# On the Space of Locally Sobolev-Slobodeckij Functions

Authors: A. Behzadan; M. Holst
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spacesWm,p or Bessel potential spaces Hs. Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces Ws,p where s does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature.


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## Body

## 1. Introduction

It is well-known that Sobolev spaces play a key role in the study of elliptic partial differential equations (PDEs) on domains inℝn. There are many resources for properties of integer order Sobolev spaces of functions and their applications in PDEs (see, e.g., [1–3]). Also, there are variety of resources for properties of real order Sobolev spaces of functions and their applications (see, e.g., classical references such as [4–8] or more recent works such as [9–13]). Likewise, the study of elliptic PDEs on manifolds naturally leads to the study of Sobolev spaces of functions and more generally Sobolev spaces of sections of vector bundles on manifolds. As it turns out, the study of certain differential operators between Sobolev spaces of sections of vector bundles on manifolds equipped with rough metric and the study of low regularity geometry on Riemannian and semi-Riemannian manifolds are closely related to the study of spaces of locally Sobolev functions on domains in the Euclidean space (see, e.g., [14–16]).In this paper, we focus on certain properties of spaces of locally Sobolev functions that are particularly useful in the study of differential operators on manifolds. Our work can be viewed as a continuation of the excellent work of Antonic and Burazin [17]; their work is mainly concerned with the properties of spaces of locally Sobolev functions with integer smoothness degree. In particular, they study the following fundamental questions for locally Sobolev spaces with integer smoothness degree:
(i)
Topology and metrizability(ii)
Density of smooth functions(iii)
Reflexivity and the nature of the dual(iv)
Continuity of differentiation between certain spaces of locally Sobolev functionsOur main goal here is to provide a self-contained manuscript in which the known results are collected and stated in the general setting of Sobolev-Slobodeckij spaces and then develop certain other results that are useful in the study of differential operators on manifolds. In particular, we will discuss the following topics:(i)
General embedding results(ii)
Pointwise multiplication(iii)
Invariance under compositionThe results of this type and other related results have been used in the literature—particularly in the study of Einstein constraint equations on manifolds equipped with rough metric—without complete proof. This paper should be viewed as a part of our efforts to fill some of the gaps. Interested readers can find other results in this direction in [13, 15, 16, 18]. Our hope is that the detailed presentation of this manuscript, along with these other four manuscripts, will help in better understanding the structure of the proofs and the properties of Sobolev-Slobodeckij spaces and locally Sobolev functions.


## 2. Notation and Conventions

Throughout this paper,ℝ denotes the set of real numbers, ℕ denotes the set of positive integers, and ℕ0 denotes the set of nonnegative integers. For any nonnegative real number s, the integer part of s is denoted by s. The letter n is a positive integer and stands for the dimension of the space. For all k∈ℕ, GLk,ℝ is the set of all k×k invertible matrices with real entries.Ω is a nonempty open set in ℝn. The collection of all compact subsets of Ω will be denoted by KΩ. If FΩ is any function space on Ω and K∈KΩ, then FKΩ denotes the collection of elements in FΩ whose support is inside K. Also,
(1)FcompΩ=∪K∈KΩFKΩ.IfΩ′⊆Ω and f:Ω′⟶ℝ, we denote the extension by zero of f to the entire Ω by extΩ′,Ω0f:Ω⟶ℝ, that is,
(2)extΩ′,Ω0fx=fx,ifx∈Ω′,0,otherwise.Lipschitz domain inℝn refers to a nonempty bounded open set in ℝn with Lipschitz continuous boundary. We say that a nonempty open set Ω⊆ℝn has the interior Lipschitz property provided that for each compact set K∈KΩ there exists a bounded open set Ω′⊆Ω with Lipschitz continuous boundary such that K⊆Ω′.Each element ofℕ0n is called a multi-index. For a multi-index α=α1,⋯,αn∈ℕ0n, we let α≔α1+⋯+αn. Also, for sufficiently smooth functions u:Ω⟶ℝ (or for any distribution u), we define the αth order partial derivative of u as follows:
(3)∂αu≔∂αu∂x1α1⋯∂xnαn.We use the notationA≼B to mean A≤cB, where c is a positive constant that does not depend on the nonfixed parameters appearing in A and B. We write A≃B if A≼B and B≼A.IfX and Y are two topological spaces, we use the notation X↪Y to mean X⊆Y, and the inclusion map is continuous.


## 3. Background Material

In this section, we collect some useful tools and facts we will need from topology and analysis. Statements without proof in this section are mainly taken from Rudin’s functional analysis [19], Grubb’s distributions and operators [20], excellent presentation of Reus [21], Treves’ topological vector spaces [22] and the reference [16], or are direct consequences of statements in the aforementioned references.


### 3.1. Topological Vector Spaces

Definition 1.
Atopological vector space is a vector space X together with a topology τ with the following properties:
(i)
For allx∈X, the singleton x is a closed set(ii)
The maps(4)x,y↦x+yfromX×XintoX,λ,x↦λxfromℝ×XintoX,are continuous where X×X and ℝ×X are equipped with the product topologyDefinition 2.
SupposeX,τ is a topological vector space and Y⊆X.
(i)
Y is said to be convex if for all y1,y2∈Y and t∈0,1 it is true that ty1+1−ty2∈Y(ii)
We sayY is bounded if for any neighborhood U of the origin (i.e., any open set containing the origin), there exits t>0 such that Y⊆tUDefinition 3.
LetX,τ be a topological vector space. X is said to be metrizable if there exists a metric d:X×X⟶0,∞ whose induced topology is τ. In this case, we say that the metric d is compatible with the topology τ.Theorem 4 ([19, 20]).
LetX,τ be a topological vector space. The following are equivalent:
(i)
X is metrizable(ii)
There exists a translation invariant metricd on X whose collection of open sets is the same as τ. Translation invariant means
(5)∀x,y,a∈Xdx+a,y+a=dx,y(iii)
X has a countable local base at the origin
(Recall that a subcollectionB of τ is said to be a local base at the origin if for any open set U containing the origin there is B∈B such that 0∈B⊆U.)Remark 5.
It can be shown that ifd1 and d2 are two translation invariant metrics that induce the same topology on X, then the Cauchy sequences of X,d1 will be exactly the same as the Cauchy sequences of X,d2.Definition 6.
LetX,τ be a topological vector space. We say X,τ is locally convex if it has a convex local base at the origin.Definition 7.
LetX,τ be a metrizable locally convex topological vector space. Let d be any translation invariant metric on X that is compatible with τ. We say that X is complete if and only if the metric space X,d is a complete metric space. A complete metrizable locally convex topological vector space is called a Frechet space.Definition 8.
Aseminorm on a vector space X is a real-valued function p:X⟶ℝ such that
(6)∀x,y∈Xpx+y≤px+py,∀x∈X∀α∈ℝpαx=αpx.
IfP is a family of seminorms on X, then we say P is separating provided that for all x≠0, there exists at least one p∈P such that px≠0 (that is, if px=0 for all p∈P, then x=0). It easily follows from the definition that any seminorm is a nonnegative function.Theorem 9.
Suppose thatX,.X is a normed space. Let p:X⟶ℝ be a seminorm on X. If p is continuous, then there exists a constant C>0 such that
(7)∀x∈Xpx≤CxX.Proof.
p is continuous at 0 so there exists δ>0 such that if xX≤δ, then px<1. If x≠0, then δx/xX has norm equal to δ, and so for all x≠0, pδx/xX<1. Hence, for all x≠0, we have
(8)px≤1δxX.
Sincep0=0, clearly the above inequality also holds for x=0.Definition 10.
SupposeP is a separating family of seminorms on a vector space X. The natural topology induced by P is the smallest topology on X that is translation invariant and with respect to which every p∈P is a continuous function from X to ℝ. (Recall that translation invariant means if U⊆X is open, then U+x is open for every x∈X.)Remark 11.
Suppose thatP and P′ are two separating family of seminorms on a vector space X. Let τ and τ′ be the corresponding natural topologies on X. It follows immediately from the definition that if (1) p:X,τ′⟶ℝ is continuous for each p∈P and (2) p′:X,τ⟶ℝ is continuous for each p′∈P′, then τ=τ′.The following theorem can be viewed as an extension of Theorem9.Theorem 12 ([21], page 157).
LetX be a vector space and suppose P is a separating family of seminorms on X. Equip X with the corresponding natural topology. Then, a seminorm q:X⟶ℝ is continuous if and only if there exist C>0 and p1,⋯,pm∈P such that for all x∈X(9)qx≤Cp1x+⋯+pmx.Theorem 13 ([19, 20]).
SupposeP is a separating family of seminorms on a vector space X and τ is the corresponding natural topology on X. Then, X,τ is a locally convex topological vector space. Moreover, if P=pkk∈ℕ is countable, then the locally convex topological vector space X,τ is metrizable, and the following translation invariant metric on X is compatible with τ:
(10)dx,y=∑k=1∞12kpkx−y1+pkx−y.Corollary 14.
SupposeP is a countable separating family of seminorms on a vector space X and τ is the corresponding natural topology on X. Then, X,τ is a Frechet space if and only if it is complete.Theorem 15 ([23], Sections 6.4 and 6.5).
LetX,τ be a locally convex topological vector space. Then, there exists a separating family of seminorms on X whose corresponding natural topology is τ.Theorem 16 ([19], page 28).
SupposeP is a separating family of seminorms on a vector space X and τ is the corresponding natural topology on X. Then, a set E⊆X is bounded if and only if pE is a bounded set in ℝ for all p∈P.Corollary 17.
SupposeP is a separating family of seminorms on a vector space X and τ is the corresponding natural topology on X. It follows from Theorem 12 and Theorem 16 that if E⊆X is bounded, then for any continuous seminorm q:X,τ⟶ℝ, qE is a bounded set in ℝ.Theorem 18 ([20], page 436, [23], Section 6.6).
LetX,τ be a topological vector space. Suppose Q is a separating family of seminorms on a vector space Y and τ′ is the corresponding natural topology on Y. Then, a linear map T:X,τ⟶Y,τ′ is continuous if and only if for each q∈Q, q∘T is continuous on X.Theorem 19 ([20]).
LetX be a Frechet space and let Y be a topological vector space. When T is a linear map of X into Y, the following two properties are equivalent:
(1)
T is continuous(2)
xn⟶0 in X⇒Txn⟶0 in YTheorem 20 ([19, 20]).
LetX and Y be two vector spaces and suppose P and Q are two separating families of seminorms on X and Y, respectively. Equip X and Y with the corresponding natural topologies. Then,
(1)
A sequencexn converges to x in X if and only if for all p∈P, pxn−x⟶0(2)
A linear operatorT:X⟶Y is continuous if and only if ∀q∈Q∃c>0,k∈ℕ,p1,⋯,pk∈Psuchthat∀x∈Xq∘Tx≤cmax1≤i≤kpix(3)
A linear operatorT:X⟶ℝ is continuous if and only if ∃c>0,k∈ℕ,p1,⋯,pk∈Psuchthat∀x∈XTx≤cmax1≤i≤kpixDefinition 21.
LetX,τ be a locally convex topological vector space.
(i)
Theweak topology on X is the natural topology induced by the separating family of seminorms pFF∈X∗ where
(11)∀F∈X∗pF:X⟶ℝ,pFx≔Fx.It can be shown that this topology is the smallest (weakest) topology with respect to which all the linear maps in X,τ∗ are continuous. A sequence xm converges to x in X with respect to the weak topology if and only if Fxm⟶Fx in ℝ for all F∈X∗. In this case, we may write xm⇀x. We denote the weak topology on X by σX,X∗. It can be shown that X,τ∗ is the same set as X,σX,X∗∗(ii)
Theweak∗topology on X∗ is the natural topology induced by the separating family of seminorms pxx∈X where
(12)∀x∈Xpx:X∗⟶ℝ,pxf≔fx.It can be shown that this topology is the weakest topology with respect to which all the linear maps f↦fxx∈X (from X∗ to ℝ) are continuous. A sequence fm converges to f in X∗ with respect to the weak∗ topology if and only if fmx⟶fx in ℝ for all x∈X. We denote the weak∗ topology on X∗ by σX∗,X(iii)
Thestrong topology on X∗ is the natural topology induced by the separating family of seminorms pBB⊆Xbounded where for any bounded subset B of X,
(13)pB:X∗⟶ℝpBf≔supfx:x∈B.(It can be shown that for any bounded subset B of X and f∈X∗, fB is a bounded subset of ℝ; see Theorem 16 and Theorem 28)Remark 22.
(1)
IfX is a normed space, then the topology induced by the norm
(14)∀f∈X∗fop=sup∥x∥X=1fxon X∗ is the same as the strong topology on X∗ ([22], page 198)(2)
In this manuscript, unless otherwise stated, we consider the topological dual of a locally convex topological vector space with the strong topology. Of course, it is worth mentioning that for many of the spaces that we will consider (includingX=EΩ or X=DΩ where Ω is an open subset of ℝn), a sequence in X∗ converges with respect to the weak∗ topology if and only if it converges with respect to the strong topology (for more details on this, see the definition and properties of Montel spaces in Section 34.4, page 356 of [22])Theorem 23.
LetX,τ be a locally convex topological vector space. Then, the evaluation map
(15)J:X,τ⟶X∗∗≔X∗,strongtopology∗,JxF≔Fx,is a well-defined injective linear map. (X∗∗ is called the bidual of X).Definition 24.
LetX,τ be a locally convex topological vector space. Let τ′ denote the strong topology on X∗∗ as the dual of X∗,strongtopology.
(i)
If the evaluation mapJ:X,τ⟶X∗∗,τ′ is bijective, then we say that X,τ is a semireflexive space(ii)
If the evaluation mapJ:X,τ⟶X∗∗,τ′ is a linear topological isomorphism, then we say that X,τ is a reflexive spaceTheorem 25 ([24], pages 16 and 17).
(i)
Strong dual of a reflexive topological vector space is reflexive(ii)
Every semireflexive space whose topology is defined by the inductive limit of a sequence of Banach spaces is reflexive(iii)
Every semireflexive Frechet space is reflexiveTheorem 26.
LetX,τX and Z,τZ be two locally convex topological vector spaces. For all x∈X, let lx:X∗⟶ℝ be the linear map defined by lxf=fx. Then,
(1)
a linear mapT:Z,τZ⟶X,σX,X∗ is continuous if and only if for all F∈X,τX∗, the linear map F∘T:Z,τZ⟶ℝ is continuous(2)
a linear mapT:Z,τZ⟶X∗,σX∗,X is continuous if and only if for all x∈X, the linear map lx∘T:Z,τZ⟶ℝ is continuousTheorem 27 ([21], page 163, [20], page 46).
LetX and Y be locally convex topological vector spaces and suppose T:X⟶Y is a continuous linear map. Either equip both X∗ and Y∗ with the strong topology or equip both with the weak∗ topology. Then,
(1)
the map(16)T∗:Y∗⟶X∗T∗y,xX∗×X=y,TxY∗×Yis well-defined, linear, and continuous. (T∗ is called the adjoint of T)(2)
IfTX is dense in Y, then T∗:Y∗⟶X∗ is injectiveTheorem 28 ([19], page 70).
LetX,τ be a locally convex topological vector space. Then, a set E⊆X is bounded with respect to τ if and only if it is bounded with respect to σX,X∗.Corollary 29.
IfX,τ is a locally convex topological vector space and xn⇀x (i.e., xn converges to x with respect to σX,X∗), then xn is bounded with respect to both τ and σX,X∗.Theorem 30.
LetX,τX and Y,τY be two locally convex topological vector spaces. If T:X,τX⟶Y∗,σY∗,Y is continuous, then T:X,σX,X∗⟶Y∗,σY∗,Y is continuous. In particular, if un⇀u (i.e., un converges to u with respect to σX,X∗), then Tun⟶Tu in Y∗,σY∗,Y.Proof.
For ally∈Y, let ly:Y∗⟶ℝ be the map lyF=Fy. By Theorem 26, T:X,σX,X∗⟶Y∗,σY∗,Y is continuous if ly∘T:X,σX,X∗⟶ℝ is continuous for all y∈Y. Let y∈Y.
(1)
By definition of the weak∗ topology on Y∗, we know that the linear map ly:Y∗⟶ℝ is continuous(2)
By assumptionT:X,τX⟶Y∗,σY∗,Y is a continuous linear map
Therefore,ly∘T belongs to X,τX∗. Since σX,X∗ is the weakest topology on X that makes all elements of X,τX∗ continuous, we can conclude that ly∘T:X,σX,X∗⟶ℝ is continuous.Theorem 31 ([25], page 13).
LetX,τ be a Frechet space. Then, X is reflexive if and only if every bounded set E in X is relatively weakly compact (i.e., the closure of E with respect to σX,X∗ is compact with respect to σX,X∗).Theorem 32 ([26], page 167).
LetX,τ be a separable Frechet space. If E⊆X is relatively weakly compact, then every infinite sequence in E has a subsequence that converges in X,σX,X∗.The next theorem is an immediate consequence of the previous theorems.Theorem 33.
Suppose thatX,τ is a separable reflexive Frechet space. Then, every bounded sequence in X,τ has a weakly convergent subsequence, that is, a subsequence that converges with respect to σX,X∗.Theorem 34 ([27], page 61).
LetX and Y be two Banach spaces. Let T:X⟶Y be a linear map. Then, T is continuous if and only if it is weak-weak continuous; that is, T:X,.X⟶Y,.Y is continuous if and only if T:X,σX,X∗⟶Y,σY,Y∗ is continuous.Theorem 35.
LetX be a Banach space and Y be a closed subspace of X with the induced norm. Suppose that ym is a sequence in Y and y∈Y. If ym⟶y in X,σX,X∗, then ym⟶y in Y,σY,Y∗.Proof.
This is a direct consequence of the fact that the following two topologies on the spaceY are the same (see [27], page 70):
(1)
The topology induced byσX,X∗(2)
The topologyσY,Y∗Definition 36.
LetX be a vector space and let Xαα∈I be a family of vector subspaces of X with the property that
(i)
for eachα∈I, Xα is equipped with a topology that makes it a locally convex topological vector space, and(ii)
∪α∈IXα=X
Theinductive limit topology on X with respect to the family Xαα∈I is defined to be the largest topology with respect to which
(1)
X is a locally convex topological vector space, and(2)
all the inclusionsXα⊆X are continuousTheorem 37 ([21]).
LetX be a vector space equipped with the inductive limit topology with respect to Xα as described above. If Y is a locally convex vector space, then a linear map T:X⟶Y is continuous if and only if TXα:Xα⟶Y is continuous for all α∈I.Definition 38.
LetX be a vector space and let Xjj∈ℕ0 be an increasing chain of subspaces of X:
(17)X0⊊X1⊊X2⊊⋯.
Suppose that(i)
eachXj is equipped with a locally convex topology τj(ii)
for eachj, the inclusion Xj,τj↪Xj+1,τj+1 is a linear topological embedding with closed image
Then, the inductive limit topology onX with respect to the family Xjj∈ℕ0 is called a strict inductive limit topology.Theorem 39 ([21]).
Suppose thatX is equipped with the strict inductive limit topology with respect to the chain Xjj∈ℕ0. Then, a subset E of X is bounded if and only if there exists m∈ℕ0 such that B is bounded in Xm.


