# Weighted Variable Sobolev Spaces and Capacity

Authors: Ismail Aydin
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We define weighted variable Sobolev capacity and discuss properties of capacity in the spaceW1,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.


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## Body

## 1. Introduction

In 1991 Kováčik and Rákosník [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 Lp(ℝn) have many common properties. A crucial difference between Lp(·)(ℝn) and the classical Lebesgue spaces Lp(ℝn) 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-Hölder continuous, that is,
(1.1)|p(x)-p(y)|≤C-ln⁡|x-y|,x,y∈Ω,|x-y|≤12.
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, see [1, 3–5]. Sobolev capacity for constant exponent spaces has found a great number of uses, see Maz′ja [6], Evans and Gariepy [7], and Heinonen et al. [8]. Also Kilpeläinen [9] introduced weighted Sobolev capacity and discussed the role of capacity in the pointwise definition of functions in Sobolev spaces involving weights 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 of [9–12] to the weighted variable exponent case.


## 2. Definition and Preliminary Results

We study weighted variable Lebesgue and Sobolev spaces in then-dimensional Euclidean space ℝn, 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 μ(A)=|A| and χA, respectively. The space Lloc1(ℝn) consists of all (classes of) measurable functions f on ℝn such that fχK∈L1(ℝn) for any compact subset K⊂ℝn. It is a topological vector space with the family of seminorms f↦∥fχK∥L1. A Banach function space (shortly BF-space) on ℝn is a Banach space (B,∥·∥B) of measurable functions which is continuously embedded into Lloc1(ℝn), that is, for any compact subset K⊂ℝn there exists some constant CK>0 such that ∥fχK∥L1≤CK∥f∥B for all f∈B. We denote it by B↪Lloc1(ℝn). The class C0∞(ℝn) 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(2.1)p-=essinf⁡x∈Rnp(x),p+=esssup⁡x∈Rnp(x).
For every measurable functions f on ℝn we define the function(2.2)ϱp(⋅)(f)=∫Rn|f(x)|p(x)dx.
The function ϱp(·) is convex modular; that is, ϱp(·)(f)≥0, ϱp(·)(f)=0 if and only if f=0, ϱp(·)(-f)=ϱp(·)(f) and ϱp(·) is convex. The variable exponent Lebesgue spaces (or generalized Lebesgue spaces) Lp(·)(ℝn) is defined as the set of all measurable functions f on ℝn such that ϱp(·)(λf)<∞ for some λ>0, equipped with the Luxemburg norm(2.3)‖f‖p(⋅)=inf⁡{λ>0:ϱp(fλ)≤1}.
If p+<∞, then f∈Lp(·)(ℝn) if and only if ϱp(·)(f)<∞. The set Lp(·)(ℝn) is a Banach space with the norm ∥·∥p(·). If p(x)=p is a constant function, then the norm ∥·∥p(·) coincides with the usual Lebesgue norm ∥·∥p [1]. In this paper we assume that p+<∞.A positive, measurable, and locally integrable functionw:ℝn→(0,∞) is called a weight function. The weighted modular is defined by(2.4)ϱp(⋅),w(f)=∫Rn|f(x)|p(x)w(x)dx.
The weighted variable exponent Lebesgue space Lp(·)(ℝn,w) consists of all measurable functions f on ℝn for which ∥f∥p(·),w=∥fw1/p(·)∥p(·)<∞. The relations between the modular ϱp(·),w(·) and ∥·∥p(·),w as follows:(2.5)min⁡{ϱp(⋅),w(f)1/p-,ϱp(⋅),w(f)1/p+}≤‖f‖p(⋅),w≤max⁡{ϱp(⋅),w(f)1/p-,ϱp(⋅),w(f)1/p+}min⁡{‖f‖p(⋅),wp+,‖f‖p(⋅),wp-}≤ϱp(⋅),w(f)≤max⁡{‖f‖p(⋅),wp+,‖f‖p(⋅),wp-},
see [13–15]. Moreover, if 0<C≤w, then we have Lp(·)(ℝn,w)↪Lp(·)(ℝn), since one easily sees that(2.6)C∫Rn|f(x)|p(x)dx≤∫Rn|f(x)|p(x)w(x)dx
and C∥f∥p(·)≤∥f∥p(·),w.The Schwartz classS=S(ℝn) consists of all infinitely differentiable and rapidly decreasing functions in ℝn. 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=C(β,k) such that(2.7)|Dβf(x)|≤C(1+|x|)k.
