Abstract
We consider weighted Bergman spaces \(A_\mu ^1\) on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of \(A_\mu ^1\). Also, as a consequence of a characterization of solid \(A_\mu ^1\)-spaces, we show that, in the case of entire functions, there indeed exist solid \(A_\mu ^1\)-spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of \(A_\mu ^1\), when \(\mu\) equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted \(A^1\)- and \(H^\infty\)-spaces, the validity of which requires the assumption that \(-\log v\) belongs to the class \(\mathcal {W}_0\), studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight \(v(z) = \exp ( -1 /(1-|z|)\) provides an example satisfying both assumptions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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 \(1 \le p < \infty\),
and let
We will also consider the weighted spaces
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
and the solid hull as
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
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
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
where
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
Proof
Applying (2.1) to \(h_j\) yields that \(\Vert h_j \Vert _1\) and
are proportional quantities. Moreover, \(T_nh = 0\), if n is not equal to \(n_j\) or \(n_j+1\) for any j, and
Hence, by another application of (2.1),
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 \(n \in \mathbb {N}\),
where \(b >5\) is some constant. Then, the numbers \(d_n\) were set to be
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]:
\(3^{\circ }\). 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,
Next let us state our result on the characterization of solid \(A_\mu ^1\)-spaces.
Theorem 2.4
The following are equivalent:
-
(i)
\(A_{\mu }^1\) is solid,
-
(ii)
\(s(A_{\mu }^1) = A_{\mu }^1\),
-
(iii)
The monomials \((z^n)_{n=0}^{\infty }\) are an unconditional basis of \(A_{\mu }^1\),
-
(iv)
The normalized monomials \((z^n/ \Vert z^n \Vert _1)_{n=0}^{\infty }\) are equivalent to the unit vector basis of \(\ell ^1\),
-
(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
We obtain
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
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
Then, (2.1) yields numbers \(\delta _k = t_{n,k}s_n^k d_n\) such that for all functions
we have, with the universal constants \(c_1, c_2\),
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
Proof
For a holomorphic function \(g(z) = \sum _{k=0}^{\infty } \hat{g}(k) z^k\) we write
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
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
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
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
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
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
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
-
(i)
the embedding \(X \hookrightarrow S_{BK}(X)\) is continuous,
-
(ii)
the norm \(\Vert \cdot \Vert _S\) satisfies (3.2),
-
(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 \(f \in S_{BK}(X)\),
-
(iv)
the normed space \(\big ( S_{BK}(X), \Vert \cdot \Vert _S \big )\) is complete, and
-
(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
such that
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
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
for all \(y \in \ell ^\infty\), \(f \in X\). This implies
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
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
and \(B_X\) denotes the unit ball of X. If X is solid and there exists a constant \(C > 0\) such that
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)
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
Then, B is the dual of A with respect to the dual pairing
From now on, we consider radial weights \(v : \mathbb {D} \rightarrow \mathbb {R}^+\) satisfying two following assumptions.
(I) We have
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
the function \(\rho\) must also satisfy the Hölder-property
as well as the Lipschitz property
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
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
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
where we denote \(\sigma _k = r_{\mu _n}^k\). Let us define for every \(k \in \mathbb {N}_0\) the number
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
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 \(f \in A_\mu ^1\)
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,
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
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
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
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
which contradicts (3.23). Now we get
Taking \(f_2 \Vert f_2\Vert _S^{-1}\) for \(\tilde{f}_{N+1}\), the claim is proved.
Finally, for every n, we set
and then, using the Hahn-Banach theorem, pick up a polynomial
which defines a bounded functional on \((T_n, \Vert \cdot \Vert _S)\) with respect to the dual pairing (3.22), such that
Then, we observe that \(g_n\) extends via (3.22) to a functional on \(S_{BK}(A_\mu ^1)=: S\) such that
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,
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
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.
Data availability
n/a.
References
J. M. Anderson and A. L. Shields, Coefficient multipliers of Bloch functions, Trans. Amer. Math. Soc. 224 (1976), 255–265.
J. Bonet and J. Taskinen, Solid hulls of weighted Banach spaces of entire functions, Rev. Mat. Iberoam. 34 (2018), no. 2, 593–608
J. Bonet, J. Taskinen: Solid hulls of weighted Banach spaces of analytic functions on the unit disc with exponential weights. Ann. Acan. Sci. Fenn. Math. 43 (2018), 521–530.
J. Bonet, W. Lusky, J. Taskinen: Solid hulls and cores of weighted \(H^\infty\)-spaces. Rev. Mat. Complutense 31 (2018), 781–804.
J. Bonet, W. Lusky, J. Taskinen, Solid cores and solid hulls of weighted Bergman spaces, Banach J. Math. Anal. 13 (2019), 468–485,
J. Bonet, W. Lusky, J. Taskinen, Unbounded Bergman projections on weighted spaces with respect to exponential weights, to appear in Integral Eq. Operator Th.
M. Buntinas, Products of sequence spaces, Analysis 7 (1987), 293–304.
M. Buntinas, N. Tanović-Miller, Absolute Boundedness and Absolute Convergence in Sequence Spaces, Proc. Amer. Math. Soc. 111 (1991), No. 4, 967–979.
P.L. Duren, Theory of \(H_p\)-spaces, Academic Press, New York and London, 1970.
A. Harutyunyan, W.Lusky, On \(L_1-\)subspaces of holomorphic functions, Studia Math. 198 (2010), 157–175
Z. Hu, X. Lv, A. Schuster, Bergman spaces with exponential weights, J. Functional Anal. 276 (2019), 1402–1429.
M. Jevtić, D. Vukotić, M. Arsenović, Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces, RSME Springer Series, Volume 2. Springer 2016.
J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I, Springer, Berlin, 1986.
W. Lusky, On the Fourier series of unbounded harmonic functions, J. Lond. Math. Soc. (2) 61 (2000), 568-580.
W. Lusky, On the isomorphism classes of weighted spaces of harmonic and holomorphic functions, Studia Math. 175 (2006), 19–45.
M. Pavlović, Function classes on the unit disc. An introduction, De Gruyter Studies in Mathematics, 52. De Gruyter, Berlin, 2014.
P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. press, Cambridge, 1991.
A. Zygmund, Trigonometric series, 2nd rev. ed., Cambridge Univ. Press, New York, 1959.
Funding
Open Access funding provided by University of Helsinki including Helsinki University Central Hospital. The research of Bonet was partially supported by the project MCIN PID2020-119457GB-I00/AEI/10.13039/501100011033.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Ethics approval
The research has been ethically conducted.
Informed consent
The research does not involve human participants and/or animals.
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bonet, J., Lusky, W. & Taskinen, J. ON SOLID CORES AND HULLS OF WEIGHTED BERGMAN SPACES \(A_{\mu }^1\). J Math Sci 266, 239–250 (2022). https://doi.org/10.1007/s10958-022-05764-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-05764-5