COMPACT EMBEDDINGS ON A SUBSPACE OF WEIGHTED VARIABLE EXPONENT SOBOLEV SPACES

Transcription

1 Adv Oper Theory ISSN: X electronic COMPACT EMBEDDINGS ON A SUBSPACE OF WEIGHTED VARIABLE EXPONENT SOBOLEV SPACES CIHAN UNAL and ISMAIL AYDIN Communicated by E A Sanchez Perez Abstract In this paper, we define an intersection space between weighted classical Lebesgue spaces and weighted Sobolev spaces with variable exponent We consider the basic properties of the space Also, we investigate some inclusions, continuous embeddings and compact embeddings under some conditions Introduction The history of potential theory begins in 7th century Its development can be traced to such greats as Newton, Euler, Laplace, Lagrange, Fourier, Green, Gauss, Poisson, Dirichlet, Riemann, Weierstrass, Poincaré We refer to the book by Kellogg [7] for references to some of the old works The study of variable exponent function spaces in higher dimensions was revealed in 99 an article by Kováčik and Rákosník [9] They present some basic properties of the variable exponent Lebesgue space L p R n and the Sobolev space W k,p R n such as reflexivity and Hölder inequalities were obtained Also, Fan and Zhao [3] present important results for the variable exponent Lebesgue and Sobolev spaces The study of electrorheological fluids is one of the important area where these spaces have found applications, see [24] As an another area, we can say the study of variational integrals with non-standard growth, see [],[27] Copyright 208 by the Tusi Mathematical Research Group Date: Received: Mar 2, 208; Accepted: Sep 8, 208 Corresponding author 200 Mathematics Subject Classification Primary 46E35; Secondary 43A5, 46E30 Key words and phrases Compact embedding, weighted variable exponent Sobolev space, weighted Lebesgue space

2 2 C UNAL, I AYDIN It is known that the boundedness of the maximal operator was an open problem in L p R n for a long time Diening [8] proved the first time this state over bounded domains if p satisfies locally log-hölder continuous condition, that is, p x p y C ln x y, x, y, x y 2 where is a bounded domain We denote by P log R n the class of variable exponents which satisfy the log-hölder continuous condition Diening later extended the result to unbounded domains by supposing, in addition, that the exponent p = p is a constant function outside a large ball After this study, many absorbing and crucial papers revealed in non-weighted and weighted variable exponent spaces, see [7], [], [26] The operator p f = div f p 2 f is called p - Laplacian The study of differential equations and variational problems with p - growth conditions arouses much interest with the development of elastic mechanics, electrorheological fluid dynamics and image processing etc In recent years, the corresponding results are new and interesting We refer the readers [6], [8], [2], [23], [25] and references therein In general, the methods used in these works are base on continuous and compact embeddings between Lebesgue and Sobolev spaces Our purpose is to define an intersection space L r R n W k,p R n We will consider the basic properties of this space Also, we investigate boundedness of Hardy-Littlewood maximal operator on A r,k,p R n and reveal some results Moreover, we give some continuous and compact embeddings considering that [8], [2], [25] 2 Notation and Preliminaries In this paper, we work on R n with Lebesgue measure dx We denote by C R n the space of all infinitely differentiable functions Also, the elements of the space C0 R n are the infinitely differentiable functions with compact support Moreover, let M R n be the set of all measurable real-valued functions defined on The space L loc Rn is to be space of all measurable functions f on R n such that fχ K L R n for any compact subset K R n A normed space X, X is called a Banach function space shortly BF- space, if Banach space X, X is continuously embedded into L loc Rn, briefly X L loc Rn, ie for any compact subset K R n there is some constant c K > 0 such that fχ K L c K f X for every f X Moreover, a normed space X is compactly embedded in a normed space Y, briefly X Y, if X Y and the identity operator I : X Y is compact, equivalently, I maps every bounded sequence x i i N into a sequence I x i i N that contains a subsequence converging in Y Suppose that X and Y are two Banach spaces and X is reflexive Then I : X Y is a compact operator if and only if I maps weakly convergent sequences in X onto convergent sequences in Y More details can be found in [2], [6]

