Journal of Function Spaces and Applications Volume 2012, Article ID 132690,17pages doi:10.1155/2012/132690
Research Article
Weighted Variable Sobolev Spaces and Capacity
Ismail Aydin
Department of Mathematics, Faculty of Arts and Sciences, Sinop University, 57000 Sinop, Turkey Correspondence should be addressed to Ismail Aydin,aydn.iso953@gmail.com
Received 7 July 2010; Accepted 21 November 2010 Academic Editor: V. M. Kokilashvili
Copyrightq 2012 Ismail Aydin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We define weighted variable Sobolev capacity and discuss properties of capacity in the space W1,p·Rn, w. We investigate the role of capacity in the pointwise definition of functions in this
space if the Hardy-Littlewood maximal operator is bounded on the space W1,p·Rn, w. Also it is
shown the relation between the Sobolev capacity and Bessel capacity.
1. Introduction
In 1991 Kov´aˇcik and R´akosn´ık 1 introduced the variable exponent Lebesgue space Lp·Rn
and Sobolev space Wk,p·Rn in higher dimensional Euclidean spaces. The spaces Lp·Rn and LpRn have many common properties. A crucial difference between Lp·Rn and the classical Lebesgue spaces LpRn is that Lp·Rn is not invariant under translation in general Example 2.9 in 1 and Lemma 2.3 in 2. The boundedness of the maximal operator was
an open problem in Lp·Rn for a long time. It was first proved by Diening 2 over bounded domains, under the assumption that p· is locally log-H¨older continuous, that is,
px − p
y ≤ C
− lnx− y, x,y ∈ Ω, x− y ≤ 12. 1.1 He later extended the result to unbounded domains by supposing, in addition, that the exponent p· is constant outside a large ball. After this paper, many interesting and important papers appeared in nonweighted and weighted variable exponent spaces. For more details and historical background, see1,3–5. Sobolev capacity for constant exponent spaces has
found a great number of uses, see Mazja 6, Evans and Gariepy 7, and Heinonen et al.
8. Also Kilpel¨ainen 9 introduced weighted Sobolev capacity and discussed the role of
Muckenhoupt’s Ap-class. Variable Sobolev capacity was introduced in the spaces W1,p·Rn by Harjulehto et al.10. They generalized the Sobolev capacity to the variable exponent case.
Our purpose is to generalize some results of9–12 to the weighted variable exponent case.
2. Definition and Preliminary Results
We study weighted variable Lebesgue and Sobolev spaces in the n-dimensional Euclidean spaceRn, n≥ 2. Throughout this paper all sets and functions are Lebesgue measurable. The Lebesgue measure and the characteristic function of a subset A ⊂ Rn will be denoted by
μA |A| and χA, respectively. The space L1locRn consists of all classes of measurable functions f onRnsuch that fχ
K∈ L1Rn for any compact subset K ⊂ Rn. It is a topological vector space with the family of seminorms f → fχK L1. A Banach function spaceshortly
BF-space on Rn is a Banach spaceB, ·
B of measurable functions which is continuously embedded into L1
locRn, that is, for any compact subset K ⊂ Rn there exists some constant
CK > 0 such that fχK L1 ≤ CK f Bfor all f ∈ B. We denote it by B → L1locRn. The class
C∞0Rn is defined as set of infinitely differentiable functions with compact support in Rn. For a measurable function p :Rn → 1, ∞ called a variable exponent on Rn, we put
p− essinf
x∈Rn px, p
esssup
x∈Rn
px. 2.1
For every measurable functions f onRnwe define the function
p·f
Rn
fxpxdx. 2.2
The function p· is convex modular; that is, p·f ≥ 0, p·f 0 if and only if f 0,
p·−f p·f and p·is convex. The variable exponent Lebesgue spacesor generalized Lebesgue spaces Lp·Rn is defined as the set of all measurable functions f on Rnsuch that
p·λf < ∞ for some λ > 0, equipped with the Luxemburg norm
fp· inf λ > 0 : p f λ ≤ 1 . 2.3
If p < ∞, then f ∈ Lp·Rn if and only if
p·f < ∞. The set Lp·Rn is a Banach space with the norm · p·. If px p is a constant function, then the norm · p·coincides with the usual Lebesgue norm · p1. In this paper we assume that p <∞.
