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Abstract and Applied Analysis
Volume 2016, Article ID 1393496, 9 pages
http://dx.doi.org/10.1155/2016/1393496
Research Article

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, Spain

Received 30 May 2016; Accepted 27 September 2016

Academic Editor: Hatem Mejjaoli

Copyright © 2016 Joaquín Motos et al. 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.

Abstract

We show that the dual of the variable exponent Hörmander space is isomorphic to the Hörmander space (when the exponent satisfies the conditions , the Hardy-Littlewood maximal operator is bounded on for some and is an open set in ) and that the Fréchet envelope of is the space . Our proofs rely heavily on the properties of the Banach envelopes of the -Banach local spaces of and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for , , are also given (e.g., all quasi-Banach subspace of is isomorphic to a subspace of , or is not isomorphic to a complemented subspace of the Shapiro space ). Finally, some questions are proposed.

Dedicated to the memory of Nigel J. Kalton

1. Introduction and Notation

The Lebesgue spaces with variable exponent and the corresponding Sobolev spaces have been the subject of considerable interest since the early 1990s. These spaces are of interest in their own right and also have applications to PDEs of nonstandard growth and to modelling electrorheological fluids and to image restoration. For a thorough discussion of these spaces and their history, see [1, 2]. Our paper lies in this field of variable exponent function spaces and is a continuation of [3] (see also [4, 5]). In [5] the (nonweighted) variable exponent Hörmander spaces , , and were introduced (recall that the classical Hörmander spaces , , and play a crucial role in the theory of linear partial differential operators (see, e.g., [610])) and there, extending a Hörmander result [6, Chapter XV] to our context, the dual of (when ) was calculated (as a consequence some results on sequence space representation of variable exponent Hörmander spaces were obtained). In [3] the dual was calculated when (with techniques necessarily different from those used in [5]) and a number of applications were given. In the current article we show that the dual is isomorphic to (when ) and that the Fréchet envelope of is . Applications to the study of the structure of complemented subspaces of are also given. The techniques used in the article (also in [3]) are based on the inequalities of the extrapolation theorems obtained by the authors in [4] and on the properties of the Banach envelopes of the -Banach local spaces of . Finally, three questions on duality and on sequence space representation of variable exponent Hörmander spaces are proposed.

1.1. Notation

(1)Let and be Hausdorff topological linear spaces over . If and are isomorphic (i.e., there exists a linear homeomorphism from onto ) we put . The (topological) dual of is denoted by and is given (unless otherwise stated) the topology of uniform convergence on all the bounded subsets of (sometimes denoted by ). The completion of is denoted by . If is metrizable and complete, is said to be an -space. A locally convex -space is said to be a Fréchet space. We put if is a linear subspace of and the canonical injection is continuous. If is a Banach space, (resp., ) is the topological product (resp., the locally convex direct sum) of a countable number of copies of . (resp., ) is denoted by (resp., ). For unexplained notation we refer to [1114].(2)If the Fourier transform of , or , is defined by . If is a function on , then for . is the closed Euclidean ball in . , , and are the usual Schwartz spaces (in the last space the norms , , are denoted by ). , , and are their corresponding duals. ( compact in ) is the set of distributions on with support contained in . The Fourier transform in is also denoted by (or ). If , is defined by for all ; thus coincides with the operator . When we consider function spaces (or distribution spaces) defined on the whole Euclidean space , we shall omit the “” of their notation. The letter will always denote a positive constant, not necessarily the same at each occurrence.(3) Throughout this paper all vector spaces are assumed complex. By definition, a quasi-normed space is a vector space with a quasi-norm satisfying (i) , , (ii) , and (iii) , for some independent of , . If is complete, we say it is a quasi-Banach space. The quasi-norm is -subadditive for some if ; in this case, if is complete, we say it is a -Banach space. Recall that if a quasi-normed space is locally convex then it becomes a normed space: Let be and let be a balanced convex open neighborhood of such that . If is such that then the Minkowski functional of , (), is a norm equivalent to sinceholds for all . (See [12, Chapter  1] and [15, Chapter  25].)(4) is the set of all measurable functions on with range in such that and . denotes the set of all complex-valued measurable functions on such that, for some , . With the norm (quasi-norm if ) defined by , becomes a Banach (quasi-Banach if ) space. If we can also define as the set of all measurable functions such that , where and . In this case we have . (See [1, 2, 16].)(5)If is a compact subset of and , then . is a quasi-Banach (Banach if ) space (see [17, Chapters  1, 2]). If then is a quasi-normed space (normed if ) linear space. From the Paley-Wiener-Schwartz theorem it follows that the elements of are entire analytic functions of exponential type. When , a constant, then with equality of quasi-norms (resp., norms). We shall denote by the collection of all such that ; obviously . The spaces have been introduced and studied in [4].(6)Let be and let be an open set in . ThenIf we put . is a quasi-normed space (a Banach space isomorphic to if ). Now consider the spaceIf every is equipped with the topology induced by , then (endowed with the corresponding inductive linear topology) becomes a strict inductive limit(Each step is a quasi-Banach space since it is isomorphic to via the Fourier transform and this space is a quasi-Banach space by [4, Theorem  3.5]. On the other hand, the bilinear mapping is continuous (see [5])). Finally,The topology of this space is generated by the seminorms (-seminorms when ; here ) .The spaces , , and are called variable exponent Hörmander spaces and have been introduced in [5]. If and , these spaces coincide with the Hörmander spaces , , and , respectively (see [6]). Throughout this paper, will denote the Hörmander space (see again [6, Chapter  X]).(7)We conclude this section recalling some basic facts about the Banach envelope of a quasi-normed space and the Fréchet envelope of a metrizable topological linear space.Let be a quasi-normed space whose dual separates the points of and let be the unit ball of . Then is a Banach space under the norm . The Banach envelope of is the completion of in the norm defined by coincides with the Minkowski functional of the convex hull of , and the inclusion is continuous with dense range (if is a Banach space then ). has the property that any bounded linear operator into a Banach space extends with preservation of norm to a bounded linear operator ; thus (and ) becomes linearly isometric to (see, e.g., [12, pp. 27, 28] and [18, Section  2]; in the last paper the Banach envelopes of some Besov and Triebel-Lizorkin spaces are computed; in [19] the Banach envelope of Paley-Wiener type spaces is also computed).Now let be a metrizable topological linear space such that its dual () separates points of . The Mackey topology of , , is the finest locally convex topology on which has as dual space. If is a base of balanced neighborhoods of zero for then , where denotes the -closed convex hull of , is a base of neighborhoods of zero for and thus this topology is metrizable. The Fréchet envelope of is the completion of ( when is a Fréchet space). coincides with the Banach envelope of when this space is quasi-normed. If is the canonical injection of into , then the transpose of is an algebraic isomorphism of onto . If and are metrizable topological linear spaces with separating duals and is a continuous linear mapping taking into , then is also continuous from into and so there is a unique extension of to a continuous linear mapping taking into . If in addition and are -spaces and , then . (See the proofs of these results in [20, 21]; furthermore, in these papers and in [12], the Fréchet envelopes of several -spaces of holomorphic and harmonic functions are computed.)

