Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

and Applied Analysis 3 The topology of this space is generated by the seminorms (p0-seminorms when p− < 1; here p0 ∈ (0, p−)) u → ‖u‖p(⋅),φ fl ‖φu‖Bp(⋅) , φ ∈ C∞0 (Ω). The spaces Bp(⋅), Bcp(⋅)(Ω), and Bloc p(⋅)(Ω) are called variable exponent Hörmander spaces and have been introduced in [5]. If p(⋅) ≡ p and p ≥ 1, these spaces coincide with the Hörmander spaces Bp,1, Bloc p,1(Ω), and Bloc p,1(Ω), respectively (see [6]). Throughout this paper, Bloc ∞ (Ω) will denote the Hörmander space Bloc ∞,1(Ω) (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 (X, ‖ ⋅ ‖X) be a quasi-normed space whose dual X󸀠 separates the points of X and let BX be the unit ball of X. Then X󸀠 is a Banach space under the norm ‖x󸀠‖ = sup{|⟨x, x󸀠⟩| : x ∈ BX}. The Banach envelope ?̂? of (X, ‖ ⋅‖X) is the completion ofX in the norm ‖ ⋅‖c defined by ‖x‖c fl sup {󵄨󵄨󵄨󵄨󵄨⟨x, x󸀠⟩󵄨󵄨󵄨󵄨󵄨 : 󵄩󵄩󵄩󵄩󵄩x󸀠󵄩󵄩󵄩󵄩󵄩 ≤ 1} . (7) ‖ ⋅ ‖c coincides with the Minkowski functional of the convex hull of BX, ‖ ⋅ ‖c ≤ ‖ ⋅ ‖X and the inclusion X 󳨅→ ?̂? is continuous with dense range (if X is a Banach space then X = ?̂?). ?̂? has the property that any bounded linear operatorL : X → Y into aBanach space extendswith preservation of norm to a bounded linear operator ?̂? : ?̂? → Y; thus (?̂?)󸀠 (and (X, ‖ ⋅ ‖c)󸀠) becomes linearly isometric toX󸀠 (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 PaleyWiener type spaces is also computed). Now letX[T] be ametrizable topological linear space such that its dual X󸀠(= (X[T])󸀠) separates points of X. The Mackey topology of X[T], m(X,X󸀠), is the finest locally convex topology on X which has X󸀠 as dual space. If {Un}∞n=1 is a base of balanced neighborhoods of zero forT then {?̃?n}∞n=1, where ?̃?n denotes the T-closed convex hull of Un, is a base of neighborhoods of zero for m(X,X󸀠) and thus this topology is metrizable. The Fréchet envelope ?̂? of X[T] is the completion of X[m(X,X󸀠)] (?̂? = X[T] when X[T] is a Fréchet space). ?̂? coincides with the Banach envelope of X[T] when this space is quasinormed. If j is the canonical injection of X[T] into ?̂?, then the transpose of j is an algebraic isomorphism of (?̂?)󸀠 onto (X[T])󸀠. If X and Y are metrizable topological linear spaces with separating duals and T is a continuous linear mapping taking X into Y, then T is also continuous from X[m(X,X󸀠)] into Y[m(Y,Y󸀠)] and so there is a unique extension ?̂? of T to a continuous linear mapping taking ?̂? into ?̂?. If in addition X and Y are F-spaces and T(X) = Y, then ?̂?(?̂?) = ?̂?. (See the proofs of these results in [20, 21]; furthermore, in these papers and in [12], the Fréchet envelopes of several F-spaces of holomorphic and harmonic functions are computed.) 2. The Dual and the Fréchet Envelope of Bloc p(⋅)(Ω) (0 < p− ≤ p+ ≤ 1) In [6], the isomorphism (Bc2,k(Ω))󸀠 ≃ Bloc 2,1/?̃?