A positive, measurable, and locally integrable function w : Rn → 0, ∞ is called a weight function. The weighted modular is defined by
p·,wf
Rn
fxpx
The weighted variable exponent Lebesgue space Lp·Rn, w consists of all measurable func-tions f onRn for which f
p·,w fw1/p· p· < ∞. The relations between the modular
p·,w· and · p·,was follows: min p·,wf1/p − , p·,wf1/p ≤fp·,w ≤ max p·,wf1/p − , p·,wf1/p min fpp ·,w, fpp−·,w ≤ p·,wf≤ max fp p·,w, f p− p·,w , 2.5 see13–15. Moreover, if 0 < C ≤ w, then we have Lp·Rn, w → Lp·Rn, since one easily sees that C Rn fxpxdx≤ Rn fxpxwxdx 2.6 and C f p·≤ f p·,w.
The Schwartz class S SRn consists of all infinitely differentiable and rapidly de-creasing functions inRn. Then f and any derivative Dβf die out faster than reciprocal of any polynomial at infinity. That is, f ∈ S if and only if for any β and k > 0 there is a constant
C Cβ, k such that Dβfx ≤ C 1 |x|k. 2.7 In particular, for β 0, fx ≤ C 1 |x|k. 2.8
Also it is well known that C∞0 Rn ⊂ S.
For x∈ Rnand r > 0 we denote an open ball with center x and radius r by Bx, r. For
f∈ L1
locRn, we denote the centered Hardy-Littlewood maximal operator Mf of f by
Mfx sup r>0 1 |Bx, r| Bx,r fydy, 2.9
where the supremum is taken over all balls Bx, r.
Let 1≤ p < ∞. A weight w satisfies Muckenhoupt’s ApRn Apcondition, or w∈ Ap, if there exist positive constants C and c such that, for every ball B⊂ Rn,
1 |B| B w dx 1 |B| B w−1/p−1dx p−1 ≤ C, 1 ≤ p < ∞, 2.10
or 1 |B| B w dx esssup B 1 w ≤ c, p 1. 2.11
The infimum over the constants C and c is called the Apand A1, respectively. Also it is known
that A∞ 1≤p<∞Ap. Let 1 < p <∞. Then Muckenhoupt proved that w ∈ Apif and only if the Hardy-Littlewood maximal operator is bounded on LpRn, w 16. Also Miller showed that the Schwartz class S is dense in LpRn, w for 1 < p < ∞ and w ∈ A
p17, Lemma 2.1. H¨ast ¨o and Diening defined the class Ap·to consist of those weights w for which
w Ap· : sup B∈B|B| −pB w L1B w1 Lp·/p·B <∞, 2.12
whereB denotes the set of all balls in Rn, p
B 1/|B|
B1/pxdx
−1and 1/p· 1/p·
1. Note that this class is ordinary Muckenhoupt class Apif p is a constant function13. We say that p· satisfies the local log-H¨older continuity condition if
px − py ≤ C
loge 1/x− y 2.13
for all x, y∈ Rn. If
px − p∞ ≤ C
loge |x| 2.14
for some p∞ > 1, C > 0 and all x ∈ Rn, then we say p· satisfies the log-H¨older decay con-ditionat infinity. We denote by PlogRn the class of variable exponents which are log-H ¨older continuous, that is, which satisfy the local log-log-H¨older continuity condition and the log-H ¨older decay condition.
Let p, q∈ PlogRn, 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 w Ap· ≤ C w Aq·13, Lemma 3.1. As a result of this Lemma we have
A1⊂ Ap− ⊂ Ap·⊂ Ap ⊂ A∞ 2.15
for p∈ PlogRn and 1 < p− ≤ p <∞.
Let p∈ PlogRn and 1 < p− ≤ p <∞. Then M : Lp·Rn, w → Lp·Rn, w if and only if w∈ Ap·13, Theorem 1.1.
