BASIC PROPERTIES OF SOBOLEV’S SPACES ON TIME SCALES

TIME SCALES

RAVI P. AGARWAL, VICTORIA OTERO–ESPINAR, KANISHKA PERERA, AND DOLORES R. VIVERO

Received 18 January 2006; Accepted 22 January 2006

We study the theory of Sobolev’s spaces of functions defined on a closed subinterval of an arbitrary time scale endowed with the LebesgueΔ-measure; analogous properties to that valid for Sobolev’s spaces of functions defined on an arbitrary open interval of the real numbers are derived.

Copyright © 2006 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Sobolev’s spaces are a fundamental tool in real analysis, for instance, in the use of vari- ational methods to solve boundary value problems in ordinary and partial differential equations and difference equations. In spite of this, theory for functions defined on an arbitrary bounded open interval of the real numbers is well known, see [2], and for func- tions defined on an arbitrary bounded subset of the natural numbers is trivial, as far as we know, for functions defined on an arbitrary time scale, it has not been studied before.

The aim of this paper is to give an introduction to Sobolev’s spaces of functions defined on a closed interval [a,b]∩Tof an arbitrary time scaleTendowed with the LebesgueΔ- measure. InSection 2, we gather together the concepts one needs to read this paper, such as theL^pspaces linked to the LebesgueΔ-measure and absolutely continuous functions on an arbitrary closed interval ofT. The most important part of this paper isSection 3 where we define the first-order Sobolev’s spaces as the space ofL_Δ^p([a,b)∩T) functions whose generalizedΔ-derivative belongs toL^p_Δ([a,b)∩T), moreover, we study some of their properties by establishing an equivalence between them and the usual Sobolev’s spaces defined on an open interval of the real numbers.Section 4is devoted to the gener- alization of Sobolev’s spaces to ordern≥2.

2. Preliminaries

The LebesgueΔ-measureμ_Δ was defined in [1, Section 5.7] or in [5, Section 5] as the Carath´eodory extension of a set function and it may be characterized in terms of

Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 38121, Pages1–14 DOI10.1155/ADE/2006/38121

well-known measures as the following result shows; we refer the reader to [6–8] for a broad introduction to measure and integration theory.

Proposition 2.1. The LebesgueΔ-measure is defined over the Lebesgue measurable subsets ofT; moreover, it satisfies the following equality:

μ_Δ=

⎧⎪

⎪⎨

⎪⎪

⎩ λ+

i∈I

σt_i−t_i·δ_ti+μ_M, if M∈T, λ+

i∈I

σt_i−t_i·δ_ti, if M∈T, (2.1)

where{ti}i∈I,I⊂N, is the set of all right-scattered points ofT,Mis the supremum ofT,λ is the Lebesgue measure,δti is the Dirac measure concentrate atti, andμM is a degenerate measure defined asμ_M(A)=0 ifM∈Aandμ_M(A)=+∞ifM∈A.

Proof. From properties of measure, one can deduce relation (2.1) for the outer measures

linked to these measures which plainly yields to (2.1).

As a straightforward consequence of equality (2.1), one can deduce the following for- mula to calculate the LebesgueΔ-integral; this formula was proved in [4], nevertheless, we remark that this argument is more simple than that.

Proposition 2.2. LetE⊂Tbe aΔ-measurable set. If f :T→RisΔ- integrable onE, then

Ef(s)Δs=

Ef(s)ds+

i∈IE

σt_i−t_i·ft_i+r(f,E), (2.2)

where

r(f,E)=

⎧⎨

⎩

μM(E)·f(M), ifM∈T,

0, ifM∈T, (2.3)

I_E:= {i∈I:t_i∈E}and{t_i}i∈I,I⊂N, is the set of all right-scattered points ofT.

Definition 2.3. LetA⊂T.Ais calledΔ-null set ifμ_Δ(A)=0. Say that a propertyPholds Δ-almost everywhere (Δ-a.e.) onA, or forΔ-almost all (Δ-a.a.)t∈Aif there is aΔ-null setE⊂Asuch thatPholds for allt∈A\E.