In particular, for β=0,(2.8)|f(x)|≤C(1+|x|)k.
Also it is well known that C0∞(ℝn)⊂S.Forx∈ℝn and r>0 we denote an open ball with center x and radius r by B(x,r). For f∈Lloc1(ℝn), we denote the (centered) Hardy-Littlewood maximal operator Mf of f by(2.9)Mf(x)=sup⁡r>01|B(x,r)|∫B(x,r)|f(y)|dy,
where the supremum is taken over all balls B(x,r).Let1≤p<∞. A weight w satisfies Muckenhoupt’s Ap(ℝn)=Ap condition, or w∈Ap, if there exist positive constants C and c such that, for every ball B⊂ℝn,(2.10)(1|B|∫Bwdx)(1|B|∫Bw-1/(p-1)dx)p-1≤C,1≤p<∞,
or(2.11)(1|B|∫Bwdx)(esssup⁡B1w)≤c,p=1.
The infimum over the constants C and c is called the Ap and A1, respectively. Also it is known that A∞=⋃1≤p<∞Ap. Let 1<p<∞. Then Muckenhoupt proved that w∈Ap if and only if the Hardy-Littlewood maximal operator is bounded on Lp(ℝn,w) [16]. Also Miller showed that the Schwartz class S is dense in Lp(ℝn,w) for 1<p<∞ and w∈Ap [17, Lemma 2.1].Hästö and Diening defined the classAp(·) to consist of those weights w for which(2.12)‖w‖Ap(⋅):=sup⁡B∈B|B|-pB‖w‖L1(B)‖1w‖Lp′(⋅)/p(⋅)(B)<∞,
where ℬ denotes the set of all balls in ℝn, pB=((1/|B|)∫B(1/p(x))dx)-1 and 1/p(·)+1/p′(·)=1. Note that this class is ordinary Muckenhoupt class Ap if p is a constant function [13].We say thatp(·) satisfies the local log-Hölder continuity condition if(2.13)|p(x)-p(y)|≤Clog⁡(e+1/|x-y|)
for all x,y∈ℝn. If(2.14)|p(x)-p∞|≤Clog⁡(e+|x|)
for some p∞>1, C>0 and all x∈ℝn, then we say p(·) satisfies the log-Hölder decay condition (at infinity). We denote by Plog(ℝn) the class of variable exponents which are log-Hölder continuous, that is, which satisfy the local log-Hölder continuity condition and the log-Hölder decay condition.Letp,q∈Plog(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 ∥w∥Ap(·)≤C∥w∥Aq(·) [13, Lemma 3.1]. As a result of this Lemma we have(2.15)A1⊂Ap-⊂Ap(⋅)⊂Ap+⊂A∞
for p∈Plog(ℝn) and 1<p-≤p+<∞.Letp∈Plog(ℝn) 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(2.16)P(Rn)={p(⋅):1<p-≤p(x)≤p+<∞,‖Mf‖p(⋅),w≤C‖f‖p(⋅),w},
that is, the maximal operator M is bounded on Lp(·)(ℝn,w). Hence we can find a sufficient condition for p(·)∈𝒫(ℝn).Proposition 2.1.
Letw be a weight function and 1<p-≤p(x)≤p+<∞. If w-1/(p(·)-1)∈Lloc1(ℝn), then Lp(·)(ℝn,w)↪Lloc1(ℝn).Proof.
Suppose thatf∈Lp(·)(ℝn,w), and let K⊂ℝn be a compact set. For 1/p(·)+1/q(·)=1, by using Hölder’s inequality for variable exponent Lebesgue spaces [1], then there exists a AK>0 such that
(2.17)∫K|f(x)|dx≤AK‖fw1/p(⋅)‖p(⋅),K‖w-1/p(⋅)‖q(⋅),K≤AK‖fw1/p(⋅)‖p(⋅)‖w-1/p(⋅)‖q(⋅),K.
It is known that ∥w-1/p(·)∥q(·),K<∞ if and only if ϱq(·),K(w-1/p(·))<∞ for q+<∞. Since w-1/(p(·)-1)∈Lloc1(ℝn), then we have
(2.18)ϱq(⋅),K(w-1/p(⋅))=∫Kw(x)-q(x)/p(x)dx=∫Kw(x)-1/(p(x)-1)dx=BK<∞.
If we use (2.17) and (2.18), then the proof is completed.Definition 2.2 (Mollifiers).