We use the notation
PRn p· : 1 < p− ≤ px ≤ p <∞, Mf
p·,w ≤ Cfp·,w
, 2.16
that is, the maximal operator M is bounded on Lp·Rn, w. Hence we can find a sufficient condition for p· ∈ PRn.
Proposition 2.1. Let w be a weight function and 1 < p−≤ px ≤ p <∞. If w−1/p·−1∈ L1
locRn,
then Lp·Rn, w → L1
locRn.
Proof. Suppose that f∈ Lp·Rn, w, and let K ⊂ Rnbe a compact set. For 1/p· 1/q· 1, by using H ¨older’s inequality for variable exponent Lebesgue spaces1, then there exists a
AK> 0 such that K fxdx≤ AKfw1/p· p·,K w−1/p· q·,K ≤ AKfw1/p· p· w−1/p· q·,K. 2.17
It is known that w−1/p· q·,K < ∞ if and only if q·,Kw−1/p· < ∞ for q < ∞. Since
w−1/p·−1∈ L1
locRn, then we have
q·,K w−1/p· K wx−qx/pxdx K wx−1/px−1dx BK<∞. 2.18
If we use2.17 and 2.18, then the proof is completed.
Definition 2.2 Mollifiers. Let ϕ : Rn → R be a nonnegative, radial, decreasing function
belonging to C0∞Rn and having the properties: i ϕx 0 if |x| ≥ 1,
iiRnϕxdx 1.
Let ε > 0. If the function ϕεx ε−nϕx/ε is nonnegative, belongs to C0∞Rn, and
satisfies
i ϕεx 0 if |x| ≥ ε and iiRnϕεxdx 1,
then ϕεis called a mollifier and we define the convolution by
ϕε∗ fx Rn ϕε x− yfydy. 2.19
The following proposition was proved in18, Proposition 2.7.
Proposition 2.3. Let ϕεbe a mollifier and f∈ L1locRn. Then
sup ε>0
ϕε∗ fx ≤ Mfx. 2.20
Proposition 2.4. If p· ∈ PRn and f ∈ Lp·Rn, w, then ϕ
Proof. Let f ∈ Lp·Rn, w and ε > 0 be given. If f is continuous, then the assertion is trivial. ByProposition 2.3, we have
ϕε∗ fp·,w≤Mfp·,w ≤ Cfp·,w 2.21
and we have ϕε∗ f ∈ Lp·Rn, w for all ε > 0. It can be proved that the class C0Rn of
continuous functions with compact support is dense in the space Lp·Rn, w. Then there is a
function g∈ C0Rn such that
f− gp·,w< ε. 2.22
Also it is well known that if g ∈ C0Rn, then ϕε∗ g ∈ C∞0 Rn for all ε > 0. It is easily seen that ϕε∗ g → g uniformly on compact sets as ε → 0 . Hence we have
ϕε∗ gx − gxpx−→ 0, p·,wϕε∗ g − g K ϕε∗ gx − gxpxwxdx ≤ εp− K wxdx, 2.23
where suppϕε∗ g ∪ supp g ⊂ K, K ⊂ Rncompact. Hence p·,wϕε∗ g − g → 0 as ε → 0 and we write
ϕε∗ g − gp·,w < ε. 2.24
Finally by using2.22 and 2.24,
f− ϕε∗ f
p·,w ≤f− gp·,w g− ϕε∗ gp·,w ϕε∗ g − ϕε∗ fp·,w
<C 2ε.
2.25
The proof is complete.
As a direct consequence ofProposition 2.4there follows.
Corollary 2.5. Let p· ∈ PRn. The class C∞
0 Rn is dense in Lp·Rn, w.
This result was proved without the assumption that the maximal operator is bounded in Lp·Rn, w by Kokilashvili and Samko 19.
Remark 2.6. Let 1 < p− ≤ px ≤ p < ∞ and w−1/p·−1 ∈ L1locRn. Then every function in Lp·Rn, w has distributional derivatives byProposition 2.1.