(Ω) is shown (being Ω an open convex set in R and k a weight satisfying the estimate k(x + y) ≤ (1 +C|x|)Nk(y), x, y ∈ R, C andN positive constants). In Theorem 4.3 of [5] this isomorphism is extended to variable exponent Hörmander spaces with 1 < p− ≤ p+ < ∞ : (Bcp(⋅)(Ω))󸀠 ≃ Bloc p󸀠(⋅)(Ω). In [3] it is shown that (Bcp(⋅)(Ω))󸀠 ≃ Bloc ∞ (Ω) when the exponent p(⋅) satisfies 0 < p− ≤ p+ ≤ 1 (the techniques used are different from those used in [5] since if p+ < 1 the dual of Lp(⋅) is trivial and the steps Bp(⋅) ∩ E󸀠(K) 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 Lp(⋅) (see [1, Corollary 2.81]) one gets the isomorphism (Bloc p(⋅)(Ω))󸀠 ≃ Bcp󸀠(⋅)(Ω) when 1 < p− ≤ p+ < ∞ and the Hardy-Littlewood maximal operator is bounded on Lp(⋅) and Lp(⋅). In this section we show the p+ ≤ 1 counterpart of this result: the dual (Bloc p(⋅)(Ω))󸀠 (equipped with the topology T of the uniform convergence on m(Bloc p(⋅)(Ω), (Bloc p(⋅)(Ω))󸀠)bounded subsets of Bloc p(⋅)(Ω)) is isomorphic to Bc∞(Ω) (and therefore to l(N) ∞ ) when 0 < p− ≤ p+ ≤ 1 and the HardyLittlewood maximal operator M is bounded on Lp(⋅)/p0 . 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 p0-Banach local spaces of Bloc p(⋅)(Ω), and on the identification of the Fréchet envelope of Bloc p(⋅)(Ω). We also give a characterization of the locally convex complemented subspaces of Bloc p(⋅)(Ω) and we show that l∞ is not isomorphic to a complemented subspace of the Shapiro space hp− (see Remark 8(1) toTheorem 7). Note that Theorem 7 can have independent interest to calculate Fréchet envelopes of F-spaces. Throughout the entire article, p(⋅) denotes a variable exponent in P such that 0 < p− ≤ p+ ≤ 1 and the HardyLittlewood maximal operator M is bounded on Lp(⋅)/p0 for some 0 < p0 < p−, Ω denotes an open set in R, {Kj}∞j=1 is a fundamental sequence of compact subsets ofΩ such that, for all j,Kj = ∘ Kj and ∘ Kj has the segment property, and {θj}∞j=1 is a C∞0 (Ω)-partition of unity on Ω such that supp θj ⊂ Kj for every j. Finally, {χj}∞j=1 denotes a sequence in C∞0 (Ω) such that χj ≡ 1 on Kj and suppχj ⊂ ∘ Kj+1 for each j. 4 Abstract and Applied Analysis Recall (see Section 1 and [5]) that Bloc p(⋅)(Ω), with the topology defined by the collection of p0-seminorms {‖ ⋅‖p(⋅),χj : j = 1, 2, . . .}, becomes an F-space (actually, a locally p0-convex space) and that ‖ ⋅ ‖p(⋅),χj ≤ Cj‖ ⋅ ‖p(⋅),χj+1 holds for all j. The family {Vj,ε : j ∈ N, ε > 0}, where Vj,ε = {u ∈ Bloc p(⋅)(Ω) : ‖u‖p(⋅),χj < ε}, is a base of neighborhoods of 0 in Bloc p(⋅)(Ω). Lemma 1. X fl (Bloc p(⋅)(Ω)/ ker ‖⋅‖p(⋅),χj , ‖ ⋅‖∗p(⋅),χj ) is an infinite dimensional p0-normed space whose dual separates points ofX (here ‖ ⋅ ‖∗p(⋅),χ is the corresponding quotient p0-norm). If p(⋅) ≡ p, 0 < p < 1, then X becomes an infinite dimensional pnormed space with separating dual. Proof. If u ∈ Bloc p(⋅)(Ω), [u]j denotes the coset of u. Then {[φ]j : φ ∈ C∞0 (Kj)} is an infinite dimensional subspace of Bloc p(⋅)(Ω)/ ker ‖ ⋅ ‖p(⋅),χj (see also [5, Theorem 3.7/2]). Now, for each φ ∈ S, put ⟨[u]j, Uφ⟩ fl ⟨φ, χju⟩. Let us see that Uφ ∈ X󸀠. Naturally, Uφ is well defined (if ] ∈ [u]j then χj] = χju). Furthermore, of the embedding L−Kj+1 p(⋅) 󳨅→ L−Kj+1 1 (see [4,Theorem 3.5/5]) and the fact that for u ∈ Bloc p(⋅)(Ω) one has χju ∈ Bp(⋅) ∩ E󸀠(Kj+1), that is, (χju)∧ ∈ L−Kj+1 p(⋅) , it follows that ⟨[u]j , Uφ⟩ = ⟨φ, χju⟩ = (2π)−n ⟨̂̃ φ, (χju)⟩ = (2π)−n ∫ R ̂̃ φ (χju) dx, 󵄨󵄨󵄨󵄨󵄨⟨[u]j , Uφ⟩󵄨󵄨󵄨󵄨󵄨 ≤ (2π)−n ∫ R 󵄨󵄨󵄨󵄨󵄨̂̃ φ󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨(χju)∧󵄨󵄨󵄨󵄨󵄨󵄨 dx ≤ (2π)−n 󵄩󵄩󵄩󵄩?̂?󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩󵄩󵄩(χju)∧󵄩󵄩󵄩󵄩󵄩󵄩1 ≤ C 󵄩󵄩󵄩󵄩?̂?󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩󵄩󵄩(χju)∧󵄩󵄩󵄩󵄩󵄩󵄩p(⋅) = C 󵄩󵄩󵄩󵄩?̂?󵄩󵄩󵄩󵄩∞ ‖u‖p(⋅),χj = (C 󵄩󵄩󵄩󵄩?̂?