2. The Dual and the Fréchet Envelope of

In [6], the isomorphism is shown (being an open convex set in and a weight satisfying the estimate , and positive constants). In Theorem  4.3 of [5] this isomorphism is extended to variable exponent Hörmander spaces with . In [3] it is shown that when the exponent satisfies (the techniques used are different from those used in [5] since if the dual of is trivial and the steps are quasi-Banach spaces instead of Banach spaces) and several applications of this result were given.

As a consequence [5, Theorem  4.3] and the reflexivity of (see [1, Corollary  2.81]) one gets the isomorphism when and the Hardy-Littlewood maximal operator is bounded on and . In this section we show the counterpart of this result: the dual (equipped with the topology of the uniform convergence on -bounded subsets of ) is isomorphic to (and therefore to ) when and the Hardy-Littlewood maximal operator is bounded on . Our proof is based on the inequalities obtained in the extrapolation theorem [4, Theorem  3.5], on the properties of the Banach envelopes of the -Banach local spaces of , and on the identification of the Fréchet envelope of . We also give a characterization of the locally convex complemented subspaces of and we show that is not isomorphic to a complemented subspace of the Shapiro space (see Remark 8() to Theorem 7). Note that Theorem 7 can have independent interest to calculate Fréchet envelopes of -spaces.

Throughout the entire article, denotes a variable exponent in such that and the Hardy-Littlewood maximal operator is bounded on for some , denotes an open set in , is a fundamental sequence of compact subsets of such that, for all , and has the segment property, and is a -partition of unity on such that for every . Finally, denotes a sequence in such that on and for each .

Recall (see Section 1 and [5]) that , with the topology defined by the collection of -seminorms , becomes an -space (actually, a locally -convex space) and that holds for all . The family , where , is a base of neighborhoods of in .

Lemma 1. is an infinite dimensional -normed space whose dual separates points of (here is the corresponding quotient -norm). If , then becomes an infinite dimensional -normed space with separating dual.

Proof. If , denotes the coset of . Then is an infinite dimensional subspace of (see also [5, Theorem  3.7/2]). Now, for each , put . Let us see that . Naturally, is well defined (if then ). Furthermore, of the embedding (see [4, Theorem  3.5/5]) and the fact that for one has , that is, , it follows thatwhich proves that . Hence the required conclusion follows easily.
The second part of lemma is obvious taking into account that is a -norm and that [5, Theorem  3.7/2] and [4, Theorem  3.5/5] are also valid when because the Hardy-Littlewood maximal operator is bounded in for all .