Letφ:ℝn→ℝ be a nonnegative, radial, decreasing function belonging to C0∞(ℝn) and having the properties:(i)
φ(x)=0 if |x|≥1,(ii)
∫ℝnφ(x)dx=1.
Letε>0. If the function φε(x)=ε-nφ(x/ε) is nonnegative, belongs to C0∞(ℝn), and satisfies(i)
φε(x)=0 if |x|≥ε and(ii)
∫ℝnφε(x)dx=1,then φε is called a mollifier and we define the convolution by
(2.19)φε*f(x)=∫Rnφε(x-y)f(y)dy.The following proposition was proved in [18, Proposition 2.7].Proposition 2.3.
Letφε be a mollifier and f∈Lloc1(ℝn). Then
(2.20)sup⁡ε>0|φε*f(x)|≤Mf(x).Proposition 2.4.
Ifp(·)∈𝒫(ℝn) and f∈Lp(·)(ℝn,w), then φε*f→f in Lp(·)(ℝn,w) as ε→0+.Proof.
Letf∈Lp(·)(ℝn,w) and ε>0 be given. If f is continuous, then the assertion is trivial. By Proposition 2.3, we have
(2.21)‖φε*f‖p(⋅),w≤‖Mf‖p(⋅),w≤C‖f‖p(⋅),w
and we have φε*f∈Lp(·)(ℝn,w) for all ε>0. It can be proved that the class C0(ℝn) of continuous functions with compact support is dense in the space Lp(·)(ℝn,w). Then there is a function g∈C0(ℝn) such that
(2.22)‖f-g‖p(⋅),w<ε.
Also it is well known that if g∈C0(ℝn), then φε*g∈C0∞(ℝn) for all ε>0. It is easily seen that φε*g→g uniformly on compact sets as ε→0+. Hence we have
(2.23)|φε*g(x)-g(x)|p(x)⟶0,ϱp(⋅),w(φε*g-g)=∫K|φε*g(x)-g(x)|p(x)w(x)dx≤εp-∫Kw(x)dx,
where supp(φε*g)∪suppg⊂K, K⊂ℝn compact. Hence ϱp(·),w(φε*g-g)→0 as ε→0+ and we write
(2.24)‖φε*g-g‖p(⋅),w<ε.
Finally by using (2.22) and (2.24),
(2.25)‖f-φε*f‖p(⋅),w≤‖f-g‖p(⋅),w+‖g-φε*g‖p(⋅),w+‖φε*g-φε*f‖p(⋅),w<(C+2)ε.
The proof is complete.As a direct consequence of Proposition2.4 there follows.Corollary 2.5.
Letp(·)∈𝒫(ℝn). The class C0∞(ℝn) is dense in Lp(·)(ℝn,w).This result was proved without the assumption that the maximal operator is bounded inLp(·)(ℝn,w) by Kokilashvili and Samko [19].Remark 2.6.
Let1<p-≤p(x)≤p+<∞ and w-1/(p(·)-1)∈Lloc1(ℝn). Then every function in Lp(·)(ℝn,w) has distributional derivatives by Proposition 2.1.


## 3. Weighted Variable Sobolev Spaces

Let1<p-≤p(x)≤p+<∞, w-1/(p(·)-1)∈Lloc1(ℝn) and k∈ℕ. We define the weighted variable Sobolev spaces Wk,p(·)(ℝn,w) by(3.1)Wk,p(⋅)(Rn,w)={f∈Lp(⋅)(Rn,w):Dαf∈Lp(⋅)(Rn,w),0≤|α|≤k}
equipped with the norm(3.2)‖f‖k,p(⋅),w=∑0≤|α|≤k‖Dαf‖p(⋅),w
where α∈ℕ0n is a multiindex, |α|=α1+α2+⋯+αn, and Dα=∂|α|/(∂x1α1⋯∂xnαn). It can be shown that Wk,p(·)(ℝn,w) is a reflexive Banach space. Throughout this paper, we will always assume that 1<p-≤p(x)≤p+<∞ and w-1/(p(·)-1)∈Lloc1(ℝn).The spaceW1,p(·)(ℝn,w) is defined by(3.3)W1,p(⋅)(Rn,w)={f∈Lp(⋅)(Rn,w):|∇f|∈Lp(⋅)(Rn,w)}.