3. Weighted Variable Sobolev Spaces
Let 1 < p−≤ px ≤ p <∞, w−1/p·−1∈ L1locRn and k ∈ N. We define the weighted variable
Sobolev spaces Wk,p·Rn, w by
Wk,p·Rn, w f ∈ Lp·Rn, w : Dαf∈ Lp·Rn, w, 0 ≤ |α| ≤ k 3.1
equipped with the norm
f k,p·,w 0≤|α|≤k Dαf p·,w 3.2 where α ∈ Nn 0 is a multiindex,|α| α1 α2 · · · αn, and Dα ∂|α|/∂αx11· · · ∂ αn xn. It can be
shown that Wk,p·Rn, w is a reflexive Banach space. Throughout this paper, we will always assume that 1 < p−≤ px ≤ p <∞ and w−1/p·−1∈ L1
locRn.
The space W1,p·Rn, w is defined by
W1,p·Rn, w f ∈ Lp·Rn, w :∇f ∈ Lp·Rn, w . 3.3
The function 1,p·,w : W1,p·Rn, w → 0, ∞ is defined as 1,p·,wf p·,wf
p·,w∇f. The norm f 1,p·,w f p·,w ∇f p·,w.
The Bessel kernel gαorder α, α > 0, is defined by
gαx π n/2 Γα/2 ∞ 0 e−s−π2|x|2/ssα−n/2ds s , x∈ R n. 3.4
Let α≥ 0. The weighted variable Bessel potential space Lα,p·Rn, w is, for α > 0, defined by Lα,p·Rn, w : h g
α∗ f; f ∈ Lp·Rn, w
, 3.5
and is equipped with the norm
h α;p·,w fp·,w. 3.6
If α 0 we put g0∗ f f and L0,p·Rn, w Lp·Rn, w.
Let p· ∈ PRn. If f ∈ Lp·Rn, w, then g
α∗f ∈ Lp·Rn, w. Indeed, since gα∈ L1Rn and gαis radial, we havegα∗fx ≤ Mfx, x ∈ Rn20, page 62. The assertion thus follows from boundedness of maximal function in Lp·Rn, w.
The unweighted variable Bessel potential space Lα,p·Rn was firstly studied by Almeida and Samko in21.
Lemma 3.1. Let p· ∈ PlogRn, 1 < p− ≤ p <∞, and w ∈ A
p·. Then
i C∞
0 Rn is dense in Wk,p·Rn, w, k ∈ N,
ii The Schwartz class S is dense in Lα,p·Rn, w, α ≥ 0.
Proof. i ByProposition 2.4the proof is complete. ii Let α 0. The class C∞
0 Rn is dense in Lp·Rn, w byCorollary 2.5. It remains
only to show that S ⊂ Lp·Rn, w. Let f ∈ S. Then there exist C Cr > 0 and r > 0 such that
fx ≤ C
1 |x|r. 3.7
Also since rpx ≥ r and 1 |x|r ≥ 1, then
p·,wf Rn fxpxwxdx ≤ max Cp−, Cp Rn wx 1 |x|rpxdx ≤ max Cp−, Cp Rn wx 1 |x|rdx. 3.8
It is known that Ap·⊂ Ap for 1 < p <∞. Also the fact that the Muckenhoupt weights with
constant p are integrable with some power weight. Then
Rn
wx
1 |x|rdx <∞, 3.9
see22, Lemma 1. If we use 3.9 in 3.8, then the Schwartz class S is dense in Lp·Rn, w. Let α > 0 and h∈ Lα,p·Rn, w. Then there is a function f ∈ Lp·Rn, w such that h
gα∗ f. By density of C∞0 Rn in f ∈ Lp·Rn, w we can find a sequence fjj∈N⊂ C0∞Rn ⊂ S converging to f in Lp·Rn, w. Since the mapping f → g
α∗f maps S onto S 20, the functions
hj gα∗ fj, j∈ N, belong to S. Moreover,
h− hjα;p·,wf− fjp·,w−→ 0 as j −→ ∞ 3.10
and the assertion follows.
The following Theorem can be proved similarly in12, Theorem 3.1.
Theorem 3.2. Let p· ∈ PRn and k ∈ N. Then Lk,p·Rn, w Wk,p·Rn, w and the
corre-sponding norms are equivalent.