### 3.2. Function Spaces and Distributions

Definition 40.
LetΩ be a nonempty open set in ℝn and m∈ℕ0.
(18)CΩ=f:Ω⟶ℝ:fiscontinuous,CmΩ=f:Ω⟶ℝ:∀α≤m∂αf∈CΩC0Ω=CΩ,BCΩ=f:Ω⟶ℝ:fiscontinuousandboundedonΩ,BCmΩ=f∈CmΩ:∀α≤m∂αfisboundedonΩ,C∞Ω=∩m∈ℕ0CmΩ,BC∞Ω=∩m∈ℕ0BCmΩ,Cc∞Ω=f∈C∞Ω:supportoffisanelementofKΩ.
Let0<λ≤1. A function F:Ω⊆ℝn⟶ℝk is called λ-Holder continuous if there exists a constant L such that
(19)Fx−Fy≤Lx−yλ∀x,y∈Ω.
Clearly, aλ-Holder continuous function on Ω is uniformly continuous on Ω. 1-Holder continuous functions are also called Lipschitz continuous functions or simply Lipschitz functions. We define
(20)BCm,λΩ=f:Ω⟶ℝ:∀α≤m∂αfisλ−Holdercontinuousandbounded=f∈BCmΩ:∀α≤m∂αfisλ−Holdercontinuous,BC∞,λΩ≔∩m∈ℕ0BCm,λΩ.Theorem 41 [20].
LetΩ be a nonempty open set in ℝn and let K∈KΩ. There is a function ψ∈Cc∞Ω taking values in 0,1 such that ψ=1 on a neighborhood containing K.Theorem 42 (exhaustion by compact sets) [20].
LetΩ be a nonempty open subset of ℝn. There exists a sequence of compact subsets Kjj∈ℕ such that ∪j∈ℕK˚j=Ω and
(21)K1⊆K˚2⊆K2⊆⋯⊆K˚j⊆Kj⊆⋯.
Moreover, as a direct consequence, ifK is any compact subset of the open set Ω, then there exists an open set V such that K⊆V⊆V¯⊆Ω.Theorem 43 [20].
LetΩ be a nonempty open subset of ℝn. Let Kjj∈ℕ be an exhaustion of Ω by compact sets. Define
(22)V0=K˚4,∀j∈ℕVj=K˚j+4\Kj.
Then,(1)
eachVj is an open bounded set and Ω=∪jVj(2)
the coverVjj∈ℕ0 is locally finite in Ω; that is, each compact subset of Ω has nonempty intersection with only a finite number of the Vj’s(3)
there is a family of functionsψj∈Cc∞Ω taking values in 0,1 such that suppψj⊆Vj and
(23)∑j∈ℕ0ψjx=1forallx∈ΩLetΩ be a nonempty open set in ℝn. For all φ∈C∞Ω, j∈ℕ, and K∈KΩ, we define
(24)φj,K≔sup∂αφx:α≤j,x∈K.For allj∈ℕ and K∈KΩ, .j,K is a seminorm on C∞Ω. We define EΩ to be C∞Ω equipped with the natural topology induced by the separating family of seminorms .j,Kj∈ℕ,K∈KΩ. It can be shown that EΩ is a Frechet space.For allK∈KΩ, we define EKΩ to be CK∞Ω equipped with the subspace topology. Since CK∞Ω is a closed subset of the Frechet space EΩ, EKΩ is also a Frechet space.We defineDΩ=∪K∈KΩEKΩ equipped with the inductive limit topology with respect to the family of vector subspaces EKΩK∈KΩ. It can be shown that if Kjj∈ℕ0 is an exhaustion by compacts sets of Ω, then the inductive limit topology on DΩ with respect to the family EKjj∈ℕ0 is exactly the same as the inductive limit topology with respect to EKΩK∈KΩ.Remark 44.
SupposeY is a topological space and the mapping T:Y⟶DΩ is such that TY⊆EKΩ for some K∈KΩ. Since EKΩ↪DΩ, if T:Y⟶EKΩ is continuous, then T:Y↪DΩ will be continuous.Theorem 45 (convergence and continuity forEΩ).
LetΩ be a nonempty open set in ℝn. Let Y be a topological vector space whose topology is induced by a separating family of seminorms Q.
(1)
A sequenceφm converges to φ in EΩ if and only if φm−φj,K⟶0 for all j∈ℕ and K∈KΩ(2)
SupposeT:EΩ⟶Y is a linear map. Then, the following are equivalent:
(i)
T is continuous(ii)
For everyq∈Q, there exist j∈ℕ and K∈KΩ, and C>0 such that
(25)∀φ∈EΩqTφ≤Cφj,K(iii)
Ifφm⟶0 in EΩ, then Tφm⟶0 in Y(3)
In particular, a linear mapT:EΩ⟶ℝ is continuous if and only if there exist j∈ℕ and K∈KΩ, and C>0 such that
(26)∀φ∈EΩTφ≤Cφj,K(4)
A linear mapT:Y⟶EΩ is continuous if and only if
(27)∀j∈ℕ,∀K∈KΩ∃C>0,k∈ℕ,q1,⋯,qk∈Qsuchthat∀yTyj,K≤Cmax1≤i≤kqiyTheorem 46 (convergence and continuity forEKΩ).
LetΩ be a nonempty open set in ℝn and K∈KΩ. Let Y be a topological vector space whose topology is induced by a separating family of seminorms Q.
(1)
A sequenceφm converges to φ in EKΩ if and only if φm−φj,K⟶0 for all j∈ℕ(2)
SupposeT:EKΩ⟶Y is a linear map. Then, the following are equivalent:
(i)
T is continuous(ii)
For everyq∈Q, there exists j∈ℕ and C>0 such that
(28)∀φ∈EKΩqTφ≤Cφj,K(iii)
Ifφm⟶0 in EKΩ, then Tφm⟶0 in YTheorem 47 (convergence and continuity forDΩ).
LetΩ be a nonempty open set in ℝn. Let Y be a topological vector space whose topology is induced by a separating family of seminorms Q.
(1)
A sequenceφm converges to φ in DΩ if and only if there is a K∈KΩ such that suppφm⊆K and φm⟶φ in EKΩ(2)
SupposeT:DΩ⟶Y is a linear map. Then, the following are equivalent:
(i)
T is continuous(ii)
For allK∈KΩ, T:EKΩ⟶Y is continuous(iii)
For everyq∈Q and K∈KΩ, there exists j∈ℕ and C>0 such that(29)∀φ∈EKΩqTφ≤Cφj,K(iv)
Ifφm⟶0 in DΩ, then Tφm⟶0 in Y(3)
In particular, a linear mapT:DΩ⟶ℝ is continuous if and only if for every K∈KΩ, there exists j∈ℕ and C>0 such that
(30)∀φ∈EKΩTφ≤Cφj,KRemark 48.
LetΩ be a nonempty open set in ℝn. Here are two immediate consequences of the previous theorems and remark:
(1)
The identity map(31)iD,E:DΩ⟶EΩis continuous (that is, DΩ↪EΩ)(2)
IfT:EΩ⟶EΩ is a continuous linear map such that suppTφ⊆suppφ for all φ∈EΩ (i.e., T is a local continuous linear map), then T restricts to a continuous linear map from DΩ to DΩ. Indeed, the assumption suppTφ⊆suppφ implies that TDΩ⊆DΩ. Moreover T:DΩ⟶DΩ is continuous if and only if for K∈KΩT:EKΩ⟶DΩ is continuous. Since TEKΩ⊆EKΩ, this map is continuous if and only if T:EKΩ⟶EKΩ is continuous (see Remark 44). However, since the topology of EKΩ is the induced topology from EΩ, the continuity of the preceding map follows from the continuity of T:EΩ⟶EΩTheorem 49.
LetΩ be a nonempty open set in ℝn. Then DΩ is separable.Definition 50.
LetΩ be a nonempty open set in ℝn. The topological dual of DΩ, denoted D′ΩD′Ω=DΩ∗, is called the space of distributions on Ω. Each element of D′Ω is called a distribution on Ω. The action of a distribution u∈D′Ω on a function φ∈DΩ is sometimes denoted by u,φD′Ω×DΩ or simply u,φ.Remark 51.
Every functionf∈Lloc1Ω defines a distribution uf∈D′Ω as follows
(32)∀φ∈DΩufφ≔∫Ωfφdx.
In particular, every functionφ∈EΩ defines a distribution uφ. It can be shown that the map i:EΩ⟶D′Ω which sends φ to uφ is an injective linear continuous map ([21], page 11). Therefore, we can identify EΩ with a subspace of D′Ω; we sometimes refer to the map i as the “identity map.”Theorem 52 ([20], page 47).
LetΩ be a nonempty open set in ℝn. Equip D′Ω with the weak∗ topology. Then, under the above identification, Cc∞Ω is dense in D′Ω.Theorem 53 ([22], page 302).
LetΩ be a nonempty open set in ℝn. Equip D′Ω with the strong topology. Then, under the identification described in Remark 51, Cc∞Ω is sequentially dense in D′Ω.Remark 54.
(i)
Clearly sequential density is a stronger notion than density. SoCc∞Ω is dense in D′Ω, strong topology)(ii)
Recall that, according to Remark22, a sequence converges in D′Ω,weak∗ if and only if it converges in D′Ω, strong topology). This together with the fact that weak∗ topology is weaker than the strong topology implies that convergent sequences in both topologies converge to the same limit. Therefore, it follows from Theorem 53 that Cc∞Ω is sequentially dense in D′Ω,weak∗. Hence, Theorem 52 can be viewed as a corollary of Theorem 53Theorem 55 ([21], page 9).
DΩ is reflexive. So D′Ω,strongtopology∗ can be identified with the topological vector space DΩ.Definition 56 (restriction of a distribution).
LetΩ be an open subset of ℝn and V be an open subset of Ω. We define the restriction map resΩ,V:D′Ω⟶D′V as follows:
(33)resΩ,Vu,φD′V×DV≔u,extV,Ω0φD′Ω×DΩ.
This is well-defined; indeed,resΩ,V:D′Ω⟶D′V is a continuous linear map as it is the adjoint of the continuous map extV,Ω0:DV⟶DΩ. Given u∈D′Ω, we sometimes write uV instead of resΩ,Vu.Definition 57 (support of a distribution).
LetΩ be a nonempty open set in ℝn. Let u∈D′Ω.
(i)
We sayu is equal to zero on some open subset V of Ω if uV=0(ii)
LetVii∈I be the collection of all open subsets of Ω such that u is equal to zero on Vi. Let V=∪i∈IVi. The support of u is defined as follows:(34)suppu≔Ω\V
Note that suppu is closed in Ω but it is not necessarily closed in ℝn.Theorem 58 ([21]).
LetΩ be a nonempty open set in ℝn and let u∈D′Ω. If φ∈DΩ vanishes on a neighborhood containing suppu, then uφ=0.Theorem 59 ([21]).
Letui be a sequence in D′Ω, u∈DΩ, and K∈KΩ such that ui⟶u in D′Ω and suppui⊆K for all i. Then, also suppu⊆K.Theorem 60 ([28], page 38).
LetΩ be a nonempty open set in ℝn. Suppose that Ti is a sequence in D′Ω with the property that for all φ∈DΩ, limi⟶∞Ti,φD′Ω×DΩ exists. Then, there exists T∈D′Ω such that
(35)∀φ∈DΩT,φD′Ω×DΩ=limi⟶∞Ti,φD′Ω×DΩ.Definition 61 (Sobolev-Slobodeckij spaces).
LetΩ be a nonempty open set in ℝn. Let s∈ℝ and p∈1,∞.
(i)
Ifs=k∈ℕ0,
(36)Wk,pΩ=u∈LpΩ:uWk,pΩ≔∑ν≤k∂νuLpΩ<∞(ii)
Ifs=θ∈0,1(37)Wθ,pΩ=u∈LpΩ:uWθ,pΩ≔∫∫Ω×Ωux−uypx−yn+θpdxdy1/p<∞(iii)
Ifs=k+θ,k∈ℕ0,θ∈0,1,
(38)Ws,pΩ=u∈Wk,pΩ:uWs,pΩ≔uWk,pΩ+∑ν=k∂νuWθ,pΩ<∞(iv)
W0s,pΩ is defined as the closure of Cc∞Ω in Ws,pΩ(v)
Ifs<0,
(39)Ws,pΩ=W0−s,p′Ω∗1p+1p′=1(vi)
For all compact setsK⊂Ω, we define
(40)WKs,pΩ=u∈Ws,pΩ:suppu⊆Kwith uWKs,pΩ≔uWs,pΩ. Note that for s<0, Ws,pΩ can be viewed as a subspace of D′Ω (see Theorem 68), and the support of u∈Ws,pΩ is interpreted as the support of a distribution.(vii)
Wcomps,pΩ≔∪K∈KΩWKs,pΩ. Wcomps,pΩ is equipped with the inductive limit topology with respect to the family of vector subspaces WKs,pΩK∈KΩ. It can be shown that if Kjj∈ℕ0 is an exhaustion by compacts sets of Ω, then the inductive limit topology on Wcomps,pΩ with respect to the family WKjs,pΩj∈ℕ0 is exactly the same as the inductive limit topology with respect to WKs,pΩK∈KΩTheorem 62.
LetΩ be a nonempty open set in ℝn, s≥1, and 1<p<∞. Then, u∈Ws,pΩ if and only if u∈LpΩ, and for all 1≤i≤n, ∂u/∂xi∈Ws−1,pΩ.Proof.
We consider two cases:
Case 1:s=k∈ℕ.
(41)u∈Wk,pΩ⟺u∈LpΩand∂αu∈LpΩ∀1≤α≤k⟺u∈LpΩand∂β∂u∂xi∈LpΩ∀0≤β≤k−1,1≤i≤n⟺u∈LpΩand∂u∂xi∈Wk−1,pΩ∀1≤i≤n
Case 2:s=k+θ, k∈ℕ, and 0<θ<1.
(42)u∈Ws,pΩ⟺u∈Wk,pΩand∂νux−∂νuyx−yn/p+θ∈LpΩ×Ω∀ν=k⟺u∈LpΩand∂u∂xi∈Wk−1,pΩ∀1≤i≤nand∂νux−∂νuyx−yn/p+θ∈LpΩ×Ω∀ν=k⟺u∈LpΩand∂u∂xi∈Wk−1,pΩand∂β∂u/∂xix−∂β∂u/∂xiyx−yn/p+θ∈LpΩ×Ω∀β=k−1∀1≤i≤n⟺u∈LpΩand∂u∂xi∈Ws−1,pΩ∀1≤i≤nRemark 63.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Clearly for s≥0, Cc∞Ω⊆Ws,pΩ. For s<0, it is easy to see that for all φ∈Cc∞Ω, the map lφ:W0−s,p′Ω⟶ℝ which sends u∈W0−s,p′Ω to ∫Ωuφdx belongs to W0−s,p′Ω∗=Ws,pΩ. The map φ↦lφ is one-to-one and we can use it to identify Cc∞Ω with a subspace of Ws,pΩ; we sometimes refer to the map that sends φ to lφ as the “identity map.” So we can talk about the identity map from Cc∞Ω to Ws,pΩ for all s∈ℝ.Theorem 64 ([16]).
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Then, Ws,pΩ is a reflexive Banach space.Corollary 65.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. A closed subspace of a reflexive space is reflexive, so W0s,pΩ is reflexive. Dual of a reflexive Banach space is a reflexive Banach space, so W−s,p′Ω is a reflexive Banach space.Remark 66.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Since W0s,pΩ is reflexive, it can be identified with W0s,pΩ∗∗ and we may write W−s,p′Ω∗=W0s,pΩ and talk about the duality pairing u,fW0s,pΩ×W−s,p′Ω. To be more precise, we notice that the identification of W0s,pΩ∗∗ and W0s,pΩ is done by the evaluation map
(43)J:W0s,pΩ⟶W0s,pΩ∗∗Juf=fu.
Therefore, for allu∈W0s,pΩ and f∈W−s,p′Ω,
(44)u,fW0s,pΩ×W−s,p′Ω=Ju,fW0s,pΩ∗∗×W−s,p′Ω=fu=f,uW−s,p′Ω×W0s,pΩ.Theorem 67.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Then, Cc∞Ω is dense in W−s,p′Ω. We may write this as W0−s,p′Ω=W−s,p′Ω.Proof.
Our proof will be based on a similar argument given in page 65 of [1]. Let φ↦lφ be the mapping introduced in Remark 63. Our goal is to show that the set
(45)V≔lφ:φ∈Cc∞Ωis dense in W−s,p′Ω. To this end, it is enough to show that if F∈W−s,p′Ω∗ is such that Flφ=0 for all φ∈Cc∞Ω, then F=0. Indeed, let F be such an element. By reflexivity of W0s,pΩ, there exists f∈W0s,pΩ such that
(46)∀v∈W−s,p′ΩFv=vf.
Thus, for allφ∈Cc∞Ω, we have
(47)0=Flφ=lφf=∫Ωfxφxdx.
So, by the fundamental lemma of the calculus of variations (see [27], page 110), we have f=0 (as an element of Ws,pΩ⊆Lloc1Ω) and therefore F=0.Theorem 68.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Equip D′Ω with weak∗ topology or strong topology. Then
(48)DΩ↪Ws,pΩ↪D′Ω.Proof.
Recall that the convergent sequences inD′Ω equipped with strong topology are exactly the same as the convergent sequences of D′Ω equipped with the weak∗ topology (see Remark 22). This together with Theorem 19 implies that in the study of the continuity of the inclusion map from Ws,pΩ to D′Ω, it does not matter whether we equip D′Ω with the strong topology or weak∗ topology. In the proof, as usual, we assume D′Ω is equipped with the strong topology. We consider two cases:
Case 1:s≥0. The continuity of the embedding DΩ↪Ws,pΩ has been studied in [16]. Also clearly Ws,pΩ↪LpΩ↪D′Ω. The former continuous embedding holds by the definition of Ws,pΩ and the latter embedding is continuous because if um⟶0 in LpΩ, then for all φ∈DΩ,
(49)um,φD′Ω×DΩ−0=∫Ωumφdx≤umpφ∞⟶0.
So,um⟶0 in D′Ω. This implies the continuity of the inclusion map from LpΩ to D′Ω by Theorem 19
Case 2:s<0. Since W0−s,p′Ω↪W−s,p′Ω, it follows from previous case that W0−s,p′Ω↪D′Ω. Also since DΩ⊆W0−s,p′Ω is dense in D′Ω (see Theorem 52, Theorem 53, and Remark 54), it follows that the inclusion map from W0−s,p′Ω to D′Ω is continuous with dense image. Thus, by Theorem 27, DΩ↪Ws,pΩ. Here, we used the facts that (1) the strong dual of the normed space W0−s,p′Ω is Ws,pΩ and that (2) the dual of D′Ω,strongtopology is DΩ (see Theorem 55). It remains to show that Ws,pΩ↪D′Ω. It follows from Case 1 that DΩ↪W0−s,p′Ω and by definition DΩ is dense in W0−s,p′Ω. So, by Theorem 27, Ws,pΩ↪D′ΩRemark 69.
Note that fors≤0, W0s,pΩ is the same as Ws,pΩ. For s>0, W0s,pΩ is a subspace of Ws,pΩ which contains Cc∞Ω. So it follows from the previous theorem that
(50)DΩ↪W0s,pΩ↪D′Ω.
To be more precise, we should note that fors<0, we identify φ∈DΩ with the corresponding distribution in D′Ω. Under this identification, for all s∈ℝ the “identity map” i:DΩ⟶W0s,pΩ is continuous with dense image, and so its adjoint i∗:W0s,pΩ∗⟶D′Ω will be an injective continuous map (Theorem 27), and we have
(51)i∗u,φD′Ω×DΩ=u,iφW0s,pΩ∗×W0s,pΩ=u,φW0s,pΩ∗×W0s,pΩ.
We usually identifyW0s,pΩ∗ with its image under i∗ and view W0s,pΩ∗ as a subspace of D′Ω. So, under this identification, we can rewrite the above equality as follows:
(52)∀u∈W0s,pΩ∗∀φ∈DΩu,φD′Ω×DΩ=u,φW0s,pΩ∗×W0s,pΩ.
Finally, noting that for alls∈ℝ and 1<p<∞, W0s,pΩ∗=W0−s,p′Ω (see Definition 61, Theorem 67, and Corollary 65), we can write
(53)∀u∈W0−s,p′Ω∀φ∈DΩu,φD′Ω×DΩ=u,φW0−s,p′Ω×W0s,pΩ.Theorem 70.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Then,
(1)
the mappingF↦FCc∞Ω is an isometric isomorphism between W−s,p′Ω and Cc∞Ω,.s,p∗(2)
supposeu∈D′Ω. If u:Cc∞Ω,.−s,p′⟶ℝ is continuous, then u∈W0s,pΩ (more precisely, there is a unique element in W0s,pΩ whose corresponding distribution is u). Moreover,(54)uW0s,pΩ=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφW−s,p′ΩProof.
The first item has been studied in [16]. Here, we will prove the second item. Since u:Cc∞Ω,.−s,p′⟶ℝ is continuous, it can be extended to a continuous linear map u~:W−s,p′Ω⟶ℝ. So u~∈W−s,p′Ω∗. However, W0s,pΩ is reflexive, therefore there exists a unique v∈W0s,pΩ such that u~=Jv where Jv:W−s,p′Ω⟶ℝ is the evaluation map defined by JvF=F,vW−s,p′Ω×W0s,pΩ. To finish the proof, it is enough to show that v=u as elements of D′Ω. For all φ∈Cc∞Ω, we have
(55)v,φD′Ω×DΩ=∫Ωvφdx=Remark63φ,vW−s,p′Ω×W0s,pΩ=Jvφ=u~φ=uφ=u,φD′Ω×DΩ.
Also,(56)uW0s,pΩ=vW0s,pΩ=JvW−s,p′Ω∗=u~W−s,p′Ω∗=sup0≡φ∈Cc∞Ωu~,φD′Ω×DΩφW−s,p′Ω=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφW−s,p′Ω.Corollary 71.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Suppose that u∈D′Ω. As a direct consequence of Theorem 70, we have the following:
(i)
Ifsup0≡φ∈Cc∞Ωu,φD′Ω×DΩ/φWs,pΩ<∞, then u∈W−s,p′Ω and
(57)uW−s,p′Ω=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφWs,pΩ(ii)
Ifsup0≡φ∈Cc∞Ωu,φD′Ω×DΩ/φW−s,p′Ω<∞, then u∈W0s,pΩ and
(58)uWs,pΩ=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφW−s,p′Ω
That is, for anye∈ℝ and 1<q<∞, in order to show that u∈D′Ω belongs to W0e,qΩ, it is enough to prove that
(59)sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφW−e,q′Ω<∞,and in fact uWe,qΩ=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩ/φW−e,q′Ω.Theorem 72.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Suppose that K∈KΩ. Then, WKs,pΩ is a closed subspace of Ws,pΩ.Proof.
It is enough to show that ifui is a sequence of elements in WKs,pΩ such that ui⟶u in Ws,pΩ, then u∈WKs,pΩ, i.e., suppu⊆K. By Theorem 68, we have ui⟶u in D′Ω. Now it follows from Theorem 59 that suppu⊆K. Note that for any s≥0, we have Ws,pΩ⊆LpΩ⊆Lloc1Ω; in this proof, we implicitly used the fact that for functions in Lloc1Ω, the usual definition of support agrees with the distributional definition of support.Next, we list several embedding theorems for Sobolev-Slobodeckij spaces.Theorem 73 ([29], Section 2.8.1).
Suppose1<p≤q<∞ and −∞<t≤s<∞ satisfy s−n/p≥t−n/q. Then, Ws,pℝn↪Wt,qℝn. In particular, Ws,pℝn↪Wt,pℝn.Theorem 74 ([30, 18]).
LetΩ be a nonempty bounded open subset of ℝn with Lipschitz continuous boundary. Suppose 1≤p,q<∞ (p does NOT need to be less than or equal to q) and 0≤t≤s satisfy s−n/p≥t−n/q. If s∈ℕ0, additionally assume that s≠t. Then, Ws,pΩ↪Wt,qΩ. Furthermore, if s>t, then the embedding Ws,pΩ↪Wt,pΩ is compact.Theorem 75 ([16]).
LetΩ⊆ℝn be an arbitrary nonempty open set.
(1)
Suppose1≤p≤q<∞ and 0≤t≤s satisfy s−n/p≥t−n/q. Then, WKs,pΩ↪WKt,qΩ for all K∈KΩ(2)
For allk1,k2∈ℕ0 with k1≤k2 and 1<p<∞, Wk2,pΩ↪Wk1,pΩ(3)
If0≤t≤s<1 and 1<p<∞, then Ws,pΩ↪Wt,pΩ(4)
If0≤t≤s<∞ are such that s=t and 1<p<∞, then Ws,pΩ↪Wt,pΩ(5)
If0≤t≤s<∞, t∈ℕ0, and 1<p<∞, then Ws,pΩ↪Wt,pΩTheorem 76 ([6]).
LetΩ be a nonempty bounded open subset of ℝn with Lipschitz continuous boundary or Ω=ℝn. If sp>n, then Ws,pΩ↪L∞Ω∩C0Ω and Ws,pΩ is a Banach algebra.In the next several theorems, we will list certain multiplication properties of Sobolev spaces. Supposeφ∈C∞Ω and u∈Ws,pΩ. If s≥0, then the product φu has a clear meaning. What if s<0? In this case, uDΩ is a distribution and by the product φu we mean the distribution φuDΩ; then φu is in Ws,pΩ if φuDΩ:Cc∞Ω,.−s,p′⟶ℝ is continuous. Because then it possesses a unique extension to a continuous linear map from W0−s,p′Ω to ℝ and so it can be viewed as an element of W0−s,p′Ω∗=Ws,pΩ. See Theorem 70 and Corollary 71. Also see Remark 89.Theorem 77 (multiplication by smooth functions I, [31], page 203).
Lets∈ℝ, 1<p<∞, and φ∈BC∞ℝn. Then, the linear map
(60)mφ:Ws,pℝn⟶Ws,pℝn,u↦φuis well-defined and bounded.Theorem 78 (multiplication by smooth functions II, [16]).
LetΩ be a nonempty bounded open set in ℝn with Lipschitz continuous boundary.
(1)
Letk∈ℕ0 and 1<p<∞. If φ∈BCkΩ, then the linear map Wk,pΩ⟶Wk,pΩ defined by u↦φu is well-defined and bounded(2)
Lets∈ℝ and 1<p<∞. If φ∈BC∞Ω, then the linear map Ws,pΩ⟶Ws,pΩ defined by u↦φu is well-defined and boundedTheorem 79 (multiplication by smooth functions III, [16]).
LetΩ be any nonempty open set in ℝn. Let p∈1,∞.
(1)
If0≤s<1 and φ∈BC0,1Ω (that is, φ∈L∞Ω and φ is Lipschitz), then
(61)mφ:Ws,pΩ⟶Ws,pΩ,u↦φuis a well-defined bounded linear map(2)
Ifk∈ℕ0 and φ∈BCkΩ, then
(62)mφ:Wk,pΩ⟶Wk,pΩ,u↦φuis a well-defined bounded linear map(3)
If−1<s<0 and φ∈BC∞,1Ω or s∈ℤ− and φ∈BC∞Ω, then
(63)mφ:Ws,pΩ⟶Ws,pΩ,u↦φuis a well-defined bounded linear mapTheorem 80 (multiplication by smooth functions IV, [16]).
LetΩ be a nonempty open set in ℝn, K∈KΩ, p∈1,∞, and −1<s<0 or s∈ℤ− or s∈0,∞. If φ∈C∞Ω, then the linear map
(64)WKs,pΩ⟶WKs,pΩ,u↦φuis well-defined and bounded.Theorem 81 (multiplication by smooth functions V, [16]).
LetΩ be a nonempty bounded open set in ℝn with Lipschitz continuous boundary. Let K∈KΩ. Suppose s∈ℝ and p∈1,∞. If φ∈C∞Ω, then the linear map WKs,pΩ⟶WKs,pΩ defined by u↦φu is well-defined and bounded.In the next definition, we introduce the notion ofsmooth multiplication triple which will play a key role in several theorems that will follow.Definition 82 (smooth multiplication triple).
LetΩ be a nonempty open set in ℝn, s∈ℝ and 1<p<∞.
(i)
We say that the triples,p,Ω is a smooth multiplication triple if for all φ∈Cc∞Ω, the map
(65)mφ:Ws,pΩ⟶Ws,pΩu↦φuis well-defined and bounded(ii)
We say that the triples,p,Ω is an interior smooth multiplication triple if for all φ∈Cc∞Ω and K∈KΩ, the map
(66)mφ:WKs,pΩ⟶WKs,pΩu↦φuis well-defined and boundedRemark 83.
(1)
Every smooth multiplication triple is also an interior smooth multiplication triple(2)
It is a direct consequence of Theorems77, 78, and 79 that
(i)
ifΩ=ℝn or Ω is bounded with Lipschitz continuous boundary, then for all s∈ℝ and 1<p<∞, s,p,Ω is a smooth multiplication triple(ii)
ifΩ is any open set in ℝn, 1<p<∞, and s∈ℝ is not a noninteger with magnitude greater than 1, then s,p,Ω is a smooth multiplication triple(3)
It is a direct consequence of Theorem80 and Theorem 81 that
(i)
ifΩ=ℝn or Ω is bounded with Lipschitz continuous boundary, then for all s∈ℝ and 1<p<∞, s,p,Ω is an interior smooth multiplication triple(ii)
ifΩ is any open set in ℝn, 1<p<∞, and s∈ℝ is not a noninteger less than −1, then s,p,Ω is an interior smooth multiplication triple(4)
Ifs,p,Ω is a smooth multiplication triple and K∈KΩ, then WKs,pΩ⊆W0s,pΩ (see the proof of Theorem 7.31 in [16]). Of course, if s<0, then Ws,pΩ=W0s,pΩ and so WKs,pΩ⊆W0s,pΩ holds for all s<0, 1<p<∞ and open sets Ω⊆ℝnTheorem 84.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. If s,p,Ω is a smooth multiplication triple so is −s,p′,Ω.Proof.
Letφ∈Cc∞Ω. For all u∈W−s,p′Ω=W0−s,p′Ω and ψ∈DΩ, we have
(67)φu,ψD′Ω×DΩ=u,φψD′Ω×DΩ=Remark69u,φψW−s,p′Ω×W0s,pΩ≤uW−s,p′ΩφψWs,pΩ≼uW−s,p′ΩψWs,pΩ.
The last inequality holds becauses,p,Ω is a smooth multiplication triple. It follows from Corollary 71 that φu∈W0−s,p′Ω and φuW−s,p′Ω≼uW−s,p′Ω; that is, mφ:W−s,p′Ω⟶W−s,p′Ω is well-defined and continuous.Theorem 85.
LetΩ be a nonempty open set in ℝn, s∈ℝ and 1<p<∞. If s<0, further assume that −s,p′,Ω is a smooth multiplication triple. Suppose that Ω′⊆Ω and K∈KΩ′. Then,
(1)
for allu∈WKs,pΩ, uWs,pΩ≃uΩ′Ws,pΩ′(2)
for allu∈WKs,pΩ′, extΩ′,Ω0uWs,pΩ≃uWs,pΩ′Proof.
The claim follows from the argument presented in the proofs of Corollary 7.39 and Theorem 7.46 in [16].Theorem 86 (([30], pages 598-605), ([6], Section 1.4)).
Lets∈ℝ, 1<p<∞, and α∈ℕ0n. Suppose Ω is a nonempty open set in ℝn. Then,
(1)
the linear operator∂α:Ws,pℝn⟶Ws−α,pℝn is well-defined and bounded(2)
fors<0, the linear operator ∂α:Ws,pΩ⟶Ws−α,pΩ is well-defined and bounded(3)
fors≥0 and α≤s, the linear operator ∂α:Ws,pΩ⟶Ws−α,pΩ is well-defined and bounded(4)
ifΩ is bounded with Lipschitz continuous boundary, and if s≥0, s−1/p≠integer (i.e., the fractional part of s is not equal to 1/p), then the linear operator ∂α:Ws,pΩ⟶Ws−α,pΩ for α>s is well-defined and boundedTheorem 87.
Assumptions:(i)
Ω=ℝn or Ω is a bounded domain with Lipschitz continuous boundary(ii)
si,s∈ℝ, si≥s≥0 for i=1,2(iii)
1<pi≤p<∞ for i=1,2(iv)
si−s≥n1/pi−1/p(v)
s1+s2−s>n1/p1+1/p2−1/p
Claim: Ifu∈Ws1,p1Ω and v∈Ws2,p2Ω, then uv∈Ws,pΩ and moreover the pointwise multiplication of functions is a continuous bilinear map
(68)Ws1,p1Ω×Ws2,p2Ω⟶Ws,pΩ.Remark 88.
A number of other results concerning the sufficient conditions on the exponentssi,pi,s,p that guarantee the multiplication Ws1,p1Ω×Ws2,p2Ω↪Ws,pΩ is well-defined and continuous are discussed in detail in [18].Remark 89.
Suppose thats,p,Ω is a smooth multiplication triple with s≥0. W−s,p′Ω=W0−s,p′Ω is the dual of W0s,pΩ and u,fW0−s,p′Ω×W0s,pΩ is the action of the functional u on the function f. As it was discussed before, if ψ is a function in Cc∞Ω, ψuDΩ is defined as a product of a smooth function and a distribution. Since s,p,Ω is a smooth multiplication triple, −s,p′,Ω will also be a smooth multiplication triple, and that means ψuDΩ:Cc∞Ω,.s,p⟶ℝ is continuous (see the note right after Theorem 76). We interpret ψu as an element of W−s,p′Ω=W0s,pΩ∗ to be the unique continuous linear extension of ψuDΩ to the entire W0s,pΩ. It is easy to see that, this unique linear extension is given by
(69)ψu,fW−s,p′Ω×W0s,pΩ≔u,ψfW−s,p′Ω×W0s,pΩ,that is, the above map is linear continuous and its restriction to DΩ is the same as ψuDΩ. (Note that since s,p,Ω is a smooth multiplication triple, ψf is indeed an element of W0s,pΩ.)Theorem 90 ([32]).
Lets∈1,∞ and 1<p<∞, and let
(70)m=s,ifsisaninteger,⌊s⌋+1,otherwise.
IfF∈Cmℝ is such that F0=0 and F,F′,⋯,Fm∈L∞ℝ (in particular, note that every F∈Cc∞ℝ with F0=0 satisfies these conditions), then the map u↦Fu is well-defined and continuous from Ws,pℝn∩W1,spℝn into Ws,pℝn.Corollary 91.
Lets, p, and F be as in the previous theorem. Moreover, suppose sp>n. Then, the map u↦Fu is well-defined and continuous from Ws,pℝn into Ws,pℝn. The reason is that when sp>n, we have Ws,pℝn↪W1,spℝn.In the remaining of this section, we will state certain useful properties of the topological vector spaceWcomps,p. The properties we will discuss here echo the ones stated in [24] for spaces Hcomps.Theorem 92.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Then, DΩ is continuously embedded in Wcomps,pΩ.Proof.
For allK∈KΩ, we have
(71)EKΩ↪DΩ↪Ws,pΩ.
This together with the fact that the image ofEKΩ under the identity map is inside WKs,pΩ implies that
(72)EKΩ↪WKs,pΩ.
Also, by the definition of the inductive limit topology onWcomps,pΩ, we have
(73)WKs,pΩ↪Wcomps,pΩ.
It follows from (72) and (73) that for all K∈KΩ,
(74)EKΩ↪Wcomps,pΩ,which, by Theorem 37, implies that DΩ↪Wcomps,pΩ.Theorem 93.
Lets,p,Ω be a smooth multiplication triple. Then, Cc∞Ω is dense in Wcomps,pΩ.Proof.
We will follow the proof given in [24] for spaces Hcomps. Let u∈Wcomps,pΩ. It is enough to show that there exists a sequence in Cc∞Ω that converges to u in Wcomps,pΩ (this proves sequential density which implies density). By Meyers-Serrin theorem, there exists a sequence φm∈C∞Ω∩Ws,pΩ such that φm⟶u in Ws,pΩ. Let χ∈Cc∞Ω be such that χ=1 on a neighborhood containing suppu (see Theorem 41). Let K=suppχ. Since s,p,Ω is a smooth multiplication triple, multiplication by χ is a linear continuous map on Ws,pΩ and so χφm⟶χu in Ws,pΩ. Now, we note that χu=u and for all m, χφm are in Cc∞Ω with support inside K. Consequently, χφm⟶u in WKs,pΩ. Now, since WKs,pΩ↪Wcomps,pΩ, we may conclude that χφm is a sequence in Cc∞Ω that converges to u in Wcomps,pΩ.Remark 94.
As a consequence, ifs,p,Ω is a smooth multiplication triple, then Wcomps,pΩ∗ (equipped with the strong topology) is continuously embedded in D′Ω. More precisely, the identity map i:DΩ⟶Wcomps,pΩ is continuous with dense image, and therefore, by Theorem 27, the adjoint i∗:Wcomps,pΩ∗⟶D′Ω is a continuous injective map. We have
(75)i∗u,φD′Ω×DΩ=u,iφWcomps,pΩ∗×Wcomps,pΩ=u,φWcomps,pΩ∗×Wcomps,pΩ.
We usually identifyWcomps,pΩ∗ with its image under i∗ and view Wcomps,pΩ∗ as a subspace of D′Ω. So, under this identification, we can rewrite the above equality as follows:
(76)∀u∈Wcomps,pΩ∗u,φD′Ω×DΩ=u,φWcomps,pΩ∗×Wcomps,pΩ.Next, we will prove that ifs,p,Ω is a smooth multiplication triple, then Wcomps,pΩ is separable. To this end, we need the following lemma.Lemma 95.
LetX,τ and Y,τ′ be two topological spaces. Suppose that
(1)
A is dense in X,τ(2)
T:X,τ⟶Y,τ′ is continuous(3)
TX is dense in Y,τ′
Then,TA is dense in Y,τ′.Proof.
It is enough to show thatTA intersects every nonempty open set in Y,τ′. So let O∈τ′ be nonempty. Since TX is dense in Y,τ′, we have O∩TX≠∅ and so T−1O is nonempty. Also, since T is continuous, T−1O∈τ. A is dense in X,τ, so A∩T−1O≠∅. Therefore,
(77)TA∩O⊇TA∩T−1O≠∅.Theorem 96.
Lets,p,Ω be a smooth multiplication triple. Then, Wcomps,pΩ is separable.Proof.
According to Theorems92 and 93, DΩ is continuously embedded in Wcomps,pΩ and it is dense in Wcomps,pΩ. Since DΩ is separable, it follows from Lemma 95 that Wcomps,pΩ is separable.Theorem 97.
Lets,p,Ω be an interior smooth multiplication triple. Let ψjj∈ℕ0 be the partition of unity introduced in Theorem 43. Let S be the collection of all sequences whose terms are nonnegative integers. For all sequences a=a0,a1,⋯∈S define qa,s,p:Wcomps,pΩ⟶ℝ by
(78)qa,s,pu=∑j=0∞ajψjuWs,pΩ.
Then,qa,s,pa∈S is a separating family of seminorms on Wcomps,pΩ and the natural topology induced by this family on Wcomps,pΩ is the same as the inductive limit topology on Wcomps,pΩ.Proof.
Note that support of everyu∈Wcomps,p is compact, so for each u, only finitely many of ψju’s are nonzero. Thus, the sum in the definition of qa,s,p is a finite sum. Now it is not hard to show that each qa,s,p is a seminorm and qa,s,pa∈S is separating. Here, we will show that the topologies are the same. Let us denote the inductive limit topology on Wcomps,pΩ by τ and the natural topology induced by the given family of seminorms τ′.
In what follows, we implicitly use the fact that both topologies are locally convex and translation invariant.
Step 1 (τ′⊆τ). We will prove that for each K∈KΩ, WKs,pΩ↪Wcomps,pΩ,τ′. This together with the definition of τ (the biggest topology with this property) implies that τ′⊆τ. Let K∈KΩ. By Theorem 18, it is enough to show that for all a∈S, qa,s,p∘Id:WKs,pΩ⟶ℝ is continuous. Since K is compact, there are only finitely may ψj’s such that K∩suppψj≠∅; let us call them ψj1,⋯,ψjl. So, for all u∈WKs,pΩ,
(79)qa,s,pu=aj1ψj1uWs,pΩ+⋯+ajlψjluWs,pΩ.
By assumption,s,p,Ω is an interior smooth multiplication triple, so for each j∈j1,⋯,jl, the mapping u↦∥ψju∥Ws,pΩ from WKs,pΩ⟶ℝ is continuous. Hence, qa,s,p∘Id:WKs,pΩ⟶ℝ must be continuous.
Step 2 (τ⊆τ′). Since Wcomps,pΩ,τ is a locally convex topological vector space, there exists a separating family of seminorms P whose corresponding natural topology is τ (see Theorem 15). We will prove that for all p~∈P, p~:Wcomps,pΩ,τ′⟶ℝ is continuous. This together with the fact that τ is the smallest topology with this property shows that τ⊆τ′. Let p~∈P. By Theorem 12, it is enough to prove that there exists a∈S such that
(80)∀u∈Wcomps,pΩp~u≤qa,s,pu.
For allu∈Wcomps,pΩ, we have p~u=p~∑jψju. Since u has compact support, only finitely many terms in the sum are nonzero, and so by the finite subadditivity of a seminorm, we get
(81)p~u=p~∑jψju≤∑jp~ψju.
Now, note thatψju belongs to the normed space Wsuppψjs,pΩ. Since p~:Wcomps,pΩ,τ⟶ℝ is continuous and Wsuppψjs,pΩ↪Wcomps,pΩ,τ, we can conclude that p~:Wsuppψjs,pΩ⟶ℝ is continuous. Thus, by Theorem 9, there exists a positive integer aj such that
(82)∀u∈Wcomps,pΩp~ψju≤ajψjuWs,pΩ.
It follows that for allu∈Wcomps,pΩ,
(83)p~u≤∑jp~ψju≤∑jajψjuWs,pΩ=qa,s,pu,
wherea=a0,a1,⋯


