
(3.8)

Thus, \(M_\theta g \in Y\).

Next we recall an elementary fact concerning Banach sequence spaces. Assume that the sequences \((\beta _k)_{k=0}^\infty\) and \((\gamma _k)_{k=0 }^\infty\) of positive numbers are given and \(\alpha _k = \gamma _k \beta _k^{-1}\) for all k. Let also \((\mu _n)_{n=0}^\infty\) be an increasing, unbounded sequence of non-negative numbers; denote \(\mu _{-1} = -1\) and let

$$\begin{aligned} A= & {} \big \{ a = (a_k)_{k=0}^\infty \, : \, \Vert a \Vert _A = \sum \limits _{n \in \mathbb {N}} \max _{\mu _{n-1}< k \le \mu _n } \alpha _k |a_k| < \infty \big \}, \end{aligned}$$
(3.9)
$$\begin{aligned} B= & {} \big \{ b = (b_k)_{k=0}^\infty \, : \, \Vert b \Vert _B = \sup \limits _{n \in \mathbb {N}} \sum _{\mu _{n-1}< k \le \mu _n } \beta _k |b_k| < \infty \big \} . \end{aligned}$$
(3.10)

Then, B is the dual of A with respect to the dual pairing

$$\begin{aligned} \langle a, b \rangle = \sum \limits _{k=0}^\infty \gamma _k b_k \overline{a_k}, \ \ \ \text {where} \ a=(a_k)_{k=0}^\infty \in A , \ b=(b_k)_{k=0}^\infty \in B . \end{aligned}$$
(3.11)

From now on, we consider radial weights \(v : \mathbb {D} \rightarrow \mathbb {R}^+\) satisfying two following assumptions.

(I) We have

$$\begin{aligned} v(z) = \exp ( - \varphi (z) ) , \end{aligned}$$
(3.12)

where \(\varphi\) belongs to the class \(\mathcal {W}_0\) of [11].

We will not need a detailed definition of \(\mathcal {W}_0\), but recall that \(\varphi \in \mathcal {W}_0\), if it is a twice continuously differentiable real valued function with \(\Delta \varphi > 0\) on \(\mathbb {D}\) and there exists a function \(\rho : \mathbb {D} \rightarrow \mathbb {R}\) and a constant \(C >0\) such that

$$\begin{aligned} \frac{1}{C}\rho (z) \le \frac{1}{\sqrt{\Delta \varphi (z)}} \le C \rho (z) \ \ \forall \, z \in \mathbb {D} ; \end{aligned}$$
(3.13)

the function \(\rho\) must also satisfy the Hölder-property

$$\begin{aligned} \sup \limits _{z,w \in \mathbb {D}, z\not =w} \frac{|\rho (z) - \rho (w)|}{|z-w|} < \infty \end{aligned}$$
(3.14)

as well as the Lipschitz property

$$\begin{aligned} \forall \, \varepsilon > 0 \, \exists \, \text { compact }\,E \subset \mathbb {D}: \ |\rho (z) - \rho (w) | \le \varepsilon |z-w| \ \forall \, z,w \in \mathbb {D} \setminus E. \end{aligned}$$
(3.15)

For more details, see [11]. Note that the considerations in [11] are not restricted to radial weights, contrary to our situation.

According to [11], Theorem 4.3, if the weight v satisfies the condition (I), then the space \(H_v^\infty\) is the dual of \(A^1_v\) with respect to the dual pairing

$$\begin{aligned} \langle f , g \rangle = \int \limits _\mathbb {D} f \overline{g} v^2 dA . \end{aligned}$$
(3.16)

The second requirement is the following:

(II) The weight v satisfies the condition (b) of [3, 4].