Remark 3.3. The equivalence of the spacesLk,p·Rn, w and Wk,p·Rn, w fails when p 1 or p ∞.
For E⊂ Rn, we denote
Sp·,wE
f∈ W1,p·Rn, w : f ≥ 1 in open set containing E . 3.11
The Sobolevp·, w-capacity of E is defined by
Cp·,wE inf f∈Sp·,wE1,p·,w f inf f∈Sp·,wE Rn fxpx ∇fxpxwxdx. 3.12
In case Sp·,wE ∅, we set Cp·,wE ∞. The Cp·,w-capacity has the following properties.
i Cp·,w∅ 0.
ii If E1⊂ E2, then Cp·,wE1 ≤ Cp·,wE2.
iii If E is a subset of Rn, then
Cp·,wE inf
Cp·,wU : E ⊂ U, U open
. 3.13
iv If E1and E2are subsets ofRn, then
Cp·,wE1∪ E2 Cp·,wE1∩ E2 ≤ Cp·,wE1 Cp·,wE2. 3.14
v If K1⊃ K2⊃ · · · are compact, then
lim i→ ∞Cp·,wKi Cp·,w ∞ i1 Ki . 3.15
Note that the assertionv above is not true in general for noncompact sets 9.
vi If E1⊂ E2⊂ · · · are subsets of Rn, then
lim i→ ∞Cp·,wEi Cp·,w ∞ i1 Ei . 3.16
vii If Ei⊂ Rnfor i 1, 2, . . ., then
Cp·,w ∞ i1 Ei ≤∞ i1 Cp·,wEi. 3.17
For the proof of these properties see8,10. Hence the Sobolev Cp·,w capacity is an outer measure. A set function which satisfies the capacity propertiesi, ii, v, and vi is called Choquet capacity; see23. Therefore we have the following result.
Corollary 3.4. The set function E → Cp·,wE, E ⊂ Rn, is a Choquet capacity. In particular, all
Suslin sets E⊂ Rnare capacitable, that is,
Cp·,wE infE⊂U
U open
Cp·,wU sup
K⊂E K compact
Cp·,wK. 3.18
Lemma 3.5. Let wx ≥ 1 for x ∈ Rn. Then every measurable set E⊂ Rnsatisfies|E| ≤ C
p·,wE.
Proof. If f ∈ Sp·,wE, then there is an open set E ⊂ U such that f ≥ 1 in U and hence |E| ≤ |U| ≤ Rn fxpx wxdx ≤ Rn fxpx ∇fxpx wxdx. 3.19
We obtain the claim by taking the infimum on Sp·,wE.
Definition 3.6Bessel Capacity. Let E ⊂ Rn, α > 0. Define that theα, p·, w-Bessel capacity inLα,p·Rn, w is the number
Bα,p·,wE inf p·,wf, 3.20
where the infimum is taken over all f ∈ Lp·Rn, w such that g
α∗ f ≥ 1 on E. Since gα is nonnegative we can assume that f ≥ 0.
Theorem 3.7. Bα,p·,wis an outer capacity defined on all subsets ofRn.
Proof. It is known that
i Bα,p·,w∅ 0;
ii if E1⊂ E2, then Bα,p·,wE1 ≤ Bα,p·,wE2;
iii if Ei⊂ Rnfor i 1, 2, . . ., then
Bα,p·,w ∞ i1 Ei ≤∞ i1 Bα,p·,wEi 3.21
by12, Lemma 4.1. We will show that
Bα,p·,wE inf
E⊂G
G open
Bα,p·,wG. 3.22
for any E ⊂ Rn. Let E ⊂ Rn be arbitrary. Obviously Bα,p·,wE ≤ inf E⊂G
G openBα,p·,wG. We assume that Bα,p·,wE < ∞. If 0 < ε < 1 there must exist a test function measurable and
nonnegative for Bα,p·,wE, call it f, such that gα∗ f ≥ 1 on E, and
Let G {x ∈ Rn : g
α∗ f > 1 − ε}. Since gα∗ f is lower semicontinuous in x, G is an open set and since gα∗ f > 1 − ε on E, G ⊃ E. Therefore 1 − ε−1f is a test function for Bα,p·,wG and
we have Bα,p·,wG ≤ p·,w f 1− ε ≤ 1 − ε−p p·,wf<1 − ε−p Bα,p·,wE ε. 3.24
This proves the theorem as ε → 0 .