## 3.1. Topological Vector Spaces

Definition 1.
Atopological vector space is a vector space X together with a topology τ with the following properties:
(i)
For allx∈X, the singleton x is a closed set(ii)
The maps(4)x,y↦x+yfromX×XintoX,λ,x↦λxfromℝ×XintoX,are continuous where X×X and ℝ×X are equipped with the product topologyDefinition 2.
SupposeX,τ is a topological vector space and Y⊆X.
(i)
Y is said to be convex if for all y1,y2∈Y and t∈0,1 it is true that ty1+1−ty2∈Y(ii)
We sayY is bounded if for any neighborhood U of the origin (i.e., any open set containing the origin), there exits t>0 such that Y⊆tUDefinition 3.
LetX,τ be a topological vector space. X is said to be metrizable if there exists a metric d:X×X⟶0,∞ whose induced topology is τ. In this case, we say that the metric d is compatible with the topology τ.Theorem 4 ([19, 20]).
LetX,τ be a topological vector space. The following are equivalent:
(i)
X is metrizable(ii)
There exists a translation invariant metricd on X whose collection of open sets is the same as τ. Translation invariant means
(5)∀x,y,a∈Xdx+a,y+a=dx,y(iii)
X has a countable local base at the origin
(Recall that a subcollectionB of τ is said to be a local base at the origin if for any open set U containing the origin there is B∈B such that 0∈B⊆U.)Remark 5.
It can be shown that ifd1 and d2 are two translation invariant metrics that induce the same topology on X, then the Cauchy sequences of X,d1 will be exactly the same as the Cauchy sequences of X,d2.Definition 6.
LetX,τ be a topological vector space. We say X,τ is locally convex if it has a convex local base at the origin.Definition 7.
LetX,τ be a metrizable locally convex topological vector space. Let d be any translation invariant metric on X that is compatible with τ. We say that X is complete if and only if the metric space X,d is a complete metric space. A complete metrizable locally convex topological vector space is called a Frechet space.Definition 8.
Aseminorm on a vector space X is a real-valued function p:X⟶ℝ such that
(6)∀x,y∈Xpx+y≤px+py,∀x∈X∀α∈ℝpαx=αpx.
IfP is a family of seminorms on X, then we say P is separating provided that for all x≠0, there exists at least one p∈P such that px≠0 (that is, if px=0 for all p∈P, then x=0). It easily follows from the definition that any seminorm is a nonnegative function.Theorem 9.
Suppose thatX,.X is a normed space. Let p:X⟶ℝ be a seminorm on X. If p is continuous, then there exists a constant C>0 such that
(7)∀x∈Xpx≤CxX.Proof.
p is continuous at 0 so there exists δ>0 such that if xX≤δ, then px<1. If x≠0, then δx/xX has norm equal to δ, and so for all x≠0, pδx/xX<1. Hence, for all x≠0, we have
(8)px≤1δxX.
Sincep0=0, clearly the above inequality also holds for x=0.Definition 10.
SupposeP is a separating family of seminorms on a vector space X. The natural topology induced by P is the smallest topology on X that is translation invariant and with respect to which every p∈P is a continuous function from X to ℝ. (Recall that translation invariant means if U⊆X is open, then U+x is open for every x∈X.)Remark 11.
Suppose thatP and P′ are two separating family of seminorms on a vector space X. Let τ and τ′ be the corresponding natural topologies on X. It follows immediately from the definition that if (1) p:X,τ′⟶ℝ is continuous for each p∈P and (2) p′:X,τ⟶ℝ is continuous for each p′∈P′, then τ=τ′.The following theorem can be viewed as an extension of Theorem9.Theorem 12 ([21], page 157).
LetX be a vector space and suppose P is a separating family of seminorms on X. Equip X with the corresponding natural topology. Then, a seminorm q:X⟶ℝ is continuous if and only if there exist C>0 and p1,⋯,pm∈P such that for all x∈X(9)qx≤Cp1x+⋯+pmx.Theorem 13 ([19, 20]).
SupposeP is a separating family of seminorms on a vector space X and τ is the corresponding natural topology on X. Then, X,τ is a locally convex topological vector space. Moreover, if P=pkk∈ℕ is countable, then the locally convex topological vector space X,τ is metrizable, and the following translation invariant metric on X is compatible with τ:
(10)dx,y=∑k=1∞12kpkx−y1+pkx−y.Corollary 14.
SupposeP is a countable separating family of seminorms on a vector space X and τ is the corresponding natural topology on X. Then, X,τ is a Frechet space if and only if it is complete.Theorem 15 ([23], Sections 6.4 and 6.5).
LetX,τ be a locally convex topological vector space. Then, there exists a separating family of seminorms on X whose corresponding natural topology is τ.Theorem 16 ([19], page 28).
SupposeP is a separating family of seminorms on a vector space X and τ is the corresponding natural topology on X. Then, a set E⊆X is bounded if and only if pE is a bounded set in ℝ for all p∈P.Corollary 17.
SupposeP is a separating family of seminorms on a vector space X and τ is the corresponding natural topology on X. It follows from Theorem 12 and Theorem 16 that if E⊆X is bounded, then for any continuous seminorm q:X,τ⟶ℝ, qE is a bounded set in ℝ.Theorem 18 ([20], page 436, [23], Section 6.6).
LetX,τ be a topological vector space. Suppose Q is a separating family of seminorms on a vector space Y and τ′ is the corresponding natural topology on Y. Then, a linear map T:X,τ⟶Y,τ′ is continuous if and only if for each q∈Q, q∘T is continuous on X.Theorem 19 ([20]).
LetX be a Frechet space and let Y be a topological vector space. When T is a linear map of X into Y, the following two properties are equivalent:
(1)
T is continuous(2)
xn⟶0 in X⇒Txn⟶0 in YTheorem 20 ([19, 20]).
LetX and Y be two vector spaces and suppose P and Q are two separating families of seminorms on X and Y, respectively. Equip X and Y with the corresponding natural topologies. Then,
(1)
A sequencexn converges to x in X if and only if for all p∈P, pxn−x⟶0(2)
A linear operatorT:X⟶Y is continuous if and only if ∀q∈Q∃c>0,k∈ℕ,p1,⋯,pk∈Psuchthat∀x∈Xq∘Tx≤cmax1≤i≤kpix(3)
A linear operatorT:X⟶ℝ is continuous if and only if ∃c>0,k∈ℕ,p1,⋯,pk∈Psuchthat∀x∈XTx≤cmax1≤i≤kpixDefinition 21.
LetX,τ be a locally convex topological vector space.
(i)
Theweak topology on X is the natural topology induced by the separating family of seminorms pFF∈X∗ where
(11)∀F∈X∗pF:X⟶ℝ,pFx≔Fx.It can be shown that this topology is the smallest (weakest) topology with respect to which all the linear maps in X,τ∗ are continuous. A sequence xm converges to x in X with respect to the weak topology if and only if Fxm⟶Fx in ℝ for all F∈X∗. In this case, we may write xm⇀x. We denote the weak topology on X by σX,X∗. It can be shown that X,τ∗ is the same set as X,σX,X∗∗(ii)
Theweak∗topology on X∗ is the natural topology induced by the separating family of seminorms pxx∈X where
(12)∀x∈Xpx:X∗⟶ℝ,pxf≔fx.It can be shown that this topology is the weakest topology with respect to which all the linear maps f↦fxx∈X (from X∗ to ℝ) are continuous. A sequence fm converges to f in X∗ with respect to the weak∗ topology if and only if fmx⟶fx in ℝ for all x∈X. We denote the weak∗ topology on X∗ by σX∗,X(iii)
Thestrong topology on X∗ is the natural topology induced by the separating family of seminorms pBB⊆Xbounded where for any bounded subset B of X,
(13)pB:X∗⟶ℝpBf≔supfx:x∈B.(It can be shown that for any bounded subset B of X and f∈X∗, fB is a bounded subset of ℝ; see Theorem 16 and Theorem 28)Remark 22.
(1)
IfX is a normed space, then the topology induced by the norm
(14)∀f∈X∗fop=sup∥x∥X=1fxon X∗ is the same as the strong topology on X∗ ([22], page 198)(2)
In this manuscript, unless otherwise stated, we consider the topological dual of a locally convex topological vector space with the strong topology. Of course, it is worth mentioning that for many of the spaces that we will consider (includingX=EΩ or X=DΩ where Ω is an open subset of ℝn), a sequence in X∗ converges with respect to the weak∗ topology if and only if it converges with respect to the strong topology (for more details on this, see the definition and properties of Montel spaces in Section 34.4, page 356 of [22])Theorem 23.
LetX,τ be a locally convex topological vector space. Then, the evaluation map
(15)J:X,τ⟶X∗∗≔X∗,strongtopology∗,JxF≔Fx,is a well-defined injective linear map. (X∗∗ is called the bidual of X).Definition 24.
LetX,τ be a locally convex topological vector space. Let τ′ denote the strong topology on X∗∗ as the dual of X∗,strongtopology.
(i)
If the evaluation mapJ:X,τ⟶X∗∗,τ′ is bijective, then we say that X,τ is a semireflexive space(ii)
If the evaluation mapJ:X,τ⟶X∗∗,τ′ is a linear topological isomorphism, then we say that X,τ is a reflexive spaceTheorem 25 ([24], pages 16 and 17).
(i)
Strong dual of a reflexive topological vector space is reflexive(ii)
Every semireflexive space whose topology is defined by the inductive limit of a sequence of Banach spaces is reflexive(iii)
Every semireflexive Frechet space is reflexiveTheorem 26.
LetX,τX and Z,τZ be two locally convex topological vector spaces. For all x∈X, let lx:X∗⟶ℝ be the linear map defined by lxf=fx. Then,
(1)
a linear mapT:Z,τZ⟶X,σX,X∗ is continuous if and only if for all F∈X,τX∗, the linear map F∘T:Z,τZ⟶ℝ is continuous(2)
a linear mapT:Z,τZ⟶X∗,σX∗,X is continuous if and only if for all x∈X, the linear map lx∘T:Z,τZ⟶ℝ is continuousTheorem 27 ([21], page 163, [20], page 46).
LetX and Y be locally convex topological vector spaces and suppose T:X⟶Y is a continuous linear map. Either equip both X∗ and Y∗ with the strong topology or equip both with the weak∗ topology. Then,
(1)
the map(16)T∗:Y∗⟶X∗T∗y,xX∗×X=y,TxY∗×Yis well-defined, linear, and continuous. (T∗ is called the adjoint of T)(2)
IfTX is dense in Y, then T∗:Y∗⟶X∗ is injectiveTheorem 28 ([19], page 70).
LetX,τ be a locally convex topological vector space. Then, a set E⊆X is bounded with respect to τ if and only if it is bounded with respect to σX,X∗.Corollary 29.
IfX,τ is a locally convex topological vector space and xn⇀x (i.e., xn converges to x with respect to σX,X∗), then xn is bounded with respect to both τ and σX,X∗.Theorem 30.
LetX,τX and Y,τY be two locally convex topological vector spaces. If T:X,τX⟶Y∗,σY∗,Y is continuous, then T:X,σX,X∗⟶Y∗,σY∗,Y is continuous. In particular, if un⇀u (i.e., un converges to u with respect to σX,X∗), then Tun⟶Tu in Y∗,σY∗,Y.Proof.
For ally∈Y, let ly:Y∗⟶ℝ be the map lyF=Fy. By Theorem 26, T:X,σX,X∗⟶Y∗,σY∗,Y is continuous if ly∘T:X,σX,X∗⟶ℝ is continuous for all y∈Y. Let y∈Y.
(1)
By definition of the weak∗ topology on Y∗, we know that the linear map ly:Y∗⟶ℝ is continuous(2)
By assumptionT:X,τX⟶Y∗,σY∗,Y is a continuous linear map
Therefore,ly∘T belongs to X,τX∗. Since σX,X∗ is the weakest topology on X that makes all elements of X,τX∗ continuous, we can conclude that ly∘T:X,σX,X∗⟶ℝ is continuous.Theorem 31 ([25], page 13).
LetX,τ be a Frechet space. Then, X is reflexive if and only if every bounded set E in X is relatively weakly compact (i.e., the closure of E with respect to σX,X∗ is compact with respect to σX,X∗).Theorem 32 ([26], page 167).
LetX,τ be a separable Frechet space. If E⊆X is relatively weakly compact, then every infinite sequence in E has a subsequence that converges in X,σX,X∗.The next theorem is an immediate consequence of the previous theorems.Theorem 33.
Suppose thatX,τ is a separable reflexive Frechet space. Then, every bounded sequence in X,τ has a weakly convergent subsequence, that is, a subsequence that converges with respect to σX,X∗.Theorem 34 ([27], page 61).
LetX and Y be two Banach spaces. Let T:X⟶Y be a linear map. Then, T is continuous if and only if it is weak-weak continuous; that is, T:X,.X⟶Y,.Y is continuous if and only if T:X,σX,X∗⟶Y,σY,Y∗ is continuous.Theorem 35.
LetX be a Banach space and Y be a closed subspace of X with the induced norm. Suppose that ym is a sequence in Y and y∈Y. If ym⟶y in X,σX,X∗, then ym⟶y in Y,σY,Y∗.Proof.
This is a direct consequence of the fact that the following two topologies on the spaceY are the same (see [27], page 70):
(1)
The topology induced byσX,X∗(2)
The topologyσY,Y∗Definition 36.
LetX be a vector space and let Xαα∈I be a family of vector subspaces of X with the property that
(i)
for eachα∈I, Xα is equipped with a topology that makes it a locally convex topological vector space, and(ii)
∪α∈IXα=X
Theinductive limit topology on X with respect to the family Xαα∈I is defined to be the largest topology with respect to which
(1)
X is a locally convex topological vector space, and(2)
all the inclusionsXα⊆X are continuousTheorem 37 ([21]).
LetX be a vector space equipped with the inductive limit topology with respect to Xα as described above. If Y is a locally convex vector space, then a linear map T:X⟶Y is continuous if and only if TXα:Xα⟶Y is continuous for all α∈I.Definition 38.
LetX be a vector space and let Xjj∈ℕ0 be an increasing chain of subspaces of X:
(17)X0⊊X1⊊X2⊊⋯.
Suppose that(i)
eachXj is equipped with a locally convex topology τj(ii)
for eachj, the inclusion Xj,τj↪Xj+1,τj+1 is a linear topological embedding with closed image
Then, the inductive limit topology onX with respect to the family Xjj∈ℕ0 is called a strict inductive limit topology.Theorem 39 ([21]).
Suppose thatX is equipped with the strict inductive limit topology with respect to the chain Xjj∈ℕ0. Then, a subset E of X is bounded if and only if there exists m∈ℕ0 such that B is bounded in Xm.


