Almost-compact and compact embeddings of variable exponent spaces
aa r X i v : . [ m a t h . F A ] J a n ALMOST-COMPACT AND COMPACT EMBEDDINGS OFVARIABLE EXPONENT SPACES
D. E. EDMUNDS, A. GOGATISHVILI, AND A. NEKVINDA
Abstract.
Let Ω be an open subset of R N , and let p, q : Ω → [1 , ∞ ] bemeasurable functions. We give a necessary and sufficient condition for theembedding of the variable exponent space L p ( · ) (Ω) in L q ( · ) (Ω) to be almostcompact. This leads to a condition on Ω , p and q sufficient to ensure thatthe Sobolev space W ,p ( · ) (Ω) based on L p ( · ) (Ω) is compactly embedded in L q ( · ) (Ω) ; compact embedding results of this type already in the literature areincluded as special cases. Introduction
Let Ω be an open subset of R N and consider the Lebesgue measure on Ω. If M ⊂ Ω is measurable we write | M | for its measure. Let p, q : Ω → [1 , ∞ ] bemeasurable. Much attention has been paid in recent years to the variable exponentspace L p ( · ) (Ω) , the space W ,p ( · ) (Ω) of Sobolev type based on L p ( · ) (Ω) and con-ditions under which W ,p ( · ) (Ω) is embedded in L q ( · ) (Ω) : we refer to [2, 4, 5] for acomprehensive account of such matters. The compactness of such an embeddingis addressed here: we give conditions that are sufficent to ensure compactness yetweak enough for much earlier work on this topic to be included. To do this wefirst establish necessary and sufficient conditions for the embedding of L p ( · ) (Ω) in L q ( · ) (Ω) to be almost compact.Let M (Ω) be the family of all measurable functions u : Ω → [ −∞ , ∞ ]; denoteby χ E the characteristic function of a set E ⊂ Ω. Given any sequence { E n } ofmeasurable subsets of Ω , we write E n → ∅ a.e. if the characteristic functions χ E n converge to 0 pointwise almost everywhere in Ω. Let the symbol | u | stand for themodulus of a function u . We recall the definition of a Banach function space: see,for example, [1]. A normed linear space ( X, k . k X ) is a Banach function space (BFS Mathematics Subject Classification.
Key words and phrases. almost-compact embeddings, Banach function spaces, variableLebesgue spaces, variable Sobolev spaces.The second and the third author of this research were supported by the grant P201-18-00580Sof the Grant Agency of the Czech Republic. The second author has been partially supported byShota Rustaveli National Science Foundation of Georgia (SRNSFG) [grant number FR17-589] andRVO:67985840. for short) if the following conditions are satisfied:the norm k u k X is defined for all u ∈ M (Ω) , and u ∈ X if and only if(1.1) k u k X < ∞ ; k u k X = k | u | k X for every u ∈ M (Ω);(1.2) if 0 u n ր u a.e. in Ω , then k u n k X ր k u k X ;(1.3) if E ⊂ Ω is a measurable set of finite measure, then χ E ∈ X ;(1.4) for every measurable set E ⊂ Ω of finite measure | E | , there exists(1.5) a positive constant C E such that Z E | u ( x ) | dx C E k u k X . If X and Y are Banach function spaces then X is said to be almost-compactlyembedded in Y and we write X ∗ ֒ → Y if, for every sequence ( E n ) n ∈ N of measurablesubsets of Ω such that E n → ∅ a.e., we havelim n →∞ sup k u k X ≤ k uχ E n k Y = 0 . We believe this notion to have independent interest. Moreover, as we know from[10], almost compactness results quickly lead to assertions concerning the compact-ness of the Sobolev embedding.To explain in a little more detail what is acheived, suppose that Ω is bounded, p ∈ C (cid:0) Ω (cid:1) and for all x ∈ Ω , < p − ≤ p ( x ) ≤ p + < N and p ( x ) = N p ( x ) N − p ( x ) ;(1.6)denote by W ,p ( · )0 (Ω) the closure of C ∞ (Ω) in W ,p ( · ) (Ω) . Let I p,q (resp. I p,q, )stand for the embedding of W ,p ( · ) (Ω) (cid:16) resp. W ,p ( · )0 (Ω) (cid:17) in L q ( · ) (Ω) . Then it isknown (see [7]) that I p,q, is compact if there exists ε > q ( x ) ≤ p ♯ ( x ) − ε for all x ∈ Ω . In [8] the compactness of I ,q, is studied under more generalassumptions: it is supposed that there exists x ∈ Ω, a small η >
0, 0 < l <
C > q ( x ) = 2 N/ ( N −
2) and q ( x ) ≤ NN − − C (cid:16) log | x − x | (cid:17) l holds for a.e. x ∈ Ω with | x − x | η . Some generalizations of these results aregiven in [6] and [9]. In [9] it is assumed that q ( x ) = p ♯ ( x ) on a compact set K, and compactness of I p,q, is established under some restrictions on K and on thebehavior of p ♯ ( x ) − q ( x ) far from K. The principal aim of this paper is to establishcompactness of I p,q for a wider class of sets K on which q is allowed to have the samevalues as p : various examples of Cantor type are given for which this is possible.First we find a necessary and sufficient condition for this embedding to be al-most compact and as an application we establish the compactness of the Sobolevembedding mentioned above under more general conditions than those previouslyavailable. LMOST-COMPACT AND COMPACT EMBEDDING . . . 3 Preliminaries
Let X and Y be Banach function spaces on an open set Ω of R N with norms k·k X , k·k Y respectively. We say that X is embedded in Y, and write X ֒ → Y, ifthere exists c > k u k Y ≤ c k u k X for all u ∈ X. The space X is said to becompactly embedded in Y, and we write X ֒ → ֒ → Y, if given any sequence { u n } n ∈ N with each k u n k X ≤ , there is a subsequence (cid:8) u n ( k ) (cid:9) ⊂ Y and a point u ∈ Y suchthat (cid:13)(cid:13) u n ( k ) − u (cid:13)(cid:13) Y → . Definition 2.1.
Let X be a BFS. The Sobolev space W ( X ) is defined to be theset of all functions u ∈ M (Ω) with k u k W ( X ) = k u k X + k∇ u k X < ∞ . The following proposition is proved in [10], see Theorem 3.2.
Proposition 2.2.