Now we give relationship between the capacities Bα,p·,wand Cp·,w12.
Lemma 3.8. Let p· ∈ PRn and E ⊂ Rn. Then
B1,p·,wE ≤ c max Cp·,wEp−/p , Cp·,wEp /p−
, Cp·,wE ≤ C max B1,p·,wEp−/p , B1,p·,wEp /p− . 3.25
Here c and C are positive constants independent of E.
Proposition 3.9. Let p· ∈ PRn.
i If f ∈ W1,p·Rn, w, then Mf ∈ W1,p·Rn, w and |∇Mfx| ≤ M|∇fx| for almost
everywhere inRn.
ii Let 1 ≤ s < ∞. Then sp· ∈ PRn and there exists a constant C > 0 such that the
inequality
Mf1,sp·,w≤ Cf1,sp·,w 3.26
holds for all f ∈ W1,sp·Rn, w.
Proof. i By Proposition 2.1 we have Lp·Rn, w → Lp·
locRn, w → L1locRn and
W1,p·Rn, w → W1,p·
loc Rn, w → W 1,1
locRn. Since f ∈ W 1,1
locRn, then we have |∇Mfx| ≤
M|∇fx| for almost everywhere in Rnby24. Since f, |∇f| ∈ Lp·Rn, w and p· ∈ PRn, then Mf, |∇Mf| ∈ Lp·Rn, w. Hence Mf ∈ W1,p·Rn, w.
ii Let f ∈ Lsp·Rn, w. By using definition of ·
p·,w, we have
fsp·,wfs1/sp·,w 3.27
and|f|s∈ Lp·Rn, w. Therefore we have
and sp· ∈ PRn. Since f ∈ W1,sp·Rn, w, then f, |∇f| ∈ Lsp·Rn, w. Hence we write Mf1,sp·,w Mfs1/sp·,w ∇Mfs1/sp·,w ≤Mfs1/sp·,w M∇fs1/sp·,w ≤Mfs1/sp·,w M∇fs1/sp·,w ≤ C1fs1/sp·,w C2∇fs1/sp·,w. 3.29
by3.28. If we set C max{C1, C2}, then
Mf
1,sp·,w≤ Cf1,sp·,w. 3.30
This completes the proof.
Proposition 3.10. Let p· ∈ PRn. Then for every λ > 0 and every f ∈ W1,p·Rn, w we have
Cp·,w x∈ Rn: Mfx > λ≤ c max fλ p 1,p·,w, fλp − 1,p·,w . 3.31
Proof. Since Mf is lower semicontinuous, the set{x ∈ Rn : Mfx > λ} is open for every
λ > 0. ByProposition 3.9we can takeMf/λ Mf/λ as a test function for the capacity. Then we have Cp·,wx∈ Rn: Mfx > λ≤ 1,p·,w Mf λ ≤ max Mfλ p 1,p·,w, Mfλp − 1,p·,w ≤ c max fλ p 1,p·,w, fλp − 1,p·,w . 3.32
We say that a property holdsp·, w-quasi everywhere if it holds except in a set of capacity zero. A function f isp·, w-quasicontinuous in Rnif for each ε > 0 there exists an open set E with Cp·,wE < ε such that f restricted to Rn\ E is continuous. The following
proof of theorem is quite similar to Theorem 4.7 in11.
