The Kolmogorov-Riesz Theorem and Some Compactness Criterions of Bounded Subsets in Weighted Variable Exponent Amalgam and Sobolev Spaces

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p(.) q In this study, we focus especially on the spaces Lw ,ℓ on R, and discuss totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. p(.) p(.) q Moreover, it is well known that the spaces Lw and Lw ,ℓ are not translation

p(.) q invariant and the map y −→ Lyf is not continuous for f ∈ Lw ,ℓ . Also, the p(.) Rn Young theorem kf ∗ gkp(.),w ≤ kfkp(.),w kgk1 is not valid for f ∈ Lw ( ) and 1 n p(.) q g ∈ L (R ) , see [23]. Hence, our concerned amalgam space Lw ,ℓ whose local component doesn’t provide conditions in [29, Deﬁnition 5.1]. In this regard, we will propose some new conditions and criterions for compactness of bounded subsets in p(.) q p(.) these spaces. In addition, Gurkanli [16] showed that Lw ,ℓ = Lw under the

p(.) q p(.) some conditions, that is, the space Lw ,ℓ is a extension of Lw . Finally, we will obtain that our main theorem provides a generalization of the corresponding results, such as Bandaliyev [5], Bandaliyev and Górka [6], Górka and Macios [12], [13], Górka and Rafeiro [14] and Rafeiro [30].

2. Notations and Preliminaries In this section, we give some essential definitions, theorems and compactness criterions about totally bounded subsets in weighted variable exponent Lebesgue spaces. Definition 1. Let (X, d) be a metric space and ε> 0. A subset K in X is called a ε-net (or ε-cover) for X if for every x ∈ X there is a xε ∈ K such that d (x, xε) <ε. Moreover, a metric space is called totally bounded if it admits a finite ε-net. It is known that a subset of a complete metric space relatively compact (i.e. its closure is compact) if and only if it is totally bounded, see [35]. Theorem 1. Assume that X is a metric space and K ⊂ X. Then the following conditions are equivalent. (i) K is totally bounded (or, precompact) and complete (ii) K is compact. Definition 2. Let X and Y be metric spaces. A family F of functions from X to Y is said to be equicontinuous if given ε> 0 there exists a number δ > 0 such that dY (f(x),f(y)) <ε for all f ∈ F and all x, y ∈ X satisfying dX (x, y) < δ. The following theorem is quite useful for several compactness results. Theorem 2. ([18])Let X be a metric space. Assume that, for every ε > 0, there exists some δ > 0, a metric space W , and a mapping Φ: X −→ W so that Φ[X] is totally bounded, and whenever x, y ∈ X are such that d (Φ(x), Φ(y)) < δ, then d (x, y) < ε. Then X is totally bounded. Definition 3. ([23])For a measurable function p (.): Rn −→ [1, ∞) (called the variable exponent on Rn by the symbol P (Rn)), we put p− = essinfp(x), p+ = esssupp(x). x∈Rn x∈Rn The variable exponent Lebesgue spaces Lp(.)(Rn) is defined as the set of all measur- Rn able functions f on such that ̺p(.)(λf) < ∞ for some λ> 0, equipped with the THE KOLMOGOROV-RIESZ COMPACTNESS THEOREM AND SOME COMPACTNESS CRITERIONS3

Luxemburg norm f kfk = inf λ> 0 : ̺ ≤ 1 , p(.) p(.) λ p(x) where ̺p(.)(f)= |f(x)| dx. ZRn + p(.) Rn p(.) Rn If p < ∞, then f ∈ L ( ) iff ̺p(.)(f) < ∞. The set L ( ) is a Banach space with the norm k.kp(.). Moreover, the norm k.kp(.) coincides with the usual Lebesgue norm k.kp whenever p(.)= p is a constant function. Definition 4. A measurable and locally integrable function w : Rn −→ (0, ∞) is called a weight function. The weighted modular is defined by

p(x) ̺p(.),w(f)= |f(x)| w(x)dx. ZRn

p(.) n The weighted variable exponent Lebesgue space Lw (R ) consists of all measurable 1 p(.) Rn p(.) Rn functions f on for which kfk p(.) Rn = fw < ∞. Also, Lw ( ) is a Lw ( ) p(.) uniformly convex Banach space, thus reﬂexive. The relations between the modular ̺ (.) and k.k p(.) n as follows p(.),w Lw (R ) 1 1 1 1 p− p+ p− p+ min ̺ (f) ,̺ (f) ≤ kfk p(.) n ≤ max ̺ (f) ,̺ (f) p(.),w p(.),w Lw (R ) p(.),w p(.),w n p+ p− o n p+ p− o min kfk p(.) n , kfk p(.) n ≤ ̺p(.),w(f) ≤ max kfk p(.) n , kfk p(.) n . Lw (R ) Lw (R ) Lw (R ) Lw (R ) n o p(.) n p(.)n n o Moreover, if 0 < C ≤ w, then we have Lw (R ) ֒→ L (R ), since one easily sees that C |f(x)|p(x) dx ≤ |f(x)|p(x) w(x)dx ZRn ZRn and C kfk ≤kfk p(.) n , see [2]. p(.) Lw (R ) Rn 1 1 Theorem 3. Let p (.) , q (.) ∈ P ( ) such that p(.) + q(.) = 1. Then for f ∈ p(.) Rn q(.) Rn 1 Rn Lw ( ) and g ∈ Lw∗ ( ), we have fg ∈ L ( ) and

|f(x)g(x)| dx ≤ C kfk p(.) Rn kgk q(.) Rn , Lw ( ) Lw∗ ( ) ZRn where w∗ = w1−q(.). Proof. By the H¨older inequality for variable exponent Lebesgue spaces, we get

1 − 1 |f(x)g(x)| dx = |f(x)g(x)| w(x) p(x) p(x) dx ZRn ZRn 1 − 1 ≤ C fw p(.) gw p(.) p(.) q(.) for some C > 0. That is the desired result.

