# Fractional Sobolev Spaces via Riemann-Liouville Derivatives

Authors: Dariusz Idczak; Stanisław Walczak
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Using Riemann-Liouville derivatives, we introduce fractional Sobolev
spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones. Next, we
prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and compactness of
some imbeddings. An application to boundary value problems is given as well.


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## Body

## 1. Introduction

Let1≤p<∞. Classical Sobolev space W1,p=W1,p((a,b),ℝm) of order one on an open bounded interval (a,b)⊂ℝ is defined by (cf. [1])
(1)W1,p={u∈Lp;∃g∈Lp∀φ∈Cc∞∫ab‍u(t)φ(1)(t)dt=-∫ab‍g(t)φ(t)dt},
where Lp=Lp((a,b),ℝm) is the space of functions g:(a,b)→ℝm that are integrable with power p, Cc∞=Cc∞((a,b),ℝm) is the set of smooth functions φ:(a,b)→ℝm with compact support suppφ⊂(a,b), and φ(1) is the classical derivative of φ. The function g satisfying the above condition is denoted by u′ and called the weak derivative of u of order one.Sobolev spaceWn,p=Wn,p((a,b),ℝm) of order n>1 is defined by (cf. [1])
(2)Wn,p={u∈Wn-1,p;u′∈Wn-1,p}.
The spaces Wn,p can be characterized as follows.Theorem 1.
Letn∈ℕ. Then, u∈Wn,p if and only if there exist functions g1,…,gn∈Lp such that
(3)∫ab‍u(t)φ(j)(t)dt=(-1)j∫ab‍gj(t)φ(t)dt
for φ∈Cc∞ and j=1,…,n, where φ(j) denotes the classical derivative of φ of order j. In such a case there exists an absolutely continuous function u~:[a,b]→ℝm such that u=u~ a.e. on (a,b), which has absolutely continuous classical derivatives u(1),…,u(n-1), the derivative u(n)=(u(n-1))(1)∈Lp, and
(4)g1=u(1),…,gn=u(n).Remark 2.
One shows thatg1=u′, g2=(u′)′,…, gn=(⋯(u′)′⋯)′ (n times).Remark 3.
In our paper, we will identify functions defined on(a,b) and [a,b] ((a,b], [a,b), resp.) that are equal a.e. on (a,b).Each of the functionsgj, j=1,…,n, is denoted by Dju and called the weak derivative of u of order j.The spaceWn,p endowed with a norm
(5)∥u∥Wn,pp=∑i=0n‍∥Diu∥Lpp,u∈Wn,p,
where D0u=u, has many useful properties such as completeness, reflexivity (for 1<p<∞), and separability. Moreover, some imbeddings of these spaces are compact (cf. [1]).In the last years, many papers and books on fractional calculus and its applications have appeared. Most of them concern fractional differential equations, including calculus of variations and optimal control. In the classical (positive integer) case the fundamental role in this field is played by the mentioned Sobolev spaces. To our best knowledge, there are no “fractional” Sobolev spaces based on the notion of fractional derivative in Riemann-Liouville sense, which seems to be the most used in the theory of fractional differential equations. Our aim is to give some systematic basics for applications of fractional calculus to differential equations. More precisely, we extend the above definitions, with the aid of the Riemann-Liouville derivatives, to the case of noninteger positive (fractional) orderα, derive a fractional counterpart of Theorem 1, and prove the basic properties of the introduced spaces. When α=n∈ℕ, the obtained results reduce to the classical ones. In the literature, some generalizations of Sobolev spaces to noninteger orders, on a domain Ω⊂ℝn, are known (cf. [2]): Gagliardo spaces Wα,p(Ω), Besov spaces Bα,p(Ω), and Nikolskii spaces Hα,p(Ω). They have been introduced with the aid of approaches different from ours and their comparison with our spaces (in the case of Ω=(a,b)) is an open problem. Let us point that only Gagliardo spaces coincide with the classical Sobolev spaces when α=n.The paper is organized as follows. In the second section, we recall some basic notions and facts from the fractional calculus including a characterization of functions possessing the left (right) Riemann-Liouville derivatives. In the third section, we derive some special cases of the fractional theorem on the integration by parts. In the fourth section, we define the fractional Sobolev spaces of any orderα>0 and characterize them. In the fifth section, we derive a fractional counterpart of Theorem 1, define the weak fractional derivatives of order α>0, and show that they coincide with the Riemann-Liouville derivatives. In the sixth section, we introduce two norms in the fractional Sobolev spaces and prove their equivalence. In the seventh section, we derive completeness, reflexivity, and separability of the introduced spaces. In the eighth section, we prove compactness of some imbeddings. In the ninth section, we present some applications of the obtained results to fractional boundary value problems using a variational approach.In the paper, we limit ourselves to the left fractional Sobolev spaces, but in an analogous way one can define right spaces and derive their appropriate properties.


