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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 178460, 6 pages
Isomorphisms on Weighed Banach Spaces of Harmonic and Holomorphic Functions
Departamento de Matemática Aplicada, E. Politécnica Superior de Alcoy, Universidad Politécnica de Valencia, Plaza Ferrándiz y Carbonell 2, 03801 Alcoy, Spain
Received 30 May 2013; Accepted 3 August 2013
Academic Editor: Miguel Martin
Copyright © 2013 Enrique Jordá and Ana María Zarco. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For an arbitrary open subset or and a continuous function we show that the space of weighed harmonic functions is almost isometric to a (closed) subspace of , thus extending a theorem due to Bonet and Wolf for spaces of holomorphic functions on open sets . Inspired by recent work of Boyd and Rueda, we characterize in terms of the extremal points of the dual of when is isometric to a subspace of . Some geometric conditions on an open set and convexity conditions on a weight on are given to ensure that neither nor are rotund.
1. Introduction, Notation, and Preliminaries
Let or be a nonempty open set. A weight on is a function which is strictly positive, continuous, and bounded. Let denote the space of complex valued harmonic functions on . For a weight the weighed Banach spaces of harmonic functions with weight are defined by
A function is said to vanish at infinity on if for each there exists a compact subset such that for each .
If then the weighed Banach spaces of holomorphic functions and are defined in the same way. The unit disc and the unit circle in are denoted by and , respectively. is said to be balanced if for each and each . A weight on a balanced open subset is said to be radial whenever for each .
Weighed Banach spaces of holomorphic functions with weighed supremum norms and composition operators between them have been studied by Bierstedt et al., Bonet et al., Contreras and Hernández-Díaz, García et al., Montes Rodríguez, and others; see [1–8] and the references therein.
Spaces of harmonic functions have been investigated by Shields and Williams , in connection with the growth of the harmonic conjugate of a function. In , they proved results of duality for weighed spaces of harmonic functions on the open unit disk. Lusky also considers weighed spaces of harmonic functions in [11, 12], where the isomorphism classes in the case of radial weights on the disk are determined. In  the authors have studied associated weights in weighed Banach spaces of harmonic functions and composition operators with holomorphic symbol between weighed spaces of pluriharmonic functions.
In this work we prove in the second section that, for any weight on an open subset of , the weighed Banach space of harmonic functions with vanishing at infinity on is always isomorphic to a subspace of thus extending a result due to Bonet and Wolf  for holomorphic functions. Further, given one can get a linear mapping such that for each ; that is, is almost isometric to a subspace of . We show that for radial weights in balanced open subsets the isomorphism cannot be isometric, generalizing [14, Theorem 16, Corollary 17] to spaces of harmonic functions defined on not necessarily bounded sets. We also show in this section that if the unit ball of has a big quantity of extremal points, then is not isometric to any subspace of .
In the third section we mainly focus ourselves on weighed spaces of holomorphic and harmonic functions defined on balanced subsets and with radial weights. We study the extremal points of the unit sphere of in the way that Boyd and Rueda initiated in [14, 15]. More precisely, we give conditions on and under which one can find in the unit sphere of a big quantity of extremal points. We show that these conditions imply that is not rotund.
Our notation for functional analysis and Banach space theory is standard. We refer to . For a Banach space we denote by and its unit ball and its unit sphere, respectively. A (real or complex) Banach space is said to be rotund whenever for and . A vector in the unit sphere of a Banach space is called extremal if there are not such that and , and it is called exposed if there is such that , , and for each . If is complex this last condition is equivalent to and for , where denotes the real part of a complex number . If is exposed, then is also extremal. If is a dual Banach space and the functional exposing can be taken in the predual, then is called weak*exposed. For (or ) we denote by its canonical scalar product. The Euclidean norm is denoted by . When we consider other norms on a finite dimensional space they are denoted by , making an abuse of notation because the modulus of a complex number is denoted also by .
For a Hausdorff locally compact topological space we denote by its Alexandroff compactification. The boundary of a subset of a topological space is denoted by . If is an open subset then the space is formed by the harmonic functions on such that can be extended continuously to by . For an open set , the isometry between and a subspace of by means of permits to use the same argument as in [14, Proposition 1] to show that the extreme points of are contained in the set where denotes the evaluation at . The following definitions are due to Boyd and Rueda [14–16]. The harmonic -boundary is defined as follows: The set of harmonic -peak points is defined as
Remark 1. The same argument as in [15, Theorem 6] shows that to check that a functional of the form is weak* exposed it is enough to show that there exists in the unit ball of such that , for each .