1 Rn Rn The space Lloc ( ) consists of all measurable functions f on such that fχK ∈ L1 (Rn) for any compact subset K ⊂ Rn. It is a topological vector space with the family of seminorms f −→ kfχK kL1 . A Banach function space (shortly, BF-space) 4 ISMAILAYDINANDCIHANUNAL

Rn on is a Banach space (B, k.kB) of measurable functions which is continuously 1 Rn 1 Rn ⊃ embedded into Lloc ( ), brieﬂy B ֒→ Lloc ( ) , i.e. for any compact subset K Rn there exists some constant CK > 0 such that kfχK kL1 ≤ CK kfkB for all f ∈ B. p(.) Rn q(.) Rn 1 1 ∗ 1−q(.) The dual space of Lw ( ) is Lw∗ ( ), where p(.) + q(.) = 1 and w = w , see [25]. Also, it is known that if X is a BF-space, then the dual space X∗ consisting of g such that

kgkX∗ = sup |g(x)f(x)| dx f∈X ZRn kfkX ≤1 is also a BF-space. If using H¨older inequality for variable Lebesgue spaces, then we have

|f(x)g(x)| dx ≤ C kfk p(.) Rn kgk q(.) Rn Lw ( ) Lw∗ ( ) ZRn and

g p . ∗ g q(.) k k ( ) Rn ≤k kL (Rn) Lw ( ) w∗

∗ for some C > 0. Therefore, the norm k.k p(.) Rn is well deﬁned. Moreover, Lw ( )

. p . ∗ . q(.) k k ( ) Rn and k kL (Rn) are equivalent by the similar methods for dual spaces Lw ( ) w∗ ∗ p(.) p(.) n of L , see [23]. Therefore, there is a isometric isomorphism between Lw (R ) ∗ q(.) Rn p(.) Rn q(.) Rn and Lw∗ ( ). This yields that Lw ( ) = Lw∗ ( ). Rn Remark 1. Let K ⊂ with |K| < ∞. Then we have kχ k p(.) n < ∞. K Lw (R ) Proof. Fix λ ≥ 1. Since weight function w is locally integrable function Rn, then p(x) χK |χK (x)| w(x) −p(x) ̺p(.),w = dx = λ w(x)dx λ λp(x) ZRn ZK −p− −1 ≤ λ w(x)dx ≤ λ CK , ZK where CK = w(x)dx < ∞. If we take λ = CK + 1 > 0, then we have ZK kχ k p(.) n ≤ CK + 1. K Lw (R ) Deﬁnition 5. For x ∈ Rn and r > 0, we denote an open ball with center x and 1 Rn radius r by B(x, r). For f ∈ Lloc ( ) , we denote the (centered) Hardy-Littlewood maximal operator Mf of f by 1 Mf(x) = sup |f(y)| dy r>0 |B(x, r)| B(Zx,r) where the supremum is taken over all balls B(x, r).

Hästöand Diening defined the class Ap(.) to consist of those weights w such that

−pB 1 kwk = sup |B| kwk 1 p < ∞, Ap(.) L (B) w p (.) B∈ß L p(.) (B)

THE KOLMOGOROV-RIESZ COMPACTNESS THEOREM AND SOME COMPACTNESS CRITERIONS5

−1 Rn 1 1 1 1 where ß denotes the set of all balls in , pB = |B| p(x) dx and p(.) + pp(.) = B 1. Note that this class is ordinary Muckenhoupt classR Ap(.) if p (.) is a constant function, see [7]. Definition 6. We say that p(.) satisfies the local log-Hölder continuity condition if C |p(x) − p(y)|≤ 1 log e + |x−y| for all x, y ∈ Rn. If the inequality C |p(x) − p |≤ ∞ log (e + |x|) n holds for some p∞ > 1, C > 0 and all x ∈ R , then we say that p(.) satisfies the log-Hölder decay condition. We denote by the symbol P log(Rn) the class of variable exponents which are log-Hölder continuous, i.e. which satisfy the local log-Hölder continuity condition and the log-Hölder decay condition. Let p (.) , q (.) ∈ P log(Rn), 1 < p− ≤ p+ < ∞ and 1 < q− ≤ q+ < ∞. If q (.) ≤ p (.), then there exists a constant C > 0 depending on the characteristics of p (.) and q (.) such that kwk ≤ C kwk . This yields that Ap(.) Aq(.)