## 2. Preliminaries

ByACn,1=ACn,1([a,b],ℝm), where n∈ℕ, we denote the set of all functions f:[a,b]→ℝm that have a representative (a.e. on [a,b]) which is absolutely continuous together with its classical derivatives of orders 1,…,n-1. Of course, such a function possesses also the classical derivative of order n, existing a.e. on [a,b] and belonging to L1. These classical derivatives are the weak derivatives D1f,…,Dnf of f. It is known (cf. [3, Lemma 2.4]) that f∈ACn,1 if and only if there exist c0,c1,…,cn-1∈ℝm and φ∈L1 such that
(6)f(t)=∑i=0n-1‍cii!(t-a)i+∫at‍∫at1‍⋯∫atn-1‍φ(τ)dτdtn-1⋯dt1,t∈[a,b]a.e.
In such a case,
(7)Dif(a)=ci,i=0,…,n-1,Dnf(t)=φ(t),t∈[a,b]a.e.ByACn,p=ACn,p([a,b],ℝm) we denote the set of all functions f:[a,b]→ℝm that have representation (6) with c0,c1,…,cn-1∈ℝm and φ∈Lp. It is known that ACn,p=Wn,p for 1≤p<∞.Letα>0, and let h∈L1. By the left Riemann-Liouville fractional integral of h on the interval [a,b] we mean (cf. [3]) a function Ia+αh given by
(8)(Ia+αh)(t)=1Γ(α)∫at‍h(τ)(t-τ)1-αdτ,t∈[a,b]a.e.Remark 4.
The above integral exists and is finite a.e. on[a,b].In view of the convergence (cf. [3, Theorem 2.7])
(9)limα→0+(Ia+αh)(t)=h(t),t∈[a,b]a.e.,
it is natural to put
(10)(Ia+0h)(t)=h(t),t∈[a,b]a.e.Remark 5.
Ifα≥1, the right side of (8) exists (and is finite) everywhere on [a,b]. So, Ia+αh can be defined everywhere on [a,b]. In such a case it is continuous on [a,b].Remark 6.
It is easy to see that ifh is essentially bounded on [a,b] and 0<α<1, then the right side of (8) is defined and bounded everywhere on (a,b]. So, in this case Ia+αh can be defined everywhere on (a,b]. From [3, Theorem 3.6] it follows that in such a case Ia+αh is equal everywhere on (a,b] to a continuous function on [a,b]. It is also known (cf. [3, Theorem 2.6]) that if h∈Lp with 1≤p<∞, then Ia+αh∈Lp.Letn-1<α≤n, where n∈ℕ. By ACa+α,1=ACa+α,1([a,b],ℝm) we denote the set of all functions f:[a,b]→ℝm that have the representation
(11)f(t)=∑i=0n-1‍ciΓ(α-n+1+i)(t-a)α-n+i+Ia+αφ(t),t∈[a,b]a.e.,
with c0,c1,…,cn-1∈ℝm and φ∈L1. Of course, ACa+n,1=ACn,1.We say thatf∈L1 possesses the left Riemann-Liouville derivative Da+αf of order α∈(n-1,n], n∈ℕ, on the interval [a,b] if Ia+n-αf∈ACn,1. By this derivative we mean the derivative Dn(Ia+n-αf). Of course, Da+nf=Dnf.The next theorem can be deduced from [3, Corollary 2.1, Lemma 2.5 (b), Lemma 2.6 (b)] but, to our best knowledge, it has not been formulated by other authors. In [4] we give a direct proof of it in the case of n=1, in [5]—when n≥2. In the case of α=n it reduces to the theorem on the integral representation of type (6) of functions belonging to ACn,1.Theorem 7.
Ifn-1<α≤n, n∈ℕ, and f∈L1, then f has the left Riemann-Liouville derivative Da+αf of order α on the interval [a,b] if and only if f∈ACa+α,1; that is, f has the representation (11). In such a case,
(12)Di(Ia+n-αf)(a)=ci,i-0,…,n-1,(Da+αf)(t)=φ(t),t∈[a,b]a.e.ByACa+α,p (1≤p<∞) we denote the set of all functions f:[a,b]→ℝm possessing representation (11) with c0,c1,…,cn-1∈ℝm, φ∈Lp.Remark 8.
It is easy to see that the above theorem implies the following one (for any1≤p<∞): f has the left Riemann-Liouville derivative Da+αf∈Lp if and only if f∈ACa+α,p; that is, f has the representation (11) with φ∈Lp.Letα>0. By the right Riemann-Liouville fractional integral of h∈L1 on the interval [a,b] we mean a function
(13)(Ib-αh)(t)=1Γ(α)∫tb‍h(τ)(τ-t)1-αdτ,t∈[a,b]a.e.
Similarly, as in the case of left integral, we put
(14)(Ib-0h)(t)=h(t),t∈[a,b]a.e.Remark 9.
Clearly,Ib-αφ has the properties analogous to those described in Remarks 5 and 6.ByACb-α,1=ACb-α,1([a,b],ℝm) we denote the set of all functions f:[a,b]→ℝm that have the representation
(15)f(t)=∑i=0n-1‍(-1)idiΓ(α-n+1+i)(b-t)α-n+i+Ib-αψ(τ),t∈[a,b]a.e.,
with d0,d1,…,dn-1∈ℝm and ψ∈L1. It is easy to see that ACb-n,1=ACn,1.We say thatf∈L1 possesses the right Riemann-Liouville derivative Db-αf of order α∈(n-1,n], n∈ℕ, on the interval [a,b] if Ib-n-αf∈ACn,1. By this derivative we mean the function (-1)nDn(Ib-1-αf). Of course, Db-nf=(-1)nDnf.We also have the following.Theorem 10.
Ifn-1<α≤n, n∈ℕ, and f∈L1, then f has the right Riemann-Liouville derivative Db-αf of order α on the interval [a,b] if and only if f∈ACb-α,1f; that is, f has the representation (15). In such a case,
(16)Di(Ib-n-αf)(b)=di,i=0,…,n-1,(Db-αf)(t)=ψ(t),t∈[a,b]a.e.ByACb-α,p, 1<p<∞, we denote the set of all functions f:[a,b]→ℝm possessing representation (15) with d0,d1,…,dn-1∈ℝm, ψ∈Lp.Remark 11.
As in the case of left Riemann-Liouville derivative, the following theorem holds true for any1≤p<∞: f has the right Riemann-Liouville derivative Db-αf∈Lp if and only if f∈ACb-α,p; that is, f has the representation (15) with ψ∈Lp.In the next section, we will use the following two theorems (cf. [6, 7]).Theorem 12.
Ifα,β>0, then
(17)ACa+α,1⊂ACa+β,1
if and only if
(18)β∈(0,1],α∈⋃i=0∞‍[β+i,1+i]
or
(19)β∈(1,∞),α=β+i,i=0,1,….Theorem 13.
(a) Ifn-1<α≤n, n∈ℕ, 0<β<α-n+1, and x∈ACa+α,1 is of the form (11), then
(20)x(t)=Ia+βψ(t),t∈[a,b]a.e.,
where ψ∈L1 is given by
(21)ψ(t)=∑i=0n-1‍ciΓ(α-n+1-β+i)(t-a)α-n-β+i+Ia+α-βφ(t),t∈[a,b]a.e.
(b) If n-1<α≤n, n∈ℕ, n≥2, β=α-j, j∈{1,…,n-1}, and x∈ACa+α,1 are of the form (11), then
(22)x(t)=∑i=0(n-j)-1‍diΓ(β-(n-j)+1+i)(t-a)β-(n-j)+i+Ia+βψ(t),t∈[a,b]a.e.,
where
(23)di=ci,i=0,…,(n-j)-1,(24)ψ(t)=∑i=0j-1‍Ia+icn-j+i(t)+Ia+jφ(t),t∈[a,b]a.e.


