Abstract
Let \(B_N\) be the Euclidean ball of \({\mathbb {C}}^N\). The space \(H^\infty (B_N)\) of bounded holomorphic functions on \(B_N\) is known to have a predual, denoted by \(G^\infty (B_N)\). We study the functions in \(H^\infty (B_N)\) that attain their norm as elements of the dual of \(G^\infty (B_N)\). We also examine similar questions for the polydisc algebra \(H^\infty ({\mathbb {D}}^N)\) and for the space of Dirichlet series \( {\mathcal {D}}^\infty ({\mathbb {C}}_+).\)
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1 Introduction
Ando [1] proved that the Banach space \(H^\infty ({\mathbb {D}})\) of bounded holomorphic functions on the unit disc \({\mathbb {D}}\) has a unique isometric predual. Let us denote it by \(G^\infty ({\mathbb {D}})\). By the Bishop-Phelps theorem, the set \(NA(G^\infty ({\mathbb {D}}))\) of functions \(f \in H^\infty ({\mathbb {D}})\) which attain their norm as elements of the dual of \(G^\infty ({\mathbb {D}})\) is a norm-dense subset of \(H^\infty ({\mathbb {D}})\). Fisher [6] showed that \(f \in H^\infty ({\mathbb {D}}), ||f||=1,\) attains its norm as an element of the dual of \(G^\infty ({\mathbb {D}})\) if and only if the radial limits \(f^*(w)\) of f in the torus \({\mathbb {T}}\) satisfy that the set \(\{w\in {\mathbb {T}}:\, |f^*(w)|=1\}\) has positive Lebesgue measure on \({\mathbb {T}}\). The aim of this article is to investigate versions of Fisher’s result for the Banach space of bounded holomorphic functions on the N-dimensional ball and the N-dimensional polydisc. Our main results are Theorems 5 and 8 and Propositions 6 and 7 in the case of the ball. The case of the polydisc is treated in Sect. 3. The final section deals with the Banach space of bounded Dirichlet series.
Let X be a complex Banach space. Its open unit ball is denoted by \(B_X\) and its closed unit ball by \(U_X\). The space of all holomorphic functions on \(B_X\) (i.e. the \({\mathbb {C}}-\)Fréchet differentiable functions \(f:B_X \rightarrow {\mathbb {C}}\)) will be denoted \(H(B_X).\) The Banach space \(H^\infty (B_X)\) of all bounded holomorphic functions f in \(H(B_X)\) is endowed with the supremum norm \(\Vert f\Vert _\infty =\sup _{x\in B_X}|f(x)|\). We denote by \(\tau _0\) the compact-open topology on \(H^\infty (B_X),\) that is, the topology of uniform convergence on compact subsets of \(B_X.\) Recall that \(\tau _0\) is Hausdorff and coarser than the norm topology. Let \(U_{H^\infty (B_X)}\) denote the closed unit ball of \(H^\infty (B_X)\). The vector space \(G^\infty (B_X),\) given by
is a Banach space when endowed with the dual norm. By using the Ng-Dixmier Theorem [12], Mujica [11], proved that the topological dual of \(G^\infty (B_X)\) is isometrically isomorphic to \(H^\infty (B_X)\). We abbreviate this fact by
For each \(x \in B_X\) we denote by \(\delta _x: H^\infty (B_X) \rightarrow {\mathbb {C}}\) the evaluation \(\delta _x(f):=f(x)\) at the point x. Clearly \(\delta _x\) is \(\tau _0\) continuous. Moreover, the vector space \(\text {span}\{\delta _x:\ x\in B_X\}\) is a norm-dense subset in \(G^\infty (B_X)\). Indeed, \(\{\delta _x:\ x\in B_X\}\) separates points of \(H^\infty (B_X)\). Hence \(\text {span}\{\delta _x\,:\ x\in B_X\}\) is a subspace of \(G^\infty (B_X)\) that is \(w(G^\infty (B_X), H^\infty (B_X))\)-dense in \(G^\infty (B_X)\). Thus it is is also norm-dense subset of \(G^\infty (B_X)\). We collect the following consequence for reference later in the paper.
Lemma 1
If \({\mathcal {F}}\) is a closed subspace of \(G^\infty (B_X)\) containing \(\{\delta _x:\ x\in B_X\}\), then \({\mathcal {F}}= G^\infty (B_X)\).