A1 ⊂ Ap− ⊂ Ap(.) ⊂ Ap+ ⊂ A∞ for p (.) ∈ P log(Rn) and 1 0. If the function ϕε(x)= ε ϕ( ε ) is nonnegative, belongs to C0 ( ), and satisﬁes

(i) ϕε(x)=0 if |x|≥ ε and

(ii) ϕε(x)dx =1, ZRn

then ϕε is called a molliﬁer and we deﬁne the convolution by

ϕε ∗ f(x)= ϕε(x − y)f(y)dy. ZRn 6 ISMAILAYDINANDCIHANUNAL

The following proposition was proved in [8, Proposition 2.7]. 1 Rn Proposition 2. Let ϕε be a molliﬁer and f ∈ Lloc ( ). Then

sup |ϕε ∗ f(x)|≤ Mf(x). ε>0 n p(.) n Proposition 3. ([2])Let p (.) ∈ P (R ) , w ∈ Ap(.) and f ∈ Lw (R ). Then p(.) Rn + ϕε ∗ f −→ f in Lw ( ) as ε −→ 0 . As a direct consequence of Proposition 3 there follows. ∞ Rn Corollary 1. The class C0 ( ) denotes continuous functions having continuous derivatives of all orders with compact support on Rn. Now, let p (.) ∈ P (Rn) and ∞ Rn p(.) Rn w ∈ Ap(.). Then C0 ( ) is dense in Lw ( ).

Deﬁnition 8. Let w = {wk} be a sequence of positive numbers. The weighted variable sequence Lebesgue spaces lpn (w) is deﬁned by ∞ pk lpn (w)= x = {xk} : ∃λ> 0, (λ |xk|) wk < ∞ k=1 equipped with the norm P ∞ pk 1 |xk| kxk := xw pn = inf λ> 0 : wk ≤ 1 . lpn (w) lpn (w) k=1 λ The following theorem was proved for weightedP variable exponent sequence spaces by [5] and [13]. Also, this theorem for a constant exponential case was obtained by [18]. + Theorem 4. Let F ⊂ lpn (w), p < ∞. Then the subset F is precompact in lpn (w) if and only if ∞ pk (i) F is bounded, i.e. ∀x = {xk} ∈ F , ∃C > 0, |xk| wk ≤ C k=1 (ii) For every ε > 0 there is a K = K(ε) > 0 suchP that for every x = {xk} in F 1 xw pk <ε lp (k>K) k or equivalently ∞ pk |xk| wk < ε. k=K+1 The following theorem is an extensionP to the weighted variable exponent Lebesgue spaces of the classical Riesz-Kolmogorov Theorem, see [5]. Theorem 5. Let p (.) ∈ P log(Rn) and 1 < p− ≤ p+ < ∞. Assume that w is a p(.) n weight function and w ∈ Ap(.). Then F ⊂ Lw (R ) is relatively compact if and only if p(.) n F L R f p(.) < (i) is bounded in w ( ), i.e. sup k kL (Rn) ∞ f∈F w (ii) For every ε> 0 there is a γ > 0 such that for all f ∈ F

kfk p(.) <ε Lw (|x|>γ) or equivalently |f(x)|p(x) w(x)dx < ε.

|xZ|>γ THE KOLMOGOROV-RIESZ COMPACTNESS THEOREM AND SOME COMPACTNESS CRITERIONS7

(iii) lim kf ∗ ϕε − fk p(.) Rn =0 uniformly for f ∈ F , where ϕε is a molliﬁer ε−→0+ Lw ( ) function.

p(.) n The following theorem can be proved for the spaces Lw (R ) as Theorem 3 and Theorem 4 in [13].

p(.) n log n − + Theorem 6. Let F ⊂ Lw (R ), p (.) ∈ P (R ), 1 < p ≤ p < ∞ and w ∈ p(.) n p(.) n Ap(.). Then the family F ⊂ Lw (R ) is precompact in Lw (R ) if and only if p(.) n F L R , w f p(.) < (i) is bounded in ( ), i.e. sup k kL (Rn) ∞ f∈F w (ii) For every ε> 0 there is a R> 0 such that for all f ∈ F

|f(x)|p(x) w(x)dx <ε

|xZ|>R (iii) For every ε> 0 there is a δ > 0 such that for all f ∈ F and ∀ |h| <δ

kfh − fk p(.) n <ε Lw (R ) or equivalently

p(x) |fh(x) − f(x)| w(x)dx<ε ZRn 1 where fh(x) = (f)B(x,h) = |B(x,h)| B(x,h) f(t)dt.

3. Weighted Variable ExponentR Amalgam Spaces p(.) Rn Rn Deﬁnition 9. The space Lloc,w ( ) is to be space of functions on such that n p(.) n f restricted to any compact subset K of R belongs to Lw (R ). Note that the p(.) Rn p(.) Rn 1 Rn .embeddings Lw ( ) ֒→ Lloc,w ( ) ֒→ Lloc ( ) hold

