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Proximinality in Banach space valued Musielak-Orlicz spaces

Journal of Inequalities and Applications20142014:146

https://doi.org/10.1186/1029-242X-2014-146

  • Received: 22 December 2013
  • Accepted: 27 March 2014
  • Published:

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 ) .

Keywords

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

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 y A ,

     
  2. (b)

    y φ ( y , t ) is measurable for all t 0 .

     

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

Definition 3 Let φ Φ ( A , μ ) . Define
ϱ φ ( f ) : = A φ ( y , f ( y ) ) d μ ( y )
for strongly μ-measurable functions f : A X . Then the Bochner-Musielak-Orlicz space L φ ( A , X ) is the collection of all strongly μ-measurable functions f : A X 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 φ Φ ( A , μ ) . The function φ is said to obey the Δ 2 -condition if there exists K 2 such that
φ ( s , 2 t ) K φ ( s , t )

for all s A and all t 0 .

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 = { x E : 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 x X there exists y Y such that
x y = 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 x X , 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 ) , Y X , 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 φ Φ ( A , μ ) . For f L φ ( A , X ) , define ϕ : A R 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 x X , f n ( s ) f ( s ) 0 implies that | dist ( f n ( s ) , Y ) dist ( f ( s ) , Y ) | 0 . Moreover, each function ϕ n : A R 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 ( λ ϕ ) ρ φ ( λ f ) < 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 ) d μ ( s ) A i φ ( s , dist ( x i , Y ) ) d μ ( 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 ) 1 ϵ . 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 φ Φ ( 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 s A . 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 s A .

Corollary 2 Let Y be a Chebyshev subspace of a Banach space X. Suppose φ Φ ( 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 φ Φ ( 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 s A . Then g is a best approximation to f from L φ ( A , Y ) .

Proof Since 0 Y , it follows that g ( s ) 2 f ( 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 ( f g ) ) ρ φ ( λ ( f h ) ) 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 Q T 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 : T H is said to be analytic measurable if f 1 ( C ) σ ( A ) for every C B ( 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 x H 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 M X , there exists an analytic measurable mapping h : M Y such that h ( M ) is separable in Y and h ( x ) is a best approximation of x in Y for any x M .

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 φ Φ ( 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 φ Φ ( 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 x X , let f ( t ) : = χ Q ( t ) x , t A . 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 s A 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 φ Φ ( 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 φ Φ ( 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 n N , 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 y Y ,
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 0 Y , it follows that g ( t ) 2 f ( 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 φ Φ ( A , μ ) satisfies E φ ( A ) = L φ ( A ) and, for each t A , 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 t A , 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 y Y ,
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 y Y ,
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 0 Y , it follows that h ( t ) 2 f ( 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 φ Φ ( A , μ ) such that E φ ( A ) = L φ ( A ) , for each t A , 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].

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Hainan Normal University, Haikou, 571158, China

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