## 3. Some Special Cases of Integration by Parts

Below,C=C([a,b],ℝm) is the set of continuous functions f:[a,b]→ℝm and Cn=Cn([a,b],ℝm) is the set of functions f:[a,b]→ℝm such that f(i)∈C, i=0,…,n (by f(0) we mean the function f). Of course, these classical derivatives coincide with the weak ones D0f,D1f,…,Dnf.Lemma 14.
Ifn-1<α<n, n∈ℕ, f∈Cn, and
(25)Dif(b)=0,i=0,…,n-1,
then Ib-n-αf∈Cn,
(26)Di(Ib-n-αf)(b)=0,i=0,…,n-1,Db-αf=Ib-n-α(Db-nf)=Ib-n-α((-1)nDnf)∈C.
Consequently, f∈Ib-α(C).Proof.
Iff∈Cn and
(27)Dif(b)=0,i=0,…,n-1,
then (cf. Remark6 and [3, formulas (2.21) and (2.58)])
(28)Ib-n-αf(t)=Ib-n-αIb-n(Db-nf)(t)=Ib-nIb-n-α(Db-nf)(t)=Ib-nIb-n-α((-1)nDnf)(t)
for all t∈[a,b]. This means (cf. Remark 6) that Ib-n-αf∈Cn and Db-αf=Ib-n-α((-1)nDnf)∈C. Equalities (28) imply also the equalities
(29)Di(Ib-n-αf)(b)=0,i=0,…,n-1.
So (cf. Theorem 7), f∈Ib-α(C).Remark 15.
Analogous lemma can be obtained for the left integralIa+n-αf and derivative Da+αf (without the term (-1)n).We have the following special case of the theorem on integration by parts.Theorem 16.
Ifn-1<α≤n, n∈ℕ, then
(30)∫ab‍f(t)(Db-αg)(t)dt=∫ab‍(Da+αf)(t)g(t)dt
for any f∈ACa+α,1 and g∈Cn satisfying boundary conditions
(31)Dig(a)=Dig(b)=0,i=0,…,n-1.Proof.
Letf∈ACa+α,1 be of the form (11) with c0,c1,…,cn-1∈ℝm and φ∈L1. We have (Lemma 14 and Remark 9)
(32)∫ab‍f(t)(Db-αg)(t)dt=∫ab‍f(t)(Ib-n-α(Db-ng))(t)dt=∫ab‍(∑i=0n-1‍ciΓ(α-n+1+i)(t-a)α-n+i+Ia+αφ(τ))×(Ib-n-α(Db-ng))(t)dt=∑i=0n-1‍ciIb-α-n+1+i(Ib-n-αDb-ng)(a)+∫ab‍(Da+αf)(t)(Ib-nDb-ng)(t)dt=∑i=0n-1‍ciIb-1+iDb-ng(a)+∫ab‍(Da+αf)(t)g(t)dt=∫ab‍(Da+αf)(t)g(t)dt.
Existence of the first integral and the first equality follows from Lemma 14; in the third equality we used an integral form of a classical fractional theorem on the integration by parts (cf. [3, formula (2.20)]), in the fourth equality we used continuity of Dng and in the fifth equality we used the equalities
(33)Ib-1(Db-ng)(t)=∫tb‍(Db-ng)(t)dt=(-1)n∫tb‍(Dng)(t)dt=(-1)n(Dn-1g(b)-Dn-1g(t))=(-1)n-1Dn-1g(t)=Db-n-1g(t).
The proof is completed.The next theorem will be used in the last section of the paper. Proof of this theorem is contained in [4] and in [5] its extension to the case of fractional derivatives of higher order is derived (the method of the proof is the same in both papers).Theorem 17.
If0<α<1, α>1/p, α>1/q, 1≤p<∞, and 1≤q<∞, then
(34)∫ab‍f(t)(Db-αg)(t)dt=(Ia+1-αf)(a)g(a)-(Ib-1-αg)(b)f(b)+∫ab‍(Da+αf)(t)g(t)dt
for f∈ACa+α,p([a,b],ℝn) and g∈ACb-α,q([a,b],ℝn).