The function ϱ1,p(·),w:W1,p(·)(ℝn,w)→[0,∞) is defined as ϱ1,p(·),w(f)=ϱp(·),w(f)+ϱp(·),w(∇f). The norm ∥f∥1,p(·),w=∥f∥p(·),w+∥∇f∥p(·),w.The Bessel kernelgα order α,α>0, is defined by(3.4)gα(x)=πn/2Γ(α/2)∫0∞e-s-(π2|x|2)/ss(α-n)/2dss,x∈Rn.
Let α≥0. The weighted variable Bessel potential space ℒα,p(·)(ℝn,w) is, for α>0, defined by(3.5)Lα,p(⋅)(Rn,w):={h=gα*f;f∈Lp(⋅)(Rn,w)},
and is equipped with the norm(3.6)‖h‖α;p(⋅),w=‖f‖p(⋅),w.
If α=0 we put g0*f=f and ℒ0,p(·)(ℝn,w)=Lp(·)(ℝn,w).Letp(·)∈𝒫(ℝn). If f∈Lp(·)(ℝn,w), then gα*f∈Lp(·)(ℝn,w). Indeed, since gα∈L1(ℝn) and gα is radial, we have (gα*f)(x)≤Mf(x), x∈ℝn [20, page 62]. The assertion thus follows from boundedness of maximal function in Lp(·)(ℝn,w).The unweighted variable Bessel potential spaceℒα,p(·)(ℝn) was firstly studied by Almeida and Samko in [21].Lemma 3.1.
Letp(·)∈Plog(ℝn), 1<p-≤p+<∞, and w∈Ap(·). Then(i)
C0∞(ℝn) is dense in Wk,p(·)(ℝn,w), k∈ℕ,(ii)
The Schwartz classS is dense in ℒα,p(·)(ℝn,w), α≥0.Proof.
(i) By Proposition2.4 the proof is complete.
(ii) Letα=0. The class C0∞(ℝn) is dense in Lp(·)(ℝn,w) by Corollary 2.5. It remains only to show that S⊂Lp(·)(ℝn,w). Let f∈S. Then there exist C=C(r)>0 and r>0 such that
(3.7)|f(x)|≤C(1+|x|)r.
Also since rp(x)≥r and (1+|x|)r≥1, then
(3.8)ϱp(⋅),w(f)=∫Rn|f(x)|p(x)w(x)dx≤max⁡{Cp-,Cp+}∫Rnw(x)(1+|x|)rp(x)dx≤max⁡{Cp-,Cp+}∫Rnw(x)(1+|x|)rdx.
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
(3.9)∫Rnw(x)(1+|x|)rdx<∞,
see [22, Lemma 1]. If we use (3.9) in (3.8), then the Schwartz class S is dense in Lp(·)(ℝn,w).
Letα>0 and h∈ℒα,p(·)(ℝn,w). Then there is a function f∈Lp(·)(ℝn,w) such that h=gα*f. By density of C0∞(ℝn) in f∈Lp(·)(ℝn,w) we can find a sequence (fj)j∈ℕ⊂C0∞(ℝn)⊂S converging to f in Lp(·)(ℝn,w). Since the mapping f↦gα*f maps S onto S [20], the functions hj=gα*fj, j∈ℕ, belong to S. Moreover,
(3.10)‖h-hj‖α;p(⋅),w=‖f-fj‖p(⋅),w⟶0asj⟶∞
and the assertion follows.The following Theorem can be proved similarly in [12, Theorem 3.1].Theorem 3.2.
Letp(·)∈𝒫(ℝn) and k∈ℕ. Then ℒk,p(·)(ℝn,w)=Wk,p(·)(ℝn,w) and the corresponding 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=∞.ForE⊂ℝn, we denote(3.11)Sp(⋅),w(E)={f∈W1,p(⋅)(Rn,w):f≥1inopensetcontainingE}.
The Sobolev (p(·),w)-capacity of E is defined by(3.12)Cp(⋅),w(E)=inf⁡f∈Sp(⋅),w(E)ϱ1,p(⋅),w(f)=inf⁡f∈Sp(⋅),w(E)∫Rn(|f(x)|p(x)+|∇f(x)|p(x))w(x)dx.