3 COMPACT EMBEDDINGS ON A SUBSPACE 3 We denote the family of all measurable functions p : R n [, called the variable exponent on R n by the symbol P R n In this paper, the function p always denotes a variable exponent For p P R n, put For each A R n we set p = ess inf px, x R p+ = ess suppx n p A = ess inf x A px, p+ A Moreover, we denote p p 2 if x R n = ess suppx x A inf p x R n 2 x p x > 0, where p, p 2 P R n A measurable and locally integrable function : R n 0, is called a weight function We say that if only if there exists c > 0 such that x c x for all x R n Two weight functions are called equivalent and written, if and The weighted modular is defined by ϱ p, f = fx px x dx R n The weighted variable exponent Lebesgue spaces L p Rn consist of all measurable functions f on R n endowed with the Luxemburg norm f p, = inf λ > 0 : R n fx λ px x dx When x =, the space L p Rn is the variable exponent Lebesgue space The space L p Rn is a Banach space with respect to p, Also, some basic properties of this space were investigated in [3], [4] If the inequality 0 < C holds, we have L p R n L p R n, see [4] We say that p is non-weaker than p 2, denoted as p p 2, if and only if there exist positive constants K, K 2 and h L R n, h 0 such that t p x K K 2 t p 2 x + hx, for ae x R n and all t 0 Moreover, the embedding L p 2 R n L p R n holds if and only if p p 2, see [8] Now, let < p p p + <, k N L loc R n Thus, the embedding L p R n L loc Rn holds and then the weighted variable exponent Sobolev spaces W k,p R n is well-defined by [4, Proposition 2] We set the weighted variable exponent Sobolev spaces R n by and p W k,p { W k,p R n = equipped with the norm f L p } R n : D α f L p Rn, 0 α k

4 4 C UNAL, I AYDIN f k,p, = D α f p, 0 α k where α N n 0 is a multiindex, α = α + α α n, and D α = is already known that W k,p R n is a reflexive Banach space In particular, the space W,p R n is defined by W,p R n = { f L p R n : f L p Rn } α α x α 2 x 2 xn αn It The function ρ,p, : W,p R n [0, is shown as ρ,p, f = ρ p, f+ ρ p, f Also, the norm f,p, = f p, + f p, makes the space W,p R n a Banach space If the exponent p satisfies locally log-hölder continuous condition, then a lot of regularities for variable exponent spaces holds R n under the circumstances, see Rn is defined in the classical way More information on the classic theory of variable exponent spaces can be found in [9] For x R n and s > 0 we denote an open ball with center x and radius s by B x, s For f L loc Rn, the Hardy-Littlewood maximal operator Mf of f given by Because, the space C0 R n is dense in W,p [4] Moreover, the local weighted variable exponent Sobolev space W,p,loc Mf x = sup s>0 B x, s Bx,s f y dy where the supremum is taken over all balls B x, s, see [20] Let r < A weight satisfies Muckenhoupt s A r R n = A r condition, briefly A r, if there are positive constants C and C 2 such that, for all ball B x, s = B R n, or B B x dx B B B B x dx r x r dx C, < r <, ess sup B C 2, r = x The infimum over the constants C and C 2 are called the A r and A, respectively Also it is known that A = A r Let < r < Then it is known that r< A r if and only if the Hardy-Littlewood maximal operator is bounded on L r Rn, see [22] In [0], the class A p was defined to consist of weight such that Ap = sup B p B L B p < B ß p B L