Let 1 ≤ p(.),q < ∞ and Jk = [k, k + 1), k ∈ Z. The weighted variable exponent p(.) q amalgam spaces Lw ,ℓ are deﬁned by p(.) q p(.) R Lw ,ℓ = f ∈ Lloc,w ( ): kfk p(.) q < ∞ , Lw ,ℓ where 1 q q kfk p(.) q = fχJ p(.) . Lw ,ℓ k Lw (R) k∈Z p(.) q P It is well known that Lw ,ℓ is a Banach space and does not depend on the particular choice of Jk, that is, Jk can be equal to [k, k + 1), [k, k +1] or (k, k + 1). p(.) q If the weight w is a constant function, then the space Lw ,ℓ coincides with Lp(.),ℓq . Moreover, If the exponent p(.) and the weightw are constant functions, then we have the usual amalgam space (Lp,ℓq), see [3], [20], [34]. The dual space of p(.) q r(.) t 1 1 1 1 Lw ,ℓ is isometrically isomorphic to Lw∗ ,ℓ where p(.) + r(.) = 1, q + t =1 ∗ 1−r(.) p(.) q and w = w . Also, the space Lw ,ℓ is reﬂexive. Moreover, it is known

p(.) q that Lw ,ℓ is a solid Banach function space, see [4]. 8 ISMAILAYDINANDCIHANUNAL

In 2014, Meskhi and Zaighum [27] proved the boundedness of maximal operator for weighted variable exponent amalgam spaces under some conditions, see [27, Theorem 3.3], [27, Theorem 3.4]. Throughout this paper, we assume that p (.) ∈ log n − + P (R ), 1 < p ≤ p (.) ≤ p < ∞, w ∈ Ap(.) and the maximal operator is bounded in weighted variable exponent amalgam spaces. 1 1 1 1 Remark 2. Let p(.) + r(.) =1 and q + s =1. Then there exists a constant C > 0 such that

kfgk 1 1 ≤ C kfk p(.) kgk r(.) (L ,ℓ ) L ,ℓq L ,ℓs w w∗ p(.) q r(.) s for f ∈ Lw ,ℓ and g ∈ Lw∗ ,ℓ . Moreover, the expression

p(.) q r(.) s 1 1 1 Lw ,ℓ Lw∗ ,ℓ ⊂ L ,ℓ = L is satisﬁed.

p(.) q r(.) s Proof. Let f ∈ Lw ,ℓ and g ∈ Lw∗ ,ℓ . Using H¨older inequality for variable exponent Lebesgue and classical sequences spaces, we have

kfgk 1 1 = fgχ (L ,ℓ ) Jk L1(R) k∈Z P ≤ C fχ p(.) gχ r(.) Jk Lw (R) Jk L (R) k∈Z w∗ P 1 1 q q s s p(.) r(.) ≤ C fχJk R gχJk R Lw ( ) Lw∗ ( ) k∈Z k∈Z

≤ C kfPk p(.) kgk r(.) .P L ,ℓq L ,ℓs w w∗ This completes the proof.

p(.) p(.) Deﬁnition 10. ([3],[31])Lc,w (R) denotes the functions f in Lw (R) such that suppf ⊂ R is compact, that is, p(.) R p(.) R Lc,w ( )= f ∈ Lw ( ): suppf compact . Let K ⊂ R be given. The cardinalityn of the set o

S(K)= {Jk : Jk ∩ K 6= ∅} is denoted by |S(K)|, where {Jk}k∈Z is a collection of intervals. p(.) Proposition 4. ([3])If g belongs to Lc,w (R), then 1 q (i) kgk p(.) q ≤ |S(K)| kgkLp(.)(R) for 1 ≤ q< ∞, Lw ,ℓ w

g p . S K g p(.) q (ii) k k ( ) ∞ ≤ | ( )|k kL R for = ∞, Lw ,ℓ w ( ) p(.) p(.) q (iii) Lc,w (R) ⊂ Lw ,ℓ for 1 ≤ q ≤ ∞, where K is the compact support of g.

p(.) p(.) q Theorem 7. Lc,w (R) is dense subspace of Lw ,ℓ for 1 ≤ p(.),q< ∞. Proof. If we use the similar techniques of Theorem 7 in [20] or Theorem 3.6 in [31], then we can prove the theorem similarly. THE KOLMOGOROV-RIESZ COMPACTNESS THEOREM AND SOME COMPACTNESS CRITERIONS9

Proposition 5. Cc (R), which consists of continuous functions on R whose support p(.) q is compact, is dense in Lw ,ℓ for 1 ≤ p(.),q< ∞. p(.) q p(.) q Proof. It is clear that Cc (R) is included in Lw ,ℓ . Let f ∈ Lw ,ℓ . By p(.) Theorem 7, given ε> 0 there exists g ∈ Lc,w (R) such that ε (3.1) kf − gk p(.) q < . Lw ,ℓ 2

If E is the compact support of g, then there exists h in Cc (E) such that ε g h p(.) < k − kL (E) 1 w 2 |S(E)| q

p(.) since Cc (E) is dense in Lw (E), see [1]. Hence by Proposition 4, we have

1 ε q (3.2) kg − hk p(.) q ≤ |S(E)| kg − hkLp(.) E < . Lw ,ℓ w ( ) 2 If we consider the (3.1) and (3.2), then

kf − hk p(.) q ≤ kf − gk p(.) q + kg − hk p(.) q Lw ,ℓ Lw ,ℓ Lw ,ℓ ε ε < + = ε. 2 2 This completes the proof.

By the Corollary 1, the proof of the following result can be obtained similar to Proposition 5.