In case Sp(·),w(E)=∅, we set Cp(·),w(E)=∞. The Cp(·),w-capacity has the following properties.(i)
Cp(·),w(∅)=0.(ii)
IfE1⊂E2, then Cp(·),w(E1)≤Cp(·),w(E2).(iii)
IfE is a subset of ℝn, then
(3.13)Cp(⋅),w(E)=inf⁡{Cp(⋅),w(U):E⊂U,Uopen}.(iv)
IfE1 and E2 are subsets of ℝn, then
(3.14)Cp(⋅),w(E1∪E2)+Cp(⋅),w(E1∩E2)≤Cp(⋅),w(E1)+Cp(⋅),w(E2).(v)
IfK1⊃K2⊃⋯ are compact, then
(3.15)lim⁡i→∞Cp(⋅),w(Ki)=Cp(⋅),w(⋂i=1∞Ki).Note that the assertion (v) above is not true in general for noncompact sets [9].(vi)
IfE1⊂E2⊂⋯ are subsets of ℝn, then
(3.16)lim⁡i→∞Cp(⋅),w(Ei)=Cp(⋅),w(⋃i=1∞Ei).(vii)
IfEi⊂ℝn for i=1,2,…, then
(3.17)Cp(⋅),w(⋃i=1∞Ei)≤∑i=1∞Cp(⋅),w(Ei).For the proof of these properties see [8, 10]. Hence the Sobolev Cp(·),w capacity is an outer measure. A set function which satisfies the capacity properties (i), (ii), (v), and (vi) is called Choquet capacity; see [23]. Therefore we have the following result.Corollary 3.4.
The set functionE↦Cp(·),w(E), E⊂ℝn, is a Choquet capacity. In particular, all Suslin sets E⊂ℝn are capacitable, that is,
(3.18)Cp(⋅),w(E)=inf⁡E⊂UUopenCp(⋅),w(U)=sup⁡K⊂EKcompactCp(⋅),w(K).Lemma 3.5.
Letw(x)≥1 for x∈ℝn. Then every measurable set E⊂ℝn satisfies |E|≤Cp(·),w(E).Proof.
Iff∈Sp(·),w(E), then there is an open set E⊂U such that f≥1 in U and hence
(3.19)|E|≤|U|≤∫Rn|f(x)|p(x)w(x)dx≤∫Rn(|f(x)|p(x)+|∇f(x)|p(x))w(x)dx.
We obtain the claim by taking the infimum on Sp(·),w(E).Definition 3.6 (Bessel Capacity).
LetE⊂ℝn, α>0. Define that the (α,p(·),w)-Bessel capacity in ℒα,p(·)(ℝn,w) is the number
(3.20)Bα,p(⋅),w(E)=inf⁡ϱp(⋅),w(f),
where the infimum is taken over all f∈Lp(·)(ℝn,w) such that gα*f≥1 on E. Since gα is nonnegative we can assume that f≥0.Theorem 3.7.
Bα,p(·),w is an outer capacity defined on all subsets of ℝn.Proof.
It is known that(i)
Bα,p(·),w(∅)=0;(ii)
ifE1⊂E2, then Bα,p(·),w(E1)≤Bα,p(·),w(E2);(iii)
ifEi⊂ℝn for i=1,2,…, then
(3.21)Bα,p(⋅),w(⋃i=1∞Ei)≤∑i=1∞Bα,p(⋅),w(Ei)by [12, Lemma  4.1]. We will show that
(3.22)Bα,p(⋅),w(E)=inf⁡E⊂GGopenBα,p(⋅),w(G).
for any E⊂ℝn. Let E⊂ℝn be arbitrary. Obviously Bα,p(·),w(E)≤infE⊂GGopenBα,p(·),w(G). We assume that Bα,p(·),w(E)<∞. If 0<ε<1 there must exist a test function (measurable and nonnegative) for Bα,p(·),w(E), call it f, such that gα*f≥1 on E, and
(3.23)ϱp(⋅),w(f)<Bα,p(⋅),w(E)+ε.
Let G={x∈ℝn:gα*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(·),w(G) and we have
(3.24)Bα,p(⋅),w(G)≤ϱp(⋅),w(f1-ε)≤(1-ε)-p+ϱp(⋅),w(f)<(1-ε)-p+(Bα,p(⋅),w(E)+ε).
This proves the theorem as ε→0+.Now we give relationship between the capacitiesBα,p(·),w and Cp(·),w [12].Lemma 3.8.
Letp(·)∈𝒫(ℝn) and E⊂ℝn. Then
(3.25)B1,p(⋅),w(E)≤cmax⁡{Cp(⋅),w(E)p-/p+,Cp(⋅),w(E)p+/p-},Cp(⋅),w(E)≤Cmax⁡{B1,p(⋅),w(E)p-/p+,B1,p(⋅),w(E)p+/p-}.
Here c and C are positive constants independent of E.Proposition 3.9.