Remark 2. Naturally in the second part of the previous lemma we could apply [17, Proposition  1.3.2, p. 17] instead of [4, Theorem  3.5/5].

Lemma 3. Let be a locally -convex space () and metrizable whose topology is defined by a family of -seminorms such that, for every , ( constants ). Let be a (complemented) quasi-normed subspace of . Then there exists such that, for each , is isomorphic to a (complemented) subspace of the local -normed space . If furthermore is complete, that is, a quasi-Banach space, then is isomorphic to a (complemented) subspace of the local -Banach space .

Proof. Let be the quasi-norm on which generates the topology of . Then the identity is an isomorphism. Thus, for every , there exists an such that for all , and there exist also an integer and so that for all . Next, fix . Then, for every , we havewhich shows that on is a -norm equivalent to . Furthermore, these inequalities prove immediately that the restriction to of the canonical mapping is an isomorphism onto .
If is complemented in and is a continuous projection in such that , there exist an integer and a constant such that for every . Then it is easy to check that, for every , the mapping defined by is a continuous projection such that .
Finally, if is complete then the extension of to , is a continuous projection in such that .

Remark 4. This lemma is well known in the locally convex case (see, e.g., [22]).

Proposition 5. Let , and . Then, consider the following:(1)The completion of is a -Banach space (-dimensional and with separating dual) isomorphic to a subspace of and contains a subspace isomorphic to .(2) is not locally convex.(3)If , then .(4)All quasi-Banach subspace of is isomorphic to a subspace of .

Proof. (1) Since the operator is an isometry, the completion of is a -Banach space isometric to a closed subspace of . Let be such that and and consider the following diagram:where is the Fourier transform, is the canonical injection, is the isomorphism defined by , and is the isomorphic embedding defined by (this property of is well known; see, e.g., [23, pp. 101, 197] for and [24, Lemma  1.8, p. 17] for ). The proof concludes composing these operators with the former isometric isomorphism. The second claim is a consequence of a result of Stiles (see, e.g., [12, Theorem  2.5]).
(2) First we observe that, for each compact , the restriction mapping (here is the natural injection from into and is the natural extension from into ) is continuous: If in thenfor every (we have used the continuity of the bilinear mapping ; see (6) in Section 1.1). Next we show that the space is not locally convex. Suppose otherwise and recall that the family is a local base of . Then, given there exist an absolutely convex neighborhood of and with such that and so we have that holds for all ( is the Minkowski functional of ). We consider now the following commutative diagramwhere (resp., ) is the completion of the -normed space (resp., ), is the completion of the normed space , (resp., , ) denotes the extension of the natural operator (resp., , ), and is an isomorphism from onto (see (1)). By a result of Stiles (see, e.g., [12, Proposition  2.9]), the operator is compact but then, by the properties of , is also compact. From this and of , it follows that is compact.
In order to complete the proof we consider a sequence with and . Obviously, this sequence lies in and it is bounded here ( for ). Thus is bounded in and so is bounded in and since is a compact operator we can find such that the subsequence converges in . By applying (1), we see that in but in (because ). Hence it follows that , that is, that . This contradiction concludes the proof of (2).
(3) If , then these spaces should be isomorphic by the open mapping theorem and so there exist positive integers and a constant such that holds for all . Hence and from the fact that on for every , it follows that holds for all and therefore that is also valid for all . Then, by using the density of in (see [25, Proposition  1.4.4]) and the embedding [17, Proposition  1.3.2, p. 17], we get for all . This and the density of in imply that (coinciding algebraically and topologically). But then, reasoning as in the proof of (1), it is found that contains a subspace isomorphic to which contradicts a result of Stiles (see, e.g., [12, Corollary  2.8]).
(4) Let be a quasi-Banach subspace of . By using Lemma 3, becomes isomorphic to a complemented subspace of the local -Banach space for all large enough . But we know by (1) that each of these spaces is isomorphic to a subspace of . This concludes the proof of (4).

Theorem 6. is algebraically isomorphic to when (in particular is algebraically isomorphic to for all ).