5 COMPACT EMBEDDINGS ON A SUBSPACE 5 where ß denotes the family of all balls in R n, p B = B B dx and p is px the conjugate exponent of p Let p, q P log R n, < p p p + < and < q q q + < If the inequality q p is satisfied, then there exists a constant C > 0 depending on the characteristics of p and q such that Ap C Aq Also, under these conditions, M : L p see [0] R n L p R n if and only if A p, Throughout this paper, we assume that p P log R n with < p p p + < and L loc Rn, p L loc R n 3 The Space A r,k,p R n We set an intersection space as A r,k,p this vector space with the norm f r,k,p R n = L r R n W k,p R n and equip = f r, + f k,p,2 for any f A r,k,p R n Now, we give some basic properties of the space A r,k,p R n Theorem 3 Let, be weight functions on R n If C > 0 and C 2 > 0, then the space A r,k,p R n is a Banach space with respect to r,k,p Proof Let {f i } i IN be a Cauchy sequence in A r,k,p R n Thus given ε > 0, there exists an n N such that for all i, j n implies f i f j r,k,p = f i f j r, + f i f j k,p,2 < ε Therefore, {f i } i IN L r R n and {f i } i IN W k,p R n are Cauchy sequences with respect to r, and k,p,, respectively Since the spaces L r R n, r, and W k,p R n, k,p, are two Banach spaces, for ε > 0 there are n 2, n 3 N such that for all i n 2, i n 3 imply f i f r, < ε 2 3 and f i g k,p,2 < ε 2 32 Since C > 0, it is known that L r R n L r R n holds This follows that f i f in L r R n Therefore, there exists a sequence {f ik } k IN {f i } i IN such that f ik f ae If we consider the definition of the space W k,p R n, then it is clear that W k,p R n L p R n Since C 2 > 0, we have L p R n L p R n, see [4] Thus f ik g in L p R n Since p + <,

6 6 C UNAL, I AYDIN convergence in L p } is necessity with convergence in measure [9] and then there is a sequence {f ikl {f ik } k IN such that f ikl g ae Now, we denote l IN { } K = x R n : f ikl x f x and K 2 = { } x R n : f ikl x g x Thus, we have K = 0, K 2 = 0 Moreover, there is n 4 N such that for all i kl n 4 implies fikl x f x ε < 2 for every x R n K Similarly, we have an n 5 N such that for all i kl n 5 implies fikl x g x ε < 2 for every x R n K 2 Now, we set n 0 = max {n 4, n 5 } There exists an n 0 N such that for all i kl n 0 implies the inequality f x g x f x f ikl x + f ikl x g x < ε holds for every x R n K K 2 Hence, we have f = g, ae Since the elements of L r R n and W k,p R n are equivalence classes, we get that f = g This follows that f A r,k,p R n Let us define k 0 = max {n 2, n 3 } If we use 3 and 32, then the inequality f i f r,k,p = f i f r, + f i f k,p,2 < ε is satisfied for all i k 0 This completes the proof Theorem 32 Assume that, are weight functions on R n If C > 0, then the space A r,k,p R n is a BF- space Proof Let f A r,k,p R n be given Then, we have f L r R n and f W k,p R n It is obvious that the embedding W k,p R n L p R n is satisfied Also, the space L p R n is a BF- space, see [4] Moreover, if we consider the embedding L p R n L p R n holds for C > 0, then we have W k,p R n L p R n L p R n L loc R n This yields that f L loc R n c f p c c 2 f p,2 c c 2 c 3 f k,p,2 max {, c c 2 c 3 } f r, + f k,p,2 for all f A r,k,p R n Hence, we get that A r,k,p R n L loc Rn Theorem 33 If p P log R n, A r and A p, then the maximal operator M is bounded from A r,,p R n to A r,,p R n