Letp(·)∈𝒫(ℝn).(i)
Iff∈W1,p(·)(ℝn,w), then Mf∈W1,p(·)(ℝn,w) and |∇Mf(x)|≤M|∇f(x)| for almost everywhere in ℝn.(ii)
Let1≤s<∞. Then sp(·)∈𝒫(ℝn) and there exists a constant C>0 such that the inequality
(3.26)‖Mf‖1,sp(⋅),w≤C‖f‖1,sp(⋅),w
holds for all f∈W1,sp(·)(ℝn,w).Proof.
(i) By Proposition2.1 we have Lp(·)(ℝn,w)↪Llocp(·)(ℝn,w)↪Lloc1(ℝn) and W1,p(·)(ℝn,w)↪Wloc1,p(·)(ℝn,w)↪Wloc1,1(ℝn). Since  f∈Wloc1,1(ℝn), then we have |∇Mf(x)|≤M|∇f(x)| for almost everywhere in ℝn by [24]. Since  f, |∇f|∈Lp(·)(ℝn,w) and p(·)∈𝒫(ℝn), then Mf,|∇Mf|∈Lp(·)(ℝn,w). Hence Mf∈W1,p(·)(ℝn,w).
(ii) Letf∈Lsp(·)(ℝn,w). By using definition of ∥·∥p(·),w, we have
(3.27)‖f‖sp(⋅),w=‖|f|s‖p(⋅),w1/s
and |f|s∈Lp(·)(ℝn,w). Therefore we have
(3.28)‖Mf‖sp(⋅),w=‖(Mf)s‖p(⋅),w1/s≤‖M(|f|s)‖p(⋅),w1/s≤c‖|f|s‖p(⋅),w1/s=c‖f‖sp(⋅),w
and sp(·)∈𝒫(ℝn). Since f∈W1,sp(·)(ℝn,w), then f,|∇f|∈Lsp(·)(ℝn,w). Hence we write
(3.29)‖Mf‖1,sp(⋅),w=‖(Mf)s‖p(⋅),w1/s+‖|∇Mf|s‖p(⋅),w1/s≤‖(Mf)s‖p(⋅),w1/s+‖(M|∇f|)s‖p(⋅),w1/s≤‖M(|f|s)‖p(⋅),w1/s+‖M(|∇f|s)‖p(⋅),w1/s≤C1‖|f|s‖p(⋅),w1/s+C2‖|∇f|s‖p(⋅),w1/s.
by (3.28). If we set C=max{C1,C2}, then
(3.30)‖Mf‖1,sp(⋅),w≤C‖f‖1,sp(⋅),w.
This completes the proof.Proposition 3.10.
Letp(·)∈𝒫(ℝn). Then for every λ>0 and every f∈W1,p(·)(ℝn,w) we have
(3.31)Cp(⋅),w({x∈Rn:Mf(x)>λ})≤cmax⁡{‖fλ‖1,p(⋅),wp+,‖fλ‖1,p(⋅),wp-}.Proof.
SinceMf is lower semicontinuous, the set {x∈ℝn:Mf(x)>λ} is open for every λ>0. By Proposition 3.9 we can take (Mf)/λ=M(f/λ) as a test function for the capacity. Then we have
(3.32)Cp(⋅),w({x∈Rn:Mf(x)>λ})≤ϱ1,p(⋅),w(Mfλ)≤max⁡{‖Mfλ‖1,p(⋅),wp+,‖Mfλ‖1,p(⋅),wp-}≤cmax⁡{‖fλ‖1,p(⋅),wp+,‖fλ‖1,p(⋅),wp-}.We say that a property holds(p(·),w)-quasi everywhere if it holds except in a set of capacity zero. A function f is (p(·),w)-quasicontinuous in ℝn if for each ε>0 there exists an open set E with Cp(·),w(E)<ε such that f restricted to ℝn∖E is continuous. The following proof of theorem is quite similar to Theorem 4.7 in [11].Theorem 3.11.
Letp(·)∈𝒫(ℝn). If f∈W1,p(·)(ℝn,w), then the limit
(3.33)f*(x)=lim⁡r→01|B(x,r)|∫B(x,r)f(y)dy
exists (p(·),w)-quasi everywhere in ℝn. The function f* is the (p(·),w)-quasicontinuous representative of f.Proof.
Since the classC0∞(ℝn) is dense in W1,p(·)(ℝn,w) by Lemma 3.1, then we can choose a sequence (fi) such that
(3.34)‖f-fi‖1,p(⋅),w≤2-2i.
