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Proximinality in Banach space valued Musielak-Orlicz spaces

Abstract

Let (A,A,μ) be a σ-finite complete measure space and let Y be a subspace of a Banach space X. Let φ be a generalized Φ-function on (A,A,μ). Denote by L φ (A,Y) and L φ (A,X) the Musielak-Orlicz spaces whose functions take values in Y and X, respectively. Firstly, let f L φ (A,X), we characterize the distance of f from L φ (A,Y). Then, if Y is weakly -analytic and proximinal in X, we show that L φ (A,Y) is proximinal in L φ (A,X). Finally, we give the connection between the proximinality of L φ (A,Y) in L φ (A,X) and the proximinality of L 1 (A,Y) in L 1 (A,X).

1 Introduction

It is well known that Musielak-Orlicz spaces include many spaces as special spaces, such as Lebesgue spaces, weighted Lebesgue spaces, variable Lebesgue spaces and Orlicz spaces; see [1]. Especially, in recent decades, variable exponent function spaces, such as Lebesgue, Sobolev, Besov, Triebel-Lizorkin, Hardy, Morrey, and Herz spaces with variable exponents, have attracted much attention; see [216] and references therein. Cheng and the author discussed geometric properties of Banach space valued Bochner-Lebesgue and Bochner-Sobolev spaces with a variable exponent in [17]. Very recently, Musielak-Orlicz-Hardy spaces have been systemically developed; see, for example, [1822]. These spaces have many applications in various fields such as PDE, electrorheological fluids, and image restoration; see [6, 2325].

In recent years, proximinality in Banach space valued Bochner-Lebesgue spaces with constant exponent have been extensively studied; see [2633]. Proximinality in Banach space valued Bochner-Lebesgue spaces with variable exponent was discussed by the author in [34]. In fact, we generalized those results in [29, 31] to Banach space valued Bochner-Lebesgue spaces with a variable exponent. Khandaqji, Khalil and Hussein considered proximinality in Orlicz-Bochner function spaces on the unit interval in [35], and Al-Minawi and Ayesh consider the same problem on finite measures in [36]. The best simultaneous approximation in Banach space valued Orlicz spaces was discussed in [37, 38]. Micherda discussed proximinality of subspaces of vector-valued Musielak-Orlicz spaces via modular in [39]. However, as usual, one considers the best approximation via the norm, so in this paper, we will discuss proximinality of subspaces of vector-valued Musielak-Orlicz spaces via the norm. To proceed, we need to recall some definitions. Our results will be given in the next section.

In what follows, (A,A,μ) will be a σ-finite complete measure space. Suppose D is a subset of A, let χ D be the indicator function on D. Let (X,) be a Banach space. The dual space of X is the vector space X of all continuous linear mappings from X to or . To avoid a double definition we let be either or .

Definition 1 A convex, left-continuous function φ:[0,)[0,] with φ(0)=0, lim t 0 + φ(t)=0, lim t φ(t)= is called a Φ-function. It is called positive if φ(t)>0 for all t>0.

It is easy to see that if φ is a Φ-function, then it is nondecreasing on [0,).

Definition 2 Let (A,A,μ) be a σ-finite complete measure space. A real function φ:A×[0,)[0,] is called a generalized Φ-function on (A,A,μ) if

  1. (a)

    φ(y,) is a Φ-function for all yA,

  2. (b)

    yφ(y,t) is measurable for all t0.

If φ is a generalized Φ-function on (A,A,μ), we write φΦ(A,μ).

Definition 3 Let φΦ(A,μ). Define

ϱ φ (f):= A φ ( y , f ( y ) ) dμ(y)

for strongly μ-measurable functions f:AX. Then the Bochner-Musielak-Orlicz space L φ (A,X) is the collection of all strongly μ-measurable functions f:AX endowed with the norm:

f L φ ( A , X ) :=inf { λ > 0 , ϱ φ ( f / λ ) 1 } .

Let

E φ (A,X):= { f L φ ( A , X ) : ρ φ ( λ f ) <  for all  λ > 0 } .

Definition 4 Let φΦ(A,μ). The function φ is said to obey the Δ 2 -condition if there exists K2 such that

φ(s,2t)Kφ(s,t)

for all sA and all t0.