∞ R p(.) q Corollary 2. The class C0 ( ) is dense in Lw ,ℓ for 1 ≤ p(.),q< ∞. p(.) q Now we give the proof of the Kolmogorov-Riesz Theorem for F ⊂ Lw ,ℓ . p(.) q Theorem 8. A subset F ⊂ Lw ,ℓ is totally bounded if and only if

p(.) q (i) F is bounded in L ,ℓ , i.e. sup kfk p(.) < ∞ w L ,ℓq f∈F w (ii) For every ε> 0 there is some γ > 0 such that for all f ∈ F

kfk p(.) q <ε Lw ,ℓ (|x|>γ)

(iii) lim kf ∗ ϕε − fk p(.) q =0 uniformly for f ∈ F , where ϕε is a molli- ε−→0+ Lw ,ℓ ﬁer function.

p(.) q Proof. Assume that F is totally bounded in Lw ,ℓ . Then, for every ε> 0 there exists a ﬁnite ε-cover for the set F . This implies plainly the boundedness of F , and then we get (i). To prove condition (ii), let ε > 0 be given. If we take the set {V1, V2, .., Vm} as an ε-cover of F , and hj ∈ Vj for j =1, ..., m, then for a γ > 0 we have

khj k p(.) q < ε. Lw ,ℓ (|x|>γ) 10 ISMAILAYDINANDCIHANUNAL

If f ∈ Vj , then we have kf − hj k p(.) q ≤ ε. This follows that Lw ,ℓ

kfk p(.) q ≤ kf − hj k p(.) q + khj k p(.) q Lw ,ℓ (|x|>γ) Lw ,ℓ (|x|>γ) Lw ,ℓ (|x|>γ)

≤ kf − hj k p(.) q + khjk p(.) q < 2ε. Lw ,ℓ Lw ,ℓ (|x|>γ) This implies the condition (ii). Finally, we show the condition (iii). Let f ∈ F be p(.) given. Then by Theorem 7, given ε> 0 there exists g ∈ Lc,w (R) such that ε (3.3) kf − gk p(.) q < . Lw ,ℓ 2max {1,c} If E is the compact support of g, then we have ε g g ϕ p(.) < (3.4) k − ∗ εkL (E) 1 w 2 |S(E)| q by Proposition 3. Also by Proposition 4, we get 1 ε q (3.5) kg − g ∗ ϕεk p(.) q ≤ |S(E)| kg − g ∗ ϕεkLp(.) E < . Lw ,ℓ w ( ) 2 If we consider the Proposition 2, the boundedness of maximal operator, (3.3) and (3.5), then we have kf − f ∗ ϕεk p(.) q ≤ kf − gk p(.) q + kg − g ∗ ϕεk p(.) q + kg ∗ ϕε − f ∗ ϕεk p(.) q Lw ,ℓ Lw ,ℓ Lw ,ℓ Lw ,ℓ

≤ max {1,c}kf − gk p(.) q + kg − g ∗ ϕεk p(.) q Lw ,ℓ Lw ,ℓ ε ε < + = ε. 2 2 p(.) q This ﬁnishes the proof of necessity. Now, we assume that F ⊂ Lw ,ℓ satisﬁes all three conditions. Let ε> 0 be given. Denote

Fε = {fε : f ∈ F , fε = f ∗ ϕε} . By Proposition 2, we have |fε(x)|≤ Mf(x), where Mf is the maximal function. Since the condition (i) hold, we have uniformly boundedness of all functions in Fε. Now, we denote

Fεε = {fεε : fε ∈ Fε, fεε = fε ∗ ϕε} . If we consider the Proposition 2 and the monotonicity of the maximal operator, then we get

|fεε (x)| ≤ |(fε ∗ ϕε) (x)|≤ M (fε) (x) ≤ M (Mf) (x) . This yields

sup kfεεk p(.) q < ∞. Lw ,ℓ fε∈Fε Therefore, we get that all functions in Fεε are uniformly bounded. If we use the techniques of Theorem 11 in [30] for the rest of proof, then we can prove the theorem similarly. The following theorem has been given us a diﬀerent characterization of precom- p(.) q pactness in Lw ,ℓ similar to Theorem 3 and Theorem 4 in [13]. THE KOLMOGOROV-RIESZ COMPACTNESS THEOREM AND SOME COMPACTNESS CRITERIONS11

p(.) q p(.) q Theorem 9. The family F ⊂ Lw ,ℓ is totally bounded in Lw ,ℓ if and only if p(.) q (i) F is bounded in L ,ℓ , i.e. sup kfk p(.) < ∞ w L ,ℓq f∈F w (ii) For every ε> 0, r −→ 0+and for all f ∈ F we have

f − (f) <ε B(.,r) p(.) q Lw ,ℓ

or equivalently lim f − (f) =0 + B(.,r) p(.) q r−→0 Lw ,ℓ 1 where (f)B(x,r) = |B(x,r )| B(x,r) f(t) dt. (iii) For every ε> 0 there is a γ > 0 such that for all f ∈ F R

kfk p(.) q < ε. Lw ,ℓ (|x|>γ)