Definition 2.4. LetE⊂Tbe aΔ-measurable set and letp∈_R¯ _≡[−∞, +∞] be such that p≥1 and letf :E→_R¯ be aΔ-measurable function. Say thatf belongs toL_Δ^p(E) provided that either

E|f|^p(s)Δs <∞ if p∈R, (2.4) or there exists a constantC∈Rsuch that

|f| ≤C Δ-a.e. onE if p=+∞. (2.5) Note that equality (2.2) guarantees that in order for f :T→Rto belong toL_Δ^p(T), p∈R, andTbounded from above, it is necessary that f(M)=0. We will work with the

L_Δ^p(J^o) spaces, whereJ=[a,b]∩T,a,b∈T,a < b, is an arbitrary closed subinterval ofT andJ^o=[a,b)∩T; we state some of their properties whose proofs can be found in [6–8].

Theorem 2.5. Letp∈_R¯ be such thatp≥1. Then, the setL^p_Δ(J^o) is a Banach space together with the norm defined for every f ∈L_Δ^p(J^o) as

f_LΔ^p:=

⎧⎪

⎪⎨

⎪⎪

⎩ ^J

o|f|^p(s)Δs^{1/ p}, ifp∈R,

infC∈R:|f| ≤CΔ-a.e. onJ^o, ifp=+∞.

(2.6)

Moreover,L²_Δ(J^o) is a Hilbert space together with the inner product given for every (f,g)∈ L²_Δ(J^o)×L²_Δ(J^o) by

(f,g)_L²_Δ:=

J^of(s)·g(s)Δs. (2.7)

Proposition 2.6. Supposep∈_R¯ andp≥1. Letp∈_R¯ be such that 1/ p+ 1/ p=1.

Then, if f ∈L_Δ^p(J^o) andg∈L_Δ^p(J^o), then f·g∈L¹_Δ(J^o) and f·gL¹_Δ≤ f_LΔ^p· g_L^p

Δ. (2.8)

This expression is called H¨older’s inequality and Cauchy-Schwarz’s inequality whenever p=2.

Proposition 2.7. Ifp∈Randp≥1, then, the setCc(J^o) of all continuous functions onJ^o with compact support inJ^ois dense inL_Δ^p(J^o).

As a consequence ofProposition 2.2, one can establish the following equivalence be- tween theL_Δ^p(J^o) spaces and the usualL^p([a,b]) spaces linked to the Lebesgue measure.

Corollary 2.8. Letp∈_R¯ withp≥1, let f :J→_R¯, and letf: [a,b]→_R¯ be the extension of f to [a,b] defined as

f(t) :=

⎧⎨

⎩

f(t), ift∈J, f(ti), ift∈

ti,σti

, for some i∈IJ, (2.9) withIJ:= {i∈I:ti∈J}and{ti}i∈I,I⊂N, is the set of all right-scattered points ofT.

Then, f ∈L_Δ^p(J^o) if and only if f ∈L^p([a,b]). In this case,

f_L^p_Δ= f_Lp. (2.10)

As we know from general theory of Sobolev’s spaces, another important class of func- tions is just the absolutely continuous functions.

Definition 2.9. A function f :J→Ris said to be absolutely continuous onJ, f ∈AC(J), if for every ε >0, there exists aδ >0 such that if{[a_k,b_k)∩T}ⁿ_k=1, witha_k,b_k∈J, is a finite pairwise disjoint family of subintervals ofJ satisfyingⁿ_k=1(bk−ak)< δ, then _n

k=1|f(bk)−f(ak)|< ε.

These functions are precisely that for which the fundamental theorem of Calculus holds.

Theorem 2.10 [3, Theorem 4.1]. A function f :J→Ris absolutely continuous onJif and only if f isΔ-differentiableΔ-a.e. onJ^o, f^Δ∈L¹_Δ(J^o) and

f(t)= f(a) +

[a,t)∩Tf^Δ(s)Δs, ∀t∈J. (2.11)

Absolutely continuous functions onTverify the integration by parts formula.

Theorem 2.11. If f,g:J→Rare absolutely continuous functions onJ, then f ·gis abso- lutely continuous onJand the following equality is valid:

J^o

f^Δg+f^σg^Δ(s)Δs= f(b)g(b)−f(a)g(a)=

J^o

f g^Δ+ f^Δg^σ(s)Δs. (2.12) They are linked to the class of absolutely continuous functions on [a,b] as the follow- ing property shows.