Proof. For each let be the normed space , where is the Minkowski functional of the convex hull of the unit ball of the -normed space . It is easily seen that the mapping is linear, injective, and continuous (for each one has for all , and so is continuous).
Let be a continuous linear functional on and let and be such that holds for all . The linear functional defined by is continuous also since it is in the dual and this space and are linearly isometric (Lemma 1 and (7), Section 1); therefore we get . Hence it follows that the linear functional is continuous on : the family of seminorms generates the product topology on andholds for all . By the Hahn-Banach theorem, can be extended to a continuous linear functional on . Then, by using the isomorphismdefined by (see, e.g., [13, p. 284]), we find such that and we obtain the following representation of :Now we shall prove that the mappingdefined by , is an algebraic isomorphism (here is the sequence which represents to and, for every , is the tempered distribution defined by for all ). Let us see that is well defined: (i) First we show that each . If in then in and so in ; therefore , that is, in . As a consequence, and becomes a tempered distribution. Furthermore, for each , we have(since ) and so . Thus (see, e.g., [26, p. 165]) coincides with the Fourier-Laplace transform of defined byTaking here absolute values and using [2, Lemma  3.2.5], we getTherefore and . (ii) If comes from another extension , then sinceholds for all .
We have proved that is well defined, and it is obvious that it is linear. If then for all , but is dense in [5, Theorem  3.7/2]; thus and is injective. Let us see that is surjective: Let be an element of . We now define the functionaland we show that it is continuous. These integrals converge because that is , that is , and [4, Theorem  3.5]. Moreover, by the properties of the -partition of unity , there exists a positive integer such that for all and . Then we have thatholds for all and so . Finally we check that : Assume that represents to and recall that and (for all ); thenholds for all .
Finally, if and then the Hardy-Littlewood maximal operator is bounded on for each and so we also have that is algebraically isomorphic to .

Now we prove a result we use to calculate the Fréchet envelope of .

Theorem 7. Let be an -space such that its dual separates points of . Assume that is a dense linear subspace of a Fréchet space , that the inclusion map is continuous, and that   , that is, the transpose of , , is an algebraic isomorphism. Assume finally that is a complemented subspace of . Then, we have the following:(1); that is, is the Fréchet envelope of , , and is also a complemented subspace of .(2)If furthermore is separable then becomes an isomorphism of onto being the topology of the uniform convergence on the -bounded subsets of .

Proof. (1) To see that it suffices to show that the induced topology by on , , coincides with the Mackey topology ( is also an -space with separating dual since it is a complemented subspace of ). To do so first we observe that () (obviously and, on the other hand, if and is an extension of then this has the form , with ; thus ) and then recall that every metrizable locally convex space has the Mackey topology (see, e.g., [13, pp. 379, 380]). In particular, we have also shown that .
It remains to prove the last claim. Let be a continuous projection in such that and let be the unique extension linear and continuous of (recall that is ). Since , is also a linear and continuous mapping from into . Furthermore, if and is a sequence in convergent to in then . Therefore, . To conclude we check that is a projection: If and is a sequence in such that in then in and so .
(2) The continuity of the mapping follows from the fact that every -bounded subset of is -bounded subset of (by (1), ). If is separable, every bounded subset of the Fréchet space is contained in the closure of a bounded subset of (apply [13, (1) p. 403]), and since , it follows that the mapping is also continuous.

Remark 8. (1) In [21] Shapiro constructs subspaces of the -space (of harmonic functions) isomorphic to and also proves that the Fréchet envelope of is the separable Fréchet space (see notations in [21]). From Theorem 7 it follows that these subspaces are not complemented in .
(2) If in Theorem 7   is a quasi-Banach space and is a Banach space not necessarily separable, then is the Banach envelope of and is an isomorphism of onto (these spaces equipped with the norms , , resp.): It suffices to take into account that if is a Banach space and a dense linear subspace; then every bounded subset of is contained in the closure of a bounded subset of .
Thus Theorem 7 recovers known results (see, e.g., [18, Theorem  5]).

In Proposition 9 and Theorem 10 we will use the same notation as in the proof of Theorem 6.

Proposition 9. Let be the topology on of the uniform convergence on the bounded subsets of . Then the mapping is open. If then becomes an isomorphism.

Proof. First we show that is continuous ( is open). For this it suffices to check, since , that, for every , is continuous from into . Fix and let be a positive integer such that, for all , for all and . Let be a bounded subset of ; then . Now we argue as in the proof of Theorem 6 and we obtain thatholds for all , which shows the continuity of . If , then is the strong topology . By a result of Vogt [10] , and thus (apply, e.g., [13, p. 287]). Hence it follows that is an -space. Since is also an -space, we can apply [14, (4) b p. 43] to and conclude that is an isomorphism.

The next theorem improves the first part of the previous result considering the topology of the uniform convergence on the -bounded subsets of instead of the topology . Our method requires the calculation of the Fréchet envelope of .

Theorem 10. (1) ; that is, is the Fréchet envelope of (in particular, for all ).
(2) If is the topology of the uniform convergence on the -bounded subsets of , then the spaces and are isomorphic (in particular, and are isomorphic for all ).

Proof. (1) is an -space on which separates points (see Theorem 6). Furthermore, is a dense linear subspace of the Fréchet space [5, Theorem  3.7] and the inclusion map is continuous (for each and each , we have in virtue of the embedding