Let
X, Y, Z be BFSs and assume W ( X ) ֒ → Y, Y ∗ ֒ → Z. Then W ( X ) ֒ → ֒ → Z. Now, define variable Lebesgue spaces. Let E (Ω) denote the set of all measurablefunctions p ( · ) : Ω → [1 , ∞ ). Let p ( · ) ∈ E (Ω). Define for a function u : Ω → R amodular m p ( · ) ( u ) = Z Ω | u ( x ) | p ( x ) dx (2.1)and define the space L p ( · ) (Ω) to be the set of all measurable functions u on Ω witha finite norm k u k p ( · ) = inf n λ > m p ( · ) ( u/λ ) ≤ o . We adopt the notation p − = inf { p ( x ); x ∈ Ω } , p + = sup { p ( x ); x ∈ Ω } and p ′ ( x ) = p ( x ) − p ( x ) . Define for a function u : Ω → R a non-increasing rearrangement u ∗ on [0 , ∞ ) by u ∗ ( t ) = inf { λ > |{ x ∈ Ω; | u ( x ) | > λ }| t } , ( t > . Lemma 2.3.
Let s : Ω → R and α > . Then ( α s ( · ) ) ∗ ( t ) = α s ∗ ( t ) for all t > .Proof. We can easily write( α s ( · ) ) ∗ ( t ) = inf { λ > |{ x ∈ Ω; α s ( x ) > λ }| t } = inf { α µ > |{ x ∈ Ω; α s ( x ) > α µ }| t } = inf { α µ > |{ x ∈ Ω; s ( x ) > µ }| t } = α inf { µ> |{ x ∈ Ω; s ( x ) >µ }| t } = α s ∗ ( t ) . (cid:3) In [7] (see Theorem 2.8) the following lemma is proved.
Lemma 2.4.
Let p, q ∈ E (Ω) . Then L p ( · ) (Ω) ֒ → L q ( · ) (Ω) if and only if q ( x ) p ( x ) a.e. in Ω . D.E.EDMUNDS, A.GOGATISHVILI, AND A.NEKVINDA
Definition 2.5.
We say that p : Ω → R satisfies a log-H¨older condition if there is c > | p ( x ) − p ( y ) | − c ln | x − y | , < | x − y | . (2.2) Definition 2.6.
We say that Ω ∈ C , if there is a finite number of balls B ( x k , r k ) , k =1 , , . . . , m and the same number of bi-Lipschitz mappings T k : [0 , N − × [ − , → B ( x k , r k ) such that for all k ∈ { , , . . . , m } ,(i) x k ∈ ∂ Ω,(ii) S mk =1 B ( x k , r k ) ⊃ ∂ Ω,(iii) T k ([0 , N − × [ − , R N \ Ω) ∩ B ( x k , r k ),(iv) T k ([0 , N − × [0 , ∩ B ( x k , r k ),(v) T k ([0 , N − × { } ) = ∂ Ω ∩ B ( x k , r k ).Let Ω ∈ C , and M ⊂ Ω be a compact set. Given p ( · ) , q ( · ) : Ω → R we findconditions on M and, moreover, determine how quickly can q ( · ) tend to p ( · ) near M while preserving the compactness of the embedding of W ,p ( · ) (Ω) in L q ( · ) (Ω).3. Almost-compact embedding between variable spaces
We fix in this section a domain Ω ⊂ R N and functions p ( · ) , q ( · ) ∈ E (Ω). Adoptthe notation r ( x ) = p ( x ) q ( x ) . Lemma 3.1.
Let Ω be bounded and q ( · ) ∈ E (Ω) , q + < ∞ . Then k u k q ( · ) ( m q ( · ) ( u )) /q + provided m q ( · ) ( u ) , (3.1) k u k q ( · ) > ( m q ( · ) ( u )) /q − provided m q ( · ) ( u ) , (3.2) k u k q ( · ) ( m q ( · ) ( u )) /q − provided m q ( · ) ( u ) > , (3.3) k u k q ( · ) > ( m q ( · ) ( u )) /q + provided m q ( · ) ( u ) > . (3.4) Proof.
Set a = m q ( · ) ( u ) and assume a
1. Then Z Ω (cid:16) | u ( x ) | a /q + (cid:17) q ( x ) dx Z Ω (cid:16) | u ( x ) | a /q ( x ) (cid:17) q ( x ) dx = Z Ω | u ( x ) | q ( x ) a dx = 1which gives k u k q ( · ) a /q + and proves (3.1). The assertions (3.2), (3.3) and (3.4)can be proved analogously. (cid:3) Lemma 3.2.
Let Ω be bounded and suppose that q ( x ) p ( x ) p + < ∞ for all x ∈ Ω . If k u k p ( · ) then k u k q + p ( · ) k | u ( · ) | q ( · ) k r ( · ) k u k q − p ( · ) . Proof.
Assume first 0 < a := k u k p ( · ) <
1. Then1 = Z Ω (cid:16) | u ( x ) | a (cid:17) p ( x ) dx > Z Ω | u ( x ) | p ( x ) dx. (3.5)Denote b = k | u ( · ) | q ( · ) k r ( · ) . Then1 = Z Ω (cid:16) | u ( x ) | q ( x ) b (cid:17) r ( x ) dx = Z Ω (cid:16) | u ( x ) | b /q ( x ) (cid:17) p ( x ) dx. LMOST-COMPACT AND COMPACT EMBEDDING . . . 5 If b > Z Ω (cid:16) | u ( x ) | b /q ( x ) (cid:17) p ( x ) dx < Z Ω | u ( x ) | p ( x ) dx (3.5) < b
1. Consequently, Z Ω (cid:16) | u ( x ) | b /q + (cid:17) p ( x ) dx Z Ω (cid:16) | u ( x ) | b /q ( x ) (cid:17) p ( x ) dx Z Ω (cid:16) | u ( x ) | b /q − (cid:17) p ( x ) dx which gives b /q + > k u k p ( · ) > b /q − and finally k u k q + p ( · ) k | u ( · ) | q ( · ) k r ( · ) k u k q − p ( · ) . Assume now 0 < k u k p ( · )
1. Choose ε >
0. Then k u ε k p ( · ) < (cid:13)(cid:13)(cid:13) | u ( · ) | ε (cid:13)(cid:13)(cid:13) q + p ( · ) (cid:13)(cid:13)(cid:13) (cid:16) | u ( · ) | ε (cid:17) q ( · ) (cid:13)(cid:13)(cid:13) r ( · ) (cid:13)(cid:13)(cid:13) | u ( · ) | ε (cid:13)(cid:13)(cid:13) q − p ( · ) . Since ε ) q − > ε ) q ( x ) > ε ) q + we have1(1 + ε ) q + (cid:13)(cid:13)(cid:13) | u ( · ) | q ( · ) (cid:13)(cid:13)(cid:13) r ( · ) (cid:13)(cid:13)(cid:13) (cid:16) | u ( · ) | ε (cid:17) q ( · ) (cid:13)(cid:13)(cid:13) r ( · ) (cid:13)(cid:13)(cid:13) | u ( · ) | ε (cid:13)(cid:13)(cid:13) q − p ( · ) ε ) q − k u k q − p ( · ) and 1(1 + ε ) q − (cid:13)(cid:13)(cid:13) | u ( · ) | q ( · ) (cid:13)(cid:13)(cid:13) r ( · ) > (cid:13)(cid:13)(cid:13) (cid:16) | u ( · ) | ε (cid:17) q ( · ) (cid:13)(cid:13)(cid:13) r ( · ) > (cid:13)(cid:13)(cid:13) | u ( · ) | ε (cid:13)(cid:13)(cid:13) q + p ( · ) > ε ) q + k u k q + p ( · ) which proves(1 + ε ) q − − q + k u k q + p ( · ) k | u ( · ) | q ( · ) k r ( · ) (1 + ε ) q + − q − k u k q − p ( · ) . Tending ε → + we obtain k u k q + p ( · ) k | u ( · ) | q ( · ) k r ( · ) k u k q − p ( · ) . (cid:3) Lemma 3.3.