󵄩󵄩󵄩󵄩∞) 󵄩󵄩󵄩󵄩󵄩[u]j󵄩󵄩󵄩󵄩󵄩∗p(⋅),χj , (8) which proves that Uφ ∈ X󸀠. Hence the required conclusion follows easily. The second part of lemma is obvious taking into account that ‖ ⋅ ‖∗p,χj is a p-norm and that [5, Theorem 3.7/2] and [4, Theorem 3.5/5] are also valid when p(⋅) ≡ p because the Hardy-Littlewood maximal operator is bounded in Lp/p0 for all 0 < p0 < p. 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 E[T] be a locally p-convex space (0 < p < 1) and metrizable whose topology is defined by a family of pseminorms {‖ ⋅ ‖n : n ≥ 1} such that, for every m < n, ‖ ⋅ ‖m ≤ Cm,n‖ ⋅ ‖n (Cm,n constants > 0). Let Q be a (complemented) quasi-normed subspace of E. Then there exists k such that, for each r ≥ k, Q is isomorphic to a (complemented) subspace of the local p-normed space Er = (E/ ker ‖ ⋅ ‖r, ‖ ⋅ ‖∗r ). If furthermore Q is complete, that is, a quasi-Banach space, then Q is isomorphic to a (complemented) subspace of the local pBanach space Er. Proof. Let ‖ ⋅ ‖Q be the quasi-norm on Q which generates the topology of Q. Then the identity idQ : (Q, ‖ ⋅ ‖Q) → Q[T] is an isomorphism. Thus, for every n, there exists anMn > 0 such that ‖x‖n ≤ Mn‖x‖Q for all x ∈ Q, and there exist also an integer m and C > 0 so that ‖x‖Q ≤ C‖x‖m for all x ∈ Q. Next, fix n ≥ m. Then, for every x ∈ Q, we have ‖x‖n ≤ Mn ‖x‖Q ≤ MnC ‖x‖m ≤ MnCCm,n ‖x‖n , (9) which shows that on Q‖ ⋅ ‖n is a p-norm equivalent to ‖ ⋅ ‖Q. Furthermore, these inequalities prove immediately that the restriction toQ of the canonical mapping πn : E → En : x → [x]n is an isomorphism onto πn(Q). If Q is complemented in E and P is a continuous projection in E such that ImP = Q, there exist an integer k ≥ m and a constant B > 0 such that ‖Px‖m ≤ B‖x‖k for every x ∈ E. Then it is easy to check that, for every r ≥ k, the mapping Pr : Er → Er defined by Pr([x]r) = [Px]r is a continuous projection such that ImPr = πr(Q). Finally, ifQ is complete then the extension of Pr to Er, Pr, is a continuous projection in Er such that ImPr = πr(Q). Remark 4. This lemma is well known in the locally convex case (see, e.g., [22]). Proposition 5. Let p(⋅) ≡ p, 0 < p < 1, and X fl (Bloc p (Ω)/ ker ‖ ⋅ ‖p,χj , ‖ ⋅ ‖∗p,χj ). Then, consider the following: (1) The completion of X is a p-Banach space (∞-dimensional and with separating dual) isomorphic to a subspace of lp and contains a subspace isomorphic to lp. (2) Bloc p (Ω) is not locally convex. (3) If 0 < p < q ≤ 1, then Bloc(Ω) ⫋ Bloc q (Ω). (4) All quasi-Banach subspace of Bloc p (Ω) is isomorphic to a subspace of lp. Proof. (1) Since the operator X → {χju : u ∈ Bloc p(⋅)(Ω)}(⊂ Bp ∩ E󸀠(Kj+1)) : [u]j → χju is an isometry, the completion of X is a p-Banach space isometric to a closed subspace of Bp ∩ E󸀠(Kj+1). Let a, b > 0 be such that b < π and Kj+1 ⊂ [−a, a]n and consider the following diagram: Bp ∩ E (Kj+1) ∧ 󳨀→ L−Kj+1 p j 󳨀→ L[−a,a] p s 󳨀→ L[−b,b] p D 󳨀→ 