7 COMPACT EMBEDDINGS ON A SUBSPACE 7 Proof Let f A r,,p R n be given Then we get f L r R n and f W,p R n Since p L loc R n, it is known that the embeddings L p R n L p, loc R n L loc Rn and W,p R n W,p, loc R n W, loc Rn are satisfied, see [4] This yields f W, loc Rn By [4], we have Mf x M f x for ae x R n Since f, f L p R n and A p, there exist c, c 2 > 0 such that Mf p,2 c f p,2 and M f p,2 c 2 f p,2 This follows Mf, Mf L p R n, that is, Mf W,p R n Since A r and A p, there exist c 3, c 4 > 0 such that Mf r,,p c 3 f r, + c 4 f,p,2 max {c 3, c 4 } f r,,p Corollary 34 Let A r, A p and s < Then the maximal operator M is bounded from A r,,sp R n to A r,,sp R n Proof Let f A r,,sp R n be given Hence we get f L r R n and f R n Suppose that s < There is a C > 0 such that the inequality W,sp Mf,sp,2 C f,sp,2 is satisfied for every f W,sp R n, see [4] This follows that That is the desired result Mf r,,sp c f r, + C f,sp,2 max {c, C} f r,,sp Definition 35 [2] Suppose that φ : R n R is a nonnegative, radial, decreasing function belonging to C0 R n and having the following properties i φ x = 0 if x, ii R n φ x dx = Let ε > 0 If the function φ ε x = ε n φ x ε is nonnegative, belongs to C 0 R n, and satisfies i φ ε x = 0 if x ε, ii R n φ ε x dx =, then φ ε is called a mollifier and we define the convolution by φ ε f x = φ ε x y f y dy R n

8 8 C UNAL, I AYDIN Theorem 36 [4]If A p and f L p as ε 0 + R n, then φ ε f f in L p R n Corollary 37 [4]Assume that A p The class C 0 R n is dense in W k,p R n By Theorem 36 and Corollary 37, we get the following theorem Theorem 38 If A r, A p and f A r,k,p R n, then φ ε f f in A r,k,p R n as ε 0 + Proof Let f A r,k,p R n be given Since A r, A p, we have f φ ε f r,k,p = f φ ε f r, + f φ ε f,p,2 This completes the proof < ε 2 + ε 2 = ε As a direct result of Theorem 38 there follows Corollary 39 Let A r, A p Then the class C0 R n is dense in A r,k,p R n It is clear that the Hölder inequality for the variable exponent Lebesgue spaces is well known, see [9] Now, we consider this inequality for the space A r,,p R n Theorem 30 Let p, q, r P R n with + = Then there p q r exists positive constant C h such that fg,r, C h f,p, g,q, for all f W,p R n, g W,q R n Proof Assume that f W,p R n and g W,q R n with + = p q r Then we have f, f L p R n and g, g L q R n By the Hölder inequality for variable exponent Lebesgue spaces, we get f fg r, c p g q = c f p, g q, p q and fg r, c 2 f p, g q, + c 3 f p, g q, max {, c 2, c 3 } f p, g q, + f p, g q, + f p, g q, This follows that fg,r, C h f p, + f p, g q, + g q, where C h = max {, c, c 2, c 3 }

9 COMPACT EMBEDDINGS ON A SUBSPACE 9 Corollary 3 Let p, q, r P R n with + = and < p q r r, s, t < with + = Then there exists positive constant r s t C h such that fg t,,r for all f A r,,p R n, g A s,,q R n Ch f r,,p g s,,q 4 Some Continuous Embeddings of A r,k,p R n In this section, we present several continuous embeddings briefly, embeddings of defined space under some conditions Theorem 4 Let,, 3, 4 be weight functions on R n Then A r,k,p R n A r,k,p 3, 4 R n holds if and only if the embedding A r,k,p R n A r,k,p 3, 4 R n is satisfied Proof The sufficient condition of the theorem is obvious by definition of embedding Now, let the inclusion A r,k,p R n A r,k,p 3, 4 R n holds Moreover, we define the sum norm = r,k,p + r,k,p 3, 4 It is easy to see that A r,k,p R n, is a Banach space Now, let us define the unit function I from R n, into R n, r,k,p Then I is continuous A r,k,p A r,k,p Because, we can obtain the inequality I f r,k,p = f r,k,p f If we consider the Banach s theorem, then I is a homeomorphism, see [5] That means the norms and r,k,p there exists k > 0 such that are equivalent Thus, for every f A r,k,p R n f k f r,k,p 4 Therefore, by using 4 and the definition of norm, we get That is the desired result f r,k,p 3, 4 f k f r,k,p Theorem 42 Let,, be weight functions on R n If, then the embedding A r,k,p, R n A r,k,p, R n holds Proof It is known that, if, then there exists c > 0 such that x c x for all x R n This yields that { } f r,k,p, c r fr, + f k,p, max, c r f r,k,p, for all f A r,k,p, R n This completes the proof Theorem 43 Let,, be weight functions on R n If, then the embedding A r,k,p, R n A r,k,p, R n holds