## 4. Fractional Sobolev Spaces

Let0<α≤1 and let 1≤p<∞. By left Sobolev space of order α we will mean the set Wa+α,p=Wa+α,p((a,b),ℝm) given by
(35)Wa+α,p={u∈Lp;∃g∈Lp∀φ∈Cc∞∫ab‍u(t)Db-αφ(t)dt=∫ab‍g(t)φ(t)dt}.
A function g given above will be called the weak left fractional derivative of order α∈(0,1] of u; let us denote it by ua+α. Uniqueness of this weak derivative follows from [1, Lemma IV.2 and Propositions IV.18, IV.21].Letn-1<α≤n, n∈ℕ, n≥2. By the left Sobolev space of order α we mean the set Wa+α,p=Wa+α,p([a,b],ℝm) given by
(36)Wa+α,p={u∈Wa+α-1,p;ua+α-(n-1)∈Wa+n-1,p}.Remark 18.
SinceDb-1φ=-D1φ=-φ(1) for φ∈Cc∞, we see that the weak left fractional derivative ua+1 of u coincides with the classical weak derivative u′=D1u of u. Consequently,
(37)Wa+n,p=Wn,p=ACn,p=ACa+n,p
for n∈ℕ.We have the following characterization ofWa+α,p.Theorem 19.
Ifn-1<α≤n, n∈ℕ, and 1≤p<∞, then
(38)Wa+α,p=ACa+α,p∩Lp.Proof.
Case ofα=n follows from (37) and from the fact that ACa+n,p⊂Lp. So, let us consider the case of n-1<α<n. We will apply the induction with respect to n∈ℕ.
Letn=1. If u∈ACa+α,p∩Lp, then from Theorem 7 it follows that u has the derivative Da+αu∈Lp. Theorem 16 implies that
(39)∫ab‍u(t)Db-αφ(t)dt=∫ab‍(Da+αu)(t)φ(t)dt
for any φ∈Cc∞. So, u∈Wa+α,p with
(40)ua+α=g=Da+αu∈Lp.
Now, let us assume thatu∈Wa+α,p, that is, u∈Lp, and there exists a function g∈Lp such that
(41)∫ab‍u(t)Db-αφ(t)dt=∫ab‍g(t)φ(t)dt
for any φ∈Cc∞. To show that u∈ACa+α,p∩Lp it is sufficient to check (cf. Theorem 7 and definition of ACa+α,p) that u possesses the left Riemann-Liouville derivative of order α, belonging to Lp, that is, that Ia+1-αu is absolutely continuous on [a,b] and its classical derivative of the first order (existing a.e. on [a,b]) belongs to Lp.
Ifφ∈Cc∞, then (cf. Lemma 14)  φ∈Ib-α(C) and Db-αφ=Ib-1-α(-D1φ). From the differential form of the classical fractional theorem on the integration by parts (cf. [3, formula (2.64)]) it follows that
(42)∫ab‍u(t)Db-αφ(t)dt=∫ab‍u(t)(Ib-1-α(-D1φ))(t)dt=∫ab‍(Da+1-αIa+1-αu)(t)(Ib-1-α(-D1φ))(t)dt=∫ab‍(Ia+1-αu)(t)(-D1φ)(t)dt=-∫ab‍(Ia+1-αu)(t)(D1φ)(t)dt.
This means (cf. (41) and (42)) that
(43)∫ab‍(Ia+1-αu)(t)(D1φ)(t)dt=-∫ab‍g(t)φ(t)dt
for any φ∈Cc∞. So, Ia+1-αu∈W1,p. Consequently, Ia+1-αu is absolutely continuous and its classical derivative is equal a.e. on [a,b] to g∈Lp.
Now, letn∈ℕ and assume that
(44)Wa+α,p=ACa+α,p∩Lp
for any α∈(n-1,n). We will show that this equality holds true for any α∈(n,n+1).
Indeed, letα∈(n,n+1). If u∈Wa+α,p, then u∈Wa+α-1,p=ACa+α-1,p∩Lp and ua+α-n∈Wa+n,p. So,
(45)u(t)=∑i=0n-1‍ciΓ(α-n+i)(t-a)α-1-n+i+Ia+α-1φ(t),t∈[a,b]a.e.,
where c0,c1,…,cn-1∈ℝm and φ∈Lp. If n=1, from the above formula it follows (cf. (40)) that
(46)φ=Da+α-1u=ua+α-1∈Wa+1,p=W1,p.
If n≥2, from Theorem 13 (b) and (40) it follows that
(47)ua+α-n=Da+α-nu=Da+(α-1)-(n-1)u=∑i=0n-2‍Ia+ic1+i+Ia+n-1φ.
This means that Ia+n-1φ∈Wa+n,p and, consequently,
(48)φ=Da+n-1Ia+n-1φ=Dn-1Ia+n-1φ∈Dn-1(Wa+n,p)=Dn-1(Wn,p)=W1,p.
So, in both cases (n=1, n≥2) there exist cn∈ℝm and λ∈Lp such that
(49)φ=cn+Ia+1λ.
To show that u∈ACa+α,p∩Lp it is sufficient to check that
(50)Ia+n+1-αu∈ACa+n+1,p.
Indeed, we have (below, we use the following elementary formula
(51)Ia+ν((·-a)δ-1)(t)=Γ(δ)Γ(ν+δ)(t-a)ν+δ-1,t∈[a,b]a.e.,
for ν>0, δ>0)
(52)Ia+n+1-αu(t)=∑i=0n-1‍ciΓ(α-n+i)Ia+n+1-α((·-a)α-1-n+i)(t)+Ia+n+1-αIa+α-1φ(t)=∑i=0n-1‍ciΓ(α-n+i)Γ(α-n+i)Γ(1+i)(t-a)i+Ia+nφ(t)=∑i=0n-1‍ciΓ(1+i)+Ia+ncn(t)+Ia+n+1λ(t)=∑i=0n‍cii!(t-a)i+Ia+n+1λ(t)
for t∈[a,b] a.e. Thus, Ia+n+1-αu∈ACa+n+1,p.
Conversely, letu∈ACa+α,p∩Lp. To show that u∈Wa+α,p we have to check that
(53)u∈Wa+α-1,p,ua+α-n∈Wa+n,p.
Theorem 13(b) implies the equality
(54)Da+α-1u=cn+Ia+1φ,
where
(55)u(t)=∑i=0n‍ciΓ(α-n+i)(t-a)α-(n+1)+i+Ia+αφ(t),t∈[a,b]a.e.,
with φ∈Lp. So (cf. Theorem 12), u∈ACa+α-1,p∩Lp=Wa+α-1,p. From Theorem 13 (b) it also follows that
(56)ua+α-n=Da+α-nu=∑i=0n-1‍Ia+ic1+i+Ia+nφ=∑i=0n-1‍c1+ii!(t-a)i+Ia+nφ∈ACa+n,p=Wa+n,p
(existence of the weak fractional derivative ua+α-n and equality ua+α-n=Da+α-nu follow from the relation u∈Wa+α-n,p and (40)).From the first part of the above proof (case ofn=1) and from the uniqueness of the weak fractional derivative the following theorem follows (cf. also [4, fractional fundamental lemma]).Theorem 20.
Ifα∈(0,1], then the weak left fractional derivative ua+α of a function u∈Wa+α,p coincides with its left Riemann-Liouville fractional derivative Da+αu a.e. on [a,b].Remark 21.
Ifn-1<α≤n and (n-α)p<1, then ACa+α,p⊂Lp and, consequently, Wa+α,p=ACa+α,p∩Lp=ACa+α,p. If (n-α)p≥1, then Wa+α,p=ACa+α,p∩Lp is the set of all functions belonging to ACa+α,p that satisfy the condition (Ia+n-αf)(a)=0.