Let Ω be bounded, p ( · ) , q ( · ) ∈ E (Ω) , and suppose that q ( x ) p ( x ) p + < ∞ for all x ∈ Ω . Assume that for any sequence { E n } n ∈ N of measurablesubsets of Ω such that | E n | → , we have k χ E k r ′ ( · ) → . Then L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω) .Proof. Let E n ⊂ Ω, | E n | →
0. Then by Lemma 3.1 we obtainlim n →∞ sup {k uχ E n k q ( · ) ; k u k p ( · ) } lim n →∞ sup { max { ( m q ( · ) ( uχ E n )) /q + , ( m q ( · ) ( uχ E n )) /q − } ; k u k p ( · ) } . If k u k p ( · ) m q ( · ) ( uχ E n ) = Z Ω | u ( x ) χ E n ( x ) | q ( x ) dx c k χ E n k r ′ ( · ) k | u ( · ) | q ( · ) k r ( · ) c k χ E n k r ′ ( · ) k u k q − p ( · ) . D.E.EDMUNDS, A.GOGATISHVILI, AND A.NEKVINDA
This giveslim n →∞ sup {k uχ E n k q ( · ) ; k u k p ( · ) } c lim n →∞ sup { max {k χ E n k /q + r ′ ( · ) k u k q − /q + p ( · ) , k χ E n k /q − r ′ ( · ) k u k p ( · ) } ; k u k p ( · ) } c lim n →∞ max {k χ E n k /q + r ′ ( · ) , k χ E n k /q − r ′ ( · ) } = 0 . (cid:3) Theorem 3.4.
Let Ω be bounded, p ( · ) , q ( · ) ∈ E (Ω) , and suppose that q ( x ) p ( x ) p + < ∞ for all x ∈ Ω . Denote s ( x ) = p ( x ) − q ( x ) . Assume Z | Ω | a s ∗ ( t ) dt < ∞ . (3.6) for all a > . Then L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω) .Proof. Let E n ⊂ Ω, | E n | →
0. Assume that there is α > k χ E n k r ′ ( · ) > α for all n . Without loss of generality we can assume α <
1. Then α inf n λ > Z Ω (cid:12)(cid:12)(cid:12) χ E n ( x ) λ (cid:12)(cid:12)(cid:12) r ′ ( x ) dx o = inf n λ > Z Ω (cid:12)(cid:12)(cid:12) χ E n ( x ) λ (cid:12)(cid:12)(cid:12) p ( x ) s ( x ) dx o . Choose 0 < β < α . Then we obtain by Lemma 2.31 < Z Ω (cid:12)(cid:12)(cid:12) χ E n ( x ) β (cid:12)(cid:12)(cid:12) p ( x ) s ( x ) dx Z E n (cid:16) β p + (cid:17) s ( x ) dx = Z | E n | (cid:16) β p + (cid:17) s ∗ ( t ) dt. Since by the assumption Z | E n | (cid:16) β p + (cid:17) s ∗ ( t ) dt → n → ∞ we have a contradiction. So, k χ E n k r ′ ( · ) →
0. By Lemma 3.3 we have L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω). (cid:3) We remark that the condition (3.6) was first introduced in Corollary 2.7 of [3].
Lemma 3.5.
Let Ω be bounded, p ( · ) , q ( · ) ∈ E (Ω) , and suppose that q ( x ) p ( x ) p + < ∞ for all x ∈ Ω . Assume that there exist an α > and a sequence E n ⊂ Ω with | E n | → such that for all n k χ E n k r ′ ( · ) > α. Then L p ( · ) (Ω) ∗ ֒ → / L q ( · ) (Ω) .Proof. Let E n ⊂ Ω, k χ E n k p ′ ( · ) > α . Fix n . Without loss of generality we canassume α
1. Set u n ( x ) = χ E n ( x ) k χ E n k r ′ ( x ) r ( x ) q ( x ) r ′ ( · ) . LMOST-COMPACT AND COMPACT EMBEDDING . . . 7
Clearly, Z Ω | u n ( x ) | p ( x ) dx = Z Ω χ E n ( x ) k χ E n k r ′ ( x ) r ( x ) q ( x ) r ′ ( · ) p ( x ) dx = Z Ω χ E n ( x ) k χ E n k r ′ ( x ) r ′ ( · ) dx = Z Ω (cid:18) χ E n ( x ) k χ E n k r ′ ( · ) (cid:19) r ′ ( x ) dx = 1 . Thus k u n k p ( · ) = 1 . Then by Lemma 3.1 we obtainsup {k uχ E n k q ( · ) ; k u k p ( · ) } > min { ( m q ( · ) ( u n χ E n )) /q + , ( m q ( · ) ( u n χ E n )) /q − } . Further m q ( · ) ( u n χ E n ) = Z Ω | u n ( x ) χ E n ( x ) | q ( x ) dx = Z Ω χ E n ( x ) χ E n ( x ) k χ E n k r ′ ( x ) r ( x ) q ( x ) r ′ ( · ) q ( x ) dx = Z Ω χ E n ( x ) k χ E n k r ′ ( x ) r ( x ) r ′ ( · ) dx = k χ E n k r ′ ( · ) Z Ω χ E n ( x ) k χ E n k r ′ ( x ) r ( x ) r ′ ( · ) dx = k χ E n k r ′ ( · ) Z Ω χ E n ( x ) k χ E n k r ′ ( x ) r ′ ( · ) dx = k χ E n k r ′ ( · ) Z Ω (cid:18) χ E n ( x ) k χ E n k r ′ ( · ) (cid:19) r ′ ( x ) dx = k χ E n k r ′ ( · ) > α. Thensup {k uχ E n k q ( · ) ; k u k p ( · ) } > min { ( m q ( · ) ( u n χ E n )) /q + , ( m q ( · ) ( u n χ E n )) /q − } > min { α /q + , α /q − } = α /q − which proves the lemma. (cid:3) Lemma 3.6.