When X is or , we simply denote L φ (A,X) by L φ (A), and E φ (A,X) by E φ (A). Usually, E φ (A,X) is a proper subspace of L φ (A,X). But when the φ satisfies the Δ 2 -condition, they are the same. It is easy to see that E φ (A,X)= L φ (A,X) is equivalent to E φ (A)= L φ (A), this means that the equality depends only on φ.

We remark that ρ φ is a semimodular on the space of all X-valued strongly μ-measurable functions on A. For a semimodular, we recommend the reader reference [6]. Let ρ be a semimodular on vector space E, E ρ ={xE:ρ(x/λ)< for some λ>0}, x ρ =inf{λ>0:ρ(x/λ)1}. We will use the following elementary result for a semimodular, which is Corollary 2.1.15 in [6].

Lemma 1 Let ρ be a semimodular on E, x E ρ .

  1. (i)

    If x ρ 1, then ρ(x) x ρ .

  2. (ii)

    If 1< x ρ , then x ρ ρ(x).

Let X be a Banach space and let Y be a closed subspace of X. Then Y is called proximinal in X if for any xX there exists yY such that

xy=dist(x,Y)=inf { x u : u Y } .

In this case y is called a best approximation of x in Y. If this best approximation is unique for any xX, then Y is said to be Chebyshev.

For simplicity, we denote L φ ( A , X ) or L φ ( A ) by φ . For f L φ (A,X), YX, let

dist φ ( f , L φ ( A , Y ) ) :=inf { f g φ : g L φ ( A , Y ) } .

2 Main results

Firstly, we estimate dist φ (f, L φ (A,Y)).

Theorem 1 Let Y be a subspace of Banach space X. Suppose φΦ(A,μ). For f L φ (A,X), define ϕ:AR by ϕ(s):=dist(f(s),Y). Then

  1. (i)

    ϕ L φ (A) and dist φ (f, L φ (A,Y)) ϕ φ ;

  2. (ii)

    dist φ (f, L φ (A,Y))= ϕ φ for f E φ (A,X).

Proof (i) Given f L φ (A,X), we see that there exists a sequence of simple functions { f n } which converges to f almost everywhere and in L φ (A,X). Since the distance function d(x,Y) is a continuous function of xX, f n (s)f(s)0 implies that |dist( f n (s),Y)dist(f(s),Y)|0. Moreover, each function ϕ n :AR defined by ϕ n (s):=dist( f n (s),Y) is a simple function; therefore we conclude that ϕ is measurable. Now, for any g L φ (A,Y) and any λ>0,

ρ φ ( λ ( f g ) ) = A φ ( s , λ f ( s ) g ( s ) ) d μ ( s ) A φ ( s , λ dist ( f ( s ) , Y ) ) d μ ( s ) = ρ φ ( λ ϕ ) .

Thus, we have

f g φ ϕ φ .

This implies ϕ L φ (A) and, by taking an infimum on g L φ (A,Y), we have

dist φ ( f , L φ ( A , Y ) ) ϕ φ .

(ii) We first assume that f is a simple function. Let f(s):= i = 1 m χ A i x i where { A i } i = 1 m are disjoint measurable subsets in A such that 0<μ( A i )< and 0 x i X for i{1,,m}. Without loss of generality, we suppose that dist φ (f, L φ (A,Y))=1. Let 0<ϵ<1. Since ϕ(s)f(s), we have ρ φ (λϕ) ρ φ (λf)< for any λ>0. Then, by the dominated convergence theorem, we find that there exists δ>0 such that

A i φ ( s , dist ( x i , Y ) + δ ) dμ(s) A i φ ( s , dist ( x i , Y ) ) dμ(s)+ ϵ m ,i{1,,m}.

Now take y i Y such that x i y i <dist( x i ,Y)+δ for i{1,,m}. Let g(s)= i = 1 m χ A i y i . Therefore f g φ dist φ (f, L φ (A,Y))=1. By Lemma 1, we see that

1 f g φ ρ φ ( f g ) = i = 1 m A i φ ( s , x i y i ) d μ ( s ) i = 1 m A i φ ( s , dist ( x i , Y ) + δ ) d μ ( s ) i = 1 m ( A i φ ( s , dist ( x i , Y ) ) d μ ( s ) + ϵ m ) = A φ ( s , dist ( f ( s ) , Y ) ) d μ ( s ) + ϵ .