## 5. Weak Fractional Derivatives

Now, we will prove an extension of Theorem1.Theorem 22.
Letn-1<α≤n, n∈ℕ, 1≤p<∞, and u∈Lp. Then, u∈Wa+α,p if and only if there exist functions g1,…,gn∈Lp such that
(57)∫ab‍u(t)Db-α-(n-i)φ(t)dt=∫ab‍gi(t)φ(t)dt,φ∈Cc∞,
for any i∈{1,…,n}. In such a case there exist the left Riemann-Liouville derivatives Da+α-(n-1)u,Da+α-(n-2)u,…,Da+αu of u and
(58)g1=Da+α-(n-1)u,g2=Da+α-(n-2)u,…,gn=Da+αu.Proof.
Case ofα=n follows from Theorem 1. So, let us consider the case of n-1<α<n. We will apply the induction with respect to n∈ℕ.
Whenn=1, it is sufficient to use the definition of Wa+α,p with α∈(0,1) and Theorem 20.
So, letn∈ℕ and assume that the theorem is true for any α∈(n-1,n). We will show that it is true for any α∈(n,n+1).
Indeed, ifu∈Wa+α,p with a fixed α∈(n,n+1), then
(59)u∈Wa+α-1,p,(60)Da+α-nu=ua+α-n∈Wa+n,p.
Relation (59) implies (by the induction assumption) the existence of functions g1,…,gn∈Lp such that (of course, α-1∈(n-1,n))
(61)∫ab‍u(t)Db-α-((n+1)-i)φ(t)dt=∫ab‍u(t)Db-(α-1)-(n-i)φ(t)dt=∫ab‍gi(t)φ(t)dt
for any φ∈Cc∞ and i∈{1,…,n}. Moreover,
(62)g1=Da+(α-1)-(n-1)u=Da+α-((n+1)-1)u,g2=Da+(α-1)-(n-2)u=Da+α-((n+1)-2)u,⋮gn=Da+(α-1)-(n-n)u=Da+α-((n+1)-n)u.
From the existence of Da+α-nu and from (60) it follows that Ia+1-(α-n)u∈Wn+1,p. In particular, there exists derivative Da+αu∈Lp. If we put gn+1=Da+αu we see that gn+1=Da+α-((n+1)-(n+1))u and
(63)∫ab‍u(t)Db-α-((n+1)-(n+1))φ(t)dt=∫ab‍u(t)Db-αφ(t)dt=∫ab‍Da+αu(t)φ(t)dt=∫ab‍gn+1(t)φ(t)dt
for any φ∈Cc∞ (the second equality follows from Theorem 16).
Now, let us assume that there exist functionsg1,…,gn+1∈Lp such that
(64)∫ab‍u(t)Db-α-((n+1)-i)φ(t)dt=∫ab‍gi(t)φ(t)dt,φ∈Cc∞,
for any i∈{1,…,n+1}. Since α-((n+1)-i)=α-1-(n-i), the above condition for i∈{1,…,n} and the induction assumption mean that
(65)u∈Wa+α-1,p.
Moreover, condition (64) for i=1, Lemma 14, and the integral form of the classical fractional theorem on the integration by parts (cf. [3, formula (2.20)]) imply that
(66)∫ab‍g1(t)φ(t)dt=∫ab‍u(t)Db-α-nφ(t)dt=∫ab‍u(t)Ib-n-α+1Db-1φ(t)dt=∫ab‍Ia+n-α+1u(t)Db-1φ(t)dt=-∫ab‍Ia+n-α+1u(t)D1φ(t)dt
for any φ∈Cc∞. This means that Ia+n-α+1u∈W1,p. So, if we fix i∈{2,…,n+1} and use once again Lemma 14 and the integral form of the classical fractional theorem on the integration by parts, we obtain
(67)∫ab‍gi(t)φ(t)dt=∫ab‍u(t)Db-α-((n+1)-i)φ(t)dt=∫ab‍u(t)Ib-(n+1)-αDb-iφ(t)dt=∫ab‍Ia+(n+1)-αu(t)Db-iφ(t)dt=(-1)i∫ab‍Ia+(n+1)-αu(t)Diφ(t)dt=(-1)i∫ab‍Ia+(n+1)-αu(t)D1Di-1φ(t)dt=(-1)i-1∫ab‍D1Ia+1-(α-n)u(t)Di-1φ(t)dt=(-1)i-1∫ab‍Da+α-nu(t)Di-1φ(t)dt.
Thus, from Theorem 1 it follows that
(68)ua+α-n=Da+α-nu∈Wn,p.
Using (65) and (68) we assert that u∈Wa+α-1,p.Functionsg1,…,gn will be called the weak left fractional derivatives of u∈Wa+α,p of orders α-(n-1),…,α, respectively. Their uniqueness follows from [1, Lemma IV.2 and Propositions IV.18,IV.21]. From the above theorem it follows that they coincide with the appropriate Riemann-Liouville derivatives.We have the following counterpart of Remark2.Theorem 23.
Ifu∈Wa+α,p, n-1<α≤n, and n∈ℕ, then
(69)gi=Di-1(Da+α-(n-1)u),i=1,…,n.Proof.
Ifu∈Wa+α,p, then Ia+n-αu∈Wn,p and from the definition of the Riemann-Liouville derivative and Theorem 22 it follows that
(70)g1=Da+α-n+1u=D1Ia+1-(α-n+1)u=D1(Ia+n-αu),g2=Da+α-n+2u=D2Ia+2-(α-n+2)u=D2Ia+n-αu=D1(D1Ia+n-αu)=D1(Da+α-n+1u),⋮gn=Da+α-n+nu=DnIa+n-(α-n+n)u=Dn-1(D1Ia+n-αu)=Dn-1(Da+α-n+1u).
The proof is completed.