10 0 C UNAL, I AYDIN Proof Suppose that f A r,k,p, R n, so we write that f L r Rn and f R n Moreover, if, then there exists c > 0 such that x W k,p c x for all x R n This follows that the inequality D α f x x c D α f x x is satisfied for 0 α k Hence, we have f k,p,2 = D α f p,2 c This follows that f r,k,p, 0 α k 0 α k D α f p, = c f k,p, f r, + c f k,p, max {, c} f r,k,p, Theorem 44 Let p, p 2 P R n satisfying p p 2 Then the embedding A r,k,p 2 R n A r,k,p R n holds Proof Let f A r,k,p 2 R n be given It is known that, if the condition p p 2 holds, then the embedding L p 2 R n L p R n is satisfied, see [8] Similarly, it can be seen that W k,p 2 R n W k,p R n holds under same condition Thus we have f r,k,p f r, + c f k,p2, max {, c} f r,k,p 2 for all f A r,k,p 2 R n That is the desired result Theorem 45 Let p, p 2 P R n satisfying < p 2 p 2 p p + < and p < Then the embedding A r,k,p p p 2,, R n A r,k,p 2, R n holds Proof Suppose that f A r,k,p, R n Then we get f L r Rn and f W k,p R n Also, it is known that L p R n L p 2 R n with p < p p 2,, see [2, Theorem 5] This yields that there exists c > 0 such that f r,k,p 2, f r, + c 0 α k D α f p, max {, c} f r,k,p, for all f A r,k,p, R n This completes the proof Theorem 46 Suppose that,, 3, 4 are weight functions on R n satisfying 3, 4 and k, t Z + with k > t Then the embedding A r,k,p R n A r,t,p 3, 4 R n holds Proof Let f A r,k,p R n be given Then we can write f L r R n and f W k,p R n Since 3, it is clear that L r R n L r 3 R n Therefore there is c > 0 such that f r,3 c f r, 42

11 COMPACT EMBEDDINGS ON A SUBSPACE Since 4 and k, t Z + with k > t, there exists c 2 > 0 such that f t,p,4 D α f p,4 + D α f p,4 0 α t c 2 0 α k t+ α k D α f p,2 = c 2 f k,p,2 43 Now, we define C = max {c, c 2 } By the inequalities 42 and 43, we have f r,t,p 3, 4 c f r, + c 2 f k,p,2 C f r,k,p That is the desired result Theorem 47 Let,, 3, 4 be weight functions on R n satisfying 3, 4 and r 2 r, p 2 p, k, t Z + with k > t Also, assume that R n with < Then the embedding A r,k,p A r 2,t,p 2 holds Proof Suppose that f A r,k,p, so we write that f L r and f W k,p Now, we set α = r r 2 with r 2 < r By the Hölder inequality, we have r 2 f r 2 r 2, f x r r 2 r r2 r β 2 r x dx β dx = β dx β f r 2 r, where + = Since α β L loc and f Lr, we have f L r 2 Also, we get that f L r 2 3, because 3 Using the inclusion W k,p L p, we obtain f L p Since < and p 2 p, the embedding L p L p 2 holds Thus, there exists C > 0 such that f k,p2, C D α f p, = C f k,p, 0 α k This follows that W k,p W k,p 2 Moreover, if we consider the fact that 4 and k > t, then it is easy to see that W k,p 2 W t,p 2 4 Thus we have W k,p W k,p 2 W t,p 2 4 This follows that f L r 2 3 W t,p 2 4 = A r 2,t,p 2 3, 4 This completes the proof The following corollary can be easily proven by Theorem 4, Theorem 42, Theorem 43, Theorem 44, Theorem 45, Theorem 46 Corollary 48 Let,, 3, 4 be weight functions on R n,p, p, p 2 P R n and k, t Z + Then The equality A r,k,p, R n = A r,k,p 2 If, then the equality A r,k,p, 3, 4, R n is satisfied if R n = A r,k,p, R n holds