## 3.2. Function Spaces and Distributions

Definition 40.
LetΩ be a nonempty open set in ℝn and m∈ℕ0.
(18)CΩ=f:Ω⟶ℝ:fiscontinuous,CmΩ=f:Ω⟶ℝ:∀α≤m∂αf∈CΩC0Ω=CΩ,BCΩ=f:Ω⟶ℝ:fiscontinuousandboundedonΩ,BCmΩ=f∈CmΩ:∀α≤m∂αfisboundedonΩ,C∞Ω=∩m∈ℕ0CmΩ,BC∞Ω=∩m∈ℕ0BCmΩ,Cc∞Ω=f∈C∞Ω:supportoffisanelementofKΩ.
Let0<λ≤1. A function F:Ω⊆ℝn⟶ℝk is called λ-Holder continuous if there exists a constant L such that
(19)Fx−Fy≤Lx−yλ∀x,y∈Ω.
Clearly, aλ-Holder continuous function on Ω is uniformly continuous on Ω. 1-Holder continuous functions are also called Lipschitz continuous functions or simply Lipschitz functions. We define
(20)BCm,λΩ=f:Ω⟶ℝ:∀α≤m∂αfisλ−Holdercontinuousandbounded=f∈BCmΩ:∀α≤m∂αfisλ−Holdercontinuous,BC∞,λΩ≔∩m∈ℕ0BCm,λΩ.Theorem 41 [20].
LetΩ be a nonempty open set in ℝn and let K∈KΩ. There is a function ψ∈Cc∞Ω taking values in 0,1 such that ψ=1 on a neighborhood containing K.Theorem 42 (exhaustion by compact sets) [20].
LetΩ be a nonempty open subset of ℝn. There exists a sequence of compact subsets Kjj∈ℕ such that ∪j∈ℕK˚j=Ω and
(21)K1⊆K˚2⊆K2⊆⋯⊆K˚j⊆Kj⊆⋯.
Moreover, as a direct consequence, ifK is any compact subset of the open set Ω, then there exists an open set V such that K⊆V⊆V¯⊆Ω.Theorem 43 [20].
LetΩ be a nonempty open subset of ℝn. Let Kjj∈ℕ be an exhaustion of Ω by compact sets. Define
(22)V0=K˚4,∀j∈ℕVj=K˚j+4\Kj.
Then,(1)
eachVj is an open bounded set and Ω=∪jVj(2)
the coverVjj∈ℕ0 is locally finite in Ω; that is, each compact subset of Ω has nonempty intersection with only a finite number of the Vj’s(3)
there is a family of functionsψj∈Cc∞Ω taking values in 0,1 such that suppψj⊆Vj and
(23)∑j∈ℕ0ψjx=1forallx∈ΩLetΩ be a nonempty open set in ℝn. For all φ∈C∞Ω, j∈ℕ, and K∈KΩ, we define
(24)φj,K≔sup∂αφx:α≤j,x∈K.For allj∈ℕ and K∈KΩ, .j,K is a seminorm on C∞Ω. We define EΩ to be C∞Ω equipped with the natural topology induced by the separating family of seminorms .j,Kj∈ℕ,K∈KΩ. It can be shown that EΩ is a Frechet space.For allK∈KΩ, we define EKΩ to be CK∞Ω equipped with the subspace topology. Since CK∞Ω is a closed subset of the Frechet space EΩ, EKΩ is also a Frechet space.We defineDΩ=∪K∈KΩEKΩ equipped with the inductive limit topology with respect to the family of vector subspaces EKΩK∈KΩ. It can be shown that if Kjj∈ℕ0 is an exhaustion by compacts sets of Ω, then the inductive limit topology on DΩ with respect to the family EKjj∈ℕ0 is exactly the same as the inductive limit topology with respect to EKΩK∈KΩ.Remark 44.
SupposeY is a topological space and the mapping T:Y⟶DΩ is such that TY⊆EKΩ for some K∈KΩ. Since EKΩ↪DΩ, if T:Y⟶EKΩ is continuous, then T:Y↪DΩ will be continuous.Theorem 45 (convergence and continuity forEΩ).
LetΩ be a nonempty open set in ℝn. Let Y be a topological vector space whose topology is induced by a separating family of seminorms Q.
(1)
A sequenceφm converges to φ in EΩ if and only if φm−φj,K⟶0 for all j∈ℕ and K∈KΩ(2)
SupposeT:EΩ⟶Y is a linear map. Then, the following are equivalent:
(i)
T is continuous(ii)
For everyq∈Q, there exist j∈ℕ and K∈KΩ, and C>0 such that
(25)∀φ∈EΩqTφ≤Cφj,K(iii)
Ifφm⟶0 in EΩ, then Tφm⟶0 in Y(3)
In particular, a linear mapT:EΩ⟶ℝ is continuous if and only if there exist j∈ℕ and K∈KΩ, and C>0 such that
(26)∀φ∈EΩTφ≤Cφj,K(4)
A linear mapT:Y⟶EΩ is continuous if and only if
(27)∀j∈ℕ,∀K∈KΩ∃C>0,k∈ℕ,q1,⋯,qk∈Qsuchthat∀yTyj,K≤Cmax1≤i≤kqiyTheorem 46 (convergence and continuity forEKΩ).
LetΩ be a nonempty open set in ℝn and K∈KΩ. Let Y be a topological vector space whose topology is induced by a separating family of seminorms Q.
(1)
A sequenceφm converges to φ in EKΩ if and only if φm−φj,K⟶0 for all j∈ℕ(2)
SupposeT:EKΩ⟶Y is a linear map. Then, the following are equivalent:
(i)
T is continuous(ii)
For everyq∈Q, there exists j∈ℕ and C>0 such that
(28)∀φ∈EKΩqTφ≤Cφj,K(iii)
Ifφm⟶0 in EKΩ, then Tφm⟶0 in YTheorem 47 (convergence and continuity forDΩ).
LetΩ be a nonempty open set in ℝn. Let Y be a topological vector space whose topology is induced by a separating family of seminorms Q.
(1)
A sequenceφm converges to φ in DΩ if and only if there is a K∈KΩ such that suppφm⊆K and φm⟶φ in EKΩ(2)
SupposeT:DΩ⟶Y is a linear map. Then, the following are equivalent:
(i)
T is continuous(ii)
For allK∈KΩ, T:EKΩ⟶Y is continuous(iii)
For everyq∈Q and K∈KΩ, there exists j∈ℕ and C>0 such that(29)∀φ∈EKΩqTφ≤Cφj,K(iv)
Ifφm⟶0 in DΩ, then Tφm⟶0 in Y(3)
In particular, a linear mapT:DΩ⟶ℝ is continuous if and only if for every K∈KΩ, there exists j∈ℕ and C>0 such that
(30)∀φ∈EKΩTφ≤Cφj,KRemark 48.
LetΩ be a nonempty open set in ℝn. Here are two immediate consequences of the previous theorems and remark:
(1)
The identity map(31)iD,E:DΩ⟶EΩis continuous (that is, DΩ↪EΩ)(2)
IfT:EΩ⟶EΩ is a continuous linear map such that suppTφ⊆suppφ for all φ∈EΩ (i.e., T is a local continuous linear map), then T restricts to a continuous linear map from DΩ to DΩ. Indeed, the assumption suppTφ⊆suppφ implies that TDΩ⊆DΩ. Moreover T:DΩ⟶DΩ is continuous if and only if for K∈KΩT:EKΩ⟶DΩ is continuous. Since TEKΩ⊆EKΩ, this map is continuous if and only if T:EKΩ⟶EKΩ is continuous (see Remark 44). However, since the topology of EKΩ is the induced topology from EΩ, the continuity of the preceding map follows from the continuity of T:EΩ⟶EΩTheorem 49.
LetΩ be a nonempty open set in ℝn. Then DΩ is separable.Definition 50.
LetΩ be a nonempty open set in ℝn. The topological dual of DΩ, denoted D′ΩD′Ω=DΩ∗, is called the space of distributions on Ω. Each element of D′Ω is called a distribution on Ω. The action of a distribution u∈D′Ω on a function φ∈DΩ is sometimes denoted by u,φD′Ω×DΩ or simply u,φ.Remark 51.
Every functionf∈Lloc1Ω defines a distribution uf∈D′Ω as follows
(32)∀φ∈DΩufφ≔∫Ωfφdx.
In particular, every functionφ∈EΩ defines a distribution uφ. It can be shown that the map i:EΩ⟶D′Ω which sends φ to uφ is an injective linear continuous map ([21], page 11). Therefore, we can identify EΩ with a subspace of D′Ω; we sometimes refer to the map i as the “identity map.”Theorem 52 ([20], page 47).
LetΩ be a nonempty open set in ℝn. Equip D′Ω with the weak∗ topology. Then, under the above identification, Cc∞Ω is dense in D′Ω.Theorem 53 ([22], page 302).
LetΩ be a nonempty open set in ℝn. Equip D′Ω with the strong topology. Then, under the identification described in Remark 51, Cc∞Ω is sequentially dense in D′Ω.Remark 54.
(i)
Clearly sequential density is a stronger notion than density. SoCc∞Ω is dense in D′Ω, strong topology)(ii)
Recall that, according to Remark22, a sequence converges in D′Ω,weak∗ if and only if it converges in D′Ω, strong topology). This together with the fact that weak∗ topology is weaker than the strong topology implies that convergent sequences in both topologies converge to the same limit. Therefore, it follows from Theorem 53 that Cc∞Ω is sequentially dense in D′Ω,weak∗. Hence, Theorem 52 can be viewed as a corollary of Theorem 53Theorem 55 ([21], page 9).
DΩ is reflexive. So D′Ω,strongtopology∗ can be identified with the topological vector space DΩ.Definition 56 (restriction of a distribution).
LetΩ be an open subset of ℝn and V be an open subset of Ω. We define the restriction map resΩ,V:D′Ω⟶D′V as follows:
(33)resΩ,Vu,φD′V×DV≔u,extV,Ω0φD′Ω×DΩ.
This is well-defined; indeed,resΩ,V:D′Ω⟶D′V is a continuous linear map as it is the adjoint of the continuous map extV,Ω0:DV⟶DΩ. Given u∈D′Ω, we sometimes write uV instead of resΩ,Vu.Definition 57 (support of a distribution).
LetΩ be a nonempty open set in ℝn. Let u∈D′Ω.
(i)
We sayu is equal to zero on some open subset V of Ω if uV=0(ii)
LetVii∈I be the collection of all open subsets of Ω such that u is equal to zero on Vi. Let V=∪i∈IVi. The support of u is defined as follows:(34)suppu≔Ω\V
Note that suppu is closed in Ω but it is not necessarily closed in ℝn.Theorem 58 ([21]).
LetΩ be a nonempty open set in ℝn and let u∈D′Ω. If φ∈DΩ vanishes on a neighborhood containing suppu, then uφ=0.Theorem 59 ([21]).
Letui be a sequence in D′Ω, u∈DΩ, and K∈KΩ such that ui⟶u in D′Ω and suppui⊆K for all i. Then, also suppu⊆K.Theorem 60 ([28], page 38).
LetΩ be a nonempty open set in ℝn. Suppose that Ti is a sequence in D′Ω with the property that for all φ∈DΩ, limi⟶∞Ti,φD′Ω×DΩ exists. Then, there exists T∈D′Ω such that
(35)∀φ∈DΩT,φD′Ω×DΩ=limi⟶∞Ti,φD′Ω×DΩ.Definition 61 (Sobolev-Slobodeckij spaces).
LetΩ be a nonempty open set in ℝn. Let s∈ℝ and p∈1,∞.
(i)
Ifs=k∈ℕ0,
(36)Wk,pΩ=u∈LpΩ:uWk,pΩ≔∑ν≤k∂νuLpΩ<∞(ii)
Ifs=θ∈0,1(37)Wθ,pΩ=u∈LpΩ:uWθ,pΩ≔∫∫Ω×Ωux−uypx−yn+θpdxdy1/p<∞(iii)
Ifs=k+θ,k∈ℕ0,θ∈0,1,
(38)Ws,pΩ=u∈Wk,pΩ:uWs,pΩ≔uWk,pΩ+∑ν=k∂νuWθ,pΩ<∞(iv)
W0s,pΩ is defined as the closure of Cc∞Ω in Ws,pΩ(v)
Ifs<0,
(39)Ws,pΩ=W0−s,p′Ω∗1p+1p′=1(vi)
For all compact setsK⊂Ω, we define
(40)WKs,pΩ=u∈Ws,pΩ:suppu⊆Kwith uWKs,pΩ≔uWs,pΩ. Note that for s<0, Ws,pΩ can be viewed as a subspace of D′Ω (see Theorem 68), and the support of u∈Ws,pΩ is interpreted as the support of a distribution.(vii)
Wcomps,pΩ≔∪K∈KΩWKs,pΩ. Wcomps,pΩ is equipped with the inductive limit topology with respect to the family of vector subspaces WKs,pΩK∈KΩ. It can be shown that if Kjj∈ℕ0 is an exhaustion by compacts sets of Ω, then the inductive limit topology on Wcomps,pΩ with respect to the family WKjs,pΩj∈ℕ0 is exactly the same as the inductive limit topology with respect to WKs,pΩK∈KΩTheorem 62.
LetΩ be a nonempty open set in ℝn, s≥1, and 1<p<∞. Then, u∈Ws,pΩ if and only if u∈LpΩ, and for all 1≤i≤n, ∂u/∂xi∈Ws−1,pΩ.Proof.
We consider two cases:
Case 1:s=k∈ℕ.
(41)u∈Wk,pΩ⟺u∈LpΩand∂αu∈LpΩ∀1≤α≤k⟺u∈LpΩand∂β∂u∂xi∈LpΩ∀0≤β≤k−1,1≤i≤n⟺u∈LpΩand∂u∂xi∈Wk−1,pΩ∀1≤i≤n
Case 2:s=k+θ, k∈ℕ, and 0<θ<1.
(42)u∈Ws,pΩ⟺u∈Wk,pΩand∂νux−∂νuyx−yn/p+θ∈LpΩ×Ω∀ν=k⟺u∈LpΩand∂u∂xi∈Wk−1,pΩ∀1≤i≤nand∂νux−∂νuyx−yn/p+θ∈LpΩ×Ω∀ν=k⟺u∈LpΩand∂u∂xi∈Wk−1,pΩand∂β∂u/∂xix−∂β∂u/∂xiyx−yn/p+θ∈LpΩ×Ω∀β=k−1∀1≤i≤n⟺u∈LpΩand∂u∂xi∈Ws−1,pΩ∀1≤i≤nRemark 63.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Clearly for s≥0, Cc∞Ω⊆Ws,pΩ. For s<0, it is easy to see that for all φ∈Cc∞Ω, the map lφ:W0−s,p′Ω⟶ℝ which sends u∈W0−s,p′Ω to ∫Ωuφdx belongs to W0−s,p′Ω∗=Ws,pΩ. The map φ↦lφ is one-to-one and we can use it to identify Cc∞Ω with a subspace of Ws,pΩ; we sometimes refer to the map that sends φ to lφ as the “identity map.” So we can talk about the identity map from Cc∞Ω to Ws,pΩ for all s∈ℝ.Theorem 64 ([16]).
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Then, Ws,pΩ is a reflexive Banach space.Corollary 65.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. A closed subspace of a reflexive space is reflexive, so W0s,pΩ is reflexive. Dual of a reflexive Banach space is a reflexive Banach space, so W−s,p′Ω is a reflexive Banach space.Remark 66.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Since W0s,pΩ is reflexive, it can be identified with W0s,pΩ∗∗ and we may write W−s,p′Ω∗=W0s,pΩ and talk about the duality pairing u,fW0s,pΩ×W−s,p′Ω. To be more precise, we notice that the identification of W0s,pΩ∗∗ and W0s,pΩ is done by the evaluation map
(43)J:W0s,pΩ⟶W0s,pΩ∗∗Juf=fu.
Therefore, for allu∈W0s,pΩ and f∈W−s,p′Ω,
(44)u,fW0s,pΩ×W−s,p′Ω=Ju,fW0s,pΩ∗∗×W−s,p′Ω=fu=f,uW−s,p′Ω×W0s,pΩ.Theorem 67.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Then, Cc∞Ω is dense in W−s,p′Ω. We may write this as W0−s,p′Ω=W−s,p′Ω.Proof.
Our proof will be based on a similar argument given in page 65 of [1]. Let φ↦lφ be the mapping introduced in Remark 63. Our goal is to show that the set
(45)V≔lφ:φ∈Cc∞Ωis dense in W−s,p′Ω. To this end, it is enough to show that if F∈W−s,p′Ω∗ is such that Flφ=0 for all φ∈Cc∞Ω, then F=0. Indeed, let F be such an element. By reflexivity of W0s,pΩ, there exists f∈W0s,pΩ such that
(46)∀v∈W−s,p′ΩFv=vf.
Thus, for allφ∈Cc∞Ω, we have
(47)0=Flφ=lφf=∫Ωfxφxdx.
So, by the fundamental lemma of the calculus of variations (see [27], page 110), we have f=0 (as an element of Ws,pΩ⊆Lloc1Ω) and therefore F=0.Theorem 68.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Equip D′Ω with weak∗ topology or strong topology. Then
(48)DΩ↪Ws,pΩ↪D′Ω.Proof.
Recall that the convergent sequences inD′Ω equipped with strong topology are exactly the same as the convergent sequences of D′Ω equipped with the weak∗ topology (see Remark 22). This together with Theorem 19 implies that in the study of the continuity of the inclusion map from Ws,pΩ to D′Ω, it does not matter whether we equip D′Ω with the strong topology or weak∗ topology. In the proof, as usual, we assume D′Ω is equipped with the strong topology. We consider two cases:
Case 1:s≥0. The continuity of the embedding DΩ↪Ws,pΩ has been studied in [16]. Also clearly Ws,pΩ↪LpΩ↪D′Ω. The former continuous embedding holds by the definition of Ws,pΩ and the latter embedding is continuous because if um⟶0 in LpΩ, then for all φ∈DΩ,
(49)um,φD′Ω×DΩ−0=∫Ωumφdx≤umpφ∞⟶0.
So,um⟶0 in D′Ω. This implies the continuity of the inclusion map from LpΩ to D′Ω by Theorem 19
Case 2:s<0. Since W0−s,p′Ω↪W−s,p′Ω, it follows from previous case that W0−s,p′Ω↪D′Ω. Also since DΩ⊆W0−s,p′Ω is dense in D′Ω (see Theorem 52, Theorem 53, and Remark 54), it follows that the inclusion map from W0−s,p′Ω to D′Ω is continuous with dense image. Thus, by Theorem 27, DΩ↪Ws,pΩ. Here, we used the facts that (1) the strong dual of the normed space W0−s,p′Ω is Ws,pΩ and that (2) the dual of D′Ω,strongtopology is DΩ (see Theorem 55). It remains to show that Ws,pΩ↪D′Ω. It follows from Case 1 that DΩ↪W0−s,p′Ω and by definition DΩ is dense in W0−s,p′Ω. So, by Theorem 27, Ws,pΩ↪D′ΩRemark 69.
Note that fors≤0, W0s,pΩ is the same as Ws,pΩ. For s>0, W0s,pΩ is a subspace of Ws,pΩ which contains Cc∞Ω. So it follows from the previous theorem that
(50)DΩ↪W0s,pΩ↪D′Ω.
To be more precise, we should note that fors<0, we identify φ∈DΩ with the corresponding distribution in D′Ω. Under this identification, for all s∈ℝ the “identity map” i:DΩ⟶W0s,pΩ is continuous with dense image, and so its adjoint i∗:W0s,pΩ∗⟶D′Ω will be an injective continuous map (Theorem 27), and we have
(51)i∗u,φD′Ω×DΩ=u,iφW0s,pΩ∗×W0s,pΩ=u,φW0s,pΩ∗×W0s,pΩ.
We usually identifyW0s,pΩ∗ with its image under i∗ and view W0s,pΩ∗ as a subspace of D′Ω. So, under this identification, we can rewrite the above equality as follows:
(52)∀u∈W0s,pΩ∗∀φ∈DΩu,φD′Ω×DΩ=u,φW0s,pΩ∗×W0s,pΩ.
Finally, noting that for alls∈ℝ and 1<p<∞, W0s,pΩ∗=W0−s,p′Ω (see Definition 61, Theorem 67, and Corollary 65), we can write
(53)∀u∈W0−s,p′Ω∀φ∈DΩu,φD′Ω×DΩ=u,φW0−s,p′Ω×W0s,pΩ.Theorem 70.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Then,
(1)
the mappingF↦FCc∞Ω is an isometric isomorphism between W−s,p′Ω and Cc∞Ω,.s,p∗(2)
supposeu∈D′Ω. If u:Cc∞Ω,.−s,p′⟶ℝ is continuous, then u∈W0s,pΩ (more precisely, there is a unique element in W0s,pΩ whose corresponding distribution is u). Moreover,(54)uW0s,pΩ=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφW−s,p′ΩProof.
The first item has been studied in [16]. Here, we will prove the second item. Since u:Cc∞Ω,.−s,p′⟶ℝ is continuous, it can be extended to a continuous linear map u~:W−s,p′Ω⟶ℝ. So u~∈W−s,p′Ω∗. However, W0s,pΩ is reflexive, therefore there exists a unique v∈W0s,pΩ such that u~=Jv where Jv:W−s,p′Ω⟶ℝ is the evaluation map defined by JvF=F,vW−s,p′Ω×W0s,pΩ. To finish the proof, it is enough to show that v=u as elements of D′Ω. For all φ∈Cc∞Ω, we have