Let E ⊂ Ω , g : Ω → [0 , ∞ ) and assume that inf { g ( x ); x ∈ E } > sup { g ( x ); x ∈ Ω \ E } . Then ( gχ E ) ∗ ( t ) = g ∗ ( t ) χ (0 , | E | ) ( t ) . Proof.
Trivial. (cid:3)
Theorem 3.7.
Let Ω be bounded, p ( · ) , q ( · ) ∈ E (Ω) , and suppose that q ( x ) p ( x ) p + < ∞ for all x ∈ Ω . Denote s ( x ) = p ( x ) − q ( x ) . Assume that there is a > suchthat Z | Ω | a s ∗ ( t ) dt = ∞ . (3.7) Then L p ( · ) (Ω) ∗ ֒ → / L q ( · ) (Ω) . D.E.EDMUNDS, A.GOGATISHVILI, AND A.NEKVINDA
Proof.
Define E n = { x ∈ Ω; s ( x ) > n } . Assume for a moment that there exists n such that | E n | = 0. Then s ( x ) = 1 p ( x ) − q ( x ) < n almost everywhere and so we have for any a > Z | Ω | a s ∗ ( t ) dt Z | Ω | a n dt = a n | Ω | < ∞ which is a contradiction with the assumption. So, | E n | > n . Fix n andassume max {k χ E n k p + r ′ ( · ) , k χ E n k p − r ′ ( · ) } a . (3.8)Then 1 = Z Ω (cid:18) χ E n ( x ) k χ E k r ′ ( · ) (cid:19) r ′ ( x ) dx = Z Ω χ E n ( x ) k χ E n k p ( x ) r ′ ( · ) s ( x ) dx (3.9) > Z E n {k χ E n k p + r ′ ( · ) , k χ E n k p − r ′ ( · ) } ! s ( x ) dx > Z E n a s ( x ) dx. Now, by the definition of E n we have that s ( x ) > n on E n and s ( x ) < n on Ω \ E n .This gives us a s ( x ) > a n on E n and a s ( x ) < a n on Ω \ E n . Then we have by Lemma3.6 ( a s ( · ) χ E n ( · )) ∗ ( t ) = ( a s ( · ) ) ∗ ( t ) χ (0 , | E n | ) ( t ) = a s ∗ ( t ) χ (0 , | E n | ) ( t )which gives with (3.9)1 > Z E n a s ( x ) dx = Z Ω a s ( x ) χ E n ( x ) dx = Z | Ω | ( a s ( · ) χ E n ( · )) ∗ ( t ) dt = Z | Ω | a s ∗ ( t ) χ (0 , | E n | ) ( t ) dt = Z | E n | a s ∗ ( t ) ( t ) dt = ∞ which is a contradiction. So, our assumption (3.8) is false and we havemax {k χ E n k p + r ′ ( · ) , k χ E n k p − r ′ ( · ) } > a which yields k χ E n k r ′ ( · ) > min { a − /p + , a − /p − } := b > . Thus, we have k χ E n k r ′ ( · ) > b > n and Lemma 3.5 gives us L p ( · ) (Ω) ∗ ֒ → / L q ( · ) (Ω). (cid:3) Consider a special case. Let K ⊂ Ω be compact with | K | = 0. Denote d K ( x ) =dist( x, K ). Set K ( t ) = { x ∈ Ω; d K ( x ) < t } . (3.10)Denote ϕ ( t ) = | K ( t ) | , t ∈ [0 , diam(Ω)] . (3.11) LMOST-COMPACT AND COMPACT EMBEDDING . . . 9
Let ω : [0 , diam(Ω)] → R be a decreasing continuous non-negative function, ω := ω (diam(Ω)). Let ω − denote the inverse function to ω . Lemma 3.8.
Let Ω be bounded, p ( · ) , q ( · ) ∈ E (Ω) , and suppose that q ( x ) p ( x ) p + < ∞ for all x ∈ Ω . Assume that there is c > such that s ( x ) = 1 p ( x ) − q ( x ) c ω (d K ( x )) , x ∈ Ω , Z ∞ ω ϕ ( ω − ( y )) a y dy < ∞ for all a > . Then L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω) .Proof. Let a >
1. Then Z | Ω | a s ∗ ( t ) dt = Z Ω a s ( x ) dx Z Ω a cω (d K ( x )) dx = Z ∞ |{ x ; a cω (d K ( x )) > λ }| dλ = Z a cω |{ x ; a cω (d K ( x )) > λ }| dλ + Z ∞ a cω |{ x ; a cω (d K ( x )) > λ }| dλ = a cω | Ω | + c ln a Z ∞ ω |{ x ; a cω (d K ( x )) > a cy }| a cy dy = a cω | Ω | + c ln a Z ∞ ω |{ x ; ω (d K ( x )) > y }| a cy dy = a cω | Ω | + c ln a Z ∞ ω |{ x ; d K ( x )) < ω − ( y ) }| a cy dy = a cω | Ω | + c ln a Z ∞ ω ϕ ( ω − ( y )) a cy dy < ∞ . By Theorem 3.4 we have L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω). (cid:3) Examples of Cantor sets
Let { a k } k ∈ N be a given sequence of positive real numbers with ∞ X k =1 a k = 1 . (4.1)Construct a generalized Cantor set by the following process. Set K = [0 , K a centered interval of length a to obtain a set K . Wewrite K = K \ (cid:16) − a , a (cid:17) = h , − a i ∪ h a , i := J ∪ J . In the second step we omit from J and J centered intervals of length a / K . Then K = K \ (cid:16)(cid:16) − a − a , − a + a (cid:17) ∪ (cid:16) a − a , a + a (cid:17)(cid:17) = h , − a − a i ∪ h − a + a , − a i ∪ h a , a − a i ∪ h a + a , i := J ∪ J ∪ J ∪ J . We follow this process step by step to obtain sets K n . Then K n consists of 2 n intervals J α , α ∈ { , } n . Clearly, | J α | = 2 − n (cid:16) − n X k =1 a k (cid:17) . (4.2)Set K = ∞ \ n =1 K n . Clearly, K is a compact set and for each n | K | | K n | which gives with (4.2) | K | X α ∈{ , } n | J α | = 2 n − n (cid:16) − n X k =1 a k (cid:17) = 1 − n X k =1 a k . Using (4.1) we have | K | = 0 . Now, we will be interested in the behavior of the function | K ( t ) | . Lemma 4.1.