## 6. Norms inWa+α,p

Let us fixα∈(n-1,n] where n∈ℕ and consider in the space Wa+α,p a norm ∥·∥Wa+α,p given by
(71)∥u∥Wa+α,pp=∥u∥Lpp+∑i=0n-1‍∥Da+α-(n-1)+iu∥Lpp,u∈Wa+α,p
(here ∥·∥Lp denotes the classical norm in Lp).We have the following theorem.Theorem 24.
Ifn∈ℕ and α∈(n-1,n], then the norm ∥·∥Wa+α,p is equivalent to a norm ∥u∥a,Wa+α,p given by
(72)∥u∥a,Wa+α,pp=∑i=0n-1‍|DiIa+n-αu(a)|p+∥Da+αu∥Lpp,u∈Wa+α,p.Proof.
We will use the induction with respect ton∈ℕ.
Assume thatn=1 and (1-α)p<1. Then, for u∈Wa+α,p given by
(73)u(t)=cΓ(α)1(t-a)1-α+Ia+αφ(t)
with c∈ℝn and φ∈Lp, we have
(74)∥u∥Lpp=∫ab‍|cΓ(α)1(t-a)1-α+Ia+αφ(t)|pdt≤2p-1(|c|pΓ(α)p∫ab‍(t-a)(α-1)pdt+∥Ia+αφ∥Lpp)≤2p-1(|c|pΓ(α)p1(α-1)p+1(b-a)(α-1)p+1+Kp∥φ∥Lpp),
where K=(b-a)α/Γ(α+1) (cf. [3, formula (2.72)]). Thus (of course, c=Ia+1-αu(a) and φ=Da+αu),
(75)∥u∥Lpp≤Lα,0(|c|p+∥φ∥Lpp)=Lα,0(|Ia+1-αu(a)|p+∥Da+αu∥Lpp)=Lα,0∥u∥a,Wa+α,pp,
where
(76)Lα,0=2p-1((b-a)1-(1-α)pΓ(α)p(1-(1-α)p)+Kp).
Consequently,(77)∥u∥Wa+α,pp=∥u∥Lpp+∥Da+αu∥Lpp≤Lα,1∥u∥a,Wa+α,pp,
where Lα,1=Lα,0+1.
Now, we will prove that there exists a constantMα,1>0 such that
(78)∥u∥a,Wa+α,pp≤Mα,1∥u∥Wa+α,pp,u∈Wa+α,p.
Indeed, let u∈Wa+α,p and consider a coordinate function (Ia+1-αu)i of Ia+1-αu with a fixed i∈{1,…,m}. The mean value theorem implies the existence of t0∈(a,b) such that
(79)(Ia+1-αu)i(t0)=1b-a∫ab‍(Ia+1-αu)i(s)ds.
From the absolute continuity of (Ia+1-αu)i it follows that
(80)(Ia+1-αu)i(t)=(Ia+1-αu)i(t0)+∫t0t‍D1(Ia+1-αu)i(s)ds
for any t∈[a,b]. Consequently,
(81)|(Ia+1-αu)i(t)|≤1b-a∥Ia+1-αu∥L1+∫t0t‍|Da+αu(s)|ds≤1b-a∥Ia+1-αu∥L1+∥Da+αu∥L1≤1b-a(b-a)1-αΓ(2-α)∥u∥L1+∥Da+αu∥L1
for t∈[a,b]. In particular,
(82)|(Ia+1-αu)i(a)|≤1b-a(b-a)1-αΓ(2-α)∥u∥L1+∥Da+αu∥L1.
So,
(83)|Ia+1-αu(a)|≤mMα,0(∥u∥L1+∥Da+αu∥L1)≤mMα,0(b-a)(p-1)/p(∥u∥Lp+∥Da+αu∥Lp),
where Mα,0=(b-a)-α/Γ(2-α)+1. Thus,
(84)|Ia+1-αu(a)|p≤mpMα,0p(b-a)p-12p-1×(∥u∥Lpp+∥Da+αu∥Lpp)
and, consequently,
(85)∥u∥a,Wa+α,pp=|Ia+1-αu(a)|p+∥Da+αu∥Lpp≤(mpMα,0p(b-a)p-12p-1+1)×(∥u∥Lpp+∥Da+αu∥Lpp)=Mα,1∥u∥Wa+α,pp,
where Mα,1=mpMα,0p(b-a)p-12p-1+1.
When(1-α)p≥1, then (cf. Remark 21) Wa+α,p=ACa+α,p∩Lp is the set of all functions u belonging to ACa+α,p that satisfy the condition (Ia+1-αu)(a)=0. Consequently, in the same way as in the case of (1-α)p<1 (putting c=0) we obtain the inequality ∥u∥Wa+α,pp≤Lα,1∥u∥a,Wa+α,pp with some Lα,1>0. The inequality ∥u∥a,Wa+α,pp≤Mα,1∥u∥Wa+α,pp with some Mα,1>0 is obvious (it is sufficient to put Mα,1=1 and use the fact that (Ia+1-αu)(a)=0).
Now, let us assume that the assertion holds true for somen∈ℕ. We will prove that it is true for n+1.
Letα∈(n,n+1]. We know that there exist constants Lα,n and Mα,n>0 such that
(86)(1Mα,n)∥u∥a,Wa+α-1,pp≤∥u∥Wa+α-1,pp≤Lα,n∥u∥a,Wa+α-1,pp
for any u∈Wa+α-1,p. If u∈Wa+α,p, then u∈Wa+α-1,p and therefore (from the induction assumption)
(87)∥u∥Wa+α,pp=∥u∥Lpp+∑i=0n‍∥Da+α-n+iu∥Lpp=∥u∥Lpp+∑i=0n-1‍∥Da+(α-1)-(n-1)+iu∥Lpp+∥Da+αu∥Lpp=∥u∥Wa+α-1,pp+∥Da+αu∥Lpp≤Lα-1,n∥u∥a,Wa+α-1,pp+∥Da+αu∥Lpp=Lα-1,n(∑i=0n-1‍|DiIa+n-(α-1)u(a)|p+∥Da+α-1u∥Lpp)+∥Da+αu∥Lpp.
Since Da+α-1u=DnIa+n-(α-1)u=DnIa+n+1-αu∈Wa+1,p (because of Ia+n+1-αu∈Wa+n+1,p), therefore (cf. case of n=1)
(88)∥Da+α-1u∥Lpp=∥DnIa+n-(α-1)u∥Lpp≤∥DnIa+n-(α-1)u∥Wa+1,pp≤L1,1∥DnIa+n-(α-1)u∥a,Wa+1,pp=L1,1(|DnIa+n-(α-1)u(a)|p+∥D1DnIa+n-(α-1)u∥Lp)=L1,1(|DnIa+n-(α-1)u(a)|p+∥Dn+1Ia+n-(α-1)u∥Lp),
where L1,1 is such that
(89)∥v∥Wa+1,pp≤L1,1∥v∥a,Wa+1,pp
for v∈Wa+1,p. Thus,
(90)∥u∥Wa+α,pp≤Lα-1,n∑i=0n-1‍|DiIa+n-(α-1)u(a)|p+Lα-1,nL1,1×(|DnIa+n-(α-1)u(a)|p+∥Dn+1Ia+n-(α-1)u∥Lp)+∥Da+αu∥Lpp≤Lα,n+1(∑i=0n‍|DiIa+n+1-αu(a)|p+∥Da+αu∥Lpp)=Lα,n+1∥u∥a,Wa+α,pp,
where Lα,n+1=Lα-1,n+Lα-1,nL1,1+1 (we used the equality Dn+1Ia+n-(α-1)u=Dn+1Ia+n+1-αu=Da+αu).
On the other hand, sinceu∈Wa+α-1,p, therefore (from the induction assumption)
(91)∑i=0n-1‍|DiIa+n-(α-1)u(a)|p+∥Da+α-1u∥Lpp≤Mα-1,n(∥u∥Lpp+∑i=0n-1‍∥Da+α-1-(n-1)+iu∥Lpp).
So,
(92)∑i=0n-1‍|DiIa+n-(α-1)u(a)|p≤Mα-1,n(∥u∥Lpp+∑i=0n-2‍∥Da+α-1-(n-1)+iu∥Lpp+∥Da+α-1u∥Lpp).
Let us recall that Ia+n+1-αu∈Wn+1,p, so Ia+n+1-αu∈Wn,p, DnIa+n+1-αu∈W1,p, and Da+αu=Dn+1Ia+n+1-αu=D1DnIa+n+1-αu. Thus,
(93)|DnIa+n+1-αu(a)|p+∥Da+αu∥Lpp=∥DnIa+n+1-αu∥a,Wa+1,pp≤M1,1∥Dn(Ia+n+1-αu)∥Wa+1,pp=M1,1(∥Dn(Ia+n-(α-1)u)∥Lpp+∥Da+αu)∥Lpp),
where M1,1 is such that
(94)∥v∥a,Wa+1,pp≤M1,1∥v∥Wa+1,pp.
Consequently,
(95)∥u∥a,Wa+α,pp=∑i=0n-1‍|DiIa+n-(α-1)u(a)|p+|DnIa+n-(α-1)u(a)|p+∥Da+αu∥Lpp≤Mα-1,n(∥u∥Lpp+∑i=0n-2‍∥Da+α-1-(n-1)+iu∥Lpp+∥Da+α-1u∥Lpp∑i=0n-2‍)+M1,1(∥Da+α-1u∥Lpp+∥Da+αu∥Lpp)≤Mα,n+1(∥u∥Lpp+∑i=0n-1‍∥Da+α-1-(n-1)+iu∥Lpp+∥Da+αu∥Lpp∑i=0n-1‍)=Mα,n+1(∥u∥Lpp+∑i=0n‍∥Da+α-n+iu∥Lpp)=Mα,n+1∥u∥Wa+α,pp,
where Mα,n+1=Mα-1,n+M1,1.