12 2 C UNAL, I AYDIN 3 If 3, 4, then A r,k,p R n A r,k,p 4 The equality A r,k,p 4 3, 4 R n R n = A r,k,p 3, 4 R n is satisfied with 3 and 5 The embedding A r,k,p R n A r,k,p 2 3, 4 R n holds if 3, 4 and p 2 p 6 If 3, 4 and p p p 2, <, then A r,k,p R n A r,k,p 2 3, 4 R n 7 If the conditions 3, 4, p 2 p and k > t are satisfied, then we have A r,k,p R n A r,t,p 2 3, 4 R n 8 If 3, 4, k > t and p <, then we get p p 2, A r,k,p R n A r,t,p 2 3, 4 R n Now, we use similar method in [8, Theorem 2] for the following theorem Theorem 49 Assume that < p, q < and R n is a bounded set and moreover, i q 2 L with q > p ii x c > 0 for all x Then we have the embedding A r,,p W,pq with p q = pq q+ Proof Let f A r,k,p be given Then it is clear that f L r and f W k,p By [8], the embedding W,p W,pq holds under the conditions i, ii and p q = pq This follows that q+ f,pq c f,p, max {, c} f r,,p for all f A r,k,p Therefore, we get A r,,p This is the desired result W,p W,pq 5 Some Compact Embeddings of the space A r,,p Let p P R n Now, we define Sobolev conjugate of p as p = { np, n p p < n, p n Theorem 5 [9], [8]Suppose that R n is an open, bounded set with Lipschitz boundary and p C +, p P log with < p p + < n If r L with r > satisfies r x p x for every x, then we obtain the embedding W,p L r Moreover, the compact embedding W,p L r holds if inf p x r x > 0 x If we consider Theorem 49 and Theorem 5, then we have the following result

13 COMPACT EMBEDDINGS ON A SUBSPACE 3 Corollary 52 Assume that all assumptions in Theorem 49 are satisfied Moreover, let, the assumptions in Theorem 5 replacing p by p q holds Then we have A r,,p L r Theorem 53 [2]Assume that p C and < p x for all x Moreover, let i 0 < L α with < α C, ii < s x < p x βx holds for all x where β x = αx αx Then compact embedding W,p L s is satisfied Corollary 54 Let all conditions in Theorem 49 be satisfied Moreover, assume that the assumptions in Theorem 53 replacing p by p q are also satisfied Thus, we obtain A r,,p L s where < s < p q in β Proof By the Theorem 49 and Theorem 53, we have A r,,p W,pq and W,pq L s, respectively Then we get the desired result Theorem 55 Suppose that p C and < p x for all x and moreover, i 0 < 3 L α with < α C, t ii p t 2 L where t C and < t < p iii x c > 0 for all x Then we get the compact embedding A r,,p C and < q < t β where α + β = Proof Let f A r,,p This follows that ρ p L t By the Hölder inequality, we have f x tx dx c h f t L q 3 for every q Then we write that f L r and f W,p f t t p 2 < and ρ p t p 2 < L p t t t p 2 p p 2 p L t L p t If we consider the [, Lemma 325] and ii, then we get and t p 2 p + p t + p L p t t p 2 p L p t < This follows that t p 2 p p + t p + t p x tx px tx dx + c p L p t <

14 4 C UNAL, I AYDIN and f x tx dx c h c f t t p 2 L p t 5 In general, we can suppose that f x tx dx > By [, Lemma 325] and 5 when f x px x dx, we have f t L t c h c f t t p 2 That means where C = c h c t obtain c h c L p t p + f x px x dx c h c fp t p + p f L t C f p + t L p L p 2 52 > 0 By similar method, if f x px x dx >, we p + t + f L t C f p t L p 53 where C = c h c t > 0 If we consider the inequalities 52 and 53, then we have f L t By t < p,, ii and iii, we get that L p L t L t Therefore, we have f L p, that is, f L t This follows that f W,t Hence, the inclusion W,p W,t is satisfied Using the Banach Theorem in [5], we get W,p W,t Then, we have f,t C f,p,2 max {, C} f r,,p for all f A r,,p This follows that By Theorem 5, we have compact embedding A r,,p W,t 54 W,t L s 55 for s < t Now, we define s = q β By the Hölder inequality for variable exponent Lebesgue space, we have f x qx f q 3 x dx c h L 3 L β α <