Thus, ρ φ (ϕ)1ϵ. Since ϵ is arbitrary, we have ρ φ (ϕ)1. By Lemma 1 again, we have ϕ φ 1. This means that dist ( f ( ) , Y ) φ dist φ (f, L φ (A,Y)). Therefore, we have proved that dist ( f ( ) , Y ) φ = dist φ (f, L φ (A,Y)) for simple functions.

Finally, let f E φ (A,X), there exists a sequence of simple functions { g n } n N convergent to f μ-almost everywhere, g n (s)f(s) μ-almost everywhere and f g n φ 0 as n tends to ∞. Let ϕ n (s)=dist( g n (s),Y). From the previous proof, we have

ϕ n φ = dist φ ( g n , L φ ( A , Y ) ) .

It is easy to see that dist φ ( g n , L φ (A,Y)) dist φ (f, L φ (A,Y)) as n. Since ϕ n (s) g n (s)f(s) μ-almost everywhere, and ϕ n (s)ϕ(s) μ-almost everywhere as n tends to ∞, by Lemma 2.3.16(c) in [6], we conclude that ϕ n ϕ in L φ (A). Hence, letting n, we see that

ϕ φ = dist φ ( f , L φ ( A , Y ) ) ,

which completes the proof of Theorem 1. □

Corollary 1 Let Y be a closed subspace of a Banach space X. Suppose φΦ(A,μ). An element g of L φ (A,Y) is a best approximation to an element f in E φ (A,X) if and only if g(s) is a best approximation in Y to f(s) for almost every sA. Furthermore, if φ satisfies the Δ 2 -condition, then an element g of L φ (A,Y) is a best approximation to an element f in L φ (A,X) if and only if g(s) is a best approximation in Y to f(s) for almost every sA.

Corollary 2 Let Y be a Chebyshev subspace of a Banach space X. Suppose φΦ(A,μ) satisfies the Δ 2 -condition. If L φ (A,Y) is proximinal in L φ (A,X), then it is a Chebyshev subspace of L φ (A,X).

Remark 1 Our results Theorem 1, Corollaries 1 and 2 cover the results for vector Orlicz spaces in [36]. Indeed, in [36] the authors only considered vector Orlicz spaces on finite measures. Analogous results for best simultaneous approximation were obtained in [37, 38] for vector Orlicz spaces on finite measures.

Next, we transfer the proximinality of Y in X to L φ (A,Y) in L φ (A,X). To do so, we need some preliminaries.

Lemma 2 Let (A,A,μ) be a σ-finite complete measure space. Suppose φΦ(A,μ). Let Y be a proximinal subspace of a Banach space X. Suppose f L φ (A,X) and g is a strongly μ-measurable function such that g(s) is a best approximation to f(s) from Y for almost everywhere sA. Then g is a best approximation to f from L φ (A,Y).

Proof Since 0Y, it follows that g(s)2f(s) μ-almost everywhere. Thus, g L φ (A,Y). For each h L φ (A,Y), by assumption we know that f(s)g(s)f(s)h(s) μ-almost everywhere. So ρ φ (λ(fg)) ρ φ (λ(fh)) for any λ>0. Thus, f g φ f h φ . This ends the proof. □

Definition 5 Let (T,τ) be a Polish space (i.e. a topological space which is separable and completely metrizable). A set QT is analytic if it is empty or if there exists a continuous mapping f: N N T satisfying f( N N )=Q, where N N denotes the space of all infinite sequences of natural numbers endowed with the Tychonoff topology.

Definition 6 Let (T,τ) be a Polish space, H a topological space and denote by σ(A) the smallest σ-algebra containing all analytic subsets of T. Then a mapping f:TH is said to be analytic measurable if f 1 (C)σ(A) for every CB(H), where B(H) is for the Borel sets of H.

Definition 7 Let H, T be topological spaces. Then a multifunction F:H 2 T is said to be upper semi-continuous if for every xH and for every open set U satisfying F(x)U, there exists an open neighborhood V of x such that F(V)U.

Definition 8 A subset C of a topological space T is -analytic if it can be written as C= σ N N F(σ) for some upper semi-continuous mapping F: N N 2 T with compact values. In the case when T is a Banach space endowed with its weak topology, C is said to be weakly -analytic.