Let Y be a Banach space. The set of norm attaining functionals is defined to be the following subset of \(Y^*:\)
The Bishop–Phelps theorem (see, e.g., Theorem 8.11 in [2]) ensures that the set NA(Y) of norm attaining functionals is a norm-dense subset of \(Y^*\). As a consequence, for each non-trivial, complex Banach space X, there exists a norm-dense subset \(NA(G^\infty (B_X))\) of \(H^\infty (B_X)\), such that for every \(f\in NA(G^\infty (B_X))\), there exists an element \(\varphi \in G^\infty (B_X)\) with \(\Vert \varphi \Vert =1\) such that
The aim of this paper is to study those functions \(f \in H^\infty (B_X)\) that attain their norm as elements of the dual of \(G^\infty (B_X)\), that is, those \(f \in NA(G^\infty (B_X))\). We mainly concentrate on the case \(X=({\mathbb {C}}^N,\Vert .\Vert _2)\) and hence, \(B_X\) is the N-dimensional Euclidean ball which henceforth will be denoted \(B_N.\)
In the one dimensional case, \(B_N = {\mathbb {D}}\) and its boundary is the torus \({\mathbb {T}}=\{z\in {\mathbb {C}}:\, |z|=1\}\). In this case, by a result by Fatou, there is an isometric isomorphism between \(H^\infty ({\mathbb {D}})\) and
The isometric isomorphism \(H^\infty ({\mathbb {D}}) \rightarrow H^\infty ({\mathbb {T}})\) is given by
where the radial limit
exists almost everywhere on \({\mathbb {T}}\) (with respect to the Lebesgue normalized measure on \({\mathbb {T}},\) denoted by \( dm_1(w)=\frac{dt}{2\pi }\), where \(w=e^{it}\).) From this point of view \(H^\infty ({\mathbb {D}}) {\mathop {=}\limits ^{1}} H^\infty ({\mathbb {T}})\) is a closed subspace of \(L^\infty ({\mathbb {T}})\), and hence it is a dual space. In fact, if \(H^1_0({\mathbb {T}})\) is the closed subspace of \(L^1({\mathbb {T}})\) given by
then
Ando in [1] proved that \(H^\infty ({\mathbb {D}})\) has a unique isometric predual. Accordingly, \(L^1({\mathbb {T}})/H_0^1({\mathbb {T}}) {\mathop {=}\limits ^{1}} G^\infty ({\mathbb {D}})\). As far as we know, it is an open question for \(N\ge 2\) whether there is a unique predual of the corresponding \(H^\infty \)-spaces in the case of the N-dimensional ball and the N-polydisc. In this paper, we will introduce another natural predual and show, in Theorems 5 and 10, that it coincides with \(G^\infty (B_X)\).
The characterization of norm attaining elements of \(f \in H^\infty ({\mathbb {D}})\) was obtained by S. Fisher in 1969.
Theorem 2
(Fisher [6, Theorem 2]). Let f be an element of norm one in \(H^\infty ({\mathbb {D}})\). The function f attains its norm as an element of the dual of \(L^1({\mathbb {T}})/H_0^1({\mathbb {T}})=G^\infty ({\mathbb {D}})\) if and only if \(f^*(w)=\lim _{r\rightarrow 1-}f(rw)\) (a.e. in \({\mathbb {T}}\)) satisfies that
has positive Lebesgue measure on \({\mathbb {T}}\).
In this paper, in Sect. 2, we explore several variable versions of Fisher’s result. We also examine, in Sects. 3 and 4, similar questions for the polydisc algebra \(H^\infty ({\mathbb {D}}^N)\) and for the space of Dirichlet series \({\mathcal {D}}^\infty ({\mathbb {C}}_+).\)
2 The Case of the Euclidean Ball
Recall that the Euclidean open unit ball in \({\mathbb {C}}^N\) is:
The unit sphere in \({\mathbb {C}}^N\) is:
(Observe that this is not completely standard notation since the usual notation for the N-dimensional real sphere in \({\mathbb {R}}^N\) is \(S_{N-1}\).)
By \(\sigma _N\) we denote the unique rotation-invariant positive Borel measure on \(S_N\) for which
In other words, \(\sigma _N\) is the Haar measure of the N-dimensional sphere.
In [15, p.84], the space \(H^\infty (B_N)\), is defined as
The ball algebra is the Banach subalgebra of \(H^\infty (B_N)\) given by
Finally, by \(A(S_N)=A(B_N)\cap C(S_N)\), we understand the restrictions of the elements of \(A(B_N)\) to the sphere \(S_N\), i.e.
By the maximum modulus theorem, the mapping \(\pi : A(B_N)\rightarrow A(S_N)\) defined by \(\pi (f):=f_{|S_N}\) is an isometry.
Hardy spaces have a dual definition. The Hardy space \(H^\infty (S_N)\) is the weak-star closure of \(A(S_N)\) in \(L^\infty (S_N,\sigma _N)\). i.e.
As the polynomials are dense in \(A(B_N)\) we have that \(\text {span}\{z^\beta :\beta \in {\mathbb {N}}_0^N\}\) is a \(\Vert .\Vert _\infty \) dense subspace of \(A(B_N)\). Hence, \(\text {span}\{w^\beta :\beta \in {\mathbb {N}}_0^N\}\) is \(\Vert .\Vert _\infty \) dense in \(A(S_N)\). Thus
At this point, we show that \(H^\infty (S_N)\) and \(H^\infty (B_N)\) are isometrically isomorphic. We need some notation and results that can be found, for example, in the books [15] and [16]. The invariant Poisson kernel of \(B_N\) is the kernel function \(P_N:B_N\times S_N\rightarrow [0,+\infty [\)
The Poisson integral P(g) of a function g in \(L^1(S_N, \sigma _N)\) is defined, for \(z\in B_N\), by
We have that \(P_N:H^\infty (S_N)\longrightarrow H^\infty (B_N)\) is a linear isometry onto.
To prove that this mapping is onto, the concept of Korányi, or K-limit, of a holomorphic function on \(B_N\) is needed. For \(\alpha >1\) and \(w\in S_N\) we set
Clearly \(D_\alpha (w) \subset B_N\). We say that a function \(F:B_N\rightarrow {\mathbb {C}}\) has K-limit \(\lambda \in {\mathbb {C}}\) at \(w\in S_N\) if the following is true: For every \(\alpha >1\) and for every sequence \((z_j)\) in \(D_\alpha (w)\) that converges to a point \(w \in S_N\), we have that \(F(z_j)\) converges to \(\lambda \) and write
The following result (see e.g. [15, Section 5.4.]) is important and very useful for our paper.