Theorem 3.11. Let p· ∈ PRn. If f ∈ W1,p·Rn, w, then the limit
f∗x lim r→ 0 1 |Bx, r| Bx,rf ydy 3.33
existsp·, w-quasi everywhere in Rn. The function f∗is thep·, w-quasicontinuous
Proof. Since the class C∞0 Rn is dense in W1,p·Rn, w byLemma 3.1, then we can choose a sequencefi such that
f− fi1,p·,w ≤ 2−2i. 3.34 For i 1, 2, . . . we denote Ai x∈ Rn : Mf− fi x > 2−i , B i ∞ ji Aj, E ∞ j1 Bj. 3.35
By usingProposition 3.10and the subadditivity of Cp·,wwe have
Cp·,wAi ≤ c max ⎧ ⎨ ⎩ Mf− fi 2−i p 1,p·,w ,M f− fi 2−i p− 1,p·,w ⎫ ⎬ ⎭ c max 1 2−i p Mf− fip 1,p·,w, 1 2−i p− Mf− fip − 1,p·,w ≤ c max 1 2−i p 2−2ip , 1 2−i p− 2−2ip − ≤ c2−i, 3.36
Cp·,wBi ≤ c21−iand Cp·,wE 0. If we follow the proof of Theorem 4.7 in 11, then this
proves the theorem.
Corollary 3.12. Let p· ∈ PRn. If f ∈ W1,p·Rn, w and f is quasicontinuous, then we have
fx lim r→ 0 1 |Bx, r| Bx,rf ydy 3.37 p·, w-quasi everywhere in Rn.
Proof. By using the Theorem in25 the proof is completed.
Now we show that every quasicontinuous function satisfies a weak type capacity inequality; the proofs follow the ideas by10.
Lemma 3.13. Let p < ∞ and E ⊂ Rn. If u ∈ W1,p·Rn, w is a nonnegative p·,
w-quasicontinuous function such that u ≥ 1 on E. Then for every ε > 0 there exists a function
h∈ Sp·,wE such that 1,p·,wu − h < ε.
Proof. Let 0 < δ < 1, and let V ⊂ Rnbe an open set such that u is continuous inRn\ V and
Cp·,wV < δ. By definition of Cp·,w there exists a v∈ Sp·,wE such that 1,p·,wv < δ. If
we set h 1 δu |v|, then it is easy to show that h ∈ W1,p·Rn, w by 10, Theorem 2.2. Since the function u is continuous and the set V is open, then the set
is open, contains E, and h≥ 1 on G, thus h ∈ Sp·,wE. It is known that for 1 ≤ p· ≤ p <∞
and a, b≥ 0, a bp· ≤ 2p −1ap· bp· and |∇|v|| |∇v|. Hence we obtain ||v| δu|p· ≤
2p −1|v|p· |δu|p· and 1,p·,wu − h Rn ||v| δu|px |∇|v| δu|pxwxdx ≤ 2p −1 Rn |v|p· |δu|p· |∇|vx||px |δ∇ux|pxwxdx ≤ 2p −1 1,p·,wv δp−1,p·,wu < 2p −1δ δp−1,p·,wu . 3.39
This completes the proof as δ → 0.
Theorem 3.14. Let p <∞. If u ∈ W1,p·Rn, w is a p·, w-quasicontinuous function and λ > 0,
then Cp·,w{x ∈ Rn:|ux| > λ} ≤ Rn uxλ px ∇ux λ px wxdx. 3.40
Proof. By10, Theorem 2.2, |u| ∈ W1,p·Rn, w and |∇|u|| |∇u|. ByLemma 3.13, there is a sequence hj∈ Sp·,w{x ∈ Rn:|ux|/λ > 1} such that
1,p·,w
|u|
λ − hj
−→ 0 as j −→ ∞. 3.41
Hence we have by10, Lemma 2.6 that
1,p·,whj −→ 1,p·,w |u| λ as j−→ ∞. 3.42 By definition of Cp·,w, we write Cp·,w{x ∈ Rn:|ux| > λ} ≤ 1,p·,w hj . 3.43 Therefore Cp·,w{x ∈ Rn:|ux| > λ} ≤ 1,p·,w |u| λ as j−→ ∞. 3.44
Proposition 3.15. Let p· ∈ PRn. If u ∈ W1,p·Rn, w, then there is a C > 0 such that
B1,p·,w{x ∈ Rn: Mux ≥ λ} ≤ C max u λ p 1,p·,w, u λ p− 1,p·,w . 3.45
Proof. For r > 0, we take h |B0, r|−1χB0,r. Choose f∈ Lp·Rn, w such that u g1∗ f and
f p·,w ≈ u 1,p·,wbyTheorem 3.2. Then
1 |Bx, r| Bx,r uydy 1 |B0, r| B0,rχB0,r x− yuydy h ∗ |u|x ≤h∗g1∗fx g1∗ h∗fx ≤ g1∗ Mf x 3.46
and Mux ≤ g1∗ Mfx. Also it is known that if E1 ⊂ E2, then B1,p·,wE1 ≤ B1,p·,wE2
by12, Lemma 4.1. Therefore we have
B1,p·,w{x ∈ Rn: Mux ≥ λ} ≤ B1,p·,wx∈ Rn :g1∗ Mf x ≥ λ ≤ p·,w Mf λ ≤ max Mfλ p p·,w ,Mf λ p − p·,w ≤ c max fλ p p·,w, fλp − p·,w ≤ C maxu λ p 1,p·,w, u λ p− 1,p·,w . 3.47
The following Theorem is obtained directly fromLemma 3.8andTheorem 3.11.