The function | K ( · ) | is non-increasing and lim t → + | K ( t ) | = 0 .Proof. The monotonicity of | K ( · ) | is clear. Moreover, K ( t ) ց K and K (1) < ∞ since K is compact. It is easily seen that lim t → + | K ( t ) | = 0. (cid:3) Lemma 4.2.
For each n ∈ N let r n , ε n be given by r n = 1 − n X k =1 a k ; ε n = 2 − n (cid:16) − n X k =1 a k (cid:17) = 2 − n r n . Then r n | K ( ε n ) | r n . Proof.
Clearly, ε n = | J α | for α ∈ { , } n and so, K ( ε n ) ⊃ [ α ∈{ , } n J α which gives | K ( ε n ) | > X α ∈{ , } n | J α | = 2 n ε n = r n . For the right-hand inequality, denote by M the set of all endpoints of intervals J α , α ∈ { , } n . The number of these points is 2(1 + 2 + 2 + · · · + 2 n − ) = 2(2 n − K ( ε n ) ⊂ [ x ∈ M ( x − ε n , x + ε n ) . LMOST-COMPACT AND COMPACT EMBEDDING . . . 11
Then | K ( ε n ) | X x ∈ M ε n = 2(2 n − ε n . n ε n = 4 r n . (cid:3) One important case is obtained by choosing a k = a k − ( a +1) k where a >
0. When a = 2 we obtain the classical Cantor set. Lemma 4.3.
Let a k = a k − ( a +1) k and set s = ln( aa +1 )ln( a a +1) ) . Then there are positive constants c , c such that c t s | K ( t ) | c t s , t ∈ [0 , diam(Ω)] . Proof.
Let q = aa +1 . Then s = ln q ln( q/ . Clearly, r n = 1 − n X k =1 a k − ( a + 1) k = (cid:16) aa + 1 (cid:17) n = q n ,ε n = 2 − n (cid:16) aa + 1 (cid:17) n = 2 − n q n and r n +1 = qr n , ε n +1 = q ε n . It is easy to see that 0 < s < t ∈ [ ε n +1 , ε n ]. By Lemmas 4.1 and 4.2 we know that qr n = r n +1 | K ( ε n +1 ) | | K ( t ) | | K ( ε n ) | r n . (4.3)Since q/ ε n t ε n , we haveln q/ n ln q/ q/ ε n ln t ln ε n = n ln q/ t ln q/ n ln t − ln q/ q/ . This implies that t ln q ln q/ = q ln t ln q/ r n q ln t − ln q/ q/ = 1 q t ln q ln q/ . By (4.3) we obtain qt s | K ( t ) | q t s . (cid:3) We recall the definition of the Riemann function ζ ( s ) = ∞ X k =1 k − s , s > . Lemma 4.4.
Let a k = k − s ζ ( s ) . Then there are positive constants c , c such that c (ln(e /t )) s − | K ( t ) | c (ln(e /t )) s − , t ∈ [0 , diam(Ω)] . Proof.
Clearly, r n = 1 − ζ ( s ) n X k =1 k − s = 1 ζ ( s ) ∞ X k = n +1 k − s . It is easy to see that 1 ζ ( s )( s − n + 1) s − = 1 ζ ( s ) Z ∞ n +1 x − s ds r n ζ ( s ) Z ∞ n x − s ds = 1 ζ ( s )( s − n s − . This gives for n > − s r n +1 r n , − s ε n +1 ε n . Fix t ∈ [ ε n +1 , ε n ]. Then2 − s r n = r n +1 | K ( ε n +1 ) | | K ( t ) | | K ( ε n ) | r n . (4.4)We know 2 − s ε n ε n +1 t ε n which gives 2 − n ( n + 1) − s ζ ( s )( s − ε n − n n − s ζ ( s )( s − s s − − n n − s ζ ( s )( s − t − n n − s ζ ( s )( s − . So, there are constants b , b such that b − n n − s t b − n n − s . It yields ln b − n ln 2 − ( s − n ln b − n ln 2 − ( s −
1) ln n ln t ln b − n ln 2 − ( s −
1) ln n ln b − n ln 2and so ln( b /t )ln 2 + s − n ln( b /t )ln 2 . Then | K ( t ) | r n = 4 ζ ( s )( s − n s − s − s − ζ ( s )( s − b /t )) s − c (ln(e /t )) s − Finally, | K ( t ) | > r n +1 > ζ ( s )( s − n + 2) s − > ζ ( s )( s − (cid:16) nn + 2 (cid:17) s − n s − > s − ζ ( s )( s − n s − > (ln 2) s − s − ζ ( s )( s − b /t )) s − > c (ln(e /t )) s − . (cid:3) Lemma 4.5.
Define a function η ( s ) = P ∞ k =1 1( k +1) ln s ( k +1) , s > . Choose a k = η ( s ) 1( k +1) ln s ( k +1) . Then there are positive constants c , c and b such that c (ln ln( b/t )) − s | K ( t ) | c (ln ln( b/t )) − s . Proof.
The proof is analogous to the previous one. Clearly, r n = ∞ X k = n +1 k + 1) ln s ( k + 1) , ε n = 2 − n r n . By the integral criterion we have estimate12( ε −
1) ln s − n r n ε −
1) ln s − n , n > . Fix t ∈ [ ε n +1 , ε n ]. Then12( ε − n +1 ln s − ( n + 1) − n − r n +1 = ε n +1 t ε n = 2 − n r n ε − n ln s − n . Since 2 n +1 ln s − ( n +1) is comparable with 2 n ln s − n for large n we can take positivecontant b , b such that b n ln s − n t b n ln s − n . (4.5)By Lemma 4.2 we have r n +1 | K ( t ) | r n (4.6)and so there are two positive constants d , d with d ln s − n | K ( t ) | d ln s − n . By (4.5) we obtainln b − n ln 2 − ( s −
1) ln ln n ln t ln b − n ln 2 − ( s −
1) ln ln n which gives for some constants L , L L n n ln 2 + ( s −
1) ln ln n ln b t , ln b t n ln 2 + ( s −
1) ln ln n L n. So 1 L ln b t n L ln b t . This implies (cid:18) ln (cid:16) L ln b t (cid:17)(cid:19) s − ln s − n (cid:18) ln (cid:16) L ln b t (cid:17)(cid:19) s − . Finally we can find c , c and b such that c (ln ln( b/t )) − s | K ( t ) | c (ln ln( b/t )) − s . (cid:3) All Cantor sets are constructed on an interval [0 ,
1] so far. But we can constructCantor sets in [0 , N as a cartesian product. But having | K ( t ) | = ϕ ( t ) we have | K N ( t ) | | K ( t ) | N ϕ N ( t ). In Lemmas 4.3, 4.4 and 4.5 we essentially get nothingnew, the behavior of ϕ ( t ) stays qualitatively the same.5. Examples of almost compact embeddings
Example 5.1.