Corollary 2.12 [3, Corollary 3.1]. Assume that f :J→Rand define ¯f : [a,b]→Ras f¯(t) :=

⎧⎪

⎪⎨

⎪⎪

⎩

f(t), ift∈J,

fti

+ fσt_i−ft_i σti

−ti

t−ti

, ift∈ ti,σti

, for some i∈IJ, (2.13) withIJ:= {i∈I:ti∈J}and{ti}i∈I,I⊂N, is the set of all right-scattered points ofT.

Then, f is absolutely continuous onJif and only if ¯f is absolutely continuous on [a,b].

Moreover, for everyn∈N,n≥1, we will denote as ACⁿ(J) :=

x∈AC(J) :x^Δj∈ACJ^κj∀j∈ {1,. . .,n}

, (2.14)

where for every j∈N,j≥1,J^κj=[a,ρ^j(b)]∩T. 3. First-order Sobolev’s spaces

The aim of this section is to study the first-order Sobolev’s spaces onJequipped with the LebesgueΔ-measure.

Definition 3.1. Letp∈_R¯ be such thatp≥1 andu:J→_R¯. Say thatubelongs toW_Δ^1,p(J) if and only ifu∈L_Δ^p(J^o) and there existsg:J^κ→_R¯ such thatg∈L_Δ^p(J^o) and

J^o

u·ϕ^Δ(s)Δs= −

J^o

g·ϕ^σ(s)Δs ∀ϕ∈C_0,rd¹ J^κ (3.1) with

C_0,rd¹ J^κ:=

f :J−→R: f ∈C_rd¹J^κ, f(a)=0= f(b) (3.2) andC¹_rd(J^κ) is the set of all continuous functions onJsuch that they areΔ-differentiable onJ^κand theirΔ-derivatives arerd-continuous onJ^κ.

The integration by parts formula for absolutely continuous functions onJestablishes that the relation

V_Δ^1,p(J) :=

x∈AC(J) : x^Δ∈L_Δ^pJ^o⊂W_Δ^1,p(J) (3.3) is true for everyp∈_R¯ withp≥1. We will show that both sets are, as class of functions, equivalent; for this purpose, we need the following lemmas.

Lemma 3.2. Let f ∈L¹_Δ(J^o) be such that the following equality is true:

J^o(f ·u)(s)Δs=0, ∀u∈Cc

J^o, (3.4)

then

f ≡0 Δ-a.e. onJ^o. (3.5) Proof. Fixε >0, the density ofC_c(J^o) inL¹_Δ(J^o) guarantees the existence of f1∈C_c(J^o) such thatf −f1_L¹_Δ< ε, and so, by (3.4), we deduce that for everyu∈Cc(J^o), it is true

that

J^o

f1·u(s)Δs≤ uC(J^o)·f −f1

L¹_Δ< εuC(J^o). (3.6) Because the sets

A1:=

s∈J^o: f1(s)≥ε, A2:=

s∈J^o:f1(s)≤ −ε (3.7) are compact and disjoint subsets ofJ^o, Urysohn’s lemma allows to construct a function u0:J^o→Rwhich belongs toCc(J^o) and it verifies

u0≡

⎧⎨

⎩

1; onA1,

−1; onA2,

u0≤1 on J^o; (3.8)

so that, by definingA:=A1∪A2, we have that

J^o

f1(s)Δs=

J^o

f1·u0

(s)Δs−

J^o\A

f1·u0

(s)Δs + J^o\A

f1(s)Δs≤ε+ 2ε(b−a).

(3.9)

As a consequence of the arbitrary choice ofε >0, we achieve (3.5).

Lemma 3.3. Letf ∈L¹_Δ(J^o). Then, a necessary and sufficient condition for the validity of the equality

J^o

f·ϕ^Δ(s)Δs=0, for every ϕ∈C_0,rd¹ J^κ, (3.10) is the existence of a constantc∈Rsuch that

f ≡c Δ-a.e. on J^o. (3.11)

Proof. The necessary condition is consequence of the fundamental theorem of Calculus.

Conversely, fixu∈Cc(J^o) arbitrary; by definingh,ϕ:J→Ras

h(t) :=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ u(t)−

J^ou(r)Δr

b−a , ift∈J^o,

−

J^ou(r)Δr

b−a , ift=b, ϕ(t) :=

[a,t)∩Th(s)Δs, ∀t∈J,

(3.12)

the fundamental theorem of Calculus establishes thatϕ∈C¹_0,rd(J^κ) and so, equality (3.10) yields to

0=

J^o

f·

u−

J^ou(r)Δr b−a

(s)Δs

= J^o

f−

J^of(r)Δr b−a

·u

(s)Δs.