## 7. Basic Properties ofWa+α,p

Now, we are in a position to prove some basic properties of the introduced spaces.Theorem 25.
The spaceWa+α,p is complete with respect to each of the norms ∥·∥Wa+α,p and ∥·∥a,Wa+α,p, for any α>0 and 1≤p<∞.Proof.
Letn∈ℕ be such that α∈(n-1,n]. Of course, it is sufficient to show that Wa+α,p with the norm ∥·∥a,Wa+α,p is complete. Let (uk)⊂Wa+α,p be a Cauchy sequence with respect to this norm. So, the sequences (DiIa+n-αuk(a)), i=0,…,n-1, are Cauchy sequences in ℝm and (Da+αuk) is the Cauchy sequence in Lp. Let c0,c1,…,cn-1∈ℝm and φ∈Lp be limits of the above sequences in ℝm and Lp, respectively. Then the function
(96)u(t)=∑i=0n-1‍ciΓ(α-n+1+i)(t-a)α-n+i+Ia+αφ(t),t∈[a,b]a.e.,
belongs to Wa+α,p and is the limit of (uk) in Wa+α,p with respect to ∥·∥a,Wa+α,p (to assert that u∈Lp it is sufficient to consider the cases (n-α)p<1 and (n-α)p≥1—in the second case Ia+n-αuk(a)=0 for any k∈ℕ and, consequently, c0=0).In the proofs of the next two theorems we use the method presented in [1].Theorem 26.
The spaceWa+α,p is reflexive with respect to each of the norms ∥·∥Wa+α,p and ∥·∥a,Wa+α,p, for any α>0 and 1<p<∞.Proof.
Let us considerWa+α,p with the norm ∥·∥Wa+α,p and define a mapping
(97)λ:Wa+α,p∋u⟼(u,Da+α-(n-1)u,…,Da+α-1u,Da+αu)∈Lp×Lp×⋯×Lp×Lp.
Since λ is the isometry, λ(Wa+α,p) is the closed linear subspace of the reflexive space Lp×⋯×Lp. So (cf. [8, Corollary 1 in Part V.73]), it is reflexive and, consequently, Wa+α,p is reflexive with respect to ∥·∥Wa+α,p.
From the equivalence of the norms it follows thatWa+α,p with the norm ∥·∥a,Wa+α,p is also reflexive (it is sufficient to consider the identity mapping i:(Wa+α,p,∥·∥Wa+α,p)→(Wa+α,p,∥·∥Wa+α,p) being the linear homeomorphism and use [8, Remark in Part V.7.3]).Theorem 27.
The spaceWa+α,p is separable with respect to each of the norms ∥·∥Wa+α,p and ∥·∥a,Wa+α,p, for any α>0 and 1≤p<∞.Proof.
Let us considerWa+α,p with the norm ∥·∥Wa+α,p and mapping λ defined in the proof of Theorem 26. Of course, λ(Wa+α,p) is separable as a subset of separable space Lp×⋯×Lp. Since λ is the isometry, Wa+α,p is also separable with respect to the norm ∥·∥Wa+α,p. Equivalence of the norms ∥·∥a,Wa+α,p and ∥·∥Wa+α,p implies separability of Wa+α,p with respect to ∥·∥Wa+α,p.