Recall that the weight v satisfies the condition (b) if there exist numbers \(b> 2\), \(K > b\) and \(0< \mu _1< \mu _2 < \ldots\) with \(\lim _{n \rightarrow \infty } \mu _n = \infty\) such that

$$\begin{aligned} b \le \left( \frac{r_{\mu _n}}{r_{\mu _{n+1}}} \right) ^{\mu _n} \frac{v(r_{\mu _n})}{v(r_{\mu _{n+1}})}, \left( \frac{r_{\mu _{n+1}}}{r_{\mu _{n}}} \right) ^{\mu _{n+1}} \frac{v(r_{\mu _{n+1}})}{v(r_{\mu _{n}})} \le K , \end{aligned}$$
(3.17)

where \(r_m \in ]0,1[\) denotes the global maximum point of the function \(r^m v(r)\) for any \(m > 0\). Theorem 2.4 of [4] states that the solid core of the space \(H_v^\infty\) equals

$$\begin{aligned}&s(H_v^\infty ) = \Big \{ (b_k)_{k=0}^\infty \, : \, \Vert b \Vert _{v,s} = \sup \limits _{n \in \mathbb {N}} v(r_{\mu _n}) \sum _{\mu _n< k \le \mu _{n+1}} |b_k| \sigma _k < \infty \Big \}, \end{aligned}$$
(3.18)

where we denote \(\sigma _k = r_{\mu _n}^k\). Let us define for every \(k \in \mathbb {N}_0\) the number

$$\begin{aligned} S_k = \frac{\int \limits _0^1 r^{2k+1} v(r)^2dr}{v(r_{\mu _n}) \sigma _k} \ , \end{aligned}$$
(3.19)

where n is the unique number such that \(\mu _n < k \le \mu _{n+1}\).

Example 3.3

According to [4], all weights \(v(r) = \exp \big ( - \alpha / (1-r^2)^\beta \big )\) with \(\alpha , \beta >0\), satisfy the condition (b), and it is easy to see that they also satisfy the assumption (I).

Theorem 3.4

Let the weight v satisfy the assumptions (I) and (II). Then, we have

$$\begin{aligned}&S_{BK}(A_{\mu }^1) = \Big \{ b = (b_k)_{k=0}^\infty \, : \, \Vert b \Vert _{\mu ,S} = \sum \limits _{n=0}^{\infty } \sup _{\mu _n< k \le \mu _{n+1}} |b_k| S_k < \infty \Big \}, \end{aligned}$$
(3.20)

and the norm \(\Vert \cdot \Vert _S\) given by Proposition 3.1 is equivalent with \(\Vert \cdot \Vert _{\mu , S}\).

Proof

Let the solid hull \(S_{BK}(A_\mu ^1)\) be endowed with the norm \(\Vert \cdot \Vert _S\) of Proposition 3.1, and let us denote the Banach space on the right-hand side of (3.20) by Z.

We note that by the duality relations explained above (see (3.18) for the definition of \(\Vert \cdot \Vert _{v,s}\)), we have for all \(f \in A_\mu ^1\)

$$\begin{aligned} \Vert f \Vert _1 = \sup \limits _{{\mathop { \Vert g \Vert _{H_v^\infty } \le 1}\limits ^{g \in H_v^\infty }}} |\langle f , g \rangle | \ \ \text { and } \ \ \Vert f \Vert _{\mu ,S} = \sup \limits _{{\mathop { \Vert g \Vert _{v,s} \le 1}\limits ^{g \in s(H_v^\infty )}}} |\langle f , g \rangle | . \end{aligned}$$
(3.21)

It is proved in [4], Eq. (2.4) and at the very end of the proof of Theorem 2.4, that \(\Vert g \Vert _{H_v^\infty } \le C \Vert g \Vert _{v,s}\) for \(g \in s(H_v^\infty )\). Therefore \(\Vert f \Vert _1 \ge C \Vert f \Vert _{\mu ,S}\) for all \(f \in A_\mu ^1\). This implies in particular that \(A_\mu ^1 \subset Z\). Clearly, Z is a solid Banach space and the coordinate functionals are continuous; thus, it contains the space \(S_{BK}(A_\mu ^1)\). Moreover, we obtain \(\Vert f \Vert _S \ge C \Vert f \Vert _{\mu ,S}\) for \(f \in S_{BK}(A_\mu ^1)\) from Proposition 3.1.(iii).