(3.13)

Therefore,Lemma 3.2allows to deduce (3.11) withc=

J^of(r)Δr/(b−a).

Now, we are able to prove the characterization of functions inW_Δ^1,p(J) in terms of functions inV_Δ^1,p(J).

Theorem 3.4. Suppose thatu∈W_Δ^1,p(J) for some p∈_R¯ with p≥1 and that (3.1) holds forg∈L^p_Δ(J^o). Then, there exists a unique functionx∈V_Δ^1,p(J) such that the equalities

x=u, x^Δ=g Δ-a.e. on J^o (3.14)

are satisfied.

Moreover, ifg∈C_rd(J^κ), then there exists a unique functionx∈C¹_rd(J^κ) such that x₌u Δ-a.e. onJ^o, x^Δ₌g on J^κ. (3.15) Proof. Definev:J→Ras

v(t) :=

[a,t)∩Tg(s)Δs, ∀t∈J; (3.16)

the fundamental theorem of Calculus guarantees thatv∈V_Δ^1,p(J) and by the integration by parts formula, we have that for everyϕ∈C¹_0,rd(J^κ),

J^o

(v−u)·ϕ^Δ(s)Δs= −

J^o

v^Δ−g·ϕ^σ(s)Δs=0; (3.17)

so that,Lemma 3.3ensures the existence of a constantc∈Rsuch thatv−u≡cΔ-almost everywhere onJ^o. As a consequence of the fundamental theorem of Calculus we conclude that functionx:J→Rdefined asx(t) :=v(t)−cfor allt∈J is the unique function in V_Δ^1,p(J) for which (3.14) is valid.

Furthermore, ifg ∈Crd(J^κ), then the fundamental theorem of Calculus establishes

thatx∈C_rd¹(J^κ) andx^Δ=gonJ^κ.

By identifying every function inW_Δ^1,p(J) with its absolutely continuous representative inV_Δ^1,p(J) for which (3.14) holds, the setW_Δ^1,p(J) can be endowed with the structure of Banach space.

Theorem 3.5. Assumep∈_R¯ andp≥1. The setW_Δ^1,p(J) is a Banach space together with the norm defined for everyx∈W_Δ^1,p(J) as

x_W^1,p

Δ := x_LΔ^p+x^Δ_L^p

Δ. (3.18)

Moreover, the setH_Δ¹(J) :=W_Δ^1,2(J) is a Hilbert space together with the inner product given for every (x,y)∈H_Δ¹(J)×H_Δ¹(J) by

(x,y)_H¹_Δ:=(x,y)_L²_Δ+x^Δ,y^Δ_L²_Δ. (3.19) Proof. Let{x_n}n∈Nbe a Cauchy sequence inW_Δ^1,p(J);Theorem 2.5guarantees the exis- tence ofu,g∈L^p_Δ(J^o) such that{xn}n∈Nand{x_n^Δ}n∈Nconverge strongly inL_Δ^p(J^o) touand g, respectively, and so, by taking limits in the equality

J^o

x_n·ϕ^Δ(s)Δs= −

J^o

x^Δ_n·ϕ^σ(s)Δs, ϕ∈C_0,rd¹ (J^κ), (3.20)

we conclude thatu∈W_Δ^1,p(J). Thereby, it follows fromTheorem 3.4, that there exists x∈W_Δ^1,p(J) such that{x_n}n∈Nconverges strongly inW_Δ^1,p(J) tox.

3.1. Some properties. We will derive some properties of the Banach spaceW_Δ^1,p(J); the first one asserts thatW_Δ^1,p(J) is continuously inmersed intoC(J) equipped with the supre- mum norm · C(J).

Proposition 3.6. Assume p∈_R¯ with p≥1, then there exists a constantK >0, only de- pendent onb−a, such that the inequality

xC(J)≤K· x_WΔ^1,p (3.21)

holds for allx∈W_Δ^1,p(J) and hence, the immersionW_Δ^1,p(J)C(J) is continuous.

Proof. Fixx∈W_Δ^1,p(J). Let t,T ∈J be such that |x(t)|:=min_s∈T|x(s)| and |x(T)|:= maxs∈T|x(s)|; there is no harm in assumingt≤T. The fundamental theorem of Calculus and H¨older’s inequality lead to

xC(J)≤ |x(t)|+

[t,T)∩T|x^Δ|(s)Δs≤K· x_W^1,p

Δ , (3.22)

for someK >0, only dependent onb−a.