For i=1,2,… we denote
(3.35)Ai={x∈Rn:M(f-fi)(x)>2-i},Bi=⋃j=i∞Aj,E=⋂j=1∞Bj.
By using Proposition 3.10 and the subadditivity of Cp(·),w we have
(3.36)Cp(⋅),w(Ai)≤cmax⁡{‖M(f-fi)2-i‖1,p(⋅),wp+,‖M(f-fi)2-i‖1,p(⋅),wp-}=cmax⁡{(12-i)p+‖M(f-fi)‖1,p(⋅),wp+,(12-i)p-‖M(f-fi)‖1,p(⋅),wp-}≤cmax⁡{(12-i)p+(2-2i)p+,(12-i)p-(2-2i)p-}≤c2-i,Cp(·),w(Bi)≤c21-i and Cp(·),w(E)=0. If we follow the proof of Theorem  4.7 in [11], then this proves the theorem.Corollary 3.12.
Letp(·)∈𝒫(ℝn). If f∈W1,p(·)(ℝn,w) and f is quasicontinuous, then we have
(3.37)f(x)=lim⁡r→01|B(x,r)|∫B(x,r)f(y)dy(p(·),w)-quasi everywhere in ℝn.Proof.
By using the Theorem in [25] the proof is completed.Now we show that every quasicontinuous function satisfies a weak type capacity inequality; the proofs follow the ideas by [10].Lemma 3.13.
Letp+<∞ 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(·),w(E) such that ϱ1,p(·),w(u-h)<ε.Proof.
Let0<δ<1, and let V⊂ℝn be an open set such that u is continuous in ℝn∖V and Cp(·),w(V)<δ. By definition of Cp(·),w there exists a v∈Sp(·),w(E) such that ϱ1,p(·),w(v)<δ. If we set h=(1+δ)u+|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
(3.38)G={x∈Rn∖V:u(x)>1}∪V
is open, contains E, and h≥1 on G, thus h∈Sp(·),w(E). It is known that for 1≤p(·)≤p+<∞ and a,b≥0, (a+b)p(·)≤2p+-1(ap(·)+bp(·)) and |∇|v||=|∇v|. Hence we obtain ||v|+δu|p(·)≤2p+-1(|v|p(·)+|δu|p(·)) and
(3.39)ϱ1,p(⋅),w(u-h)=∫Rn(||v|+δu|p(x)+|∇(|v|+δu)|p(x))w(x)dx≤2p+-1∫Rn(|v|p(⋅)+|δu|p(⋅)+|∇|v(x)||p(x)+|δ∇u(x)|p(x))w(x)dx≤2p+-1(ϱ1,p(⋅),w(v)+δp-ϱ1,p(⋅),w(u))<2p+-1(δ+δp-ϱ1,p(⋅),w(u)).
This completes the proof as δ→0.Theorem 3.14.
Letp+<∞. If u∈W1,p(·)(ℝn,w) is a (p(·),w)-quasicontinuous function and λ>0, then
(3.40)Cp(⋅),w({x∈Rn:|u(x)|>λ})≤∫Rn(|u(x)λ|p(x)+|∇u(x)λ|p(x))w(x)dx.Proof.
By [10, Theorem  2.2], |u|∈W1,p(·)(ℝn,w) and |∇|u||=|∇u|. By Lemma 3.13, there is a sequence hj∈Sp(·),w({x∈ℝn:|u(x)|/λ>1}) such that
(3.41)ϱ1,p(⋅),w(|u|λ-hj)⟶0asj⟶∞.
Hence we have by [10, Lemma  2.6] that
(3.42)ϱ1,p(⋅),w(hj)⟶ϱ1,p(⋅),w(|u|λ)asj⟶∞.
By definition of Cp(·),w, we write
(3.43)Cp(⋅),w({x∈Rn:|u(x)|>λ})≤ϱ1,p(⋅),w(hj).
Therefore
(3.44)Cp(⋅),w({x∈Rn:|u(x)|>λ})≤ϱ1,p(⋅),w(|u|λ)asj⟶∞.Proposition 3.15.
Letp(·)∈𝒫(ℝn). If u∈W1,p(·)(ℝn,w), then there is a C>0 such that
(3.45)B1,p(⋅),w({x∈Rn:Mu(x)≥λ})≤Cmax⁡{‖uλ‖1,p(⋅),wp+,‖uλ‖1,p(⋅),wp-}.Proof.