p(.) q Proof. Assume that F ⊂ Lw ,ℓ is totally bounded. Then, for every ε> 0 there exists a ﬁnite ε-cover for the set F. Thus, we take {fl}l=1,...,m ε-cover in F such that m F ⊂ B (fl,ε) . l=1 The totally boundedness of F impliesS plainly the boundedness of F. Hence we get p(.) (i). By Theorem 7, given ε> 0 there exists g ∈ Lc,w (R) such that ε (3.6) kf − gk p(.) q < Lw ,ℓ 2max {1,c} where c> 0. Let E be the compact support of g. Now, we will show that

g − (g)B(.,r) p(.) −→ 0 Lw (E)

or equivalently p(x) g(x) − (g)B(x,r) w(x)dx −→ 0 ZE

+ p(.) 1 as r −→ 0 . By the Proposition 1, we have Lw ֒→ Lloc. Therefore, we have + (g)B(x,r) −→ g(x) for x ∈ E as r −→ 0 by the Lebesgue diﬀerentiation theorem, see [19]. If we use the boundedness of the Hardy-Littlewood maximal operator Mg p(.) for g ∈ Lw (E), then we get

p(x) p(x) p+−1 p(x) g (x) − (g)B(x,r) w (x) ≤ 2 |g (x)| + (g)B(x,r) w (x) p+−1 p(x) p(x) 1 ≤ 2 |g (x)| + | M (g) (x) | w (x) ∈ L (E) . By the Lebesgue dominated convergence theorem, we have p(x) ε g(x) − (g)B(x,r) w(x)dx < 1 . 2 |S(E)| q ZE

12 ISMAILAYDINANDCIHANUNAL for suﬃciently small r> 0. Also by Proposition 4, we get 1 ε (3.7) g − (g) ≤ |S(E)| q g − (g) < . B(.,r) p(.) q B(.,r) p(.) Lw ,ℓ Lw (E) 2

By Proposition 2, (3.6) and (3.7), we have f − (f) B(.,r) p(.) q Lw ,ℓ

≤ k f − gk p(.) q + g − (g) + (g) − (f) Lw ,ℓ B(.,r) p(.) q B(.,r) B(.,r) p(.) q Lw ,ℓ Lw ,ℓ

≤ kf − gk p(.) + kM (g − f)k p(.) + g − (g) L ,ℓq L ,ℓq B(.,r) p(.) q w w Lw ,ℓ

≤ max {1,c}kf − gk p(.) q + g − (g) < ε. Lw ,ℓ B(.,r) p(.) q Lw ,ℓ

This completes the proof of (ii). If we use similar method in Theorem 8, then we get (iii). Now, we assume that the conditions (i), (ii) and (iii) are satisﬁed. Since p(.) q Lw ,ℓ is a solid Banach function space, the proof is completed by [14, Theorem 3.1]. Rn p(.) Remark 3. Let Ω ⊂ be an open set. The set Lloc,w (Ω) is deﬁned by

p(.) p(.) Lloc,w (Ω) = f : fχK ∈ Lw (Ω) for any compact subset K ⊂ Ω with the usual identiﬁcationn of functions that are equal almost everywhere.o More- over, by [15, Lemma 2.2] it is well known that there exists a sequence of compact subsets {Kj }j∈N such that

K1 ⊂ K2 ⊂ ... ⊂ Kj ⊂ ...and Ω= Kj j∈N S ∁ 1 where Kj = x ∈ Ω: |x|≤ j and dist x, Ω ≥ j . Here, the complement of Ω is ∁ denoted by Ωn. o p(.) p(.) Lloc,w (Ω) is equipped with topology of Lw (Ω) convergence on compact subsets of Ω. In addition, any compact subset of Ω is contained in some Kj , and then the p(.) space Lloc,w (Ω) is a topological vector space with the countable family of seminorms,

pj(f)= fχKj p(.) , j =1, 2, ... Lw (Ω)

p(.) Moreover, the space Lloc,w (Ω) is a complete with respect to the metric (f,g) −→ ∞ −j p(.) min (2 ,pj (f − g) . Hence it is obtained that Lloc,w (Ω) is a Frˇechet space. j=1 X The following theorem is proved by [18] for constant exponent.

p(.) Theorem 10. A subset F ⊂ Lloc,w (Ω) is totally bounded if and only if (i) For every compact K ⊂ Ω there is some C > 0 such that

p(x) |fK (x)| w(x)dx

f(x), x ∈ K where f (x)= . K 0, otherwise (ii) For every ε> 0 and every compact K ⊂ Ω there is some r> 0 such that

kfK ∗ ϕ − fK k p(.) <ε, f ∈ F ε Lw (Ω) f(x), x ∈ K where f (x)= . K 0, otherwise p(.) p(.) Proof. The subset F ⊂ Lloc,w (Ω) is totally bounded in Lloc,w (Ω) if and only if Fj = fKj : f ∈ F is totally bounded for every j, with Kj as defined above. q Remark 4. Let {Ak}k∈Z be a family of Banach spaces. We define the space ℓ (Ak) given by q ℓ (Ak)= {x = (xk): xk ∈ Ak, kxk < ∞} , 1 q where kxk = kx kq . It can be seen that ℓq (A ) is a Banach space with k Ak k k∈Z P p(.) q q respect to the norm k.k . Moreover, Lw ,ℓ is a particular case of ℓ (Ak). Indeed, if we define the amalgam space as

p(.) q p(.) R q Lw ,ℓ = f ∈ Lloc,w ( ): fχJ p(.) R ∈ ℓ k Lw ( ) k∈Z n o p(.) take Ak = Lw (Jk) , Jk = [k, k + 1), then the map f −→ (fk), fk = fχJk is an p(.) q q p(.) isometric isomorphism from Lw ,ℓ to ℓ Lw (Jk) , see [2], [10]. Hence, for p(.) q the totally boundedness of F ⊂ Lw ,ℓ we can use the Theorem 4. Note that F is

p(.) q totally bounded in Lw ,ℓ if and only if the set fχJ p(.) R : f ∈ F k Lw ( ) k∈Z q p(.) Z n o is totally bounded in ℓ Lw (Jk) for k ∈ with 1 ≤ q< ∞ . 4. Weighted Variable Exponent Sobolev Spaces