We show that the norms \(\Vert \cdot \Vert _{\mu ,S}\) and \(\Vert \cdot \Vert _S\) are equivalent in \(S_{BK}(A_\mu ^1)\). For this purpose, we prove that \(C \Vert f \Vert _{\mu ,S} \ge \Vert f \Vert _S\). Note that the space (3.18) is the dual space of Z in the dual pairing (3.16). Indeed, if \(f=\sum _k \hat{f}(k) z^k\) and \(g = \sum _k \hat{g}(k) z^k\) are polynomials, then, by a direct calculation,

$$\begin{aligned} \langle f, g \rangle = \sum \limits _{k=0}^\infty \hat{f}(k) \overline{\hat{g}(k)} \int \limits _0^1 r^{2k +1} v(r)^2 dr . \end{aligned}$$
(3.22)

The result follows from (3.9)–(3.10), in addition to the definitions (3.16)–(3.20).

Let us suppose that by antithesis \(\Vert \cdot \Vert _{\mu ,S}\) and \(\Vert \cdot \Vert _S\) are non-equivalent norms so that we can find a sequence \((f_n)_{n=1}^\infty \subset S_{BK}(A_\mu ^1)\) such that

$$\begin{aligned} \Vert f_n \Vert _{\mu ,S} \le 2^{-n} \Vert f_n \Vert _S \ \ \text {and} \ \ \Vert f_n \Vert _S = 1 \ \forall \, n \in \mathbb {N}. \end{aligned}$$
(3.23)

By property (v) in Proposition 3.1, we can assume that \(f_n\)’s are polynomials. We claim that it is possible to find polynomials \(\tilde{f}_n\), \(n \in \mathbb {N}\), with property (3.23) such that they have distinct degrees, more precisely

$$\begin{aligned} \tilde{f}_n(z) = \sum _{k=K_n}^{K_{n+1} -1 } \hat{f} (n,k) z^k , \ \ n \in \mathbb {N}, \end{aligned}$$
(3.24)

for some unbounded sequence \(0 = K_0< K_1 < \ldots\) and some \(\hat{f} (n,k) \in \mathbb {C}\). Assume that \(N \in \mathbb {N}\) and that such polynomials \(\tilde{f}_n\) have been found for \(n \le N\), and let \(M \in \mathbb {N}\) be the highest degree of these polynomials. Since \(\mathcal {P}_M\) (the \(M+1\)-dimensional space of polynomials of degree at most M) is finite dimensional, all norms are equivalent there and we thus find a constant \(K = K(M) > 0\) such that

$$\begin{aligned} \Vert f \Vert _S \le K \Vert f \Vert _{\mu ,S} \end{aligned}$$
(3.25)

for all \(f \in \mathcal {P}_M\). We pick up the polynomial \(f_L\) as in (3.23) with \(L = M+K\) and write \(f_1 = P_M f_L\), \(f_2 = f_L- f_1\), where \(P_M\) is the Mth Dirichlet projection from \(S_{BK}(A_\mu ^1)\) onto \(\mathcal {P}_M\) (see Section Introduction and preliminaries). Then, we have \(\Vert f_2\Vert _S \ge \frac{1}{2} \Vert f_L \Vert _S\), since otherwise we get by (3.25) and the triangle inequality

$$\begin{aligned} \Vert f_L \Vert _{\mu ,S} \ge \Vert f_1\Vert _{\mu ,S} \ge \frac{1}{K} \Vert f_1\Vert _S \ge \frac{1}{2K} \Vert f_L \Vert _S > \frac{1}{2L} \Vert f_L \Vert _S \end{aligned}$$

which contradicts (3.23). Now we get

$$\begin{aligned} \Vert f_2 \Vert _{\mu ,S} \le \Vert f_L \Vert _{\mu ,S} \le 2^{-L} \Vert f_L \Vert _S \le 2^{-L + 1 } \Vert f_2 \Vert _S. \end{aligned}$$
(3.26)

Taking \(f_2 \Vert f_2\Vert _S^{-1}\) for \(\tilde{f}_{N+1}\), the claim is proved.