For the theory of -analytic sets, we recommend [40]. Specially, all reflexive and all separable Banach spaces are weakly -analytic. The following lemma is just Theorem 3.3 in [31].

Lemma 3 Let (X,) be a real Banach space and let Y be a proximinal, weakly -analytic convex subset of X. Then, for each closed and separable set MX, there exists an analytic measurable mapping h:MY such that h(M) is separable in Y and h(x) is a best approximation of x in Y for any xM.

Thus, following the argument of the proof of (i) → (ii) in [[39], p.185], we have the following conclusion, the details being omitted.

Theorem 2 Let (A,A,μ) be a σ-finite complete measure space. Suppose φΦ(A,μ), and Y is a weakly -analytic linear subspace of a real Banach space X. If Y is proximinal in X, then L φ (A,Y) is proximinal in L φ (A,X).

Theorem 3 Let (A,A,μ) be a σ-finite measure space. Suppose φΦ(A,μ) such that E φ (A)= L φ (A). Let Y be a linear subspace of a real Banach space X. If L φ (A,Y) is proximinal in L φ (A,X), then Y is proximinal in X.

Proof Since the measure μ is σ-finite, let us choose positive measure set Q such that χ Q L φ (A). For any xX, let f(t):= χ Q (t)x, tA. Then f L φ (A,X). By the assumption, we know that there is a g in L φ (A,Y) which is a best approximation element of f. Consequently, g(s) is a best approximation to f(s) in Y for almost every sA by Corollary 1. Therefore, there is a best approximation element of x in Y. Thus, Y is proximinal in X. □

From Theorems 2 and 3, we deduce the following corollary.

Corollary 3 Let (A,A,μ) be a σ-finite complete measure space. Suppose φΦ(A,μ) such that E φ (A)= L φ (A). Let Y be a weakly -analytic linear subspace of a real Banach space X. Then the following conditions are equivalent:

  1. (i)

    Y is proximinal in X;

  2. (ii)

    L φ (A,Y) is proximinal in L φ (A,X).

Remark 2 An analog to Corollary 3 in terms of a modular was obtained in [39].

Finally, we give a characterization of proximinity of L φ (A,Y) in L φ (A,X) via the proximinity of L 1 (A,Y) in L 1 (A,X). When L φ (A,X) is a Bochner-Lebesgue space, which was obtained in [29] and [32, 33] on finite measure spaces and σ finite measure spaces, respectively. L φ (A,X) is a Bochner-Orlicz space, which was discussed in [35].

Theorem 4 Let (A,A,μ) be a σ-finite complete measure space. Suppose φΦ(A,μ) such that the set of simple functions, S(A,μ), satisfies S(A,μ) L φ (A,μ), where φ is the conjugate function of φ (see [6]). Let Y be a closed subspace of a Banach space X. If L 1 (A,Y) is proximinal in L 1 (A,X), then L φ (A,Y) is proximinal in L φ (A,X).

Proof Since A is σ-finite, we may write A= i = 1 A i , where { A i } is a sequence of disjoint measurable sets each of finite measure. Let f L φ (A,X). For any nN, since μ( A n )<, then χ A n L φ (A). Thus, by the norm conjugate formula (see Corollary 2.7.5 in [6]), we find that f χ A n L 1 (A,X). By assumption, we know that there exists g n L 1 (A,Y) such that

f χ A n g n L 1 f χ A n h L 1 ,h L 1 (A,Y).

By Corollary 1, we have, for all yY,

f ( t ) χ A n g n ( t ) f ( t ) χ A n y

μ-almost everywhere. Therefore g n (t)=0 μ-almost every t A n c . Let g= n = 1 g n . Since f= n = 1 f χ A n , it follows that for all h L φ (A,Y),

f ( t ) g ( t ) f ( t ) h ( t )

μ-almost everywhere. Because 0Y, it follows that g(t)2f(t). Thus, g L φ (A,Y) and

f g φ f h φ

for all h L φ (A,Y). This finishes the proof. □

Theorem 5 Let (A,A,μ) be a σ-finite complete measure space. Suppose φΦ(A,μ) satisfies E φ (A)= L φ (A) and, for each tA, φ(t,) is strictly increasing. Let Y be a closed subspace of a Banach space X. If L φ (A,Y) is proximinal in L φ (A,X), then L 1 (A,Y) is proximinal in L 1 (A,X).