If then the holomorphic -boundary and the set of holomorphic -peak points are defined analogously, and they are denoted by and , respectively. Since , in this case, we have and also
We finish this section with this lemma also due to Boyd and Rueda.
Lemma 2 (see [14, 15]). Let be a nonempty open set, and let be a weight which vanishes at infinity on . (a)For any and the functional is extremal in the unit sphere of (is extremal in the unit sphere of ) if and only if .(b)If is balanced, is radial and then for each for each .
2. Isomorphisms on
We extend below the main theorem of  to weighed spaces of harmonic functions. The proof is analogous to the proof given by Bonet and Wolf. One only has to take in account that there are Cauchy type inequalities which are valid for the derivatives of harmonic functions [19, 2.4]. We include it for the sake of completeness. The argument was inspired by one used by Kalton and Werner [20, Corollary 4.9] to show that the little Bloch space embeds almost isometrically in . Lusky showed in  that whenever is a radial weight on , then is isomorphic to a subspace of .
Theorem 3. Let be a nonempty open set, let be a weight on , and let . There exists an isomorphism such that for each .
Proof. Let be a fundamental sequence of compact subsets of , and let . We claim that there exists a sequence of pairwise disjoint finite subsets such that and satisfying that for each with
This implies that if we denote , then we have and consequently, if we write as a sequence , then the linear mapping , is injective; it has closed range and norm no greater than one, and its inverse defined on a closed subspace of has norm less than or equal to . Since this is true for each , this is equivalent to our statement.
We proceed to show the claim. For each we define and we select such that . For each with and each , we have If and (here denotes the closed ball with center and radius ), then from the definition of we deduce that , and hence, together with the previous inequality, By Cauchy’s inequalities for harmonic functions given in [19, 2.4], there exists a positive number (independent of ) such that, for each , the th derivative of satisfies By compactness we can get a finite subset of such that where has been chosen satisfying and Consequently, for each , there exists with and . On the other hand, for each . We write to get We apply the mean value theorem to each difference and (12) to get
Now, since ,
Again as in , since each infinite dimensional subspace of contains a complemented subspace isomorphic to , the following result follows.
Corollary 4. There are not infinite dimensional subspaces of which are reflexive.
In order to get examples where the isomorphism above cannot be an isometry, we will consider a condition which is inspired by the results of Boyd and Rueda [14, Theorem 16, Corollary 17] for weighed spaces defined on bounded open sets of . First we observe that for any weight for which contains the polynomials it happens that the functionals are linearly independent. If is bounded this is the case for any weight which tends to zero at the boundary. If , then the condition is satisfied when , with being a norm in and being a rapidly decreasing continuous function, that is, with for any . First we extend easily [14, Lemma 10] to unbounded domains.
Lemma 5. Let be a nonempty open set, and let be a continuous strictly positive weight which vanishes at infinity on . If contains the polynomials, then the map , and is a homeomorphism onto its image.
Proof. Since the polynomials belong to , we get that are linearly independent functionals. We consider the map , if and . The very definition of implies that is continuous, and it is injective by the linear independence of the evaluations. By the compactness of it follows that is a homeomorphism, and hence also its restriction to .
Proposition 6. Let be a nonempty open set, and let be a weight for which contains the polynomials. Assume that is not discrete. Then cannot be isometric to any subspace of .
Proof. We give only the proof for the harmonic case. Denote by the Alexandroff compactification of . Let be an isometry onto its image. Let . Since is the subspace of formed by the functions which vanish at , the extremal points of the unit sphere of are contained in (cf. [22, Lemma V.8.6]). Let be the inverse isometry of . The transpose linear mapping is also an isometry, and it maps extremal points to extremal points. Thus for each there exists and such that . From the linear independence of in , we conclude that for with , and then is countable and is discrete for the weak* topology, since it is a sequence in the sphere of which is convergent to 0. But is a weak*-weak* homeomorphism. Hence we conclude from Lemma 5 that is discrete.
Now we apply the set of extremal points of the unit ball of a dual Banach space which is never empty and Lemma 2 to get the following result.
Corollary 7. If is a balanced open set and is a radial weight which vanishes at infinity on , then neither nor is isometric to a subspace of .
Although we do not have an example of weight on such that is a discrete subset of , we see below that we can reformulate the problem of finding an example of space which can be isometrically embedded in in terms of the possible existence of an open set and a weight such that is not discrete. Of course the analogue result is also true for the holomorphic case. In view of Lemma 2 this is not possible for radial weights in balanced domains.