(55)v,φD′Ω×DΩ=∫Ωvφdx=Remark63φ,vW−s,p′Ω×W0s,pΩ=Jvφ=u~φ=uφ=u,φD′Ω×DΩ.
Also,(56)uW0s,pΩ=vW0s,pΩ=JvW−s,p′Ω∗=u~W−s,p′Ω∗=sup0≡φ∈Cc∞Ωu~,φD′Ω×DΩφW−s,p′Ω=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφW−s,p′Ω.Corollary 71.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. Suppose that u∈D′Ω. As a direct consequence of Theorem 70, we have the following:
(i)
Ifsup0≡φ∈Cc∞Ωu,φD′Ω×DΩ/φWs,pΩ<∞, then u∈W−s,p′Ω and
(57)uW−s,p′Ω=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφWs,pΩ(ii)
Ifsup0≡φ∈Cc∞Ωu,φD′Ω×DΩ/φW−s,p′Ω<∞, then u∈W0s,pΩ and
(58)uWs,pΩ=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφW−s,p′Ω
That is, for anye∈ℝ and 1<q<∞, in order to show that u∈D′Ω belongs to W0e,qΩ, it is enough to prove that
(59)sup0≡φ∈Cc∞Ωu,φD′Ω×DΩφW−e,q′Ω<∞,and in fact uWe,qΩ=sup0≡φ∈Cc∞Ωu,φD′Ω×DΩ/φW−e,q′Ω.Theorem 72.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Suppose that K∈KΩ. Then, WKs,pΩ is a closed subspace of Ws,pΩ.Proof.
It is enough to show that ifui is a sequence of elements in WKs,pΩ such that ui⟶u in Ws,pΩ, then u∈WKs,pΩ, i.e., suppu⊆K. By Theorem 68, we have ui⟶u in D′Ω. Now it follows from Theorem 59 that suppu⊆K. Note that for any s≥0, we have Ws,pΩ⊆LpΩ⊆Lloc1Ω; in this proof, we implicitly used the fact that for functions in Lloc1Ω, the usual definition of support agrees with the distributional definition of support.Next, we list several embedding theorems for Sobolev-Slobodeckij spaces.Theorem 73 ([29], Section 2.8.1).
Suppose1<p≤q<∞ and −∞<t≤s<∞ satisfy s−n/p≥t−n/q. Then, Ws,pℝn↪Wt,qℝn. In particular, Ws,pℝn↪Wt,pℝn.Theorem 74 ([30, 18]).
LetΩ be a nonempty bounded open subset of ℝn with Lipschitz continuous boundary. Suppose 1≤p,q<∞ (p does NOT need to be less than or equal to q) and 0≤t≤s satisfy s−n/p≥t−n/q. If s∈ℕ0, additionally assume that s≠t. Then, Ws,pΩ↪Wt,qΩ. Furthermore, if s>t, then the embedding Ws,pΩ↪Wt,pΩ is compact.Theorem 75 ([16]).
LetΩ⊆ℝn be an arbitrary nonempty open set.
(1)
Suppose1≤p≤q<∞ and 0≤t≤s satisfy s−n/p≥t−n/q. Then, WKs,pΩ↪WKt,qΩ for all K∈KΩ(2)
For allk1,k2∈ℕ0 with k1≤k2 and 1<p<∞, Wk2,pΩ↪Wk1,pΩ(3)
If0≤t≤s<1 and 1<p<∞, then Ws,pΩ↪Wt,pΩ(4)
If0≤t≤s<∞ are such that s=t and 1<p<∞, then Ws,pΩ↪Wt,pΩ(5)
If0≤t≤s<∞, t∈ℕ0, and 1<p<∞, then Ws,pΩ↪Wt,pΩTheorem 76 ([6]).
LetΩ be a nonempty bounded open subset of ℝn with Lipschitz continuous boundary or Ω=ℝn. If sp>n, then Ws,pΩ↪L∞Ω∩C0Ω and Ws,pΩ is a Banach algebra.In the next several theorems, we will list certain multiplication properties of Sobolev spaces. Supposeφ∈C∞Ω and u∈Ws,pΩ. If s≥0, then the product φu has a clear meaning. What if s<0? In this case, uDΩ is a distribution and by the product φu we mean the distribution φuDΩ; then φu is in Ws,pΩ if φuDΩ:Cc∞Ω,.−s,p′⟶ℝ is continuous. Because then it possesses a unique extension to a continuous linear map from W0−s,p′Ω to ℝ and so it can be viewed as an element of W0−s,p′Ω∗=Ws,pΩ. See Theorem 70 and Corollary 71. Also see Remark 89.Theorem 77 (multiplication by smooth functions I, [31], page 203).
Lets∈ℝ, 1<p<∞, and φ∈BC∞ℝn. Then, the linear map
(60)mφ:Ws,pℝn⟶Ws,pℝn,u↦φuis well-defined and bounded.Theorem 78 (multiplication by smooth functions II, [16]).
LetΩ be a nonempty bounded open set in ℝn with Lipschitz continuous boundary.
(1)
Letk∈ℕ0 and 1<p<∞. If φ∈BCkΩ, then the linear map Wk,pΩ⟶Wk,pΩ defined by u↦φu is well-defined and bounded(2)
Lets∈ℝ and 1<p<∞. If φ∈BC∞Ω, then the linear map Ws,pΩ⟶Ws,pΩ defined by u↦φu is well-defined and boundedTheorem 79 (multiplication by smooth functions III, [16]).
LetΩ be any nonempty open set in ℝn. Let p∈1,∞.
(1)
If0≤s<1 and φ∈BC0,1Ω (that is, φ∈L∞Ω and φ is Lipschitz), then
(61)mφ:Ws,pΩ⟶Ws,pΩ,u↦φuis a well-defined bounded linear map(2)
Ifk∈ℕ0 and φ∈BCkΩ, then
(62)mφ:Wk,pΩ⟶Wk,pΩ,u↦φuis a well-defined bounded linear map(3)
If−1<s<0 and φ∈BC∞,1Ω or s∈ℤ− and φ∈BC∞Ω, then
(63)mφ:Ws,pΩ⟶Ws,pΩ,u↦φuis a well-defined bounded linear mapTheorem 80 (multiplication by smooth functions IV, [16]).
LetΩ be a nonempty open set in ℝn, K∈KΩ, p∈1,∞, and −1<s<0 or s∈ℤ− or s∈0,∞. If φ∈C∞Ω, then the linear map
(64)WKs,pΩ⟶WKs,pΩ,u↦φuis well-defined and bounded.Theorem 81 (multiplication by smooth functions V, [16]).
LetΩ be a nonempty bounded open set in ℝn with Lipschitz continuous boundary. Let K∈KΩ. Suppose s∈ℝ and p∈1,∞. If φ∈C∞Ω, then the linear map WKs,pΩ⟶WKs,pΩ defined by u↦φu is well-defined and bounded.In the next definition, we introduce the notion ofsmooth multiplication triple which will play a key role in several theorems that will follow.Definition 82 (smooth multiplication triple).
LetΩ be a nonempty open set in ℝn, s∈ℝ and 1<p<∞.
(i)
We say that the triples,p,Ω is a smooth multiplication triple if for all φ∈Cc∞Ω, the map
(65)mφ:Ws,pΩ⟶Ws,pΩu↦φuis well-defined and bounded(ii)
We say that the triples,p,Ω is an interior smooth multiplication triple if for all φ∈Cc∞Ω and K∈KΩ, the map
(66)mφ:WKs,pΩ⟶WKs,pΩu↦φuis well-defined and boundedRemark 83.
(1)
Every smooth multiplication triple is also an interior smooth multiplication triple(2)
It is a direct consequence of Theorems77, 78, and 79 that
(i)
ifΩ=ℝn or Ω is bounded with Lipschitz continuous boundary, then for all s∈ℝ and 1<p<∞, s,p,Ω is a smooth multiplication triple(ii)
ifΩ is any open set in ℝn, 1<p<∞, and s∈ℝ is not a noninteger with magnitude greater than 1, then s,p,Ω is a smooth multiplication triple(3)
It is a direct consequence of Theorem80 and Theorem 81 that
(i)
ifΩ=ℝn or Ω is bounded with Lipschitz continuous boundary, then for all s∈ℝ and 1<p<∞, s,p,Ω is an interior smooth multiplication triple(ii)
ifΩ is any open set in ℝn, 1<p<∞, and s∈ℝ is not a noninteger less than −1, then s,p,Ω is an interior smooth multiplication triple(4)
Ifs,p,Ω is a smooth multiplication triple and K∈KΩ, then WKs,pΩ⊆W0s,pΩ (see the proof of Theorem 7.31 in [16]). Of course, if s<0, then Ws,pΩ=W0s,pΩ and so WKs,pΩ⊆W0s,pΩ holds for all s<0, 1<p<∞ and open sets Ω⊆ℝnTheorem 84.
LetΩ be a nonempty open set in ℝn, s≥0, and 1<p<∞. If s,p,Ω is a smooth multiplication triple so is −s,p′,Ω.Proof.
Letφ∈Cc∞Ω. For all u∈W−s,p′Ω=W0−s,p′Ω and ψ∈DΩ, we have
(67)φu,ψD′Ω×DΩ=u,φψD′Ω×DΩ=Remark69u,φψW−s,p′Ω×W0s,pΩ≤uW−s,p′ΩφψWs,pΩ≼uW−s,p′ΩψWs,pΩ.
The last inequality holds becauses,p,Ω is a smooth multiplication triple. It follows from Corollary 71 that φu∈W0−s,p′Ω and φuW−s,p′Ω≼uW−s,p′Ω; that is, mφ:W−s,p′Ω⟶W−s,p′Ω is well-defined and continuous.Theorem 85.
LetΩ be a nonempty open set in ℝn, s∈ℝ and 1<p<∞. If s<0, further assume that −s,p′,Ω is a smooth multiplication triple. Suppose that Ω′⊆Ω and K∈KΩ′. Then,
(1)
for allu∈WKs,pΩ, uWs,pΩ≃uΩ′Ws,pΩ′(2)
for allu∈WKs,pΩ′, extΩ′,Ω0uWs,pΩ≃uWs,pΩ′Proof.
The claim follows from the argument presented in the proofs of Corollary 7.39 and Theorem 7.46 in [16].Theorem 86 (([30], pages 598-605), ([6], Section 1.4)).
Lets∈ℝ, 1<p<∞, and α∈ℕ0n. Suppose Ω is a nonempty open set in ℝn. Then,
(1)
the linear operator∂α:Ws,pℝn⟶Ws−α,pℝn is well-defined and bounded(2)
fors<0, the linear operator ∂α:Ws,pΩ⟶Ws−α,pΩ is well-defined and bounded(3)
fors≥0 and α≤s, the linear operator ∂α:Ws,pΩ⟶Ws−α,pΩ is well-defined and bounded(4)
ifΩ is bounded with Lipschitz continuous boundary, and if s≥0, s−1/p≠integer (i.e., the fractional part of s is not equal to 1/p), then the linear operator ∂α:Ws,pΩ⟶Ws−α,pΩ for α>s is well-defined and boundedTheorem 87.
Assumptions:(i)
Ω=ℝn or Ω is a bounded domain with Lipschitz continuous boundary(ii)
si,s∈ℝ, si≥s≥0 for i=1,2(iii)
1<pi≤p<∞ for i=1,2(iv)
si−s≥n1/pi−1/p(v)
s1+s2−s>n1/p1+1/p2−1/p
Claim: Ifu∈Ws1,p1Ω and v∈Ws2,p2Ω, then uv∈Ws,pΩ and moreover the pointwise multiplication of functions is a continuous bilinear map
(68)Ws1,p1Ω×Ws2,p2Ω⟶Ws,pΩ.Remark 88.
A number of other results concerning the sufficient conditions on the exponentssi,pi,s,p that guarantee the multiplication Ws1,p1Ω×Ws2,p2Ω↪Ws,pΩ is well-defined and continuous are discussed in detail in [18].Remark 89.
Suppose thats,p,Ω is a smooth multiplication triple with s≥0. W−s,p′Ω=W0−s,p′Ω is the dual of W0s,pΩ and u,fW0−s,p′Ω×W0s,pΩ is the action of the functional u on the function f. As it was discussed before, if ψ is a function in Cc∞Ω, ψuDΩ is defined as a product of a smooth function and a distribution. Since s,p,Ω is a smooth multiplication triple, −s,p′,Ω will also be a smooth multiplication triple, and that means ψuDΩ:Cc∞Ω,.s,p⟶ℝ is continuous (see the note right after Theorem 76). We interpret ψu as an element of W−s,p′Ω=W0s,pΩ∗ to be the unique continuous linear extension of ψuDΩ to the entire W0s,pΩ. It is easy to see that, this unique linear extension is given by
(69)ψu,fW−s,p′Ω×W0s,pΩ≔u,ψfW−s,p′Ω×W0s,pΩ,that is, the above map is linear continuous and its restriction to DΩ is the same as ψuDΩ. (Note that since s,p,Ω is a smooth multiplication triple, ψf is indeed an element of W0s,pΩ.)Theorem 90 ([32]).
Lets∈1,∞ and 1<p<∞, and let
(70)m=s,ifsisaninteger,⌊s⌋+1,otherwise.
IfF∈Cmℝ is such that F0=0 and F,F′,⋯,Fm∈L∞ℝ (in particular, note that every F∈Cc∞ℝ with F0=0 satisfies these conditions), then the map u↦Fu is well-defined and continuous from Ws,pℝn∩W1,spℝn into Ws,pℝn.Corollary 91.
Lets, p, and F be as in the previous theorem. Moreover, suppose sp>n. Then, the map u↦Fu is well-defined and continuous from Ws,pℝn into Ws,pℝn. The reason is that when sp>n, we have Ws,pℝn↪W1,spℝn.In the remaining of this section, we will state certain useful properties of the topological vector spaceWcomps,p. The properties we will discuss here echo the ones stated in [24] for spaces Hcomps.Theorem 92.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Then, DΩ is continuously embedded in Wcomps,pΩ.Proof.
For allK∈KΩ, we have
(71)EKΩ↪DΩ↪Ws,pΩ.
This together with the fact that the image ofEKΩ under the identity map is inside WKs,pΩ implies that
(72)EKΩ↪WKs,pΩ.
Also, by the definition of the inductive limit topology onWcomps,pΩ, we have
(73)WKs,pΩ↪Wcomps,pΩ.
It follows from (72) and (73) that for all K∈KΩ,
(74)EKΩ↪Wcomps,pΩ,which, by Theorem 37, implies that DΩ↪Wcomps,pΩ.Theorem 93.
Lets,p,Ω be a smooth multiplication triple. Then, Cc∞Ω is dense in Wcomps,pΩ.Proof.
We will follow the proof given in [24] for spaces Hcomps. Let u∈Wcomps,pΩ. It is enough to show that there exists a sequence in Cc∞Ω that converges to u in Wcomps,pΩ (this proves sequential density which implies density). By Meyers-Serrin theorem, there exists a sequence φm∈C∞Ω∩Ws,pΩ such that φm⟶u in Ws,pΩ. Let χ∈Cc∞Ω be such that χ=1 on a neighborhood containing suppu (see Theorem 41). Let K=suppχ. Since s,p,Ω is a smooth multiplication triple, multiplication by χ is a linear continuous map on Ws,pΩ and so χφm⟶χu in Ws,pΩ. Now, we note that χu=u and for all m, χφm are in Cc∞Ω with support inside K. Consequently, χφm⟶u in WKs,pΩ. Now, since WKs,pΩ↪Wcomps,pΩ, we may conclude that χφm is a sequence in Cc∞Ω that converges to u in Wcomps,pΩ.Remark 94.
As a consequence, ifs,p,Ω is a smooth multiplication triple, then Wcomps,pΩ∗ (equipped with the strong topology) is continuously embedded in D′Ω. More precisely, the identity map i:DΩ⟶Wcomps,pΩ is continuous with dense image, and therefore, by Theorem 27, the adjoint i∗:Wcomps,pΩ∗⟶D′Ω is a continuous injective map. We have
(75)i∗u,φD′Ω×DΩ=u,iφWcomps,pΩ∗×Wcomps,pΩ=u,φWcomps,pΩ∗×Wcomps,pΩ.
We usually identifyWcomps,pΩ∗ with its image under i∗ and view Wcomps,pΩ∗ as a subspace of D′Ω. So, under this identification, we can rewrite the above equality as follows:
(76)∀u∈Wcomps,pΩ∗u,φD′Ω×DΩ=u,φWcomps,pΩ∗×Wcomps,pΩ.Next, we will prove that ifs,p,Ω is a smooth multiplication triple, then Wcomps,pΩ is separable. To this end, we need the following lemma.Lemma 95.
LetX,τ and Y,τ′ be two topological spaces. Suppose that
(1)
A is dense in X,τ(2)
T:X,τ⟶Y,τ′ is continuous(3)
TX is dense in Y,τ′
Then,TA is dense in Y,τ′.Proof.
It is enough to show thatTA intersects every nonempty open set in Y,τ′. So let O∈τ′ be nonempty. Since TX is dense in Y,τ′, we have O∩TX≠∅ and so T−1O is nonempty. Also, since T is continuous, T−1O∈τ. A is dense in X,τ, so A∩T−1O≠∅. Therefore,
(77)TA∩O⊇TA∩T−1O≠∅.Theorem 96.
Lets,p,Ω be a smooth multiplication triple. Then, Wcomps,pΩ is separable.Proof.
According to Theorems92 and 93, DΩ is continuously embedded in Wcomps,pΩ and it is dense in Wcomps,pΩ. Since DΩ is separable, it follows from Lemma 95 that Wcomps,pΩ is separable.Theorem 97.
Lets,p,Ω be an interior smooth multiplication triple. Let ψjj∈ℕ0 be the partition of unity introduced in Theorem 43. Let S be the collection of all sequences whose terms are nonnegative integers. For all sequences a=a0,a1,⋯∈S define qa,s,p:Wcomps,pΩ⟶ℝ by
(78)qa,s,pu=∑j=0∞ajψjuWs,pΩ.
Then,qa,s,pa∈S is a separating family of seminorms on Wcomps,pΩ and the natural topology induced by this family on Wcomps,pΩ is the same as the inductive limit topology on Wcomps,pΩ.Proof.
Note that support of everyu∈Wcomps,p is compact, so for each u, only finitely many of ψju’s are nonzero. Thus, the sum in the definition of qa,s,p is a finite sum. Now it is not hard to show that each qa,s,p is a seminorm and qa,s,pa∈S is separating. Here, we will show that the topologies are the same. Let us denote the inductive limit topology on Wcomps,pΩ by τ and the natural topology induced by the given family of seminorms τ′.
In what follows, we implicitly use the fact that both topologies are locally convex and translation invariant.
Step 1 (τ′⊆τ). We will prove that for each K∈KΩ, WKs,pΩ↪Wcomps,pΩ,τ′. This together with the definition of τ (the biggest topology with this property) implies that τ′⊆τ. Let K∈KΩ. By Theorem 18, it is enough to show that for all a∈S, qa,s,p∘Id:WKs,pΩ⟶ℝ is continuous. Since K is compact, there are only finitely may ψj’s such that K∩suppψj≠∅; let us call them ψj1,⋯,ψjl. So, for all u∈WKs,pΩ,
(79)qa,s,pu=aj1ψj1uWs,pΩ+⋯+ajlψjluWs,pΩ.
By assumption,s,p,Ω is an interior smooth multiplication triple, so for each j∈j1,⋯,jl, the mapping u↦∥ψju∥Ws,pΩ from WKs,pΩ⟶ℝ is continuous. Hence, qa,s,p∘Id:WKs,pΩ⟶ℝ must be continuous.
Step 2 (τ⊆τ′). Since Wcomps,pΩ,τ is a locally convex topological vector space, there exists a separating family of seminorms P whose corresponding natural topology is τ (see Theorem 15). We will prove that for all p~∈P, p~:Wcomps,pΩ,τ′⟶ℝ is continuous. This together with the fact that τ is the smallest topology with this property shows that τ⊆τ′. Let p~∈P. By Theorem 12, it is enough to prove that there exists a∈S such that
(80)∀u∈Wcomps,pΩp~u≤qa,s,pu.
For allu∈Wcomps,pΩ, we have p~u=p~∑jψju. Since u has compact support, only finitely many terms in the sum are nonzero, and so by the finite subadditivity of a seminorm, we get
(81)p~u=p~∑jψju≤∑jp~ψju.
Now, note thatψju belongs to the normed space Wsuppψjs,pΩ. Since p~:Wcomps,pΩ,τ⟶ℝ is continuous and Wsuppψjs,pΩ↪Wcomps,pΩ,τ, we can conclude that p~:Wsuppψjs,pΩ⟶ℝ is continuous. Thus, by Theorem 9, there exists a positive integer aj such that
(82)∀u∈Wcomps,pΩp~ψju≤ajψjuWs,pΩ.
It follows that for allu∈Wcomps,pΩ,
(83)p~u≤∑jp~ψju≤∑jajψjuWs,pΩ=qa,s,pu,
wherea=a0,a1,⋯