Proof We use the idea from [35]. Indeed in [35] the authors only considered Banach space valued Orlicz spaces on the unit interval. Since, for each tA, φ(t,) is strictly increasing, let φ 1 (t,) be its inverse function, which means, for each s[0,), φ(t, φ 1 (t,s))=s. Define the map J: L 1 (A,X) L φ (A,X) by setting

J(f)(t):={ φ 1 ( t , f ( t ) ) f ( t ) f ( t ) , f ( t ) 0 ; 0 , f ( t ) = 0 .

Then J(f)(t)= φ 1 (t,f(t)). Therefore ρ φ (J(f))= f L 1 . So J is injective. Moreover, if g L φ (A,X), let

f(t):={ φ ( t , g ( t ) ) g ( t ) g ( t ) , g ( t ) 0 ; 0 , g ( t ) = 0 .

Then f(t)X and f(t)=φ(t,g(t)). Thus, f L 1 (A,X). In addition, for g(t)0,

J(f)(t)= φ 1 ( t , φ ( t , g ( t ) ) ) φ ( t , g ( t ) ) f(t)= g ( t ) φ ( t , g ( t ) ) f(t)=g(t).

If g(t)=0, then f(t)=0 also, thus J(f)(t)=0=g(t). Hence J is surjective and J( L 1 (A,Y))= L φ (A,Y) also.

Now, let f L 1 (A,X). Without loss of generality we may suppose that f(t)0 μ-almost everywhere, for otherwise we can restrict our measure to the support of f. Since J(f) L φ (A,X), by the assumption, we know that there exists some g L 1 (A,Y) such that

J ( f ) J ( g ) φ J ( f ) J ( v ) φ

for all v L 1 (A,Y). By Corollary 1, we see that, for all yY,

J ( f ) ( t ) J ( g ) ( t ) J ( f ) ( t ) y

μ-almost everywhere. Multiplying both sides of the last inequality by f ( t ) φ 1 ( t , f ( t ) ) , we obtain, for all yY,

f ( t ) f ( t ) φ 1 ( t , f ( t ) ) φ 1 ( t , g ( t ) ) g ( t ) g ( t ) f ( t ) y .

Let h(t)= f ( t ) φ 1 ( t , f ( t ) ) φ 1 ( t , g ( t ) ) g ( t ) g(t). Since h(t) is a best approximation of f(t) in Y, and 0Y, it follows that h(t)2f(t). Therefore, h L 1 (A,Y). Thus, for all w L 1 (A,Y),

f ( t ) h ( t ) f ( t ) w ( t )

μ-almost everywhere. Thus, by Corollary 1 h is a best approximation of f in L 1 (A,Y). This finishes the proof. □

From Theorems 4 and 5, we deduce the following corollary.

Corollary 4 Let (A,A,μ) be a σ-finite complete measure space. Suppose φΦ(A,μ) such that E φ (A)= L φ (A), for each tA, φ(t,) is strictly increasing and the set of simple functions S(A,μ) satisfies S(A,μ) L φ (A,μ). Let Y be a closed subspace of a Banach space X. Then the following conditions are equivalent:

  1. (i)

    L 1 (A,Y) is proximinal in L 1 (A,X);

  2. (ii)

    L φ (A,Y) is proximinal in L φ (A,X).

Remark 3 When (A,μ) is a finite measure and φ is a Orlicz function that satisfies the Δ 2 -condition, the result of Corollary 4 was obtained in [36]. While (A,μ) is the unit interval and φ is a Young function that satisfies the Δ 2 -condition, the result of Corollary 4 was obtained in [35].

References

  1. 1.

    Musielak J: Orlicz Spaces and Modular Spaces. Springer, Berlin; 1983.

    Google Scholar 

  2. 2.

    Kováčik O, Rákosník J:On spaces L p ( x ) and W k , p ( x ) . Czechoslov. Math. J. 1991, 41: 592-618.

    Google Scholar 

  3. 3.