Let Ω be bounded, p ( · ) , q ( · ) ∈ E (Ω), and suppose that q ( x ) p ( x ) p + < ∞ for all x ∈ Ω. Let K ⊂ Ω be compact and let ϕ be given by(3.11). Suppose that ϕ ( t ) Ct s for some C > s ∈ (0 , N ]. Assume that ψ : [ ω , ∞ ) → (0 , ∞ ) satisfies(i) ψ ( t )ln t is decreasing;(ii) lim t →∞ ψ ( t ) = ∞ ;(iii) s ( x ) := p ( x ) − q ( x ) ln(1 / d K ( x )) ψ (1 / d K ( x )) .Then L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω). Proof.
Let a >
1. Set ω ( t ) = ln(1 /t ) ψ (1 /t ) . Clearly, s ( x ) ω (d K ( x )) by (iii). Using (ii)we have ω ( t )ln(1 /t ) = 1 ψ (1 /t ) → t → + . Since ω is decreasing by (i) on (0 , /ω ) we can take an inverse function and write t = ω − ( y ), y ∈ [ ω (1 /ω ) , ∞ ). Then ψ (1 /ω − ( y )) = y ln(1 /ω − ( y )) → y → ∞ . (5.1)This gives us ln 1 ω − ( y ) = yψ (1 /ω − ( y )) ⇒ ω − ( y ) = e − yψ (1 /ω − y )) and consequently Z ∞ ω ϕ ( ω − ( y )) a y dy C Z ∞ ω e − syψ (1 /ω − y )) e y ln a dy = C Z ∞ ω e y (ln a − sψ (1 /ω − y )) ) dy = I. By (5.1) we have ψ (1 /ω − ( y )) → y → ∞ and so, ln a − sψ (1 /ω − ( y )) − y which implies I < ∞ . Now, Lemma 3.8 gives L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω). (cid:3) Example 5.2.
Let Ω be bounded, p ( · ) , q ( · ) ∈ E (Ω), and suppose that q ( x ) p ( x ) p + < ∞ for all x ∈ Ω. Let K ⊂ Ω be compact and let ϕ be givenby (3.11) and ϕ ( t ) C (ln(e /t )) − s for some C > s >
1. Assume that ψ : [ ω , ∞ ) → (0 , ∞ ) satisfies(i) ψ ( t )ln ln t is decreasing; LMOST-COMPACT AND COMPACT EMBEDDING . . . 15 (ii) lim t →∞ ψ ( t ) = ∞ ;(iii) s ( x ) := p ( x ) − q ( x ) ln ln(1 / d K ( x )) ψ (1 / d K ( x )) .Then L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω). Proof.
Let a >
1. Set ω ( t ) = ln ln(1 /t ) ψ (1 /t ) . Clearly, s ( x ) ω (d K ( x )) by (iii). Using (ii)we have ω ( t )ln ln(1 /t ) = 1 ψ (1 /t ) → t → + . Since ω is decreasing by (i) on (0 , /ω ) we can take an inverse function and write t = ω − ( y ), y ∈ [ ω (1 /ω ) , ∞ ). Then ψ (1 /ω − ( y )) = y ln ln(1 /ω − ( y )) → y → ∞ . (5.2)It gives usln ln 1 ω − ( y ) = yψ (1 /ω − ( y )) ⇒ ω − ( y ) = exp(exp( y/ψ (1 /ω − ( y ))))and consequently Z ∞ ω ϕ ( ω − ( y )) a y dy = Z ∞ ω (cid:16) ln 1 ω − ( y ) (cid:17) − s a y dy = Z ∞ ω e (1 − s ) ln ln ω − y ) e y ln a dy c Z ∞ ω e (1 − s ) yψ (1 /ω − y )) e y ln a dy = c Z ∞ ω e y (ln a + − sψ (1 /ω − y )) ) dy = I. By (5.2) we have ψ (1 /ω − ( y )) → y → ∞ and so, ln a + − sψ (1 /ω − ( y )) − y which implies I < ∞ . Now, Lemma 3.8 gives L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω). (cid:3) Example 5.3.
Let Ω ∈ C , , p ( · ) , q ( · ) ∈ E (Ω), and suppose that 1 p ( x ) p +
1. Assume that ψ : [ ω , ∞ ) → (0 , ∞ )satisfies(i) ψ ( t )ln ln ln t is decreasing;(ii) lim t →∞ ψ ( t ) = ∞ ;(iii) s ( x ) := p ( x ) − q ( x ) ln ln ln(1 / d K ( x )) ψ (1 / d K ( x )) .Then L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω). Proof.
Let a >
1. Set ω ( t ) = ln ln ln(1 /t ) ψ (1 /t ) . Clearly, s ( x ) ω (d K ( x )) by (iii). Using(ii) we have ω ( t )ln ln ln(1 /t ) = 1 ψ (1 /t ) → t → + . Since ω is strictly monotone by (i) on (0 , /ω ) we can take an inverse function andwrite t = ω − ( y ), y ∈ [ ω (1 /ω ) , ∞ ). Then ψ (1 /ω − ( y )) = y ln ln ln(1 /ω − ( y )) → y → ∞ . (5.3) Thusln ln ln 1 ω − ( y ) = yψ (1 /ω − ( y )) ⇒ ω − ( y ) = exp(exp(exp( y/ψ (1 /ω − ( y )))))and consequently Z ∞ ω ϕ ( ω − ( y )) a y dy = Z ∞ ω (cid:16) ln ln 1 ω − ( y ) (cid:17) − s a y dy = Z ∞ ω e (1 − s ) ln ln ln ω − y ) e y ln a dy c Z ∞ ω e (1 − s ) yψ (1 /ω − y )) e y ln a dy = c Z ∞ ω e y (ln a + − sψ (1 /ω − y )) ) dy = I. By (5.3) we have ψ (1 /ω − ( y )) → y → ∞ and so, ln a + − sψ (1 /ω − ( y )) − y which implies I < ∞ . Now, Lemma 3.8 gives L p ( · ) (Ω) ∗ ֒ → L q ( · ) (Ω). (cid:3) Compact embeddings between variable Sobolev and variableLebesgue spaces
First of all we establish a necessary condition for an embedding to be compact.