Theorem 3
If f is a function in \(H^\infty (B_N)\) then f has finite K-limits \(f^*\) \(\sigma _N\)-almost everywhere on \(S_N\). Moreover, \(f^*\in H^\infty (S_N)\), \(\Vert f^*\Vert _\infty =\Vert f\Vert _\infty \) and
In other words, the mapping \(f\rightarrow f^*\) is a linear isometry from \(H^\infty (B_N)\) onto \(H^\infty (S_N)\).
We also need the following well known fact, a proof of which is given for the sake of completeness.
Lemma 4
Let X be a Banach space and let Y be a weak-star closed subspace of \(X^*\). The subspace
satisfies
and Y is isometrically isomorphic to \((X/Y_{\perp })^*\).
Proof
Clearly, by the definition, \(Y\subset Y_{\perp }^\perp \). Assume that the reverse inclusion is not true. Hence there exists \(x_0^*\in Y_{\perp }^\perp {\setminus } Y\).
Since Y is \(w(X^*,X)\) closed and convex we can find \(\varphi :X^*\rightarrow {\mathbb {C}},\) \(w(X^*,X)\)-continuous, such that
for all \(y^*\in Y\). Since \(\varphi \) is weak-star continuous, there exists \(x_0\in X\) such that
for all \(x^*\in X^*\). Thus, \(x_0^*(x_0)=1\) and \(y^*(x_0)=0\) for all \(y^*\in Y\). Hence \(x_0\) belongs \(Y_{\perp }\). But, \(x_0^*\in Y_{\perp }^\perp \), which, by definition implies
This is a contradiction.
Finally, we have \((X/Y_{\perp })^*{\mathop {=}\limits ^{1}} Y_{\perp }^\perp =Y\), as follows from [10, Theorem 1.10.17] for example. \(\square \)
Now we define
Since
the subspace \(H^\infty (S_N)\subset L_\infty (S_N)\) is \({w(L_\infty (S_N),L_1(S_N))}\)-closed in \(L_\infty (S_N)\) and
In the notation of Lemma 4, with \(X=L_1(S_N)\), \(X^*=L_\infty (S_N)\) and \(Y=H^\infty (S_N)\) (which is weak-star closed in \(X^*\)), we have
Lemma 4 implies the isometric isomorphism
Next we show that \(G^\infty (B_N)\) and \(L^1(S_N)/ H_0^1(S_N)\) are isometrically isomorphic. Thus, these two natural preduals of \(H^\infty (B_N)\) coincide, and so the extension of Ando’s result on the uniqueness of the predual of \(H^\infty ({\mathbb {D}})\) to several variables is still open.
Theorem 5
For every \(N\in {\mathbb {N}}\) we have that
isometrically.
Proof
First we prove that \(L^1(S_N)/ H_0^1(S_N)\subset G^\infty (B_N)\).
Let \([\varphi ]\in L^1(S_N)/ H_0^1(S_N)\) and \(g\in H^\infty (S_N)\). The duality is given by
for every \(\varphi \in L_1(S_N)\) and every \(\eta \in H_0^1(S_N)\).
We identify \(L^1(S_N)/ H_0^1(S_N)\) as a subspace of the dual of \(H^\infty (S_N)\) in the following natural way. Define \(T_{[\varphi ]}: H^\infty (B_N)\longrightarrow {\mathbb {C}}\) by
We check that \(T_{[\varphi ]}\) belongs to \(G^\infty (B_N)\) for every equivalence class \([\varphi ]\in L^1(S_N)/ H_0^1(S_N)\).
Clearly
Hence, \(T_{[\varphi ]}\) belongs to \(H^\infty (B_N)^*\). This fact and the equality \(\Vert T_{[\varphi ]}\Vert =\Vert [\varphi ]\Vert \) are consequences of the isometric isomorphism \(H^\infty (S_N){\mathop {=}\limits ^{1}} \big (L_1(S_N)/ H_0^1(S_N)\big )^*\) and Theorem 3.
Let us check that \(T_{[\varphi ]}\) is \(\tau _0\)-continuous when restricted to the closed unit ball \(U_{ H^\infty (B_N)}\) of \(H^\infty (B_N)\).
By Theorem 3, we know that if \(f\in H^\infty (B_N)\) and \(f^*\in H^\infty (S_N)\) is its K-limit that exists a.e. in \(S_N\), then
for all \(z\in B_N.\) Conversely, if \(h\in H^\infty (S_N)\), then \(P_N(h)\in H^\infty (B_N)\) and we have
a.e. on \(S_N\).
For each \(z\in B_N\) the mapping \(P_N(z,.):S_N\rightarrow ]0,+\infty [\) is continuous on \(S_N\). Hence \(P_N(z,.)\in L^1(S_N)\).