− 1 p(.) p(.)−1 1 Rn Rn Let ϑ ∈ Lloc ( ) . Since every function in Lϑ ( ) has distributional derivatives by Proposition 1, we get that the weighted variable exponent Sobolev k,p(.) Rn spaces Wϑ ( ) are well deﬁned.

− 1 − + p(.)−1 1 Rn N Deﬁnition 11. Let 1 < p ≤ p(x) ≤ p < ∞, ϑ ∈ Lloc ( ) and k ∈ . k,p(.) Rn We deﬁne the weighted variable Sobolev spaces Wϑ ( ) by k,p(.) Rn p(.) Rn α p(.) Rn Wϑ ( )= f ∈ Lϑ ( ): D f ∈ Lϑ ( ), 0 ≤ |α|≤ k equipped with the norm n o

α kfk k,p(.) Rn = kD fk p(.) Rn , Wϑ ( ) Lϑ ( ) 0≤|Xα|≤k Nn α ∂|α| where α ∈ 0 is a multi-index, |α| = α1 + α2 + ... + αn and D = α1 αn . ∂x1 ...∂xn k,p(.) Rn Moreover, the space Wϑ ( ) is a reﬂexive Banach space. 14 ISMAILAYDINANDCIHANUNAL

1,p(.) Rn The space Wϑ ( ) is deﬁned by 1,p(.) Rn p(.) Rn p(.) Rn Wϑ ( )= f ∈ Lϑ ( ): |∇f| ∈ Lϑ ( ) . n o 1,p(.) Rn −1,q(.) Rn ∗ 1−q(.) The dual space of Wϑ ( ) is denoted by Wϑ∗ ( ) where ϑ = ϑ . The 1,p(.) Rn function ̺1,p(.),ϑ : Wϑ ( ) −→ [0, ∞) is deﬁned as ̺1,p(.),ϑ(f) = ̺p(.),ϑ(f)+ 1,p(.) Rn ̺p(.),ϑ(∇f) for every f ∈ Wϑ ( ). Also, the norm kfk 1,p(.) Rn = kfk p(.) Rn + Wϑ ( ) Lϑ ( ) 1,p(.) Rn k∇fk p(.) Rn makes the space Wϑ ( ) a Banach space, see [23]. Lϑ ( ) k,p(.) Rn Theorem 11. A subset F ⊂ Wϑ ( ) is totally bounded if and only if k,p(.) Rn (i) F is bounded in Wϑ ( ), i.e. there is a C > 0 such that for f ∈ F and 0 ≤ |α|≤ k |Dαf|p(x) ϑ(x)dx < C. ZΩ (ii) For every ε> 0 there is a γ > 0 such that for all f ∈ F and 0 ≤ |α|≤ k α kD fkp(.),ϑ(|x|>γ) <ε or equivalently

|Dαf(x)|p(x) ϑ(x)dx < ε.

|xZ|>γ

α α (iii) lim kD (f ∗ ϕε) − D fkLp(.) Rn =0 uniformly for f ∈ F and 0 ≤ |α|≤ ε−→0+ ϑ ( ) k where ϕε is a molliﬁer function. k,p(.) Rn α Proof. Note that F is totally bounded in Wϑ ( ) if and only if D [F ] = α p(.) Rn {D f : f ∈ F } is totally bounded in Lϑ ( ) for every multi-index α with 0 ≤ |α|≤ k by Theorem 5.

Using [13, Theorem 5] we have an extension of [18, Corollary 9] to weighted k,p(.) Rn variable exponent Sobolev spaces Wϑ ( ). k,p(.) Rn Theorem 12. Let F ⊂ Wϑ ( ) be given. If the following conditions are satisﬁed k,p(.) Rn (i) F is bounded in Wϑ ( ), i.e. there is C > 0 such that for f ∈ F and 0 ≤ |α|≤ k |Dαf|p(x) ϑ(x)dx < C. ZΩ (ii) for every ε> 0 there is a γ > 0 such that for all f ∈ F and 0 ≤ |α|≤ k α kD fk p(.) <ε Lϑ (|x|>γ) or equivalently

|Dαf(x)|p(x) ϑ(x)dx < ε.

|xZ|>γ THE KOLMOGOROV-RIESZ COMPACTNESS THEOREM AND SOME COMPACTNESS CRITERIONS15

(iii) for every ε> 0 there is a ρ> 0 such that

5. Applications In this section, using a compact embedding theorem for weighted variable expo- 1,p(.) Rn q(.) Rn nent Sobolev spaces Wϑ ( ) , we discuss totally bounded subsets of Lϑ ( ).