Theorem 3.16. Let p· ∈ PRn. If u ∈ W1,p·Rn, w and u is quasicontinuous, then the limit
ux lim r→ 0 1 |Bx, r| Bx,r uydy 3.48
exists1, p·, w-quasi everywhere in Rn.
The following proposition can be proved similarly as in12, Proposition 5.1.
Proposition 3.17. Let p· ∈ PRn. Every u ∈ L1,p·Rn, w is quasicontinuous. That is, for every
ε > 0, there exists a set F⊂ Rn, B
1,p·,wF ≤ ε, so that u restricted to Rn\ F is continuous.
Proposition 3.18. Let 1 < p−≤ p <∞. Then for all f ∈ Lp·Rn, w and 0 < λ < ∞ we have
Bα,p·,wx∈ Rn:gα∗ f x ≥ λ≤ max fλ p p·,w, fλp − p·,w . 3.49
Proof. We first note that by definition of B1,p·,w-capacity, λ−1f is a test function for the Bessel capacity. Hence Bα,p·,wx∈ Rn:gα∗ f x ≥ λ≤ p·,w f λ ≤ max fλ p p·,w, fλp − p·,w . 3.50
Proposition 3.19. Let 1 < p−≤ p <∞. If f ∈ Lp·Rn, w and
Ex∈ Rn:gα∗ f
x ∞, 3.51
then Bα,p·,wE 0.
Proof. ByProposition 3.18, we write Bα,p·,wE 0 as λ → ∞.
Proposition 3.20. Let 1 < p−≤ p <∞. If f ∈ Lp·Rn, w, then
lim r→ 0 1 |Bx, r| Bx,r gα∗ f ydygα∗ f x 3.52 for Bα,p·,w-q.e. x∈ Rn.
Proof. Let χ be the characteristic function for the unit ball B0, 1, and define for r > 0, χrx 1/|B0, 1|χx/r, x ∈ Rn. Then 1 |Bx, r| Bx,r gα∗ f ydy χr ∗ gα∗ f x χr∗ gα ∗ fx Bx,rχr∗ gα yfx− ydy. 3.53
As r → 0, χr ∗ gαy → gαy for every y ∈ Rn. This implies that, for fixed x ∈ Rn, χr ∗
gαyfx − y → gαyfx − y for a.e. y ∈ Rn. It was shown that χr ∗ gαy ≤ Cgαy for 0 < r≤ 1 and y ∈ Rn26, page 161. ByProposition 3.19, the integrand in3.53 is dominated by a constant times gαy|fx − y|, which is L1Rn function for Bα,p·,w-q.e. x∈ Rn. If we use
the Lebesgue’s dominated convergence theorem, then the proof is completed.
Acknowledgments
The author would like to thank Petteri Harjulehto for his significant suggestions, comments, and corrections to the original version of this paper. He is also grateful to Stefan Samko for his attention to a gap in the proof ofLemma 3.1. He also thanks the referee for carefully reading the paper and useful comments.
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