    Almeida A, Drihem D: Maximal, potential and singular type operators on Herz spaces with variable exponents. J. Math. Anal. Appl. 2012, 394: 781-795. 10.1016/j.jmaa.2012.04.043

    MathSciNet  Article  Google Scholar 

  4. 4.

    Almeida A, Hästö P: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 2010, 258: 1628-1655. 10.1016/j.jfa.2009.09.012

    MathSciNet  Article  Google Scholar 

  5. 5.

    Diening L, Hästö P, Roudenko S: Function spaces of variable smoothness and integrability. J. Funct. Anal. 2009, 256: 1731-1768. 10.1016/j.jfa.2009.01.017

    MathSciNet  Article  Google Scholar 

  6. 6.

    Diening L, Harjulehto P, Hästö P, Růžička M: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Berlin; 2011.

    Google Scholar 

  7. 7.

    Harjulehto P, Hästö P, Le UV, Nuortio M: Overview of differential equations with non-standard growth. Nonlinear Anal. 2010, 72: 4551-4574. 10.1016/j.na.2010.02.033

    MathSciNet  Article  Google Scholar 

  8. 8.

    Izuki M: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal. Math. 2010, 36: 33-50. 10.1007/s10476-010-0102-8

    MathSciNet  Article  Google Scholar 

  9. 9.

    Izuki M: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 2010, 59: 199-213. 10.1007/s12215-010-0015-1

    MathSciNet  Article  Google Scholar 

  10. 10.

    Izuki M: Vector-valued inequalities on Herz spaces and characterizations of Herz-Sobolev spaces with variable exponent. Glas. Mat. 2010, 45: 475-503. 10.3336/gm.45.2.14

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kempka H: 2-Microlocal Besov and Triebel-Lizorkin spaces of variable integrability. Rev. Mat. Complut. 2009, 22: 227-251.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kempka H: Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces. J. Funct. Spaces Appl. 2010, 8: 129-165. 10.1155/2010/469513

    MathSciNet  Article  Google Scholar 

  13. 13.

    Nakai E, Sawano Y: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 2012, 262: 3665-3748. 10.1016/j.jfa.2012.01.004

    MathSciNet  Article  Google Scholar 

  14. 14.

    Samko S: Variable exponent Herz spaces. Mediterr. J. Math. 2013, 10: 2007-2025. 10.1007/s00009-013-0285-x

    MathSciNet  Article  Google Scholar 

  15. 15.

    Xu J: Variable Besov and Triebel-Lizorkin spaces. Ann. Acad. Sci. Fenn., Math. 2008, 33: 511-522.

    MathSciNet  Google Scholar 

  16. 16.

    Xu J: The relation between variable Bessel potential spaces and Triebel-Lizorkin spaces. Integral Transforms Spec. Funct. 2008, 19: 599-605. 10.1080/10652460802030631

    MathSciNet  Article  Google Scholar 

  17. 17.

    Cheng C, Xu J: Geometric properties of Banach space valued Bochner-Lebesgue spaces with variable exponent. J. Math. Inequal. 2013, 7: 461-475.

    MathSciNet  Article  Google Scholar 

  18. 18.

    Cao J, Chang DC, Yang D, Yang S: Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems. Trans. Am. Math. Soc. 2013, 365: 4729-4809. 10.1090/S0002-9947-2013-05832-1

    MathSciNet  Article  Google Scholar 

  19. 19.

    Liang Y, Huang J, Yang D: New real-variable characterizations of Musielak-Orlicz Hardy spaces. J. Math. Anal. Appl. 2012, 395: 413-428. 10.1016/j.jmaa.2012.05.049

    MathSciNet  Article  Google Scholar 

  20. 20.

    Yang D, Yang S:Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of R n . Indiana Univ. Math. J. 2012, 61: 81-129. 10.1512/iumj.2012.61.4535

    MathSciNet  Article  Google Scholar 

  21. 21.

    Yang D, Yang S: Local Hardy spaces of Musielak-Orlicz type and their applications. Sci. China Math. 2012, 55: 1677-1720. 10.1007/s11425-012-4377-z

    MathSciNet  Article  Google Scholar 

  22. 22.

    Yang D, Yang S: Musielak-Orlicz-Hardy spaces associated with operators and their applications. J. Geom. Anal. 2014, 24: 495-570. 10.1007/s12220-012-9344-y

    MathSciNet  Article  Google Scholar 

  23. 23.