Lemma 6.1.
Let B r = B (0 , r ) denote the ball in R N centered at with radius r . Assume M ⊂ B r and s ∈ R are such that | B r \ B s | | M | . Suppose that ϕ : (0 , r ] → R is non-negative and non-increasing and set ψ ( x ) = ϕ ( | x | ) , x ∈ B r .Then Z M ψ ( x ) dx > Z B r \ B s ψ ( x ) dx. Proof.
By the assumption | B r \ B s | | M | we have | ( B r \ B s ) \ M | + | ( B r \ B s ) ∩ M | = | ( B r \ B s ) | | M | = | ( B r \ B s ) ∩ M | + | M ∩ B s | . and consequently | M ∩ B s | > | ( B r \ B s ) \ M | . By the assumptions on ψ we have ψ ( x ) > ψ ( y ) for every x ∈ B s and every y ∈ B r \ B s . This implies Z M ψ ( x ) dx = Z M ∩ B s ψ ( x ) dx + Z ( B r \ B s ) ∩ M ψ ( x ) dx > | M ∩ B s || ( B r \ B s ) \ M | Z ( B r \ B s ) \ M ψ ( x ) dx + Z ( B r \ B s ) ∩ M ψ ( x ) dx > Z ( B r \ B s ) \ M ψ ( x ) dx + Z ( B r \ B s ) ∩ M ψ ( x ) dx = Z B r \ B s ψ ( x ) dx. which finishes the proof. (cid:3) Theorem 6.2.
Let p, q ∈ E (Ω) , p ( x ) p + < N on Ω and let p ( · ) satisfy (2.2) , q ( x ) p ( x ) and let M = { x ∈ Ω; p ( x ) = q ( x ) } . Assume W ,p ( · ) (Ω) ֒ → ֒ → L q ( · ) (Ω) . Then | M | = 0 . LMOST-COMPACT AND COMPACT EMBEDDING . . . 17
Proof.
Suppose | M | >
0. Let x ∈ Ω be a point of Lebesgue density of M . Given ε > p ε − = inf { p ( x ); x ∈ B ( x , ε ) } , p ε + = sup { p ( x ); x ∈ B ( x , ε ) } and define a function u ε by u ε ( x ) = ε pε −− Npε − (1 − | x | /ε ) χ B ( x ,ε ) ( x ) . Clearly, |∇ u ε ( x ) | = ε pε −− Npε − /ε χ B ( x ,ε ) ( x ) = ε − Npε − χ B ( x ,ε ) ( x ) . First we prove that the set { u ε } is bounded in W ,p ( · ) (Ω) for ε
1. Plainly, Z Ω | u ε ( x ) | p ( x ) dx = Z B ( x ,ε ) ε pε −− Npε − p ( x ) (1 − | x | /ε ) p ( x ) dx Z B ( x ,ε ) ε p ( x ) ε − Npε − p ( x ) dx Z B ( x ,ε ) ε − Npε − p ( x ) dx := I ε . Moreover, Z Ω |∇ u ε ( x ) | p ( x ) dx Z B ( x ,ε ) ε − Npε − p ( x ) dx := I ε . Now, I ε = Z B ( x ,ε ) ε − Npε − ( p ( x ) − p ε − ) ε − N dx = Z B ( x ,ε ) e − Npε − ( p ( x ) − p ε − ) ln ε ε − N dx = Z B ( x ,ε ) e Npε − ( p ( x ) − p ε − ) ln(1 /ε ) ε − N dx. From the log-Lipschitz condition (2.2) we have( p ( x ) − p ε − ) ln(1 /ε ) C and so I ε Z B ( x ,ε ) e CNpε − ε − N dx e CN Z B ( x ,ε ) ε − N dx := A. This immediately implies that k u ε k W ,p ( · ) (Ω) is bounded for ε ε such that for all ε ε we have | B ( x , ε ) ∩ M | > | B ( x , ε/ | . Fix for a moment ε ε . Then A ε := Z ( B ( x ,ε ) \ B ( x , / ε ) ∩ M (1 − | x | /ε ) p ( x ) (cid:16) ε pε −− Npε − (cid:17) p ( x ) dx > Z ( B ( x ,ε ) \ B ( x , / ε ) ∩ M (1 − | x | /ε ) p ( x ) ε pε −− Npε − ( p ε − ) dx = Z ( B ( x ,ε ) \ B ( x , / ε ) ∩ M (1 − | x | /ε ) p ( x ) ε pε −− Npε − Npε − N − pε − dx = Z ( B ( x ,ε ) \ B ( x , / ε ) ∩ M (1 − | x | /ε ) p ( x ) ε − N dx > Z ( B ( x ,ε ) \ B ( x , / ε ) ∩ M (1 − | x | /ε ) ( p ε + ) ε − N dx := B ε . By Lemma 6.1 we have B ε > Z ( B ( x ,ε ) \ B ( x , / ε ) (1 − | x | /ε ) ( p ε + ) ε − N dx = σ N ε − N Z ε / ε (1 − r/ε ) ( p ε + ) r N − dr > σ N ε − N Z / ε / ε (1 − (15 ε/ /ε ) ( p ε + ) r N − dr = σ N (1 / ( p ε + ) ε − N Z / ε / ε r N − dr := K.σ N denotes the area of N -dimensional unit sphere S N .Denote ε n = (3 / n ε and consider the corresponding sequence u ε n ( x ). Let m > n . Then u m ( x ) = 0 for x ∈ B ( x , ε n ) \ B ( x , ε m ) and so, Z Ω | u m ( x ) − u n ( x ) | q ( x ) dx = Z B ( x ,ε n ) | u m ( x ) − u n ( x ) | q ( x ) dx > Z B ( x ,ε n ) \ B ( x ,ε m ) | u n ( x ) | q ( x ) dx = Z ( B ( x ,ε n ) \ B ( x , / ε n )) ∩ M | u n ( x ) | q ( x ) dx = Z ( B ( x ,ε n ) \ B ( x , / ε n )) ∩ M | u n ( x ) | p ( x ) dx > K. Hence, there is a constant
L > k u m − u n k L q ( · ) (Ω) > L and the embedding W ,p ( · ) (Ω) ֒ → ֒ → L q ( · ) (Ω) is not compact. (cid:3) The next lemma is proved in [2] (see Corollary 8.3.2.).