The strong compactness criterion inC(J) andProposition 3.6allow to prove the fol- lowing compactness property inC(J).

Proposition 3.7. Letp∈_R¯ be such thatp≥1. Then, the following statements are true.

(1) Ifp >1, then the immersionW_Δ^1,p(J)C(J) is compact.

(2) Ifp=1, then the immersionW_Δ^1,p(J)C(J) is compact if and only if every point of Jis isolated.

Proof. Denote byᏲ^pthe closed unit ball inW_Δ^1,p(J); we know fromTheorem 3.4thatᏲ^p is closed and bounded inC(J).

If p >1, then the fundamental theorem of Calculus and H¨older’s inequality ensure thatᏲ^pis equicontinuous.

On the other hand, ifp=1, then it is clear thatᏲ^pis equicontinuous whenever every point ofJis isolated, while if there existst0∈Tsuch thatt0is not isolated, then we will prove thatᏲ^pis not equicontinuous.

LetS:=1/(b−a+ 1), letδ >0 be arbitrary and letsδ∈(t0−δ,t0+δ)∩Tbe such that sδ=t0; it is not a loss of generality assumingsδ< t0.

Define f_δ:J→Ras

f_δ:=

⎧⎪

⎨

⎪⎩ S

t0−s_δ, ift∈ sδ,t0

∩J,

0, ift∈

s_δ,t0

∩J;

(3.23) the fundamental theorem of Calculus asserts thatF_δ:J→Rgiven by

Fδ(t) :=

[a,t)∩Tfδ(s)Δs, t∈J, (3.24)

belongs toᏲ^p; so that, as F_δt0

−F_δs_δ=

[sδ,t0)∩Tf_δ(s)Δs=S, (3.25) we conclude thatᏲ^pis not equicontinuous.

Therefore, Arzel`a-Ascoli theorem establishes our claims.

As a consequence ofProposition 3.6, we achieve the following sufficient condition for strong convergence inC(J).

Corollary 3.8. Letp∈_R¯ be such thatp >1, let{xm}m∈N⊂W_Δ^1,p(J), and letx∈W_Δ^1,p(J).

If {xm}m∈N converges weakly in W_Δ^1,p(J) tox, then{xm}m∈N converges strongly in C(J) tox.

Proof. Suppose{x_m}m∈N converges weakly inW_Δ^1,p(J) to x;Proposition 3.6establishes that {xm}m∈N converges weakly in C(J) to x and so, as {xm}m∈N is equicontinuous,

{xm}m∈Nconverges strongly inC(J) tox.

Moreover, Proposition 3.6 allows to deduce the following equivalence between the Sobolev’s spaces onJ,W_Δ^1,p(J), and the usual Sobolev’s spaces on (a,b),W^1,p((a,b)).

Corollary 3.9. Suppose thatp∈_R¯ and p≥1,x:J→Rand ¯x: [a,b]→Ris the exten- sion ofxto [a,b] defined in (2.13). Then,xbelongs toW_Δ^1,p(J) if and only if ¯xbelongs to W^1,p((a,b)).

Moreover, there exist two constantsK1,K2>0 which only depend on (b−a) such that the inequalities

K1· x¯W^1,p≤ x_WΔ^1,p≤K2· x¯W^1,p (3.26) are satisfied for everyx∈W_Δ^1,p(J) andp∈_R¯ withp≥1.

Proof. Let ¯x,x^Δ: [a,b]→Rbe the extensions ofxandx^Δto [a,b] defined in (2.13) and (2.9), respectively; it is not difficult to deduce the following equality:

x^Δ=x¯ a.e. on [a,b]. (3.27)

Therefore, Corollaries2.8and2.12andProposition 3.6yield to the result.

As an application of the previous result, we will prove that some properties known forW^1,p((a,b)) are directly transferred toW_Δ^1,p(J); in order to do this, we will use the following result.