Given \((f_n)\cup \{f\}\subset U_{ H^\infty (B_N)}\) such that \((f_n)\) converges to f with respect to the compact-open topology on \(B_N\), we have \((f^*_n)\cup \{f^*\}\subset U_{ H^\infty (S_N)}\). But \(U_{ H^\infty (S_N)}\) is a weak-star closed subset of \(U_{ L^\infty (S_N)}\) which, in turn, is a \(w(L^\infty (S_N),L^1(S_N))\)-compact set. Since \(L^1(S_N)\) is separable, it follows that \(U_{ H^\infty (S_N)}\) is a metrizable compact set with the weak-star topology. Consider now any subsequence \((f^*_{n_k})\) that is \(w(L^\infty (S_N),L^1(S_N))\)-convergent to some \(h\in U_{ H^\infty (S_N)}\). We will have
for all \(z\in B_N\). Hence,
a.e in \(S_N\). We have just proved that the only weak-star adherent point of \((f_n^*)\) is \(f^*\). Thus \((f^*_n)\) weak-star converges to \(f^*\). In particular
and \(T_{[\varphi ]}\) is continuous with the compact-open topology when restricted to the closed unit ball of \(H^\infty (B_N);\) i.e. \(T_{[\varphi ]}\in G^\infty (B_N)\).
For the other inclusion observe that
for every \(z\in B_N\) and every \(f\in H^\infty (B_N)\). Thus
The conclusion follows from Lemma 1. \(\square \)
Theorem 5 permits us to get a sufficient condition for a function on \(H^\infty (B_N)\) to attain the norm.
Proposition 6
If f is an element of \(H^\infty (B_N)\) of norm one such that the set
has positive \(\sigma _N\) measure in \(S_N\), then f attains its norm as an element of the dual of \(L^1(S_N)/ H_0^1(S_N) = G^\infty (B_N)\).
Proof
Define \(\varphi :S_N\longrightarrow {\mathbb {C}}\) by
We have that \(\varphi \) is a bounded measurable function on \(S_N\). Thus \(\varphi \in L^1(S_N)\) and
Define \(T_{[\varphi ]}: H^\infty (B_N)\longrightarrow {\mathbb {C}}\) by
By Theorem 5, \(T_{[\varphi ]} \in L^1(S_N)/ H_0^1(S_N)= G^\infty (B_N)\) and
for every \(g\in H^\infty (B_N)\). Hence
But
and f in the dual of \(G^\infty (B_N)\) attains its norm at \(T_{[\varphi ]}\). \(\square \)
A partial converse to the above proposition is the following.
Proposition 7
If f is an element of \(H^\infty (B_N)\) of norm one such that there exists \(\varphi \in L^1(S_N)\) with \(\Vert \varphi \Vert _1=1\) and \(T_{[\varphi ]}(f)=1\), then
Proof
We denote \(E=\{w\in S_N:\, |f^*(w)|=1\}\).
Assume that \(\sigma _N(E) = 0.\)
Let
Clearly \(S_N\setminus E=\cup _{n=1}^\infty K_n.\)
We have that \(T_{[\varphi ]}\in G^\infty (B_N)\) and is of norm one since
For each n, we get
Thus, \(\int _{K_n}|\varphi (w)|d\sigma _N(w)=0\). Since n is arbitrary, we get
But, by hypothesis \(\sigma _N(E)=0\) and finally we arrive at the contradiction
\(\square \)
A subset E of \(S_N\) is called a peak set if there exists \(f \in A(B_N)\) such that \(f(z)=1\) for every \(z\in E\) and \(|f(z)|<1\) for every \(z\in {\overline{B}}_N\setminus E\). Every peak set is a null set.
A result by Fatou states that every compact subset of \({\mathbb {T}}\) of Lebesgue measure zero is a peak set of \(A({\mathbb {D}})\), a fact which is instrumental in the proof of Fisher’s Theorem 2. On the other hand, there are null sets on \(S_N\) (respectively in \({\mathbb {T}}^N\)), which are not peak sets [15, 10.1.1 and 11.2.5] (respectively [14, Theorem 6.3.4, p. 149-150]). We do not know if the converse of Proposition 6 is true or not. But, if we restrict ourselves to functions in \(A(B_N)\) that attain their norm, we get the following characterization in terms of peak sets.
Theorem 8
Let f be an element of \(A(B_N)\) of norm one. The function f attains its norm as an element of \(H^\infty (B_N)\) if and only if the set
is not a peak set.
Before presenting the proof we need some notation and a lemma.
We recall that a complex Borel measure \(\mu \) on \(S_N\) is a Henkin measure (See [15, 9.1.5, p. 186]) if
for every sequence \((f_n)\) contained in the closed unit ball \(U_{A(B_N)}\) of \(A(B_N)\) that converges uniformly to 0 on the compact subsets of \(B_N\), that is, converges to 0 in the \(\tau _0\) topology in \(B_N\). (By the Montel theorem, a sequence \((f_n)\) contained in \(U_{A(B_N)}\) converges to 0 in \(\tau _0\) if and only if converges to 0 pointwise on \(B_N\)).
Lemma 9
-
(1)
For every Henkin measure \(\mu \) there is \(T\in G^\infty (B_N)\) such that
$$\begin{aligned} T(f)=\int _{S_N}f(w)d\mu (w) \end{aligned}$$for each \(f\in A(B_N),\) and \(\Vert \mu \Vert \ge \Vert T\Vert \).
-
(2)
If \(T\in G^\infty (B_N)\), then there is a Henkin measure \(\mu \) on \(S_N\) such that
$$\begin{aligned} T(f)=\int _{S_N}f(w)d\mu (w) \end{aligned}$$for each \(f\in A(B_N),\) and \(\Vert \mu \Vert = \Vert T\Vert \).