Lemma 6.3.
Let Ω ∈ C , , p, q ∈ E (Ω) and p ( · ) satisfies the log -H¨older condition (2.2) . Assume that for all x ∈ Ω1 p ( x ) p + < N. Then W ,p ( · ) (Ω) ֒ → L p ( · ) (Ω) where p ( x ) is given in (1.6) . LMOST-COMPACT AND COMPACT EMBEDDING . . . 19
Theorem 6.4.
Let Ω ∈ C , , p, q ∈ E (Ω) and let p ( · ) satisfy the log -H¨older condi-tion (2.2) . Assume that for all x ∈ Ω , p ( x ) p + < N, q ( x ) p ( x ) where p ( x ) is given in (1.6) . Let K ⊂ Ω be compact, | K | = 0 and denote ϕ ( t ) = | K ( t ) | . Let ω : [0 , diam(Ω)] → R be a decreasing continuous non-negative function, ω := ω (diam(Ω)) . Suppose that ω ( · ) satisfies p ( x ) − q ( x ) c ω (d K ( x )) , x ∈ Ω , Z ∞ ω ϕ ( ω − ( y )) a y dy < ∞ for all a > . Then W ,p ( · ) (Ω) ֒ → ֒ → L q ( · ) (Ω) .Proof. Lemmas 6.3 and 3.8 give W ,p ( · ) (Ω) ֒ → L p (Ω) ∗ ֒ → L q ( · ) (Ω) . Now Proposition 2.2 finishes the proof. (cid:3)
As an application we introduce the following several examples. The first one isin fact proved in [9] (see Theorem 3.4) but we obtain it as an easy consequence ofthe previous theorem. Let ϕ and q denote in following three examples the same asin Theorem 6.4. Example 6.5.
Let Ω ∈ C , and p ( · ) : Ω → R satisfy (2.2). Assume1 p ( x ) p + < N. Let K ⊂ Ω and ϕ ( t ) Ct s for some C > s ∈ (0 , N ]. Assume that ψ :[ ω , ∞ ) → (0 , ∞ ) satisfies(i) ψ ( t )ln t is decreasing;(ii) lim t →∞ ψ ( t ) = ∞ ;(iii) s ( x ) := p ( x ) − q ( x ) ln(1 / d K ( x )) ψ (1 / d K ( x )) .Then W ,p ( · ) (Ω) ֒ → ֒ → L q ( · ) (Ω). Proof.
It suffices to choose ω ( t ) = ln(1 /t ) ψ (1 /t )and then to use Theorem 6.4. (cid:3) Example 6.6.
Let Ω ∈ C , and p ( · ) : Ω → R satisfy (2.2). Assume1 p ( x ) p + < N. Let K ⊂ Ω and ϕ ( t ) C (ln(e /t )) − s for some C > s >
1. Assume that ψ : [ ω , ∞ ) → (0 , ∞ ) satisfies(i) ψ ( t )ln ln t is decreasing;(ii) lim t →∞ ψ ( t ) = ∞ ; (iii) s ( x ) := p ( x ) − q ( x ) ln ln(1 / d K ( x )) ψ (1 / d K ( x )) .Then W ,p ( · ) (Ω) ֒ → ֒ → L q ( · ) (Ω). Proof.
Take ω ( t ) = ln ln(1 /t ) ψ (1 /t )and use Theorem 6.4. (cid:3) Example 6.7.
Let Ω ∈ C , and p ( · ) : Ω → R satisfy (2.2). Assume1 p ( x ) p + < N. Let K ⊂ Ω and ϕ ( t ) C (ln ln(e /t )) − s for some C > s >
1. Assume that ψ : [ ω , ∞ ) → (0 , ∞ ) satisfies(i) ψ ( t )ln ln ln t is decreasing;(ii) lim t →∞ ψ ( t ) = ∞ ;(iii) s ( x ) := p ( x ) − q ( x ) ln ln ln(1 / d K ( x )) ψ (1 / d K ( x )) .Then W ,p ( · ) (Ω) ֒ → ֒ → L q ( · ) (Ω). Proof.
Take ω ( t ) = ln ln ln(1 /t ) ψ (1 /t )and use Theorem 6.4. (cid:3) To conclude we remark that the construction of Cantor sets could be refined,adding some more logarithms to give additional examples.
References [1] C. Bennett and R. Sharpley,
Interpolations of operators . Pure and Apl. Math., vol. 129,Academic Press, New York, 1988.[2] L. Diening, P. Harjulehto, P. H¨ast¨o and M. R˚uˇziˇcka,
Lebesgue and Sobolev spaces with variableexponents . Lecture Notes in Mathematics, Springer, 2011.[3] D. E. Edmunds, J. Lang and A. Nekvinda, On L p ( x ) norms. Proc. Roy. Soc. London SeriesA , no. 1981, (1999), 219–225.[4] D. E. Edmunds and J. R´akosn´ık, Sobolev embeddings with variable exponent.
Studia Math , no. 3, (2000), 267—293.[5] D. E. Edmunds and J. R´akosn´ık, Sobolev embeddings with variable exponent II.
Math. Nachr. (2002), 53—67.[6] G. Juanjuan, Z. Peihao and Z. Yong, Compact Sobolev embedding theorems involving sym-metry and its application.
Nonlinear Differential Equations Appl. , no. 2, (2010), 161-–180.[7] O. Kov´aˇcik and J. R´akosn´ık, On spaces L p ( x ) (Ω) and W k,p ( x ) . Czechoslovak Math. J. , no. 4, (1991), 592—618.[8] K. Kurata and N. Shioji, Compact embedding from W , (Ω) to L q ( x ) (Ω) and its applicationto nonlinear elliptic boundary value problem with variable critical exponent. J. Math. Anal.Appl. , no. 2, (2008), 1386—1394.[9] Y. Mizuta, T. Ohno, T. Shimomura and N. Shioji, Compact embeddings for Sobolev spacesof variable exponents and existence of solutions for nonlinear elliptic problems involving the p ( x )-Laplacian and its critical exponent. Ann. Acad. Sci. Fenn. Math. , no. 1, (2010),115–130.[10] L. Slav´ıkov´a , Almost-compact embeddings. Math. Nachr. , no. 11-12, (2012), 1500—1516.
LMOST-COMPACT AND COMPACT EMBEDDING . . . 21
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