    Chen Y, Levine S, Rao R: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 2006, 66: 1383-1406. 10.1137/050624522

    MathSciNet  Article  Google Scholar 

  24. 24.

    Li F, Li Z, Pi L: Variable exponent functionals in image restoration. Appl. Math. Comput. 2010, 216: 870-882. 10.1016/j.amc.2010.01.094

    MathSciNet  Article  Google Scholar 

  25. 25.

    Růžička M: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin; 2000.

    Google Scholar 

  26. 26.

    Abu-Sirhan E: Simultaneous approximation in function spaces. In Approximation Theory XIII: San Antonio Edited by: Neamtu M, Schumaker L. 2010, 321-329.

    Google Scholar 

  27. 27.

    Abu-Sirhan E: Best simultaneous approximation in function and operator spaces. Turk. J. Math. 2012, 36: 101-112.

    MathSciNet  Google Scholar 

  28. 28.

    Khalil R:Best approximation in L p (I,X). Math. Proc. Camb. Philos. Soc. 1983, 94: 277-279. 10.1017/S0305004100061120

    MathSciNet  Article  Google Scholar 

  29. 29.

    Khalil R, Deeb W:Best approximation in L p (I,X). J. Approx. Theory 1989, 59: 296-299. 10.1016/0021-9045(89)90094-4

    MathSciNet  Article  Google Scholar 

  30. 30.

    Khalil R, Saidi F:Best approximation in L 1 (I,X). Proc. Am. Math. Soc. 1995, 123: 183-190.

    MathSciNet  Google Scholar 

  31. 31.

    Light WA:Proximinality in L p (S,Y). Rocky Mt. J. Math. 1989, 19: 251-259. 10.1216/RMJ-1989-19-1-251

    MathSciNet  Article  Google Scholar 

  32. 32.

    Mendoza J:Proximinality in L p (μ,X). J. Approx. Theory 1998, 93: 331-343. 10.1006/jath.1997.3163

    MathSciNet  Article  Google Scholar 

  33. 33.

    You ZY, Guo TX:Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in L p (μ,X). J. Approx. Theory 1994, 78: 314-320. 10.1006/jath.1994.1081

    MathSciNet  Article  Google Scholar 

  34. 34.

    Xu, J: Proximinality in Banach space valued Bochner-Lebesgue spaces with variable exponent (submitted)

  35. 35.

    Khandaqji M, Khalil R, Hussein D: Proximinality in Orlicz-Bochner function spaces. Tamkang J. Math. 2003, 34: 71-75.

    MathSciNet  Google Scholar 

  36. 36.

    Al-Minawi H, Ayesh S: Best approximation in Orlicz spaces. Int. J. Math. Math. Sci. 1991, 14: 245-252. 10.1155/S0161171291000273

    MathSciNet  Article  Google Scholar 

  37. 37.

    Khandaqji M, Al-Sharif S: Best simultaneous approximation in Orlicz spaces. Int. J. Math. Math. Sci. 2007., 2007: Article ID 68017

    Google Scholar 

  38. 38.

    Shen ZS, Yang ZY:Best simultaneous approximation in L Φ (I,X). J. Math. Res. Expo. 2010, 30: 863-868.

    MathSciNet  Google Scholar 

  39. 39.

    Micherda B: On proximinal subspaces of vector-valued Orlicz-Musielak spaces. J. Approx. Theory 2013, 174: 182-191.

    MathSciNet  Article  Google Scholar 

  40. 40.

    Kakol J, Kubiś W, López-Pellicer M: Descriptive Topology in Selected Topics of Functional Analysis. Springer, Berlin; 2011.

    Google Scholar 

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Acknowledgements

The author would like to thank the referee for carefully reading which made the presentation more readable and for his or her suggestion for references [3638]. The author was supported by the National Natural Science Foundation of China (Grant No. 11361020) and the National Natural Science Foundation of Hainan Providence (113004).

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Correspondence to Jingshi Xu.

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Xu, J. Proximinality in Banach space valued Musielak-Orlicz spaces. J Inequal Appl 2014, 146 (2014). https://doi.org/10.1186/1029-242X-2014-146

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Keywords

  • proximinality
  • Musielak-Orlicz space
  • best approximation
  • weakly -analytic