15 This follows that W,p [5], then we get W,p for all f A r,,p COMPACT EMBEDDINGS ON A SUBSPACE 5 L q 3 If we consider the Banach Theorem in L q 3 Hence, we obtain f q,3 C f,p,2 max {, C} f r,,p This implies that the embedding A r,,p L q 3 such that f i 0 in is satisfied Now, we take a sequence f i i N A r,,p as i This follows that f i 0 in W,t by 54 Moreover, A r,,p if we consider 55, then we get that f i 0 in L s Hence, we have f i x qx fi 3 x dx c h q L 3 L β α 0 that is, f i 0 in L q 3 This completes the proof Corollary 56 Assume that all assumptions of Theorem 55 are satisfied Then there exist C, C 2 > 0 such that C f x qx f r,,p q+, if f r,,p > 3 x dx C 2 f r,,p q, if f r,,p < for all f A r,,p Proof If we consider the Theorem 55, then we have A r,,p L q+ 3 and A r,,p L q 3 for < q q q + < t Therefore, there β are c, c 2 > 0 such that and f L q + 3 = f L q 3 = f x q+ 3 x dx f x q 3 x dx q + q c f r,,p c 2 f r,,p for all f A r,,p This implies that f x qx 3 x dx f x q+ + f x q 3 x dx c q+ f r,,p q+ + c q 2 C f r,,p C 2 f r,,p q+ q f r,,p q, if f r,,p >, if f r,,p <

16 6 C UNAL, I AYDIN Pucci and Zhang [23] introduced a main variable exponent space E, E defined by E = f M Rn : f x px a x + f x px dx <, R n equip with the norm f E = inf λ > 0 : R n f x λ where a L loc Rn satisfying a x px a x + f x px dx λ c + x px for some constant c 0, ] and all x R n It is clear that E is well-defined, because C 0 R n E Theorem 57 Suppose that,, 3 are weight functions on R n satisfying x for all x R n Moreover, let 3 L p q R n, p + < q q p and a which a satisfies the condition above Then the compact embedding A r,,p R n L q 3 R n holds Proof Let f A r,,p We denote and B = λ > 0 : p R n Then we write that f L r R n and f W,p R n R n C = λ > 0 : R n f x λ f x λ px px a x + f x px dx λ + f x px x dx λ Since x and a x x for all x R n, we have C B If we consider the definitions of the norms E and,p,2, then we obtain f E f,p,2 f r,,p This yields A r,,p R n E Moreover, since 3 L p p q R n and p + < q q p, the compact embedding E L q 3 R n is satisfied, see [23, Lemma 26] Hence we write that A r,,p R n E L q 3 R n This completes the proof Ho and Sim [5], [6] revealed several continuous and compact embedding under some conditions They originally deal with existence and multiplicity of solutions for some types of elliptic boundary value problems but also investigate several embedding from the weighted variable exponent Sobolev spaces into weighted variable exponent Lebesgue spaces Now, we give some statements according to [5] and [6] with the space A r,,p For this, we present some conditions below