Proposition 3.10. If y: [a,b]→Rbelongs toW^1,p((a,b)) for some p∈_R¯ with p≥1, theny_|J belongs toW_Δ^1,p(J). Moreover, there exists a constantT >0 which only depends on (b−a) such that

y_|JW^1,p

Δ ≤T· y_W1,p, ∀y∈W^1,p(a,b), p∈_R¯, p≥1. (3.28) Proof. LetR= {ti}i∈I,I⊂N, be the set of all right-scattered points ofT, letIJ^o= {i∈I, t_i∈J^o}and suppose y∈W^1,p((a,b)) for some p∈_R¯ withp≥1. The classical funda- mental theorem of Calculus allows to assert that

y_|JΔ ti

=

[ti,σ(ti)]y(s)ds σti

−ti , for every i∈IJ^o, y_|J

_Δ

=y a.e. on J^o∩(T\R).

(3.29)

Therefore, if p=+∞, then it is clear thaty_|J ∈W_Δ^1,p(J) and (3.28) holds while ifp∈R, then, by (2.2), we have that

y_|J^Δ_L^pp

Δ≤

J^o∩(_T\R)

y^p(s)ds+

i∈IJo [ti,σ(ti)]

y^p(s)ds≤ y_W^p1,p, (3.30)

moreover, as we know that y_|JL^p

Δ≤(b−a)^{1/ p}· y_C([a,b])≤C·(b−a)^{1/ p}· y_W1,p, (3.31) for someC >0, it turns out thaty_|J∈W_Δ^1,p(J) and (3.28) is true.

Next, we deduce some properties inW_Δ^1,p(J) from the analogous ones inW^1,p((a,b)).

Corollary 3.11. Letp∈_R¯ be such thatp≥1. Then, for everyq∈[1, +∞), the inmersion W_Δ^1,p(J)L^q_Δ(J^o) is compact.

Proof. Fixq∈[1, +∞); as a consequence ofProposition 3.7and the fact that the inmer- sionC(J)L^q_Δ(J^o) is continuous, it only remains to prove thatᏲ¹is compact inL^q_Δ(J^o) wheneverJhas at least one not isolated point.

Assume the existence of a not isolated point t0∈J and let {xn}n∈N be a sequence inᏲ¹.Corollary 3.9ensures that{xn}n∈N, defined in (2.13), is a bounded sequence in W^1,1((a,b)) and hence, there exist{x_nk}_k∈N andy∈L^q([a,b]) such that{x_nk}_k∈Ncon- verges strongly inL^q([a,b]) to y. By defining x:=y_|J, it is not difficult to prove that

{xnk}_k∈Nconverges strongly inL^q_Δ(J^o) tox.

Corollary 3.12. The Banach spaceW_Δ^1,p(J) is reflexive for everyp∈(1, +∞) and separable for allp∈[1, +∞).

Proof. Letp∈_R¯ be such thatp≥1. We know, fromCorollary 3.9, that the operatorTp: W_Δ^1,p(J)→W^1,p((a,b)) given for everyx∈W_Δ^1,p(J) byTp(x) :=x, defined in (2.13), is lin-¯ ear and continuous. It follows fromCorollary 3.9andProposition 3.10thatT_p(W_Δ^1,p(J)) is a closed subspace ofW^1,p((a,b)). Therefore, sinceW^1,p((a,b)) is reflexive whenever p∈(1, +∞) and separable wheneverp∈[1, +∞),T_p(W_Δ^1,p(J)) satisfies the same proper-

ties.

Corollary 3.13. Ifx∈W_Δ^1,p(J) for somep∈[1, +∞), then there exists a sequence of in- finitely differentiable functions with compact support inR,{yn}_n∈N such that{yn_|J}_n∈N converges strongly inW_Δ^1,p(J) tox.

Proof. Corollary 3.9asserts that ¯x: [a,b]→R, defined in (2.13), belongs toW^1,p((a,b));

so that, there exists a sequence{yn}_n∈Nof infinitely differentiable functions with compact support in Rsuch that {y_n|[a,b]}_n∈N converges to ¯x in W^1,p((a,b)). Hence, our claim

follows from equality ¯x_|J =xandProposition 3.10.

3.2. The spacesW_0,Δ^1,p(J). Corollary 3.13 guarantees the density of the set C_rd¹ (J^κ) in W_Δ^1,p(J) for everyp∈[1, +∞); however, for an arbitrary bounded time scale it is not true that the set of test functions defined in (3.2),C_0,rd¹ (J^κ), is dense inW_Δ^1,p(J); this section is devoted to prove some properties concerning the closure ofC_0,rd¹ (J^κ) inW_Δ^1,p(J).