Proof
-
(1)
Define \(T_1:A(B_N)\longrightarrow {\mathbb {C}}\) by
$$\begin{aligned} T_1(g):=\int _{S_N}g(w)d\mu (w). \end{aligned}$$Clearly, \(T_1\) is a continuous linear form on \(A(B_N)\) which is \(\tau _0\)-continuous on \(U_{A(B_N)}\) and
$$\begin{aligned} \Vert T_1\Vert \le \Vert \mu \Vert . \end{aligned}$$Given \(f\in H^\infty (B_N)\), the function \(f_r(z):=f(rz)\), \(0\le r<1\), belongs to \(A(B_N).\) In addition, \((f_r)\) converges to f uniformly on the compact subsets of \(B_N\) and
$$\begin{aligned} \Vert f_r\Vert \le \Vert f\Vert ,\ \ \ \Vert f\Vert =\sup _r\Vert f_r\Vert . \end{aligned}$$(1)By [15, 11.3.1], since \(\mu \) is a Henkin measure, the limit
$$\begin{aligned} \lim _{r\rightarrow 1-}\int _{S_N}f_r(w)d\mu (w)=\lim _{r\rightarrow 1-}T_1(f_r)\in {\mathbb {C}}, \end{aligned}$$exists for every \(f\in H^\infty (B_N)\).
We define \(T:H^\infty (B_N)\longrightarrow {\mathbb {C}}\), by
T is linear and \(T\in (H^\infty (B_N))^*\), since
for every \(f\in H^\infty (B_N)\). Moreover, \(\Vert T\Vert =\Vert T_1\Vert \).
We claim that the restriction of \(T_1\) to \(U_{A(B_N)}\) is \(\tau _0\)-uniformly continuous. Indeed, given \(\varepsilon >0\) there are a compact subset K of \(B_N\) and \(\delta >0\) such that \(|T_1(g)|<\varepsilon \) if \(g\in U_{A(B_N)}\) and \(\sup _{z\in K}|g(z)|<\delta \). Hence, if \(g,h\in U_{A(B_N)}\) and \(\sup _{z\in K}|g(z)-h(z)|<\delta \), then
Since \(U_{A(B_N)}\) is \(\tau _0\)-dense in \(U_{H^\infty (B_N)},\) there exists a unique \(\widetilde{T_1}: U_{H^\infty (B_N)}\longrightarrow {\mathbb {C}}\) that is \(\tau _0\)-continuous and such that
for all \(g\in U_{A(B_N)}\). Given \(f\in U_{H^\infty (B_N)}\), then \((f_r)\subset U_{A(B_N)}\), and \((f_r)\) converges to f in \(\tau _0\) as \(r\rightarrow 1-\). Thus
in \({\mathbb {C}}\). But \(\widetilde{T_1}(f_r)=T_1(f_r)\) for each \(r\in [0,1[\) and \(T_1(f_r)\) converges to T(f) by definition. This implies \(\widetilde{T_1}(f)=T(f)\) for each \(f\in U_{H^\infty (B_N)}\). We have obtained that the restriction of T to \(U_{H^\infty (B_N)}\) is \(\tau _0\) continuous. This, by definition, implies that T belongs to \(G^\infty (B_N)\).
-
(2)
If \(T\in G^\infty (B_N)\), the restriction of T to \(U_{H^\infty (B_N)}\) is \(\tau _0\) continuous. If we define \(T_1:A(B_N)\longrightarrow {\mathbb {C}}\), by \(T_1(g):=T(g)\), then \(T_1\) is continuous for the sup-norm, \(\Vert T_1\Vert \le \Vert T\Vert \) and \(T_1\) is \(\tau _0\)-continuous on \(U_{A(B_N)}\). By (1), we have
$$\begin{aligned} |T(f)|=\lim _{r\rightarrow 1-}|T(f_r)|=\lim _{r\rightarrow 1-}|T_1(f_r)|\le \Vert T_1\Vert \sup _{r<1}\Vert f_r\Vert =\Vert T_1\Vert \Vert f\Vert , \end{aligned}$$for every \(f\in H^\infty (B_N)\). Thus, \(\Vert T_1\Vert = \Vert T\Vert \). We can consider \(T_1:A(S_N)\longrightarrow {\mathbb {C}}\). By the Riesz theorem, there is a complex Borel measure \(\mu \) on \(S_N\) such that
$$\begin{aligned} T_1(h):=\int _{S_n}h(w)d\mu (w), \end{aligned}$$for every \(h\in A(B_N)\) with \(\Vert T_1\Vert =|\mu |(S_N)=\Vert \mu \Vert .\)
The properties of \(T_1\) imply that \(\mu \) is a Henkin measure. \(\square \)
Now we give the proof of Theorem 8:
Proof
Assume that \(f\in A(B_N)\), \(\Vert f\Vert =1\) and that E(f) is not a peak set. By [15, 10.1.1] there exists a Borel measure \(\rho \), such that \(\rho (E(f))\ne 0\) such that
for every \(h\in A(B_N)\).
Define
Since g is bounded and measurable, \(g\in L^1(|\rho |)\). Hence, the measure \(g|\rho |\) defined by \(g|\rho |(M)=\int _{M}gd|\rho |\) for Borel measurable sets M is absolutely continuous with respect to \(\rho \). The measure \(\rho \) is Henkin (this fact is a direct consequence of (2) and the definition of Henkin measure as given in [15, p. 187]), and so \(g|\rho |\) is also a Henkin measure by [15, 9.3.1]. We set \(T_1:A(B_N)\longrightarrow {\mathbb {C}}\).