Forr>0, we take h=|B(0,r)|-1χB(0,r). Choose f∈Lp(·)(ℝn,w) such that u=g1*f and ∥f∥p(·),w≈∥u∥1,p(·),w by Theorem 3.2. Then
(3.46)1|B(x,r)|∫B(x,r)|u(y)|dy=1|B(0,r)|∫B(0,r)χB(0,r)(x-y)|u(y)|dy=(h*|u|)(x)≤(h*(g1*|f|))(x)=(g1*(h*|f|))(x)≤(g1*Mf)(x)
and Mu(x)≤(g1*Mf)(x). Also it is known that if E1⊂E2, then B1,p(·),w(E1)≤B1,p(·),w(E2) by [12, Lemma  4.1]. Therefore we have
(3.47)B1,p(⋅),w({x∈Rn:Mu(x)≥λ})≤B1,p(⋅),w({x∈Rn:(g1*Mf)(x)≥λ})≤ϱp(⋅),w(Mfλ)≤max⁡{‖Mfλ‖p(⋅),wp+,‖Mfλ‖p(⋅),wp-}≤cmax⁡{‖fλ‖p(⋅),wp+,‖fλ‖p(⋅),wp-}≤Cmax⁡{‖uλ‖1,p(⋅),wp+,‖uλ‖1,p(⋅),wp-}.The following Theorem is obtained directly from Lemma3.8 and Theorem 3.11.Theorem 3.16.
Letp(·)∈𝒫(ℝn). If  u∈W1,p(·)(ℝn,w) and u is quasicontinuous, then the limit
(3.48)u(x)=lim⁡r→01|B(x,r)|∫B(x,r)u(y)dy
exists (1,p(·),w)-quasi everywhere in ℝn.The following proposition can be proved similarly as in [12, Proposition 5.1].Proposition 3.17.
Letp(·)∈𝒫(ℝn). Every u∈ℒ1,p(·)(ℝn,w) is quasicontinuous. That is, for every ε>0, there exists a set F⊂ℝn, B1,p(·),w(F)≤ε, so that u restricted to ℝn∖F is continuous.Proposition 3.18.
Let1<p-≤p+<∞. Then for all f∈Lp(·)(ℝn,w) and 0<λ<∞ we have
(3.49)Bα,p(⋅),w({x∈Rn:(gα*f)(x)≥λ})≤max⁡{‖fλ‖p(⋅),wp+,‖fλ‖p(⋅),wp-}.Proof.
We first note that by definition ofB1,p(·),w-capacity, λ-1f is a test function for the Bessel capacity. Hence
(3.50)Bα,p(⋅),w({x∈Rn:(gα*f)(x)≥λ})≤ϱp(⋅),w(fλ)≤max⁡{‖fλ‖p(⋅),wp+,‖fλ‖p(⋅),wp-}.Proposition 3.19.
Let1<p-≤p+<∞. If f∈Lp(·)(ℝn,w) and
(3.51)E={x∈Rn:(gα*f)(x)=∞},
then Bα,p(·),w(E)=0.Proof.
By Proposition3.18, we write Bα,p(·),w(E)=0 as λ→∞.Proposition 3.20.
Let1<p-≤p+<∞. If f∈Lp(·)(ℝn,w), then
(3.52)lim⁡r→01|B(x,r)|∫B(x,r)(gα*f)(y)dy=(gα*f)(x)
for Bα,p(·),w-q.e. x∈ℝn.Proof.
Letχ be the characteristic function for the unit ball B(0,1), and define for r>0, χr(x)=(1/|B(0,1)|)χ(x/r), x∈ℝn. Then
(3.53)1|B(x,r)|∫B(x,r)(gα*f)(y)dy=χr*(gα*f)(x)=(χr*gα)*f(x)=∫B(x,r)χr*gα(y)f(x-y)dy.
As r→0, χr*gα(y)→gα(y) for every y∈ℝn. This implies that, for fixed x∈ℝn, χr*gα(y)f(x-y)→gα(y)f(x-y) for a.e. y∈ℝn. It was shown that χr*gα(y)≤Cgα(y) for 0<r≤1 and y∈ℝn [26, page 161]. By Proposition 3.19, the integrand in (3.53) is dominated by a constant times gα(y)|f(x-y)|, which is L1(ℝn) function for Bα,p(·),w-q.e. x∈ℝn. If we use the Lebesgue’s dominated convergence theorem, then the proof is completed.



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*Source: 132690-2012-01-03.xml*