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  (⋅) ,   (⋅) (Ω), and  loc (⋅) (Ω) were introduced (recall that the classical Hörmander spaces  , ,   , (Ω), and  loc , (Ω) play a crucial role in the theory of linear partial differential operators (see, e.g., [6][7][8][9][10])) and there, extending a Hörmander result [6, Chapter XV] to our context, the dual of   (⋅) (Ω) (when 1 <  − ≤  + < ∞) was calculated (as a consequence some results on sequence space representation of variable exponent Hörmander spaces were obtained).In [3] the dual (  (⋅) (Ω))  was calculated when 0 <  − ≤  + ≤ 1 (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 ( loc (⋅) (Ω))  is isomorphic to   ∞ (Ω) (when 0 <  − ≤  + ≤ 1) and that the Fréchet envelope of  loc (⋅) (Ω) is  loc 1 (Ω).Applications to the study of the structure of complemented subspaces of  loc (⋅) (Ω) 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  0 -Banach local spaces of  loc (⋅) (Ω).Finally, three questions on duality and on sequence space representation of variable exponent Hörmander spaces are proposed.
Let (, ‖ ⋅ ‖  ) be a quasi-normed space whose dual   separates the points of  and let   be the unit ball of .
‖ ⋅ ‖  coincides with the Minkowski functional of the convex hull of   , ‖ ⋅ ‖  ≤ ‖ ⋅ ‖  and the inclusion  → X is continuous with dense range (if  is a Banach space then  = X).X has the property that any bounded linear operator  :  →  into a Banach space extends with preservation of norm to a bounded linear operator L : X → ; thus ( X)  (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 𝑋[T]
be a metrizable topological linear space such that its dual   (= ([T])  ) separates points of .The Mackey topology of [T], (,   ), is the finest locally convex topology on  which has , where Ũ denotes the T-closed convex hull of   , is a base of neighborhoods of zero for (,   ) and thus this topology is metrizable.The Fréchet envelope X of when [T] is a Fréchet space).X coincides with the Banach envelope of [T] when this space is quasinormed.If  is the canonical injection of [T] into X, then the transpose of  is an algebraic isomorphism of ( X)  onto ([T])  .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 T of  to a continuous linear mapping taking X into Ŷ.
If in addition  and  are -spaces and () = , then T( X) = Ŷ.(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.)

The Dual and the Fréchet Envelope of
In [6], the isomorphism and  a weight satisfying the estimate ( + ) ≤ (1 + ||)  (), ,  ∈ R  ,  and  positive constants).In Theorem 4.3 of [5] this isomorphism is extended to variable exponent Hörmander spaces with when the exponent (⋅) satisfies 0 <  − ≤  + ≤ 1 (the techniques used are different from those used in [5] since if  + < 1 the dual of  (⋅) is trivial and the steps  (⋅) ∩ E  () are quasi-Banach spaces instead of Banach spaces) and several applications of this result were given.
Proof.Let ‖ ⋅ ‖  be the quasi-norm on  which generates the topology of .Then the identity id  : (, ‖ ⋅ ‖  ) → [T] is an isomorphism.Thus, for every , there exists an   > 0 such that ‖‖  ≤   ‖‖  for all  ∈ , and there exist also an integer  and  > 0 so that ‖‖  ≤ ‖‖  for all  ∈ .Next, fix  ≥ .Then, for every  ∈ , we have which 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 Im  = , there exist an integer  ≥  and a constant  > 0 such that ‖‖  ≤ ‖‖  for every  ∈ .Then it is easy to check that, for every  ≥ , the mapping   :   →   defined by Finally, if  is complete then the extension of   to Ẽ , P , is a continuous projection in Ẽ such that Im P =   ().
In order to complete the proof we consider a sequence =1 converges in   .By applying (1), we see that Hence it follows that  = 0, that is, that ‖ ⟨⋅, ] ⟩ ‖   → 0. This contradiction concludes the proof of (2).
holds for all  ∈ .