We have
and
for every \(h\in A(B_N)\) with \(\Vert h\Vert \le 1\), and we have \(g|\rho |\) a Henkin measure such that
By Lemma 9.(1), there is \(T\in G^\infty (B_N)\) with \(\Vert T\Vert =\Vert T_1\Vert =1\) such that
for every \(h\in A(B_N)\).
In particular, \(T(f)=1\) and f attains its norm on \(G^\infty (B_N)\).
Suppose now that \(f\in A(B_N)\), \(\Vert f\Vert =1\), satisfies that E(f) is a peak set and that there is \(T\in G^\infty (B_N)\), \(\Vert T\Vert =1\) with \(T(f)=1\). To get a contradiction we are going to give an argument that follows closely the one given in Proposition 7:
By Lemma 9.(2) there exists \(\mu \) a Henkin measure such that
for every \(h\in A(B_N)\) and
By [15, 9.3.1] \(|\mu |\) is also a Henkin measure. Hence, by [15, Lemma 11.3.3] (see also [15, Lemma 11.3.1]), \(|\mu |(E(f))=0\). Let
Clearly \(S_N\setminus E(f)=\cup _{n=1}^\infty K_n.\)
We have that, for each n,
This implies \(|\mu |(K_n)=0\) for each n and \(|\mu |(S_N{\setminus } E(f))=0\).
Therefore, \(1=|\mu |(S_N)=|\mu |(S_N\setminus E(f))+|\mu |(E(f))=0\), a contradiction. \(\square \)
3 The Case of the Polydisc
For a fixed \(N\in {\mathbb {N}}\), the N-dimensional Poisson kernel [14, p. 17] \(P_N:{\mathbb {D}}^N \times {\mathbb {T}}^N \rightarrow (0,\infty )\) is defined as
It is well known ( [14, Theorem 3.3.3, p.45]) that if \(f\in H^\infty ({\mathbb {D}}^N)\) then the limit
exists almost everywhere in \({\mathbb {T}}^N\), and
for all \(z \in {\mathbb {D}}^N.\) As a consequence there exists an isometric isomorphism
where \(H^\infty ({\mathbb {T}}^N):=\overline{A({\mathbb {T}}^N)}^{w(L_\infty ({\mathbb {T}}^N),L_1({\mathbb {T}}^N))}\),
and
By the maximum modulus theorem \(A({\mathbb {D}}^N)\) and \(A({\mathbb {T}}^N)\) are isometrically isomorphic. By Fejer’s theory for the polydisc we have
On the other hand, by applying Lemma 4,
where
Very similar arguments to the ones given for the N-dimensional Euclidean ball can be given for the N-polydisc to obtain the following results.
Theorem 10
For every \(N\in {\mathbb {N}}\) we have
Proposition 11
Let f be an element of \(H^\infty ({\mathbb {D}}^N)\) of norm one such that the set
has positive Lebesgue measure (in \({\mathbb {T}}^N\)). Then f attains its norm as an element of the dual of \(L^1({\mathbb {T}}^N)/ H_0^1({\mathbb {T}}^N)\).
Proposition 12
If f is an element of \(H^\infty ({\mathbb {D}}^N)\) of norm one such that there exists \(\varphi \in L^1({\mathbb {T}}^N)\) with \(\Vert \varphi \Vert _1=1\) and \(T_{[\varphi ]}(f)=1\), then the set
has positive Lebesgue measure in the polytorus \({\mathbb {T}}^N.\)
Example 13
The following example, which is inspired by [3, Theorem 3.1], shows that a polydisc (for \(N>1\)) version of Theorem 8 does not hold. Let \(f: {\mathbb {D}}\times {\mathbb {D}}\rightarrow {\mathbb {C}}\) be the function \(f(z,w):=(1/2)(1+z)\), which belongs to \(A({\mathbb {D}}\times {\mathbb {D}})\). This function does not attain its norm on \(G^\infty ({\mathbb {D}}\times {\mathbb {D}})\). Indeed, if it did, the function \(g(z)=(1/2)(1+z)\), as an element of \(H^\infty ({\mathbb {D}})\), would attain its norm on \(G^\infty ({\mathbb {D}})\), because \(H^\infty ({\mathbb {D}})\) is canonically isometrically contained in \(H^\infty ({\mathbb {D}}\times {\mathbb {D}})\). But the function g does not attain its norm on \(G^\infty ({\mathbb {D}})\) by Fisher’s Theorem 2, because \(\{z \in {\mathbb {T}}\ | \ |g(z)| = 1 \} = \{ 1 \}\). On the other hand, \(E(f)= \{(z,w) \in {\mathbb {T}}\times {\mathbb {T}}\ | \ |f(z,w)|=1 \} = \{ 1 \} \times {\mathbb {T}}\), as it is easy to check. The set E(f) is not a peak set of \(A({\mathbb {D}}\times {\mathbb {D}})\). Otherwise, it would be a zero set; see [14, 6.1.2. Theorem, p.132]. But if a function \(h \in A({\mathbb {D}}\times {\mathbb {D}})\) vanishes on E(f), then \(h(1,w) \in A({\mathbb {D}})\) vanishes on \({\mathbb {T}}.\) The maximum principle then implies that h vanishes on \(\{ 1 \} \times {\mathbb {D}}\), and therefore, there is no function \(h \in A({\mathbb {D}}\times {\mathbb {D}})\) vanishing only in E(f). Observe that E(f) is a null set in \({\mathbb {T}}\times {\mathbb {T}}\) which is not a peak set.