Theorem 13. ([26])Let p(.), q(.), ϑ1 and ϑ2 satisfy hypotheses in [26, Corollary .Then the continuous embedding W 1,p(.) (Rn) ֒→ Lq(.)(Rn) is satisﬁed .[3.1 ϑ1 ϑ2

Theorem 14. Assume that hypotheses of Theorem 13 hold. Also, let ϑ2 ∈ Aq(.) and F be a bounded subset of W 1,p(.) (Rn). If, for every ε > 0, there is a γ > 0 ϑ1 such that for all f ∈ F

p(x) p(x) ̺ 1,p(.) f f x f x ϑ x dx < ε, (5.1) W (|x|>γ) ( )= | ( )| + |∇ ( )| 1( ) ϑ1 |xZ|>γ then F is a totally bounded subset of Lq(.)(Rn). ϑ2 Proof. For the proof, we will show that F satisﬁes the hypotheses of Theorem 6 with p (.) replaced by q (.) . By Theorem 13, there exists a C > 0 such that

f q(.) C f 1,p(.) (5.2) k kL (Rn) ≤ k kW (Rn) ϑ2 ϑ1 for all f ∈ F ⊂ W 1,p(.) (Rn) . This yields the condition (i) of Theorem 6. Now, ϑ1 0, |.|≤ γ we set a function u (.) = f (.) χ (|.|− γ) where χ (|.|− γ) = . If we 1, |.| >γ consider the (5.2) and [26, Proposition 2.3], then we have

1 1 p− p+ (5.3) kuk q(.) Rn ≤ C max ̺ 1,p(.) Rn (u) , ̺ 1,p(.) Rn (u) . Lϑ ( ) Wϑ ( ) Wϑ ( ) 2 ( 1 1 ) By the expression (5.1), we get

p(x) p(x) ̺ 1,p(.) u u x u x ϑ x dx W (Rn) ( ) = | ( )| + |∇ ( )| 1 ( ) ϑ1 |xZ|>γ

p(x) p(x) + |u (x)| + |∇u (x)| ϑ1 (x) dx |xZ|≤γ

p(x) p(x) = |f (x)| + |∇f (x)| ϑ1 (x) dx < ε. |xZ|>γ Since ε is suﬃciently small, we obtain ∗ ̺ q(.) u u q(.) <ε . L (Rn) ( ) ≤k kL (Rn) ϑ2 ϑ2 16 ISMAILAYDINANDCIHANUNAL

Therefore, we have

q(x) q(x) ̺ q(.) u u x ϑ x dx u x ϑ x dx L (Rn) ( ) = | ( )| 2 ( ) + | ( )| 2 ( ) ϑ2 |xZ|>γ |xZ|≤γ

q(x) ∗ = |f (x)| ϑ2 (x) dx<ε

|xZ|>γ for all f ∈ F. This completes the condition of (ii) of Theorem 6. Now, we will consider the rest of proof. Let f ∈ F. Since the space C (Rn) is dense in Lq(.) (Rn) , c ϑ2 n given ε> 0 there exists g ∈ Cc (R ) such that ε (5.4) kf − gkLq(.) (Rn) < . ϑ2 2 Now, we will show that

(5.5) g − (g) −→ 0 B(.,h) q(.) Rn Lϑ ( ) 2 or equivalently q(x) g(x) − (g)B(x,h) ϑ2(x)dx −→ 0 ZRn as h −→ 0+. By the Proposition 1, we have Lq(.) ֒→ L1 . Therefore, we have ϑ2 loc Rn + (g)B(x,h) −→ g(x) for x ∈ as h −→ 0 by the Lebesgue diﬀerentiation theorem, see [19]. If we use the boundedness of the Hardy-Littlewood maximal operator Mg for g ∈ Lq(.) (Rn), then we get ϑ2 q(x) q(x) q+−1 q(x) g (x) − (g)B(x,h) ϑ2 (x) ≤ 2 |g (x)| + (g)B(x,h) ϑ2 (x) q+−1 q(x) q(x) 1 Rn ≤ 2 |g (x)| + | M (g) (x) | ϑ2 (x) ∈ L ( ) . By the Lebesgue dominated convergence theorem, we have q(x) ε g(x) − (g) ϑ (x)dx < . B(x,h) 2 2 ZRn for suﬃciently small h> 0. Therefore, if we consider (5.4) and (5.5), then we obtain

f − (f) B(.,h) q(.) Rn Lϑ ( ) 2

f g q(.) g g g f ≤ k − k Rn + − ( )B(.,h) q(.) + ( )B(.,h) − ( )B(.,h) q(.) Lϑ ( ) Rn Rn 2 Lϑ ( ) Lϑ ( ) 2 2

q(.) q(.) ≤ kf − gk Rn + kM (g − f)k Rn + g − (g)B(.,h) q(.) <ε Lϑ ( ) Lϑ ( ) Rn 2 2 Lϑ ( ) 2

This completes the proof. THE KOLMOGOROV-RIESZ COMPACTNESS THEOREM AND SOME COMPACTNESS CRITERIONS17

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Sinop University Faculty of Arts and Sciences Department of Mathematics, Sinop, TURKEY E-mail address: [email protected]