17 COMPACT EMBEDDINGS ON A SUBSPACE 7 O p C +, p P log, p + < n; w P +, w s L for some s C [ n such that s,, for all p p x ; there is a Lipschitz domain 0, c w x c 2 ae on 0 for some positive constants c, c 2, and w L α 0 such that p x < αx p αx s x for all x 0 C Let p satisfies O Moreover, q C +, p x < q x p x for all x, = A = { x : p s x q x p x } 0 ; b P +, c b x c 2 ae on 0 ; b L β 0 and q x < βx p βx s x for all x 0 and some β C + 0 S Assume that p, w, q and b satisfy O and C Also, r C +, a L + γ for some γ C+ such that r x < γx p γx s x for all x Now, we are ready to reveal embeddings Theorem 58 [6] Assume that O holds Then we have Ww,p L p w If we consider the norms of the space A r,,p that A r,,p below W,p and W,p, then we find under the condition O This follows the corollary Corollary 59 Assume that p and w satisfy the condition O Then we get the compact embedding A r,,p,w L p w Theorem 50 [6] Assume that O and C hold Then we have Ww,p L q b Corollary 5 Let p, w and q, b satisfy the conditions O and C, respectively This follows A r,,p,w L q b Theorem 52 [5] Assume that O and S hold Then we have Ww,p L r a The following corollary is an obvious result of the previous theorem Corollary 53 Let p, w and r, a satisfy the conditions O and S, respectively Then we have A r,,p,w L r a References E Acerbi and G Mingione, Regularity results for a class of functionals with non-standard growth, Arch Ration Mech and Anal , R A Adams and J J F Fournier, Sobolev spaces 2 nd Ed, Academic Press, New York, I Aydın, On variable exponent Amalgam spaces, An Ştiinţ Univ Ovidius Constanţa Ser Mat , no 3, I Aydın, Weighted variable Sobolev spaces and capacity, J Funct Spaces Appl

18 8 C UNAL, I AYDIN 5 H Cartan, Differential calculus, Hermann, Paris-France, 97 6 P G Ciarlet, Linear and nonlinear functional analysis with applications, Society for Industrial and Applied Mathematics SIAM, Philadelphia, D V Cruz-Uribe and A Fiorenza, Variable Lebesgue spaces-foundations and harmonic analysis, Birkhäuser/Springer, New York, L Diening, Maximal function on generalized Lebesgue spaces L p, Math Inequal Appl , no 2, L Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p and W k,p, Math Nachr , L Diening and P Hästö, Muckenhoupt weights in variable exponent spaces, preprint L Diening, P Harjulehto, P Hästö, and M Růžička, Lebesgue and Sobolev spaces with variable exponents, Springer-Verlag, Berlin, 20 2 E Edmunds, A Fiorenza, and A Meskhi, On a measure of non-compactness for some classical operators, Acta Math Sin , no 6, X Fan and D Zhao, On the spaces L px and W k,px, J Math Anal Appl , no 2, P Hajlasz and J Onninen, On boundedness of maximal functions in Sobolev spaces, Ann Acad Sci Fenn Math , no, K Ho and I Sim, Existence and multiplicity of solutions for degenerate p x Laplace equations involving concave-convex type nonlinearities with two parameters, Taiwanese J Math 9 205, no 5, K Ho and I Sim, On degenerate p x Laplace equations involving critical growth with two parameters, Nonlinear Anal , O D Kellogg, Foundations of potential theory, Springer, Berlin, Y Kim, L Wang, and C Zhang, Global bifurcation for a class of degenerate elliptic equations with variable exponents, J Math Anal Appl , no2, O Kováčik and J Rákosník, On spaces L px and W k,px, Czechoslovak Math J 4 99, no4, S Lu, Y Ding, and D Yan, Singular integrals and related topics, World Scientific Publishing Company, R A Mashiyev, S Oğraş, Z Yucedag, and M Avci, The Nehari manifold approach for Dirichlet problem involving the p x- Laplacian equation, J Korean Math Soc , no 4, B Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans Amer Math Soc , P Pucci and Q Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J Differ Equations , no 5, M Růžička, Electrorheological fluids: Modelling and mathematical theory, Springer, Berlin, S Saiedinezhad and M B Ghaemi, The fibering map approach to a quasilinear degenerate p x- Laplacian equation, Bull Iran Math Soc 4 205, no 6, S Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec Funct , no 5-6, V V Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math USSR, Izv , no, Department of Mathematics, Sinop University, 57000, Sinop, Turkey address: cihanunal88@gmailcom address: iaydin@sinopedutr