## 4. Spaces of Locally Sobolev Functions

Lets∈ℝ, 1<p<∞. Let Ω be a nonempty open set in ℝn. We define
(84)Wlocs,pΩ≔u∈D′Ω:∀φ∈Cc∞Ωφu∈Ws,pΩ.We equipWlocs,pΩ with the natural topology induced by the separating family of seminorms .φ,s,pφ∈Cc∞Ω (see Definition 10) where
(85)∀u∈Wlocs,pΩ,φ∈Cc∞Ωuφ,s,p≔φuWs,pΩ.Whens and p are clear from the context, we may just write uφ or pφu instead of uφ,s,p. It is easy to show that for all φ∈Cc∞Ω, .φ,s,p is a seminorm on Wlocs,pΩ. The fact that the family of seminorms .φ,s,pφ∈Cc∞Ω is separating will be proved in Theorem 103.Remark 98.
Note that, by item (1) of Theorem20, ui⟶u in Wlocs,pΩ if and only if φui⟶φu in Ws,pΩ for all φ∈Cc∞Ω.Remark 99.
Clearly ifs,p,Ω is a smooth multiplication triple, then Ws,pΩ⊆Wlocs,pΩ.An equivalent description of locally Sobolev functions is described in the following theorem.Theorem 100.
Suppose thats,p,Ω is a smooth multiplication triple. Then, u∈D′Ω is in Wlocs,pΩ if and only if for every precompact open set V with V¯⊆Ω there is w∈Ws,pΩ such that wV=uV.Proof.
⇒: suppose u∈Wlocs,pΩ and let V be a precompact open set such that V¯⊆Ω. Let φ∈Cc∞Ω be such that φ=1 on a neighborhood containing V¯. Let w=φu. u is a locally Sobolev function, so w∈Ws,pΩ; also clearly wV=uV.
⇐: suppose u∈D′Ω has the property that for every precompact open set V with V¯⊆Ω there is w∈Ws,pΩ such that wV=uV. Let φ∈Cc∞Ω. We need to show that φu∈Ws,pΩ. Note that suppφ is compact, so there exists a bounded open set V such that
(86)suppφ⊆V⊆V¯⊆Ω.
By assumption, there existsw∈Ws,pΩ such that wV=uV. It follows from the hypothesis of the theorem that φw∈Ws,pΩ. Clearly φw=φu on Ω. Therefore, φu∈Ws,pΩ.