Proposition 8. If is an open set and is a weight on such that contains the polynomials and is discrete, then embeds isometrically in .
Proof. Since the closed unit ball of is the weak*closure of the absolutely convex hull of the extremal points of the sphere, Lemma 2(a) implies that is in fact the closure of the absolutely convex hull of . Thus this implies that cannot be finite, and since it is discrete we have that with tending to infinity on when goes to infinity. We get now that is the weak* closure of the absolutely convex hull of , and hence, for each , it holds: Since tends to infinity on , we have that the linear map is an isometry.
3. Geometry of Weighed Banach Spaces of Holomorphic and Harmonic Functions and Their Duals
In [15, Theorem 29] Boyd and Rueda showed that if is a balanced bounded open set and is a radial weight vanishing at infinity on such that then is not rotund and then neither is. This condition is trivially satisfied when . To the best of our knowledge there are not so far concrete examples of open subsets and weights on such that . In the concrete examples of spaces given in  where is calculated the equality is always satisfied. With a similar proof to the one of [15, Theorem 29], we present below a new condition for spaces with balanced and radial depending only on the size of which ensures that is not rotund.
Proposition 9. Let be the open unit ball for a norm in , and let for be a nonincreasing continuous function with . If there exists such that the set , then is not rotund and then neither is.
Proof. Let be a linear map on such that . Since is continuous on and vanishes at infinity, there exists , such that
Since , we apply the theorem of Hahn-Banach to get such that and for each . Let . We define
Since is linear we also get For arbitrary, we use that , , and is not increasing to compute Thus is in the unit ball of and Since is assumed to be a peak point we can find in the unit ball of different from such that and for each . Hence we conclude from that is not rotund.
Let be a function such that , is decreasing, is twice differentiable in , and is convex. In [15, Corollary 12] the authors showed that for a weight defined in the Euclidean unit ball of by the equality holds, and then cannot be rotund by [15, Theorem 29]. Also if and are the Euclidean unit balls and and satisfy this last condition for certain and , then , also satisfies (cf. [15, Proposition 26]). We present below new examples of spaces which cannot be rotund. In the required assumptions, besides considering a wider class of open sets, we remove from the weight the condition of differentiability given by Boyd and Rueda as we did in [13, Theorem 1] to give conditions under which , being the associated weight introduced by Bierstedt et al. in . The same technique permits us to get concrete radial weights defined in the Euclidean ball of with odd such that the spaces are not isometric to any subspace of . If , then the assertion is a consequence of Corollary 7 since is the Euclidean unit ball of and .
Proposition 10. Let be a continuous decreasing function such that and is strictly convex in .(a)Let be the unit ball for a norm and , . Then is not rotund (and then neither is).(b)Let be the Euclidean unit ball and for . Then is not isometric to any subspace of .
Proof. Let , . Since is increasing and strictly convex, for each , the map
is increasing. Hence for each there exists which depends on such that
for all . Now we compute
We prove (a). Straszewicz’s theorem [24, Theorem 7.89] together with the Krein-Milman theorem imply that there exists which is exposed. Let such that and Re if . Let be arbitrary, and let be as above. The function defined by is holomorphic and . It follows immediately from (30) that if ,
If , then we use (30) and to get Hence peaks . We conclude that since is arbitrary. We apply Proposition 9 to get that is not rotund.
To prove (b) we consider with and get again as above. Let be an orthogonal transformation such that . For we take the harmonic function and we define for , which is harmonic; see [19, Chapter 1]. A similar argument as in (a) permits us to conclude that and for each . Hence , and therefore is not isometrically embedded in by Proposition 6.
Remark 11. (a) An inspection of the proofs of Propositions 9 and 10 shows that the statements remain true when and is a decreasing continuous function which satisfies that is strictly convex and when goes to infinity for each .
(b) For any increasing function which is strictly convex and satisfies (or with when goes to infinity for each ), the function satisfies the hypothesis of Propositions 9 and 10. The examples given in [15, Example 13] of weights in the Euclidean unit ball can be obtained using this general method.
The authors are grateful to J. Bonet for a lot of ideas and discussions during all this work. They are also indebted to P. Rueda, who besides giving them some references provided them some unpublished work which has inspired a big part of this paper. They also thank M. Maestre for his careful reading of the paper and helpful discussions. Finally, they thank the referees for the careful analysis of the paper and the important suggestions which they gave us. The research of the first author was supported by MICINN andFEDER, Project MTM2010–15200. The research of both authors is partially supported by Programa de Apoyo a la Investigación y Desarrollo de la UPV PAID-06-12.
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