4 The Case of the Space of Dirichelt Series \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\)
Let \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\) denote the Banach space of the Dirichlet series \(D(s)=\sum _{n=1}^{\infty }\frac{a_n}{n^s}\) convergent and bounded on the right half plane \({\mathbb {C}}_+\) endowed with the supremum norm. We refer the reader to [4] and [13] for detailed information about this space.
The space \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\) is a closed subspace of the Banach space \({H}^\infty ({\mathbb {C}}_+)\) of all bounded holomorphic functions in the right half plane \({\mathbb {C}}_+\) endowed with the supremum norm. Since, by the Montel theorem, the closed unit ball of \({H}^\infty ({\mathbb {C}}_+)\) is \(\tau _0\)-compact, we can apply the Dixmier-Ng theorem [12] to obtain that
is a predual of \({H}^\infty ({\mathbb {C}}_+)\).
It is well-known that the spaces \({H}^\infty ({\mathbb {C}}_+)\) and \({H}^\infty ({\mathbb {D}})\) are isometrically isomorphic. We are going to show that their preduals are also isometrically isomorphic.
Proposition 14
\({H}^\infty ({\mathbb {C}}_+)\) is isometrically isomorphic to \({H}^\infty ({\mathbb {D}})\), and \(G^\infty ({\mathbb {C}}_+)\) is isometrically isomorphic to \(G^\infty ({\mathbb {D}})\).
Proof
It is enough to consider the Cayley transformation \(\varphi :{\mathbb {C}}\setminus \{1\}\rightarrow {\mathbb {C}}\setminus \{-1\}\) defined by
The Cayley transformation is a biholomorphic mapping with inverse
Actually it is also biholomorphic from \({\mathbb {D}}\) onto \({\mathbb {C}}_+\), and it is a homeomorphism from \({\mathbb {T}}\setminus \{1\}\) onto \(\{ti:\ t\in {\mathbb {R}}\}\). Clearly the composition operator \(T_\varphi : {H}^\infty ({\mathbb {C}}_+)\rightarrow {H}^\infty ({\mathbb {D}})\) defined by
for \(g\in {H}^\infty ({\mathbb {C}}_+)\) is an isometry with inverse \((T_\varphi )^{-1}=T_{\varphi ^{-1}}\). Its adjoint \(T_\varphi ^*: {H}^\infty ({\mathbb {D}})^*\rightarrow {H}^\infty ({\mathbb {C}}_+)^*\) is also an isometric isomorphism with
It is enough to check that \(T_\varphi ^*(G^\infty ({\mathbb {D}}))=G^\infty ({\mathbb {C}}_+)\) to prove that \(G^\infty ({\mathbb {D}})\) and \(G^\infty ({\mathbb {C}}_+)\) are isometrically isomorphic.
Let \(R\in G^\infty ({\mathbb {D}}).\) We have
for all \(g \in {H}^\infty ({\mathbb {C}}_+)\). Let \(K\subset {\mathbb {D}}\) be a compact set. The set \(\varphi (K)\) is a compact subset of \({\mathbb {C}}_+\). Take \((g_n)\) and g in the closed unit ball of \({H}^\infty ({\mathbb {C}}_+)\) such that \((g_n)\) converges with respect to the compact open topology on \({\mathbb {C}}_+\) to g. Since \((g_n)\) converges to g uniformly on \(\varphi (K)\), we have \((g_n\circ \varphi )\) converges to \(g\circ \varphi \) uniformly on \(K=\varphi ^{-1}(\varphi (K))\), for every K. Thus, \((g_n\circ \varphi )\) converges to \(g\circ \varphi \) with respect to the compact open topology on \({\mathbb {D}}\). Hence, \((R(g_n\circ \varphi ))\) converges to \(R(g\circ \varphi )\) and we get
Analogously we obtain \(T_{\varphi ^{-1}}^*(G^\infty ({\mathbb {C}}_+))\subset G^\infty ({\mathbb {D}})\), from which it follows that
\(\square \)
Remark 15
Recall that for any fixed \(\alpha >1\) and \(w\in {\mathbb {T}}\) the Stolz region is \(S(\alpha , w)=\{ z\in {\mathbb {D}}\,:\, |z-w|<\alpha (|1-|z|)\} \) ( [9, Definition 8.1.9. ]). Since w is an accumulation point of \(S(\alpha , w)\) it makes sense to speak about the limit at w of any function \(f:S(\alpha , w)\rightarrow {\mathbb {C}}\). Actually, in [9, Theorem 8.1.11], it is proved that if \(f\in H^\infty ({\mathbb {D}})\) the following equality holds on \({\mathbb {T}}\)
almost everywhere with respect to the Lebesgue measure.
In [4, p. 286 and 287] it is observed that if \(g \in {H}^\infty ({\mathbb {C}}_+)\), then there exists a Lebesgue null set \(A\subset {\mathbb {R}}\) such that the limit
exists for every \(t\in {\mathbb {R}}\setminus A\) and actually that
In other words, the “horizontal” limits of g exist a.e. and coincide with the Fatou radial limits of its associated function \(T_{\varphi }(g)\) belonging to \(H^\infty ({\mathbb {D}})\).