## 5. Overview of the Basic Properties

Material of this section is mainly an adaptation of the material presented in the excellent work of Antonic and Burazin [17], which is restricted to integer order Sobolev spaces, and Peterson [24], which is restricted to Hilbert spaces Hs. We have added certain details to the statements of the theorems and their proofs to ensure all the arguments are valid for both integer and noninteger order Sobolev-Slobodeckij spaces.Definition 101.
IfA is a subset of Cc∞Ω with the property that,
(87)∀x∈Ω∃φ∈Asuchthatφ≥0andφx≠0,then we say A is an admissible family of functions.Remark 102.
Note that ifA is an admissible family of functions, then for all m∈ℕ, the set φm:φ∈A is also an admissible family of functions.Theorem 103.
Lets,p,Ω be an interior smooth multiplication triple. If A is an admissible family of functions then
(1)
Wlocs,pΩ=u∈D′Ω:∀φ∈Aφu∈Ws,pΩ(2)
The collection.φ:φ∈A is a separating family of seminorms on Wlocs,pΩ(3)
The natural topology induced by the separating family of seminorms.φ:φ∈A is the same as the topology of Wlocs,pΩProof.
(1)
Letu∈D′Ω be such that φu∈Ws,pΩ for all φ∈A. We need to show that if ψ∈Cc∞Ω, then ψu∈Ws,pΩ. By the definition of A, for all x∈suppψ, there exists φx∈A such that φxx>0. Define
(88)Ux≔y∈Ω:φxy>0.Clearly, x∈Ux, and since φx is continuous, Ux is an open set. Uxx∈suppψ is an open cover of the compact set suppψ. So there exist points x1,⋯,xk such that suppψ⊆U≔Ux1∪⋯∪Uxk. If y∈U, then there exists 1≤i≤k such that y∈Uxi and so φxiy>0. So the smooth function ∑i=1kφxi is nonzero on U. Thus, on U we have
(89)ψu=ψ∑i=1kφxi∑i=1kφxiu.Indeed, if we define
(90)ξz=ψz∑i=1kφxiz,ifz∈U,0,otherwise,then ξ is smooth with compact support in U and
(91)ψu=ξ∑i=1kφxiuon the entire Ω. Now, note that for each i, φxiu is in Ws,pΩ (because by assumption φu∈Ws,pΩ for all φ∈A). So ∑i=1kφxiu∈Ws,pΩ. Since ξ∈Cc∞Ω and ∑i=1kφxiu have compact support and s,p,Ω is an interior smooth multiplication triple, it follows that ξ∑i=1kφxiu∈Ws,pΩ(2)
Now, we prove that.φ:φ∈A is a separating family of seminorms. We need to show that if u∈Wlocs,pΩ has the property that for all φ∈Auφ=φuWs,pΩ=0, then u=0. By definition of locally Sobolev functions, u is an element of D′Ω. So, in order to show that u=0, it is enough to prove that for all η∈Cc∞Ω, u,ηD′Ω×DΩ=0. We consider two cases:Case 1.A=Cc∞Ω. Let φ∈A be such that φ=1 on a neighborhood containing suppη. By assumption φu=0 in Ws,pΩ and so it is zero in D′Ω. Now we have
(92)u,ηD′Ω×DΩ=u,φηD′Ω×DΩ=φu,ηD′Ω×DΩ=0which is exactly what we wanted to proveCase 2.A⊂Cc∞Ω. We claim that if φuWs,pΩ=0 for all φ∈A, then for any ψ∈Cc∞Ω, ψuWs,pΩ=0, and so this case reduces to the previous case. Indeed, if ψ is an arbitrary element of Cc∞Ω, then by what was proved in item (1),
(93)ψu=ξ∑i=1kφxiu,where by assumption, for each i, φxiu is zero as an element of Ws,pΩ. Hence, ψu=0 in Ws,pΩ(3)
Finally, we show that the natural topologyτP induced by P=.φ:φ∈A is the same as the natural topology τQ induced by Q=.φ:φ∈Cc∞Ω. Obviously P is a subset of Q, so it follows from the definition of natural topology induced by a family of seminorms (see Definition 10) that τP⊆τQ. In order to show that τQ⊆τP, it is enough to prove that for all ψ∈Cc∞Ω, the map .ψ:Wlocs,pΩ,τP⟶ℝ is continuous. By what was shown in item (1), we can write
(94)∀u∈Wlocs,pΩuψ=ψuWs,pΩ=ξ∑i=1kφxiuWs,pΩ≼∑i=1kφxiuWs,pΩ=∑i=1kuφxi,where the implicit constant does not depend on u. In the last inequality, we used the assumption that s,p,Ω is an interior smooth multiplication triple. Now, it follows from Theorem 20 that .ψ:Wlocs,pΩ,τP⟶ℝ is continuousLemma 104.
There exists an admissible familyA⊆Cc∞Ω that has only countably many elements.Proof.
LetKjj∈ℕ be an exhaustion by compact sets for Ω. For each j∈ℕ, let φj∈Cc∞Ω be a nonnegative function such that φj=1 on Kj and φj=0 outside K˚j+2. Clearly, A=φjj∈N is a countable admissible family of functions.Corollary 105.
Lets,p,Ω be an interior smooth multiplication triple. Considering Theorem 13, it follows from the previous lemma and Theorem 103 that Wlocs,pΩ is metrizable. Indeed, if A=φjj=1∞ is a countable admissible family, then
(95)du,v=∑j=1∞12ju−vφj1+u−vφjis a compatible translation invariant metric on Wlocs,pΩ.Theorem 106.
Lets,p,Ω be an interior smooth multiplication triple. Then, Wlocs,pΩ is a Frechet space.Proof.
By Corollary14, it is enough to show that Wlocs,pΩ equipped with the metric in (95) is complete. Note that all admissible families result in equivalent topologies in Wlocs,pΩ. So we can choose the functions φj’s in the definition of d to be the partition of unity introduced in Theorem 43. Now, suppose um is a Cauchy sequence with respect to d. In what follows, we will prove that um converges to a distribution u in D′Ω. For now, let us assume this is true. We need to show that u is an element of Wlocs,pΩ; that is, we need to show that for all j, φju∈Ws,pΩ.
It follows from the definition ofd that for each j∈ℕ, φjumm∈ℕ is a Cauchy sequence in Ws,pΩ. Since Ws,pΩ is a Banach space, there exists fj in Ws,pΩ such that φjum⟶fj in Ws,pΩ. Note that Ws,pΩ↪D′Ω, so φjum⟶fj in D′Ω, and thus, for all ψ∈DΩ we have
(96)fj,ψD′Ω×DΩ=limm⟶∞φjum,ψD′Ω×DΩ=limm⟶∞um,φjψD′Ω×DΩ=u,φjψD′Ω×DΩ=φju,ψD′Ω×DΩ.
Hence,φju=fj in D′Ω. Since fj∈Ws,pΩ we can conclude that φju∈Ws,pΩ.
It remains to show thatum converges in D′Ω. To this end, it is enough to show that for all ψ∈DΩ, the sequence um,ψ converges in ℝ (see Theorem 60). Let ψ∈DΩ. Since suppψ is compact, there are only finitely many of φj’s that are nonzero on the support of ψ (see Theorem 43) which we denote by φj1,⋯,φjl. So for each x∈suppψ, φj1x+⋯+φjlx=1. We have
(97)um,ψ=um,φj1+⋯+φjlψ=φj1+⋯+φjlum,ψ=φj1um,ψ+⋯+φjlum,ψ.
limm⟶∞φj1um,ψ,⋯,limm⟶∞φjlum,ψ all exist (since φjrum is Cauchy in Ws,pΩ, it is convergent in Ws,pΩ, and so it is convergent in D′Ω). Therefore, limm⟶∞um,ψ exists.Theorem 107.
Lets,p,Ω be a smooth multiplication triple (so we know that Ws,pΩ⊆Wlocs,pΩ and Wlocs,pΩ is metrizable). Then, Ws,pΩ↪Wlocs,pΩ.Proof.
Since both spaces are metrizable, it suffices to show that ifui⟶u in Ws,pΩ, then ui⟶u in Wlocs,pΩ. To this end, let φ be an arbitrary element of Cc∞Ω. We need to show that if ui⟶u in Ws,pΩ, then φui⟶φu in Ws,pΩ. But this is a consequence of the fact that s,p,Ω is a smooth multiplication triple.Theorem 108.
LetΩ be a nonempty open set in ℝn, s∈ℝ, and 1<p<∞. Then, EΩ is continuously embedded in Wlocs,pΩ; i.e., the “identity map” from EΩ to Wlocs,pΩ is continuous.Proof.
By Theorem45, it is enough to show that if φm⟶0 in EΩ, then φm⟶0 in Wlocs,pΩ; that is, for all ψ∈Cc∞Ω, ψφm⟶0 in Ws,pΩ.
Letψ∈Cc∞Ω and let mψ denote multiplication by ψ. Multiplication by smooth functions is a continuous linear operator on EΩ ([21]). So mψ:EΩ⟶EΩ is continuous. The range of this map is in the subspace EsuppψΩ. So mψ:EΩ⟶EsuppψΩ is continuous. However, EsuppψΩ↪DΩ. Hence, mψ:EΩ⟶DΩ is continuous. As a consequence, since φm⟶0 in EΩ, ψφm⟶0 in DΩ. Finally, since DΩ↪Ws,pΩ, we can conclude that ψφm⟶0 in Ws,pΩ.Corollary 109.
SinceDΩ↪EΩ, it follows that under the hypotheses of Theorem 108, DΩ is continuously embedded in Wlocs,pΩ.Theorem 110.
Lets,p,Ω be a smooth multiplication triple. Then, Cc∞Ω is dense in Wlocs,pΩ.Proof.
Letu∈Wlocs,pΩ. It is enough to show that there exists a sequence ψj in Cc∞Ω such that ψj⟶u in Wlocs,pΩ, i.e.,
(98)∀ξ∈Cc∞Ωξψj⟶ξuinWs,pΩ.
First, note that, sinces,p,Ω is a smooth multiplication triple, for all ξ∈Cc∞Ω, there exists a constant Cξ,s,p,Ω such that
(99)∀v∈Ws,pΩξvWs,pΩ≤Cξ,s,p,ΩvWs,pΩ.
Letφjj∈N be the admissible family introduced in the proof of Lemma 104. For each ξ∈Cc∞Ω, there exists a number Jξ such that for all j≥Jξ, φj=1 on suppξ. So,
(100)∀j≥Jξφjξ=ξ.
Clearly, by definition ofWlocs,pΩ, for each j, φju∈Ws,pΩ, also φju has compact support, so φju∈W0s,pΩ (see Remark 83). Hence, for each j, there exists ψj∈Cc∞Ω such that ψj−φju<1/j. We claim that ξψj⟶ξu in Ws,pΩ. Indeed, given ε>0 and ξ∈Cc∞Ω, let J>Jξ be such that 1/J<ε/Cξ,s,p,Ω. Then, for j≥J, we have
(101)ξψj−ξuWs,pΩ=ξψj−ξφjuWs,pΩ=ξψj−φjuWs,pΩ≤Cξ,s,p,Ωψj−φjuWs,pΩ<Cξ,s,p,Ω1J<ε.Remark 111.
As a consequence, ifs,p,Ω is a smooth multiplication triple, then Wlocs,pΩ∗ (equipped with the strong topology) is continuously embedded in D′Ω. More precisely, the identity map i:DΩ⟶Wlocs,pΩ is continuous with dense image, and therefore, by Theorem 27, the adjoint i∗:Wlocs,pΩ∗⟶D′Ω is a continuous injective map. We have
(102)i∗u,φD′Ω×DΩ=u,iφWlocs,pΩ∗×Wlocs,pΩ=u,φWlocs,pΩ∗×Wlocs,pΩ.
We usually identifyWlocs,pΩ∗ with its image under i∗ and view Wlocs,pΩ∗ as a subspace of D′Ω. So, under this identification, we can rewrite the above equality as follows:
(103)∀u∈Wlocs,pΩ∗u,φD′Ω×DΩ=u,φWlocs,pΩ∗×Wlocs,pΩ.Theorem 112.
Lets,p,Ω be a smooth multiplication triple. Then, Wlocs,pΩ is separable.Proof.
DΩ is continuously embedded in Wlocs,pΩ and it is dense in Wlocs,pΩ. Since DΩ is separable, it follows from Lemma 95 that Wlocs,pΩ is separable.As a direct consequence of the definitions, locally Sobolev functions and Sobolev functions with compact support are both subsets of the space of distributions. The next two theorems establish a duality connection between the two spaces. But first, we need to state a simple lemma.Lemma 113.
LetX and Y be two topological spaces. Suppose that Y is Hausdorff. Let f:X⟶Y and g:X⟶Y be two continuous functions that agree on a dense subset A of X. Then, f=g everywhere. (So, in particular, in order to show that two continuous mappings from X to Y are equal, we just need to show that they agree on some dense subset.)Proof.
Suppose that there existsx0∈X such that fx0≠gx0. Since Y is Hausdorff, there exist open neighborhoods U and V of fx0 and gx0, respectively, such that U∩V=∅. f−1U∩g−1V is a nonempty (x0 is in it) open set in X so its intersection with A is nonempty. Let z be a point in the intersection of f−1U∩g−1V and A. Clearly, fz∈U and gz∈V; but since z∈A, we have fz=gz. This contradicts the assumption that U∩V=∅.Theorem 114.
Suppose thats,p,Ω and −s,p′,Ω are smooth multiplication triples. Define the mapping T:Wloc−s,p′Ω⟶Wcomps,pΩ∗ by
(104)∀u∈Wloc−s,p′Ω∀f∈Wcomps,pΩTuf≔ψfu,fW0−s,p′Ω×W0s,pΩ,where ψf is any function in Cc∞Ω that is equal to 1 on a neighborhood containing the support of f. Then,
(1)
Tuf does not depend on the choice of ψf(2)
For allu∈Wloc−s,p′Ω, Tu is indeed an element of Wcomps,pΩ∗(3)
T:Wloc−s,p′Ω⟶Wcomps,pΩ∗ is bijective(4)
SupposeWcomps,pΩ∗ is equipped with the strong topology. Then, the bijective linear map T:Wloc−s,p′Ω⟶Wcomps,pΩ∗ is a topological isomorphism; i.e., it is continuous with continuous inverse. So Wcomps,pΩ∗ can be identified with Wloc−s,p′Ω as topological vector spacesProof.
(1)
For the first item, it is enough to show that ifψ∈Cc∞Ω is equal to zero on a neighborhood U containing suppf, then ψu,fW0−s,p′Ω×W0s,pΩ=0. Note that f is not necessarily in Cc∞Ω, so we cannot directly apply the duality pairing identity stated in Remark 69. Let fm be sequence in Cc∞Ω such that fm⟶f in W0s,pΩ. Let ξ∈Cc∞Ω be such that ξ=1 on suppf and ξ=0 outside U. By assumption, s,p,Ω is a smooth multiplication triple and so ξfm⟶ξf=f in W0s,pΩ. Since elements of dual are continuous, we have
(105)ψu,fW0−s,p′Ω×W0s,pΩ=limm⟶∞ψu,ξfmW0−s,p′Ω×W0s,pΩ=Remark69limm⟶∞ψu,ξfmD′Ω×DΩ=limm⟶∞u,ψξfmD′Ω×DΩ=limm⟶∞u0=0.Note that ξfm is zero outside U and ψ=0 in U, so ψξfm=0 everywhere(2)
In order to show thatTu is an element of Wcomps,pΩ∗, we need to prove that Tu:Wcomps,pΩ⟶ℝ is linear and continuous. Linearity is obvious. In order to prove continuity, we need to show that for all K∈KΩ, TuWKs,pΩ is continuous (see Theorem 37). Let K∈KΩ and fix a function ψ∈Cc∞Ω which is equal to 1 on a neighborhood containing K. For all f∈WKs,pΩ we have
(106)Tuf=ψu,fW0−s,p′Ω×W0s,pΩ≤ψuW−s,p′Ω×fWs,pΩ,which proves the continuity of the linear map Tu(3)
In order to prove thatT is bijective, we give an explicit formula for the inverse. Recall that by definition, Wloc−s,p′Ω is a subspace of D′Ω and by Remark 94, Wcomps,pΩ∗ can also be viewed as a subspace of D′Ω. More precisely, if we let i:DΩ⟶Wcomps,pΩ be the “identity map” and i∗:Wcomps,pΩ∗⟶D′Ω be the adjoint of i, then i∗ is a continuous injective linear map and
(107)∀u∈Wcomps,pΩ∗∀φ∈DΩi∗u,φD′Ω×DΩ=u,φWcomps,pΩ∗×Wcomps,pΩ.Moreover, if K∈KΩ, then WKs,pΩ↪Wcomps,pΩ, and therefore, if u∈Wcomps,pΩ∗, then uWKs,pΩ∈WKs,pΩ∗ and
(108)∀g∈WKs,pΩu,gWcomps,pΩ∗×Wcomps,pΩ=uWKs,pΩ,gWKs,pΩ∗×WKs,pΩ.Now, we claim that the image of i∗ is in Wloc−s,p′Ω and in fact i∗ is the inverse of T. Let us first prove that the image of i∗ is in Wloc−s,p′Ω. Let u∈Wcomps,pΩ∗. We need to show that for all φ∈Cc∞Ω, φi∗u∈W−s,p′Ω. To this end, we make use of Corollary 71. Let φ∈Cc∞Ω and let K=suppφ. For all ψ∈DΩ, we have
(109)φi∗u,ψD′Ω×DΩ=i∗u,φψD′Ω×DΩ=u,φψWcomps,pΩ∗×Wcomps,pΩ=uWKs,pΩ,φψWKs,pΩ∗×WKs,pΩ≤uWKs,pΩWKs,pΩ∗φψWKs,pΩ=uWKs,pΩWKs,pΩ∗φψWs,pΩ≼uWKs,pΩWKs,pΩ∗ψWs,pΩ,which, by Corollary 71, proves that φi∗u∈W−s,p′Ω. Now, we prove i∗ is the inverse of T. Note that for all u∈Wloc−s,p′Ω⊆D′Ω and φ∈DΩ,
(110)i∗∘Tu,φD′Ω×DΩ=Equation107Tu,φWcomps,pΩ∗×Wcomps,pΩ=DefinitionofTψφu,φW0−s,p′Ω×W0s,pΩ=Remark69ψφu,φD′Ω×DΩ=u,ψφφD′Ω×DΩ=u,φD′Ω×DΩ. Therefore, i∗∘T is identity. Next, we show that for all v∈Wcomps,pΩ∗, T∘i∗v=v. Note that T∘i∗v and v both are in Wcomps,pΩ∗ and so they are continuous functions from Wcomps,pΩ to ℝ. Since DΩ is dense in Wcomps,pΩ, according to Lemma 113, it is enough to show that for all f∈DΩ, we have T∘i∗vf=vf.
(111)T∘i∗v,fWcomps,pΩ∗×Wcomps,pΩ=DefinitionofTψfi∗v,fW0−s,p′Ω×W0s,pΩ=Remark69ψfi∗v,fD′Ω×DΩ=i∗v,ψffD′Ω×DΩ=i∗v,fD′Ω×DΩ=Equation107v,fWcomps,pΩ∗×Wcomps,pΩ(4)
Let us denote the topology ofWloc−s,p′Ω by τ and the strong topology on Wcomps,pΩ∗ by τ′. Our goal is to show that T:Wloc−s,p′Ω,τ⟶Wcomps,pΩ∗,τ′ and T−1=i∗:Wcomps,pΩ∗,τ′⟶Wloc−s,p′Ω,τ are both continuous maps. To this end, we make use of Theorem 20. Recall that τ is induced by the family of seminorms pφ:Wloc−s,p′Ω⟶ℝφ∈Cc∞Ω where pφu=∥φu∥W−s,p′Ω. Also τ′ is induced by the family of seminorms pB′:Wcomps,pΩ∗⟶ℝ where B varies over all bounded sets in Wcomps,pΩ and pB′u=supf∈BufStep 1. LetB be a bounded subset of Wcomps,pΩ. Since B is bounded, there exists K∈KΩ such that B is bounded in WKs,pΩ (see Theorem 39; note that the topology of Wcomps,pΩ can be constructed as the inductive limit of WKjs,pΩ where Kj is an increasing chain of compact subsets of Ω). So there exists a constant C such that for all f∈B, fWs,pΩ≤C. Let ψ be a function in Cc∞Ω which is equal to 1 on a neighborhood containing K. For all u∈Wloc−s,p′Ω, we have
(112)pB′∘Tu=supf∈BTuf=DefinitionofTsupf∈Bψu,fW0−s,p′Ω×W0s,pΩ≤supf∈BψuW−s,p′ΩfWs,pΩ≤Cpψu.It follows from Theorem 20 that T:Wloc−s,p′Ω,τ⟶Wcomps,pΩ∗,τ′ is continuousStep 2. Letφ∈Cc∞Ω. Let K be a compact set whose interior contains suppφ. Since s,p,Ω is a smooth multiplication triple, there exists a constant Cφ>0 such that for all f∈Ws,pΩ, we have φfWs,pΩ≤CφfWs,pΩ.We have(113)pφ∘i∗u=φi∗uW0s,pΩ=φi∗uW0−s,p′Ω∗=supξ∈Cc∞Ω,∥ξ∥Ws,pΩ≤1φi∗u,ξW0s,pΩ×W0−s,p′Ω=supξ∈Cc∞Ω,∥ξ∥Ws,pΩ≤1φi∗u,ξD′Ω×DΩ=supξ∈Cc∞Ω,∥ξ∥Ws,pΩ≤1i∗u,φξD′Ω×DΩ≤supη∈Csuppφ∞Ω,∥η∥Ws,pΩ≤Cφi∗u,ηD′Ω×DΩ=Equation107supη∈Csuppφ∞Ω,∥η∥Ws,pΩ≤Cφu,ηWcomps,pΩ∗×Wcomps,pΩ.So, if we let B be the ball of radius 2Cφ centered at 0 in WKs,pΩ (clearly B is a bounded set in Wcomps,pΩ), we get
(114)pφ∘i∗u≤supf∈Bu,fWcomps,pΩ∗×Wcomps,pΩ=pB′uCorollary 115.
Suppose thats,p,Ω and −s,p′,Ω are both smooth multiplication triples. By the previous theorem, Wcomp−s,p′Ω∗ can be identified with Wlocs,pΩ. Also, by Remark 94, Wcomp−s,p′Ω∗ is continuously embedded in D′Ω. Therefore, Wlocs,pΩ is continuously embedded in D′Ω. Since Wlocs,pΩ is a Frechet space, it follows from Theorem 19 and Remark 22 that the preceding statement remains true even if we consider D′Ω equipped with the weak∗ topology. So,
(115)Wlocs,pΩ↪D′Ω,strongtopologyandWlocs,pΩ↪D′Ω,weak∗topology.Theorem 116.
Suppose thats,p,Ω and −s,p′,Ω are smooth multiplication triples. Define the mapping R:Wcomp−s,p′Ω⟶Wlocs,pΩ∗ by
(116)∀u∈Wcomp−s,p′Ω∀f∈Wlocs,pΩRuf≔u,ψufW0−s,p′Ω×W0s,pΩ,where ψu is any function in Cc∞Ω that is equal to 1 on a neighborhood containing the support of u. Then,
(1)
Ruf does not depend on the choice of ψu(2)
For allu∈Wcomp−s,p′Ω, Ru is indeed an element of Wlocs,pΩ∗(3)
R:Wcomp−s,p′Ω⟶Wlocs,pΩ∗ is bijective(4)
SupposeWlocs,pΩ∗ is equipped with the strong topology. Then, the linear map R is bijective and continuous. In particular, Wlocs,pΩ∗ and Wcomp−s,p′Ω are isomorphic vector spacesProof.
(1)
Note that sinces,p,Ω is a smooth multiplication triple, ψuf is in W0s,pΩ. Also by assumption, −s,p′,Ω is a smooth multiplication triple. Therefore, for each K∈KΩ, WK−s,p′Ω↪W0−s,p′Ω, and hence, Wcomp−s,p′Ω↪W0−s,p′Ω. So the pairing in the definition of Ruf makes sense. The fact that the output is independent of the choice of ψu follows directly from Theorem 58(2)
Clearly,Ru is linear. Also Ru is continuous (so it is an element of Wlocs,pΩ∗). The reason is as follows: for all f∈Wlocs,pΩ, we have
(117)Ruf=u,ψufW0−s,p′Ω×W0s,pΩ≤uW0−s,p′ΩψufW0s,pΩ.That is, for all f∈Wlocs,pΩ, we have Ruf≼ψufWs,pΩ. It follows from Theorem 20 that Ru:Wlocs,pΩ⟶ℝ is continuous(3)
In order to prove thatR is bijective we give an explicit formula for the inverse. Recall that by Remark 111, Wlocs,pΩ∗ can also be viewed as a subspace of D′Ω. More precisely, if we let i:DΩ⟶Wlocs,pΩ be the “identity map” and i∗:Wlocs,pΩ∗⟶D′Ω be the adjoint of i, then i∗ is a continuous injective linear map and
(118)∀u∈Wlocs,pΩ∗∀φ∈DΩi∗u,φD′Ω×DΩ=u,φWlocs,pΩ∗×Wlocs,pΩ.Now, we claim that the image of i∗ is in Wcomp−s,p′Ω and in fact i∗ is the inverse of R. Let us first prove that the image of i∗ is in Wcomp−s,p′Ω. EΩ is continuously and densely embedded in Wlocs,pΩ (continuity is proved in Theorem 108 and density is a direct consequence of the density of Cc∞Ω in Wlocs,pΩ). Therefore, i∗Wlocs,pΩ∗ is indeed a subspace of E′Ω⊆D′Ω and so elements of i∗Wlocs,pΩ∗ can be identified with distributions in D′Ω that have compact support. It remains to show that if u∈Wlocs,pΩ∗, then i∗u∈W−s,p′Ω. To this end, we make use of Corollary 71. For all φ∈DΩ, we have
(119)i∗u,φD′Ω×DΩ=Equation118u,φWlocs,pΩ∗×Wlocs,pΩ=uWs,pΩ,φW0s,pΩ∗×W0s,pΩ≤uW0s,pΩW0s,pΩ∗φWs,pΩ.So, by Corollary 71, we can conclude that u∈W−s,p′Ω. In the above, we used the fact that W0s,pΩ↪Ws,pΩ↪Wlocs,pΩ, and so for u∈Wlocs,pΩ∗, we have uW0s,pΩ∈W0s,pΩ∗.Now, we prove thati∗:Wlocs,pΩ∗⟶Wcomp−s,p′Ω is the inverse of R. For all u∈Wcomp−s,p′Ω and φ∈DΩ, we have
(120)i∗∘Ru,φD′Ω×DΩ=Equation118Ru,φWlocs,pΩ∗×Wlocs,pΩ=DefinitionofRu,ψuφW0−s,p′Ω×W0s,pΩ=Remark69u,ψuφD′Ω×DΩ=ψuu,φD′Ω×DΩ=u,φD′Ω×DΩ.Therefore,i∗∘Ru=u for all u∈Wcomp−s,p′Ω.Now, we prove thatR∘i∗ is also the identity map. Considering Lemma 113, since DΩ is dense in Wlocs,pΩ∗, it is enough to show that for all v∈Wlocs,pΩ∗ and f∈DΩ, R∘i∗vf=vf. We have
(121)R∘i∗v,fWlocs,pΩ∗×Wlocs,pΩ=DefinitionofRi∗v,ψi∗vfW0−s,p′Ω×W0s,pΩ=Remark69i∗v,ψi∗vfD′Ω×DΩ=ψi∗vi∗v,fD′Ω×DΩ=i∗v,fD′Ω×DΩ=Equation118v,fWlocs,pΩ∗×Wlocs,pΩ, which shows R∘i∗v=v(4)
Let us denote the topology ofWcomp−s,p′Ω by τ and the strong topology on Wlocs,pΩ∗ by τ′. Our goal is to show that R:Wcomp−s,p′Ω,τ⟶Wlocs,pΩ∗,τ′ is continuous. To this end, we make use of Theorem 20. Recall that τ is induced by the family of seminorms qa,−s,p′:Wcomp−s,p′Ω⟶ℝa∈S where qa,−s,p′u=∑jajψjuW−s,p′Ω (here, we are using the notation introduced in Theorem 97). Also τ′ is induced by the family of seminorms pB:Wlocs,pΩ∗⟶ℝ where B varies over all bounded sets in Wlocs,pΩ and pBu=supf∈Buf.LetB be a bounded subset of Wlocs,pΩ. Since B is bounded, for all φ∈Cc∞Ω, the set φfWs,pΩ:f∈B is bounded in ℝ (see Theorem 16). Thus, for all φ∈Cc∞Ω, there exists a positive integer aφ such that for all f∈B, φfWs,pΩ<aφ. Recall that ψj in the definition of qa,−s,p′ denotes a fixed partition of unity. For each j, let φj be a function in Cc∞Ω which is equal to 1 on a neighborhood containing the support of ψj. For all u∈Wcomp−s,p′Ω, we have
(122)pB∘Ru=pB∘R∑jψju≤∑jpB∘Rψju=supf∈BRψjuf=DefinitionofR∑jsupf∈Bψju,φjfW0−s,p′Ω×W0s,pΩ≤∑jsupf∈BψjuW−s,p′ΩφjfWs,pΩ≤∑jaφjψjuW−s,p′Ω=qa,−s,p′u,where a=aφ1,aφ2,⋯. Note that the inequality pB∘R∑jψju≤∑jpB∘Rψju holds because u has compact support and so only finitely many terms in the sum are nonzero, so we can use the subadditivity of the seminorm and linearity of R.It follows from Theorem20 that R:Wcomp−s,p′Ω,τ⟶Wlocs,pΩ∗ is continuousRemark 117.
According to the previous two theorems, we have the following:(i)
Whenu∈Wloc−s,p′Ω is viewed as an element of Wcomps,pΩ∗, we have
(123)∀f∈Wcomps,pΩuf=ψfu,fW0−s,p′Ω×W0s,pΩ,where ψf is any function in Cc∞Ω that is equal to 1 on a neighborhood containing suppf(ii)
Whenu∈Wcomp−s,p′Ω is viewed as an element of Wlocs,pΩ∗, we have
(124)∀f∈Wlocs,pΩuf=u,ψfW0−s,p′Ω×W0s,pΩ,where ψ is any function in Cc∞Ω that is equal to 1 on a neighborhood containing suppuCorollary 118.
Suppose thats,p,Ω and −s,p′,Ω are both smooth multiplication triples. As a direct consequence of the previous theorems, the bidual of Wcomps,pΩ is itself. So Wcomps,pΩ is semireflexive. It follows from Theorem 25 that Wcomps,pΩ is reflexive and subsequently its dual Wloc−s,p′Ω is reflexive.Now, we put everything together to build general embedding theorems for spaces of locally Sobolev-Slobodeckij functions.Theorem 119 (embedding theorem I).
LetΩ be a nonempty open set in ℝn. If s1,s2∈ℝ and 1<p1,p2<∞ are such that Ws1,p1Ω↪Ws2,p2Ω, then Wlocs1,p1Ω↪Wlocs2,p2Ω.Proof.
We have(125)u∈Wlocs1,p1Ω⟺∀φ∈Cc∞Ω,φu∈Ws1,p1Ω,⇒∀φ∈Cc∞Ω,φu∈Ws2,p2Ω,⟺u∈Wlocs2,p2Ω.
So,Wlocs1,p1Ω⊆Wlocs2,p2Ω. Now, note that for all φ∈Cc∞Ω,
(126)uφ,s2,p2=φuWs2,p2Ω≼φuWs1,p1Ω=uφ,s1,p1.
So, it follows from Theorem20 that the inclusion map from Wlocs1,p1Ω to Wlocs2,p2Ω is continuous.Theorem 120 (embedding theorem II).
LetΩ be a nonempty open set in ℝn that has the interior Lipschitz property. Suppose that s1,s2∈ℝ and 1<p1,p2<∞ are such that Ws1,p1U↪Ws2,p2U for all bounded open sets U with Lipschitz continuous boundary. If s1<0, further assume that −s1,p1′,Ω is a smooth multiplication triple. If s2<0, further assume that −s2,p2′,Ω is a smooth multiplication triple. Then, Wlocs1,p1Ω↪Wlocs2,p2Ω.Proof.
Supposeu∈Wlocs1,p1Ω and φ∈Cc∞Ω. Let Ω′ be an open set in Ω that contains suppφ and has Lipschitz continuous boundary. We have
(127)u∈Wlocs1,p1Ω⇒φu∈Ws1,p1Ω⇒Theorem85φuΩ′∈Ws1,p1Ω′⇒φuΩ′∈Ws2,p2Ω′⇒Theorem85φu∈Ws2,p2Ω.
Sinceφ can be any element of Cc∞Ω, we can conclude that if u∈Wlocs1,p1Ω, then u∈Wlocs2,p2Ω. In order to prove the continuity of the inclusion map, we can proceed as follows: let φ∈Cc∞Ω and choose Ω′ as before.
(128)uφ,s2,p2=φuWs2,p2Ω≃Theorem85φuWs2,p2Ω′≼φuWs1,p1Ω′≃Theorem85φuWs1,p1Ω=uφ,s1,p1.
So, it follows from Theorem20 that the inclusion map from Wlocs1,p1Ω to Wlocs2,p2Ω is continuous.A version of compact embedding for spacesHlocm with integer smoothness degree has been studied in [17]. In what follows, we will state the corresponding theorem and its proof for spaces of locally Sobolev-Slobodeckij functions.Lemma 121.
Suppose thats,p,Ω and −s,p′,Ω are smooth multiplication triples. If um converges weakly to u in Wlocs,pΩ, then
(129)∀φ∈Cc∞Ωφum⇀φuinWs,pΩ.Proof.
The proof is based on the following well-known fact:
Fact 1. LetX be a topological space and suppose that x is a point in X. Let xm be a sequence in X. If every subsequence of xm contains a subsequence that converges to x, then xm⟶x.
Letφ∈Cc∞Ω. By Fact 1, it is enough to show that every subsequence of φum has a subsequence that converges weakly to φu in Ws,pΩ. Let φum′ be a subsequence of φum. We have
(130)um′⇀uinWlocs,pΩ⇒Corollary29um′isboundedinWlocs,pΩ⇒Corollary17φum′isboundedinWs,pΩ.
SinceWs,pΩ is reflexive, there exists a subsequence φum″ that converges weakly to some F∈Ws,pΩ. To finish the proof, it is enough to show that F=φu. We have
(131)um″⇀uinWlocs,pΩ⇒um″⟶uinD′Ω,weak∗⇒φum″⟶φuinD′Ω,weak∗.
In the first line, we used Theorem30 and the fact that Wlocs,pΩ↪D′Ω,weak∗ (see Corollary 115). In the second line, we used the fact that multiplication by smooth functions is a continuous operator on D′Ω,weak∗.
Similarly, sinceWs,pΩ↪D′Ω,weak∗, it follows from Theorem 30 that
(132)φum″⇀FinWs,pΩ⇒φum″⟶FinD′Ω,weak∗.
Consequently,φu=F as elements of D′Ω and subsequently as elements of W0s,pΩ.Theorem 122 (compact embedding).
LetΩ be a nonempty open set in ℝn that has the interior Lipschitz property. Suppose that s1,p1,Ω and −s1,p1′,Ω are smooth multiplication triples. If s2<0, further assume that −s2,p2′,Ω is a smooth multiplication triple. Moreover, suppose that s1,s2,p1, and p2 are such that Ws1,p1U is compactly embedded in Ws2,p2U for all bounded open sets U with Lipschitz continuous boundary. Then, every bounded sequence in Wlocs1,p1Ω has a convergent subsequence in Wlocs2,p2Ω.Proof.
The proof makes use of the following well-known fact:
Fact 2. LetX and Y be Banach spaces. Suppose that T:X⟶Y is a linear compact operator. If the sequence xm converges weakly (i.e., with respect to the weak topology) to x in X, then Txn converges to Tx (with respect to the norm of Y) in Y.
Letum be a bounded sequence in Wlocs1,p1Ω. By Theorem 33, since Wlocs1,p1Ω is a separable reflexive Frechet space, there exists u∈Wlocs1,p1Ω and a subsequence um′ such that um′⇀u in Wlocs1,p1Ω. We claim that um′ converges to u in Wlocs2,p2Ω, that is, for all φ∈Cc∞Ω, φum′⟶φu in Wlocs2,p2Ω. Suppose that φ∈Cc∞Ω and let K≔suppφ. By Lemma 121, we have
(133)φum′⇀φuinWs1,p1Ω.
So, by Theorem35,
(134)φum′⇀φuinWKs1,p1Ω.
LetΩ′ be an open bounded set in Ω with Lipschitz continuous boundary such that K⊆Ω′. By Theorem 85, the restriction map from WKs1,p1Ω to Ws1,p1Ω′ is well-defined and continuous. It follows from Theorem 34 that this restriction map is weak-weak continuous. So φum′⇀φu in WKs1,p1Ω implies that φum′⇀φu in Ws1,p1Ω′. By assumption the identity map from Ws1,p1Ω′ to Ws2,p2Ω′ is compact, so it follows from Fact 2 that φum′⟶φu in Ws2,p2Ω′ which subsequently implies φum′⟶φu in Ws2,p2Ω by Theorem 85.