Finally, for every n, we set

$$\begin{aligned} T_n := \Big (P^{(n)} \big ( S_{BK}(A_\mu ^1) \big ), \Vert \cdot \Vert _S \Big ) \ \ \text{ with } \ \ P^{(n)} = P_{K_{n+1}-1} - P_{K_n} \end{aligned}$$
(3.27)

and then, using the Hahn-Banach theorem, pick up a polynomial

$$\begin{aligned} g_n = \sum \limits _{k=K_n}^{K_{n+1} -1 } \hat{g} (n,k) z^k \end{aligned}$$

which defines a bounded functional on \((T_n, \Vert \cdot \Vert _S)\) with respect to the dual pairing (3.22), such that

$$\begin{aligned} \langle \tilde{f}_n , g_n \rangle = 1 , \ \ \Vert g_n \Vert _{n,*} := \sup \limits _{{\mathop { \Vert f \Vert _S \le 1}\limits ^{f \in T_n}}} |\langle f , g_n \rangle |= 1 \end{aligned}$$
(3.28)

Then, we observe that \(g_n\) extends via (3.22) to a functional on \(S_{BK}(A_\mu ^1)=: S\) such that

$$\begin{aligned} \sup \limits _{{\mathop { \Vert f \Vert _S \le 1}\limits ^{f \in S}}} |\langle f , g_n \rangle |= \sup \limits _{{\mathop { \Vert f \Vert _S \le 1}\limits ^{f \in S}}} |\langle P^{(n)} f , g_n \rangle |= \sup \limits _{{\mathop { \Vert f \Vert _S \le 1}\limits ^{f \in T_n}}} |\langle f , g_n \rangle | = 1, \end{aligned}$$
(3.29)

since the norm \(\Vert \cdot \Vert _S\) of \(S_{BK}(A_\mu ^1)\) satisfies (ii) of Proposition 3.1 and thus \(\Vert P^{(n)} f \Vert _S \le \Vert f \Vert _S\) for all \(f \in S_{BK}(A_\mu ^1)\). Consequently,

$$\begin{aligned} g = \sum \limits _{n \in \mathbb {N}} \frac{1}{n^2} g_n \end{aligned}$$
(3.30)

is an analytic function which also is a bounded functional on \(\big ( S_{BK}(A_\mu ^1), \Vert \cdot \Vert _S \big )\) in the dual pairing (3.22). However, g is not a bounded functional on Z, since

$$\begin{aligned} \langle 2^n \tilde{f}_n , g \rangle = \frac{2^n}{n^2} \langle \tilde{f}_n , g_n \rangle = \frac{2^n}{n^2} \end{aligned}$$
(3.31)

and by (3.23), the Z-norm \(\Vert 2^n \tilde{f}_n \Vert _{\mu ,S}\) is still at most 1.

The space Y of all analytic functions on \(\mathbb {D}\), which also are bounded functionals on \(\big (S_{BK}(A_\mu ^1), \Vert \cdot \Vert _S \big )\) in the dual pairing (3.22), equals the space Y in Lemma 3.2, when \(X:= S_{BK} (A_\mu ^1)\). Hence, Y is solid. Due to the characterization of \(H_v^\infty\) as the dual of \(A_v^1\), see (3.16), we also have \(Y \subset H_v^\infty\). On the other hand, at the beginning of the proof, we observed that the solid core \(s(H_v^\infty )\) (see (3.18)) equals the dual of Z in the pairing (3.22). The properties of the function g, (3.30), show that \(s(H_v^\infty ) \subsetneq Y\), which contradicts the definition of a solid core. We conclude that \(C \Vert f \Vert _{\mu ,S} \ge \Vert f \Vert _S\) for all \(f \in S_{BK}(A_\mu ^1)\).

We come to the conclusion that the norms \(\Vert \cdot \Vert _S\) and \(\Vert \cdot \Vert _{\mu ,S}\) are equivalent; hence, the spaces \(S_{BK}(A_\mu ^1)\) and Z coincide, since they both are complete.