Definition 3.14. Letp∈Rbe such thatp≥1, define the setW_0,Δ^1,p(J) as the closure of the setC¹_0,rd(J^κ) inW_Δ^1,p(J). Denote asH_0,¹_Δ(J) :=W_0,^1,2_Δ(J).

The spacesW_0,Δ^1,p(J) andH_0,Δ¹ (J) are endowed with the norm induced by · _WΔ^1,p, de- fined in (3.18), and the inner product induced by (·,·)_H¹_Δ, defined in (3.19), respectively.

SinceW_0,^1,p_Δ(J) is closed inW_Δ^1,p(J),Theorem 3.5andCorollary 3.12ensure thatW_0,^1,p_Δ(J) is a separable Banach space and reflexive wheneverp >1 andH_0,Δ¹ (J) is a separable Hilbert space. The spaceW_0,Δ^1,p(J) is characterized in the following result.

Proposition 3.15. Assumex∈W_Δ^1,p(J). Then,x∈W_0,Δ^1,p(J) if and only ifx(a)=0=x(b).

Proof. Firstly, suppose thatx∈W_0,^1,p_Δ(J), so that there exists a sequence{x_n}n∈N⊂C¹_0,rd(J^κ) such that{xn}n∈Nconverges strongly inW_Δ^1,p(J) tox. Therefore, inequality (3.21) allows to assert thatx(a)=0=x(b).

Conversely, assume thatx(a)=0=x(b). We know fromCorollary 3.9that ¯x: [a,b]→ R, defined in (2.13), belongs to W₀^1,p((a,b)) and so, there exists a sequence{y_n}_n∈N⊂ C¹_c((a,b)) which converges strongly inW^1,p((a,b)) to ¯x. By definingx_n:=y_n|J,n∈N, one can deduce thatx_n∈C¹_0,rd(J^κ) for everyn∈N and{x_n}n∈N converges strongly in

W_Δ^1,p(J) tox.

As a straightforward consequence of the previous result,Corollary 3.9, and the char- acterization ofW₀^1,p((a,b)) we obtain the following criterion for belonging toW_0,Δ^1,p(T).

Corollary 3.16. Letp∈Rbe such thatp≥1, letx:J→R, and let ¯x: [a,b]→Rbe the extension ofxto [a,b] defined in (2.13). Then,x∈W_0,Δ^1,p(J) if and only if ¯x∈W0^1,p((a,b)).

By usingProposition 3.15, we are able to prove the validity of Poincar´e’s inequality.

Proposition 3.17. Letp∈Rbe such thatp≥1. Then, there exists a constantL >0, only dependent on (b−a), such that

x_WΔ^1,p≤L·x^Δ_L^p

Δ, ∀x∈W_0,Δ^1,p(J), (3.32) that is, inW_0,Δ^1,p(J), the norm defined for everyx∈W_0,Δ^1,p(J) asx^Δ_LΔ^p is equivalent to the norm · _W^1,p

Δ .

Proof. Choosex∈W_0,Δ^1,p(J); the fundamental theorem of Calculus andProposition 3.15 allow to assert that the following inequality

x(t)= x(a) +

[a,t)∩Tx^Δ(s)Δs=

[a,t)∩Tx^Δ(s)Δs≤x^Δ_L¹_Δ (3.33) is valid for everyt∈T. Thus, (3.32) follows from H¨older’s inequality.

Remark 3.18. One can check that the function defined for everyx,y∈H_0,¹_Δ(J) as(x^Δ,y^Δ)_L²_Δ is an inner product inH_0,¹_Δ(J) and its associated norm is equivalent to the norm associated to (·,·)_HΔ¹.

4. Generalization to ordern≥2

The aim of this section is to define recursively thenth-order Sobolev’s spaces onJ for n≥2,W_Δ^n,p(J), which consist in theΔ-antiderivatives of functions inW_Δ^n−1,p(J^κ).

Definition 4.1. Letn∈N,n≥2, letp∈_R¯,p≥1, and letu:J→_R¯. Say thatubelongs to W_Δ^n,p(J) if and only ifu∈W_Δ^n−1,p(J) and there existsg1:J^κ→Rsuch thatg1∈W_Δ^n−1,p(J^κ) and

J^o

u·ϕ^Δ(s)Δs= −

J^o

g1·ϕ^σ(s)Δs, ∀ϕ∈C_0,rd¹ J^κ. (4.1)