## 6. Other Properties

The main results of this section do not appear to be in the literature in the generality appearing here, and they play a fundamental role in the study of the properties of differential operators between Sobolev spaces of sections of vector bundles on manifolds equipped with nonsmooth metric (see [15, 16]).Theorem 123.
LetΩ be a nonempty open set in ℝn, s≥1, and 1<p<∞. Then, u∈Wlocs,pΩ if and only if u∈LlocpΩ and for all 1≤i≤n, ∂u/∂xi∈Wlocs−1,pΩ.Proof.
(135)u∈Wlocs,pΩ⟺∀φ∈Cc∞Ωφu∈Ws,pΩ⟺Theorem62∀φ∈Cc∞Ωφu∈LpΩandforall1≤i≤n,∂φu∂xi∈Ws−1,pΩ.
Note that∂φu/∂xi=∂φ/∂xiu+φ∂u/∂xi. Since ∂φ/∂xiu∈Ws,pΩ↪Ws−1,pΩ, we have
(136)∂φu∂xi∈Ws−1,pΩ⟺φ∂u∂xi∈Ws−1,pΩ.
Therefore,(137)u∈Wlocs,pΩ⟺∀φ∈Cc∞Ωφu∈LpΩandforall1≤i≤n,φ∂u∂xi∈Ws−1,pΩ⟺u∈LlocpΩandforall1≤i≤n,∂u∂xi∈Wlocs−1,pΩ.Theorem 124.
LetΩ be a nonempty open set in ℝn, k∈ℕ, and 1<p<∞. Then, u∈Wlock,pΩ if and only if ∂αu∈LlocpΩ for all α≤k.Proof.
We prove the claim by induction onk. For k=1, we have
(138)u∈Wloc1,pΩ⟺Theorem123u∈LlocpΩ,∀1≤i≤n∂u∂xi∈LlocpΩ⟺∀α≤1∂αu∈LlocpΩ.
Now, suppose the claim holds fork=m. For k=m+1, we have
(139)u∈Wlocm+1,pΩ⟺Theorem123u∈LlocpΩ,∀1≤i≤n∂u∂xi∈Wlocm,pΩ⟺inductionhypothesisu∈LlocpΩ,∀1≤i≤n∀0≤α≤m∂α∂u∂xi∈LlocpΩ⟺u∈LlocpΩ,∀0<β≤m+1∂βu∈LlocpΩ⟺∀β≤m+1∂βu∈LlocpΩ.Theorem 125.
Lets∈ℝ, 1<p<∞, and α∈ℕ0n. Suppose Ω is a nonempty bounded open set in ℝn with Lipschitz continuous boundary. Then,
(1)
the linear operator∂α:Wlocs,pℝn⟶Wlocs−α,pℝn is well-defined and continuous(2)
fors<0, the linear operator ∂α:Wlocs,pΩ⟶Wlocs−α,pΩ is well-defined and continuous(3)
fors≥0 and α≤s, the linear operator ∂α:Wlocs,pΩ⟶Wlocs−α,pΩ is well-defined and continuous(4)
ifs≥0, s−1/p≠integer (i.e., the fractional part of s is not equal to 1/p), then the linear operator ∂α:Wlocs,pΩ⟶Wlocs−α,pΩ for α>s is well-defined and continuousProof.
This is the counterpart of Theorem86 for locally Sobolev functions. Here, we will prove the first item. The remaining items can be proved using a similar technique.
Step 1. First we prove by induction onα that if u∈Wlocs,pℝn, then ∂αu∈Wlocs−α,pℝn. Let φ∈Cc∞ℝn; we need to show that φ∂αu∈Ws−α,pℝn. If α=0, there is nothing to prove. If α=1, there exists 1≤i≤n such that ∂α=∂/∂xi. We have
(140)φ∂αu=φ∂u∂xi=∂φu∂xi−∂φ∂xiu.
By assumption,φu∈Ws,pℝn, and so it follows from Theorem 86 that the first term on the right hand side is in Ws−1,pℝn. Also, since u∈Wlocs,pℝn, the second term on the right hand side is in Ws,pℝn↪Ws−1,pℝn. Hence, φ∂αu∈Ws−1,pℝn. Now, suppose the claim holds for all α≤k. Suppose α is a multi-index such that α=k+1. Clearly, there exists 1≤i≤n such that ∂α=∂/∂xi∂β where β is a multi-index with β=k. By the induction hypothesis, ∂βu∈Wlocs−β,pℝn and so by the argument that was presented for the base case, we have ∂/∂xi∂βu∈Wlocs−β−1,pℝn=Wlocs−α,pℝn
Step 2. In this step, we prove the continuity. Again, we use induction onα. Let α=1. Choose i as in the previous step. For every φ∈Cc∞ℝn, we have
(141)φ∂∂xius−1,p=∂φu∂xi−∂φ∂xius−1,p≤∂φu∂xis−1,p+∂φ∂xius−1,p≼φus,p+∂φ∂xius,p.
On the right hand side, we have sum of two of the seminorms that define the topology ofWlocs,pℝn. It follows from item (2) of Theorem 20 that ∂α:Wlocs,pℝn⟶Wlocs−1,pℝn is continuous. Now, suppose the claim holds for all α≤k. Suppose α is a multi-index such that α=k+1. Clearly, there exists 1≤i≤n such that ∂α=∂/∂xi∂β where β is a multi-index with β=k. We have
(142)φ∂αus−α,p=φ∂∂xi∂βus−α,p≼argumentofthebasecaseφ∂βus−α+1,p+∂φ∂xi∂βus−α+1,p≼φ∂βus−β,p+∂φ∂xi∂βus−β,p≼inductionhypothesis;Theorem20maxφ1us,p,⋯,φkus,p+∂φ∂xi∂βus−β,p≼inductionhypothesis;Theorem20maxφ1us,p,⋯,φkus,p+maxψ1us,p,⋯,ψlus,p≼maxφ1us,p,⋯,φkus,p,ψ1us,p,⋯,ψlus,p
for someφ1,⋯,φk and ψ1,⋯,ψl in Cc∞ℝn. It follows from item (2) of Theorem 20 that ∂α:Wlocs,pℝn⟶Wlocs−α,pℝn is continuousNext, we want to establish a counterpart of Theorem76 for locally Sobolev-Slobodeckij spaces. To this end, first we state and prove a simple lemma.Lemma 126.
LetΩ be a nonempty open subset of ℝn. Suppose u:Ω⟶ℝ and u~:Ω⟶ℝ are such that u=u~a.e. If u~ is continuous, then suppu~⊆suppu.Proof by Contradiction.
Supposex∈suppu~\suppu. Since x belongs to the complement of suppu, which is an open set, there exists ε>0 such that Bεx⊆Ω and Bεx∩suppu=∅. Since x∈suppu~, there exists y∈Bε/4x such that u~y≠0. u~ is continuous, therefore there exists 0<δ<ε/4 such that u~z≠0 for all z∈Bδy⊆Bεx. But u=0 a.e. on Bεx. This contradicts the fact that u=u~a.e.Theorem 127.
LetΩ be a nonempty bounded open set in ℝn with Lipschitz continuous boundary or Ω=ℝn. Suppose u∈Wlocs,pΩ where sp>n. Then, u has a continuous version.Proof.
LetVjj∈ℕ0 and ψjj∈ℕ0 be as in Theorem 43. Note that u=∑jψju. For all j, ψju∈Ws,pΩ so by Theorem 76, there exists u~j∈CΩ such that ψju=u~j on Ω\Aj where Aj is a set of measure zero. Also by Lemma 126suppu~j⊆suppψj. Therefore for any x∈Ω, only finitely many of u~jx’s are nonzero. So we may define u~:Ω⟶ℝ by u~=∑ju~j. Clearly, u~=u on Ω\A where A=∪Aj (so A is a set of measure zero). Consequently u~=ua.e. It remains to show that u~:Ω⟶ℝ is indeed continuous. To this end, suppose am⟶a in Ω. We need to prove that u~am⟶u~a. Let ε>0 be such that Bεa¯⊆Ω. So Bεa intersects only finitely many of suppu~j’s; let us denote them by u~r1,⋯,u~rl. Also since am⟶a, there exists M such that for all m≥M, am∈Bεa. Hence,
(143)u~a=∑ju~ja=u~r1a+⋯+u~rla,∀m≥Mu~am=u~r1am+⋯+u~rlam.
Recall thatu~r1+⋯+u~rl is a finite sum of continuous functions and so it is continuous. Thus,
(144)limm⟶∞u~am=limm⟶∞u~r1+⋯+u~rlam=u~r1a+⋯+u~rla=u~a.Remark 128.
In the above proof, the only place we used the assumption ofΩ being Lipschitz was in applying Theorem 76. We can replace this assumption by the weaker assumption that Ω has the interior Lipschitz property. Then, since suppψju is compact, there exists Ω′ with Lipschitz boundary that contains suppψju. Then, by Theorem 85, ψju∈Ws,pΩ′ and so it has a continuous version u^j∈CΩ′. Since ψju=u^j almost everywhere on Ω′ and ψju=0 outside of the compact set suppψj, we can conclude that extΩ′,Ω0u^j is in CΩ and it is almost everywhere equal to ψju. We set u~j=extΩ′,Ω0u^j. The rest of the proof will be exactly the same as before.Theorem 129.
LetΩ=ℝn or Ω be a bounded open set in ℝn with Lipschitz continuous boundary. Suppose s1,s2,s∈ℝ and 1<p1,p2,p<∞ are such that
(145)Ws1,p1Ω×Ws2,p2Ω↪Ws,pΩ.
Then,(1)
Wlocs1,p1Ω×Wlocs2,p2Ω↪Wlocs,pΩ(2)
for allK∈KΩ, Wlocs1,p1Ω×WKs2,p2Ω↪Ws,pΩ. In particular, if f∈Wlocs1,p1Ω, then the mapping u↦fu is a well-defined continuous linear map from WKs2,p2Ω to Ws,pΩRemark 130.
In the above theorem, since the locally Sobolev spaces onΩ are metrizable, the continuity of the mapping
(146)Wlocs1,p1Ω×Wlocs2,p2Ω⟶Wlocs,pΩ,u,v↦uvcan be interpreted as follows: if ui⟶u in Wlocs1,p1Ω and vi⟶v in Wlocs2,p2Ω, then uivi⟶uv in Wlocs,pΩ. Also since WKs2,p2Ω is considered as a normed subspace of Ws2,p2Ω, we have a similar interpretation of the continuity of the mapping in item (2).Proof.
(1)
Supposeu∈Wlocs1,p1Ω and v∈Wlocs2,p2Ω. First, we show that uv is in Wlocs,pΩ. Clearly, the set A=φ2:φ∈Cc∞Ω is an admissible family of test functions. So in order to show that uv∈Wlocs,pΩ, it is enough to show that for all φ∈Cc∞Ω, φ2uv=φuφv is in Ws,pΩ. This is clearly true because φu∈Ws1,p1Ω, φv∈Ws2,p2Ω, and by assumption
(147)Ws1,p1Ω×Ws2,p2Ω↪Ws,pΩIn order to prove the continuity of the map u,v↦uv, suppose ui⟶u in Wlocs1,p1Ω and vi⟶v in Wlocs2,p2Ω. We need to show that uivi⟶uv in Wlocs,pΩ. That is, we need to prove that for all φ∈Cc∞Ω,
(148)φ2uivi⟶φ2uvinWs,pΩ.We have
(149)ui⟶uinWlocs1,p1Ω⇒φui⟶φuinWs1,p1Ω,vi⟶vinWlocs2,p2Ω⇒φvi⟶φvinWs2,p2Ω.By assumption, Ws1,p1Ω×Ws2,p2Ω↪Ws,pΩ, so
(150)φuiφvi⟶φuφvinWs,pΩ.(2)
Supposeu∈Wlocs1,p1Ω and v∈WKs2,p2Ω. First, we show that uv is in Ws,pΩ. To this end, let φ∈Cc∞Ω be such that φ=1 on a neighborhood containing K. We have
(151)uv=uφv=φu⏟∈Ws1,p1Ωv⏟∈Ws2,p2Ω∈Ws,pΩNow, we prove the continuity. Suppose ui⟶u in Wlocs1,p1Ω and vi⟶v in WKs2,p2Ω. Let φ be as before. We have
(152)ui⟶uinWlocs1,p1Ω⇒φui⟶φuinWs1,p1Ω,vi⟶vinWs2,p2Ω.This together with the assumption that Ws1,p1Ω×Ws2,p2Ω↪Ws,pΩ implies φuivi⟶φuv in Ws,pΩ. Since φv=v and φvi=vi, we conclude that uivi⟶uv in Ws,pΩ.Remark 131.
In the above theorem, the assumption thatΩ is Lipschitz or ℝn was used only to ensure that we can apply Theorem 103 and to make sure that the locally Sobolev spaces involved are metrizable. For item (1), we can use the weaker assumption that s1,p1,Ω, s2,p2,Ω, and s,p,Ω are interior smooth multiplication triples. For item (2), we just need to assume that s1,p1,Ω is an interior smooth multiplication triple.Corollary 132.
LetΩ be the same as the previous theorem. If sp>n, then Wlocs,pΩ is closed under multiplication. Moreover, if
(153)f1m⟶f1inWlocs,pΩ,⋯,flm⟶flinWlocs,pΩ,then
(154)f1m⋯flm⟶f1⋯flinWlocs,pΩ.The next theorem plays a key role in the study of differential operators on manifolds equipped with nonsmooth metrics (see [15]).Theorem 133.
LetΩ=ℝn or let Ω be a nonempty bounded open set in ℝn with Lipschitz continuous boundary. Let s∈ℝ and p∈1,∞ be such that sp>n. Let B:Ω⟶GLk,ℝ. Suppose for all x∈Ω and 1≤i,j≤k, Bijx∈Wlocs,pΩ. Then,
(1)
detB∈Wlocs,pΩ(2)
moreover if for eachm∈ℕBm:Ω⟶GLk,ℝ and for all 1≤i,j≤kBmij⟶Bij in Wlocs,pΩ, then detBm⟶detB in Wlocs,pΩProof.
(1)
By the Leibniz formula, we have(155)detBx=∑σ∈SnsgnσBσ1,1⋯Bσk,k.By assumption, for all 1≤i≤k, Bσi,i is in Wlocs,pΩ. Since sp>n, it follows from Corollary 132 that detB∈Wlocs,pΩ(2)
SinceBmij⟶Bij in Wlocs,pΩ, it again follows from Corollary 132 that for all σ∈Sn,
(156)Bmσ1,1⋯Bmσk,k⟶Bσ1,1⋯Bσk,kinWlocs,pΩ.Thus, detBm⟶detB in Wlocs,pΩTheorem 134.
LetΩ=ℝn or let Ω be a nonempty bounded open set in ℝn with the Lipschitz continuous boundary. Let s≥1 and p∈1,∞ be such that sp>n.
(1)
Suppose thatu∈Wlocs,pΩ and that ux∈I for all x∈Ω where I is some interval in ℝ. If F:I⟶ℝ is a smooth function, then Fu∈Wlocs,pΩ(2)
Suppose thatum⟶u in Wlocs,pΩ and that for all m≥1 and x∈Ω, umx,ux∈I where I is some open interval in ℝ. If F:ℝ⟶ℝ is a smooth function, then Fum⟶Fu in Wlocs,pΩ(3)
IfF:⟶ℝ is a smooth function, then the map taking u to Fu is continuous from Wlocs,pΩ to Wlocs,pΩProof.
The proof of part (1) generalizes the argument given in [33]. Let k=⌊s⌋. First, we show that Fu∈Wlock,pΩ. To this end, we fix a multi-index α=m≤k and we show that ∂αFu∈LlocpΩ (see Theorem 124).
It follows from the chain rule (and induction) that∂αFu is a sum of the terms of the form
(157)Flu∂β1u⋯∂βru,where l∈ℕ and ∑i=1rβi=m. It is a consequence of Theorem 129 that if s1,s2≥s3≥0 and s1+s2−s3>n/p, then Wlocs1,pΩ×Wlocs2,pΩ↪Wlocs3,pΩ. As a consequence,
(158)Wlocs−β1,pΩ×Wlocs−β2,pΩ↪Wlocs−β1−β2,pΩ,Wlocs−β1−β2,pΩ×Wlocs−β3,pΩ↪Wlocs−β1−β2−β3,pΩ,⋮Wlocs−β1−⋯−βr−1,pΩ×Wlocs−βr,pΩ↪Wlocs−β1−⋯−βr,pΩ=Wlocs−m,pΩ.
Considering this and the fact that∂βiu∈Wlocs−βi,pΩ, we have
(159)∂β1u⋯∂βru∈Wloct,pΩfor all 0≤t≤s−m. In particular, ∂β1u⋯∂βru∈Wloc0,pΩ=LlocpΩ. Also, since F is smooth and u is continuous, Flu∈Lloc∞Ω. Therefore,
(160)Flu∂β1u⋯∂βru∈LlocpΩ.
So,Fu∈Wlock,pΩ where k=⌊s⌋. Now, for noninteger s, we use a bootstrapping argument to show that Fu in fact belongs to Wlocs,pΩ.
F′ is smooth; therefore, F′u∈Wlock,pΩ. Also ∂u/∂xi∈Wlocs−1,pΩ (note that s−1≥0). By Theorem 129, we have
(161)Wlock,pΩ×Wlocs−1,pΩ↪Wloct−1,pΩprovided that
(162)k≥t−1≥0,s−1≥t−1≥0,k+s−1−t−1>np.
Therefore,∂/∂xiFu=F′u∂u/∂xi∈Wloct−1,pΩ for all 1≤t≤s such that t<k+s−n/p. Consequently, Fu∈Wloct,pΩ for all 1≤t≤s such that t<k+s−n/p (see Theorem 123). Now, we can repeat this argument by starting with “F′ is smooth, therefore F′u∈Wloct,pΩ for all 1≤t≤s such that t<k+s−n/p.” This results in Fu∈Wloct,pΩ for all 1≤t≤s such that t<k+2s−n/p. Repeating this, a finite number of times shows that Fu∈Wlocs,pΩ.
Our next goal is to prove items (2) and (3). First, we note that if0∈I then without loss of generality (WLOG) we may assume that F0=0. Indeed, the constant function F0 is an element of Wlocs,pΩ. So,
(163)Fum⟶FuinWlocs,pΩ⟺F~um⟶F~uinWlocs,pΩ,where F~t=Ft−F0. Thus WLOG we may assume that F0=0.
LetKjj∈ℕ0, Vjj∈ℕ0, and ψjj∈ℕ0 be as in Theorem 43. Clearly, ψj is an admissible family of functions. Therefore, in order to show that Fum⟶Fu in Wlocs,pΩ, it is enough to prove that
(164)∀r∈ℕ0ψrFum−Fu⟶0inWs,pΩasm⟶∞.
Letψr1,⋯,ψrk be those admissible test functions whose support intersects the support of ψr. So,
(165)∀x∈suppψr∑j∈ℕ0ψju=ψr1u+⋯+ψrku.
Consequently,(166)ψrFum−Fu=ψrFψr1um+⋯+ψrkum−ψrFψr1u+⋯+ψrku.
Sinceum⟶u in Wlocs,pΩ, for all 1≤i≤k, we have
(167)ψrium⟶ψriuinWs,pΩ,and so,
(168)ψr1um+⋯+ψrkum⟶ψr1u+⋯+ψrkuinWs,pΩ.
SinceWs,pΩ↪L∞Ω, we have
(169)ψr1um+⋯+ψrkum⟶ψr1u+⋯+ψrkuinL∞Ω.
Consequently, for the continuous representatives ofψr1um+⋯+ψrkum and ψr1u+⋯+ψrku, we have uniform convergence. From this point, we work with these continuous versions. The continuous function ψr1u+⋯+ψrku attains its max and min on the compact set suppψr which we denote by Amax and Amin, respectively. Note that
(170)∀x∈suppψrψr1u+⋯+ψrkux=ux∈I.
So,Amax and Amin are in I (that is, Amin,Amax⊆I). Let ε>0 be such that Amin−2ε,Amax+2ε⊆I. By (169) there exists M such that
(171)∀m≥M,∀x∈suppψrψr1um+⋯+ψrkumx∈Amin−ε,Amax+ε⊆I.
Letξ∈Cc∞ℝ be such that ξ=1 on Amin−ε,Amax+ε and ξ=0 outside of Amin−2ε,Amax+2ε⊆I. Define F^:ℝ⟶ℝ by
(172)F^t=ξtFt,ift∈I,0,ift∈I.
Clearly,F^:ℝ⟶ℝ is a smooth function and F^0=0. Moreover, F^=F on Amin−ε,Amax+ε. Also, for all x∈Ω and m≥M, we have
(173)ψrFum−Fu=ψrFψr1um+⋯+ψrkum−ψrFψr1u+⋯+ψrku=ψrF^ψr1um+⋯+ψrkum−ψrF^ψr1u+⋯+ψrku.
Indeed, ifx∈suppψr, then both sides are equal to zero. If x∈suppψr, then
(174)ψr1u+⋯+ψrkux∈Amin,Amax,ψr1um+⋯+ψrkumx∈Amin−ε,Amax+ε,and so,
(175)Fψr1u+⋯+ψrkux=F^ψr1u+⋯+ψrkux,Fψr1um+⋯+ψrkumx=F^ψr1um+⋯+ψrkumx.
F^ is a smooth function and its value at 0 is 0. Also, by assumption, sp>n. Therefore, the mapping v⟶ψrF^v from Ws,pΩ to Ws,pΩ is continuous. Hence,
(176)ψrF^ψr1um+⋯+ψrkum⟶ψrF^ψr1u+⋯+ψrkuinWs,pΩ.
That is,(177)ψrFum−Fu⟶0inWs,pΩ.
So, we proved item (2). Finally, we note thatWlocs,pΩ is metrizable. So continuity of the mapping u⟶Fu is equivalent to sequential continuity which was proved in item (2).


## 7. Conclusion

Sobolev-Slobodeckij spaces play a key role in the study of elliptic differential operators in nonsmooth setting. The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In the present paper, we provided a self-contained rigorous study of certain fundamental properties of locally Sobolev-Slobodeckij spaces. In particular, by introducing notions such as “smooth multiplication triple” and “interior smooth multiplication triple,” we rigorously studied completeness, separability, nature of the dual space, general embedding results, continuity of differentiation, and invariance under composition by smooth functions for locally Sobolev-Slobodeckij spaces.



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*Source: 9094502-2022-07-18.xml*