We can now get the following consequence of Ando’s Theorem [1] and Fisher’s Theorem 2.
Corollary 16
The space \(H^\infty ({\mathbb {C}}_+)\) has a unique predual. Moreover, \(g \in {H}^\infty ({\mathbb {C}}_+)\) with \(\Vert g\Vert _{{\mathbb {C}}_+}=1\) is norm attaining if and only if the set
has positive (including \(+\infty \)) Lebesgue measure.
Proposition 17
\({\mathcal {D}}^\infty ({\mathbb {C}}_+)\) is a dual space.
Proof
By a result of F. Bayart (see e.g. [4, Theorem 3.11]), it is known that if \((D_n)\) is a bounded sequence in \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\) then there exists a subsequence \((D_{n_k})\) and a Dirichlet series \(D\in {\mathcal {D}}^\infty ({\mathbb {C}}_+)\) such that for every \(\sigma >0\) the sequence \((D_{n_k})\) converges to D uniformly on \({\mathbb {C}}_\sigma :=\{s\in {\mathbb {C}}\,;\, \text {Re}{s}\ge \sigma \)}. Thus, if we denote by \(\tau _+\) the topology of uniform convergence on these half planes \({\mathbb {C}}_\sigma \), Bayart’s result says that the closed unit ball of \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\) is a compact set. Now the Dixmier-Ng theorem [12] implies that
endowed with the topology induced by \({\mathcal {D}}^\infty ({\mathbb {C}}_+)^*\) is a predual of \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\).
\(\square \)
We can now get a positive result about norm attaining elements of \( {\mathcal {D}}^\infty ({\mathbb {C}}_+)\) with respect to that predual.
Proposition 18
Consider the space \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\) as the dual of \({\mathcal {G}}^\infty ({\mathbb {C}}_+).\) Given \(D \in {\mathcal {D}}^\infty ({\mathbb {C}}_+)\) of norm one, if the set
has positive (including \(+\infty \)) Lebesgue measure, then D is norm attaining.
Proof
As \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\) is a closed subspace of \(H^\infty ({\mathbb {C}}_+)\), we can consider \(D\in H^\infty ({\mathbb {C}}_+)\). By Corollary 16, we know that there exists \(R\in G^\infty ({\mathbb {C}}_+)\) such that
Recall that \(R\in {H}^\infty ({\mathbb {C}}_+)^*\) and satisfies that the restriction of R to \(U_{{H}^\infty ({\mathbb {C}}_+)}\) is \(\tau _0\) continuous. We denote by S the restriction of R to \({\mathcal {D}}^\infty ({\mathbb {C}}_+)\). Since \(U_{{\mathcal {D}}^\infty ({\mathbb {C}}_+)}\subset U_{{H}^\infty ({\mathbb {C}}_+)}\) we have that S is \(\tau _0\) continuous when restricted to \(U_{{\mathcal {D}}^\infty ({\mathbb {C}}_+)}\). The theorem of Bayart [4, Theorem 3.11] implies that \(U_{{\mathcal {D}}^\infty ({\mathbb {C}}_+)}\) is a compact set with respect to \(\tau _+\). The compact open topology \(\tau _0\) on \({\mathbb {C}}_+\) is Hausdorff and weaker than \(\tau _+\) on that ball. Hence both topologies coincide on \(U_{ {\mathcal {D}}^\infty ({\mathbb {C}}_+)}\) and \(S\in {\mathcal {G}}^\infty ({\mathbb {C}}_+)\). Moreover
and D attains its norm. \(\square \)
It is natural to ask whether the converse of Proposition 18 holds. Actually, by the Hahn-Banach theorem one can extend R in \({\mathcal {G}}^\infty ({\mathbb {C}}_+)\) to an element T belonging to \(H^\infty ({\mathbb {C}}_+)^*\) with the same norm. But we don’t know if it is possible to choose an extension T in \({G}^\infty ({\mathbb {C}}_+)\).
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Acknowledgements
The authors are very grateful to the referee for the careful reading of our manuscript and for the suggestions which improved our article. The research of R. Aron was partially supported by the project PID2021-122126NB-C33/MCIN/AEI/ 10.13039/ 501100011033 (FEDER). The research of J. Bonet was partially supported by the project PID2020-119457GB-100 funded by MCIN/AEI/10.13039/501100011033 and by “ERFD A way of making Europe” and by the project GV AICO/2021/170. The research of M. Maestre was partially supported by the project PID2021-122126NB-C33/MCIN/AEI/10.13039/501100011033 (FEDER) and the project GV PROMETEU/2021/070.
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The research of R. Aron was partially supported by the project PID2021-122126NB-C33/MCIN/AEI/ 10.13039/ 501100011033 and by “ERFD A way of making Europe”. The research of J. Bonet was partially supported by the project PID2020-119457GB-100 funded by MCIN/AEI/10.13039/501100011033 and by “ERFD A way of making Europe” and by the project GV AICO/2021/170. The research of M. Maestre was partially supported by the project PID2021-122126NB-C33/MCIN/AEI/ 10.13039/ 501100011033 and by “ERFD A way of making Europe” and the project GV PROMETEU/2021/070.
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Aron, R.M., Bonet, J. & Maestre, M. Norm Attaining Elements of the Ball Algebra \(H^\infty (B_N)\). Results Math 79, 82 (2024). https://doi.org/10.1007/s00025-023-02111-1
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DOI: https://doi.org/10.1007/s00025-023-02111-1