## 8. Imbeddings

We have the following extension of Theorem12.Theorem 28.
(a) If0<β<α≤1, then
(98)ACa+α,p⊂Ia+β(Lq)⊂ACa+β,q
for 1≤q≤p<∞ and 1≤q<1/(1-α+β); consequently,
(99)Wa+α,p⊂Wa+β,q
for 1≤q≤p<∞ and 1≤q<1/(1-α+β).
(b) Ifn-1<α≤n, n∈ℕ, n≥2, and 0<β<α-(n-1), then
(100)ACa+α,p⊂Ia+β(Lq)⊂ACa+β,q
for 1≤p<∞ and 1≤q<1/(n-α+β); consequently,
(101)Wa+α,p⊂Wa+β,q
for 1≤p<∞ and 1≤q<(1/(n-α+β)).
(c) Ifn-1<α≤n, n∈ℕ, n≥2, β=α-i, and i∈{1,…,n-1}, then
(102)ACa+α,p⊂ACa+β,q
for 1≤p<∞ and 1≤q<∞; consequently,
(103)Wa+α,p⊂Wa+β,q
for 1≤p<∞, (n-α)p<1, 1≤q<∞, and (n-α)q<1 or 1≤p<∞, (n-α)p≥1, and 1≤q<∞.
(d) If0<α=β, then
(104)ACa+α,p⊂ACa+β,q
for 1≤q≤p<∞; consequently,
(105)Wa+α,p⊂Wa+α,q
for 1≤q≤p<∞.Proof.
(a) Let us fixp∈[1,∞). Theorems 12 and 13 (a) imply that
(106)ACa+α,p⊂ACa+α,1⊂Ia+β(Lq)⊂ACa+β,q
provided that (α-β-1)q>-1 and 1≤q≤p (cf. (20), (21)); that is, 1≤q<1/(1-α+β) and 1≤q≤p. Consequently, for such p and q,
(107)Wa+α,p=ACa+α,p∩Lp⊂ACa+β,q∩Lq=Wa+β,q.
(b) Let us fixp∈[1,∞). In this case α-β>1 and, consequently, Ia+α-βφ=Ia+1Ia+α-β-1φ∈C for any φ∈L1. So (cf. Theorems 12 and 13 (a))
(108)ACa+α,p⊂ACa+α,1⊂Ia+β(Lq)⊂ACa+β,q
provided that (α-β-n)q>-1 and 1≤q<∞ (cf. (20), (21)); that is, 1≤q<1/(n-α+β). Consequently, for such q and 1≤p<∞,
(109)Wa+α,p=ACa+α,p∩Lp⊂Ia+β(Lq)=Ia+β(Lq)∩Lq⊂ACa+β,q∩Lq=Wa+β,q
(we used here the inclusion Ia+β(Lq)⊂Lq).
(c) Let us fixp∈[1,∞). Theorems 12 and 13 (b) imply that
(110)ACa+α,p⊂ACa+α,1⊂ACa+β,q
for any q∈[1,∞) (cf. (22), (24)).
If(n-α)p<1, then (cf. Remark 21) ACa+α,p∩Lp=ACa+α,p. So,
(111)Wa+α,p=ACa+α,p∩Lp=ACa+α,p⊂ACa+β,q⊂ACa+β,q∩Lq=Wa+β,q
provided that (cf. Theorem 13 (b)) (n-α)q<1. If (n-α)p≥1, then (cf. Remark 21) ACa+α,p∩Lp is the set of all functions belonging to ACa+α,p that satisfy the condition (Ia+n-αf)(a)=0. Consequently (cf. Theorem 13 (b)), ACa+α,p∩Lp⊂ACa+β,q∩Lq, that is, Wa+α,p⊂Wa+β,q for any 1≤q<∞.
(d) These facts are obvious.The first part of the point (c) of the above theorem implies the following.Corollary 29.
Ifn-1<α≤n, n∈ℕ, n≥2, β=α-i, and i∈{1,…,n-1}, then
(112)Wa+α,p⊂Wa+β,q
for 1≤q≤p<∞.In the next section we will use the following important result obtained in [9, Lemma 1.1].Theorem 30.
Ifα>0 and 1≤p<∞, then the operator Ia+α:Lp→Lp is completely continuous, that is, it maps the bounded sets onto relatively compact ones.Now, we are in a position to prove theorems on compactness of some imbeddings.Theorem 31.
The imbedding(113)Wa+α,p⊂Wa+β,q
for 0<β<α≤1, 1≤q≤p<∞, and 1≤q<1/(1-α+β), given in Theorem 28 (a), is compact.Proof.
Let0<β<α≤1, 1≤q≤p<∞, 1≤q<1/(1-α+β), and (uk), where
(114)uk(t)=ckΓ(α)1(t-a)1-α+Ia+αφk(t),t∈[a,b]a.e.,k∈ℕ,
be a bounded sequence in Wa+α,p. We will show that it contains a subsequence which is convergent in Wa+β,q.
Since(uk) is bounded (cf. Theorem 24), the sequences (ck) and (φk) are bounded in ℝm and Lp, respectively. So, one can choose a subsequence (kj)j∈ℕ of positive integers such that (ckj) is convergent to some c0 in ℝm and (cf. Theorem 30) such that (Ia+α-βφkj) is convergent to some ψ¯0 in Lp. Of course, (Ia+α-βφkj) is convergent to ψ¯0 in Lq. Moreover, the sequence ((ckj/Γ(α-β))(·-a)α-1-β) converges in Lq to (c0/Γ(α-β))(·-a)α-1-β. This means (cf. (20), (21)) that the sequence (ukj) converges in Wa+β,q to u0 given by
(115)u0(t)=Ia+βψ0(t),t∈[a,b]a.e.,
where ψ0(t)=(c0/Γ(α-β))(t-a)α-1-β+ψ¯0(t). The proof is completed.The above theorem implies the following.Corollary 32.
The imbedding(116)Wa+α,p⊂Lq
for 0<α<1, 1≤q≤p<∞, 1≤q<1/(1-α) or α=1, and 1≤q≤p<∞ is compact.Theorem 33.
The imbedding(117)Wa+α,p⊂Wa+β,q,
for n-1<α≤n, n∈ℕ, n≥2, 0<β<α-(n-1), 1≤p<∞, and 1≤q<1/(n-α+β), given in Theorem 28 (b), is compact.Proof.
Let us consider a bounded sequence(uk) in Wa+α,p where
(118)uk(t)=c0kΓ(α-n+1)(t-a)α-n+c1kΓ(α-n+2)(t-a)α-n+1+⋯+cn-1kΓ(α)(t-a)α-1+Ia+αφk(t),t∈[a,b]a.e.,
with c0k,c1k,…,cn-1k∈ℝm and φk∈Lp. We will show that it contains a subsequence which is convergent in Wa+β,q.
Since(uk) is bounded, the sequences (c0k),(c1k),…,(cn-1k)∈ℝm and (φk) are bounded in ℝm and Lp, respectively. So, one can choose a subsequence (kj)j∈ℕ of positive integers such that (c0kj),(c1kj),…,(cn-1kj) are convergent to some c0,c1,…,cn-1 in ℝm and (cf. Theorem 30) such that the sequence (Ia+γφkj) is convergent in Lp to some ω0 where γ=(α-β)-1>0. Consequently, (Ia+γφkj) is convergent in L1 to ω0 and
(119)Ia+α-βφkj=Ia+1Ia+γφkj⟶Ia+1ω0
in Lq, because
(120)∫ab‍|Ia+1Ia+γφkj(t)-Ia+1ω0(t)|qdt≤∫ab‍(∫at‍|Ia+γφkj(s)-ω0(s)|ds)qdt≤∥Ia+γφkj-ω0∥L1q(b-a).
So, the sequence (Da+βukj), where
(121)Da+βukj(t)=c0kjΓ(α-n+1-β)(t-a)α-n-β+c1kjΓ(α-n+2-β)(t-a)α-n-β+1+⋯+cn-1kjΓ(α-β)(t-a)α-β-1+Ia+α-βφkj(t),
converges to the function
(122)ψ0(t)=c0Γ(α-n+1-β)(t-a)α-n-β+c1Γ(α-n+2-β)(t-a)α-n-β+1+⋯+cn-1Γ(α-β)(t-a)α-β-1+Ia+1ω0(t)
inLq. Thus, the sequence (ukj) converges in Wa+β,q to Ia+βψ0.
The proof is completed.The above theorem implies the following.Corollary 34.
The imbedding(123)Wa+α,p⊂Lq
for n-1<α<n, n∈ℕ, n≥2, 1≤p<∞, 1≤q<1/(n-α) or α=n, n∈ℕ, n≥2, 1≤p<∞, and 1≤q<∞ is compact.Theorem 35.
The imbedding(124)Wa+α,p⊂Wa+β,q,
for n-1<α≤n, n∈ℕ, n≥2, β=α-i, i∈{1,…,n-1}, and 1≤q≤p<∞, given in Corollary 29, is compact.Proof.
Let us consider a bounded sequence(uk) in Wa+α,p where
(125)uk(t)=c0kΓ(α-n+1)(t-a)α-n+c1kΓ(α-n+2)(t-a)α-n+1+⋯+cn-1kΓ(α)(t-a)α-1+Ia+αφk(t),t∈[a,b]a.e.,
with c0k,c1k,…,cn-1k∈ℝm and φk∈Lp. We will show that it contains a subsequence which is convergent in Wa+β,q.
Since(uk) is bounded, the sequences (c0k),(c1k),…,(cn-1k)∈ℝm and (φk) are bounded in ℝm and Lp, respectively. So, one can choose a subsequence (kj)j∈ℕ of positive integers such that (c0kj),(c1kj),…,(cn-1kj) are convergent to some c0,c1,…,cn-1 in ℝm and (cf. Theorem 30) such that
(126)(Ia+1cn-i+1kj),…,(Ia+i-1cn-1kj),(Ia+iφkj)
are convergent to some ψ¯1,0,…,ψ¯i-1,0,ψ¯0 in Lp and, consequently, in Lq. So (cf. Theorem 13 (b)),
(127)Da+βukj=cn-ikj+Ia+1cn-i+1kj+⋯+Ia+i-1cn-1kj+Ia+iφkj⟶ψ0
in Lq, where ψ0∈Lq is given by
(128)ψ0(t)=cn-i+ψ¯1,0(t)+⋯+ψ¯i-1,0(t)+ψ¯0(t),t∈[a,b]a.e.
This means (cf. (22), (23), and (24)) that the sequence (ukj) converges in Wa+β,q to u0 given by
(129)u0(t)=d0Γ(β-(n-i)+1)(t-a)β-(n-i)+d1Γ(β-(n-i)+2)(t-a)β-(n-i)+1+⋯+dn-(i+1)Γ(β)(t-a)β-1+Ia+βψ0(t),t∈[a,b]a.e.,
where(130)d0=c0,d1=c1,…,dn-(i+1)=cn-(i+1).
The proof is completed.The above theorem implies the following.Corollary 36.
The imbedding(131)Wa+α,p⊂Lq
for n-1<α≤n, n∈ℕ, n≥2, and 1≤q≤p<∞ is compact.Remark 37.
It is easy to see that imbeddingsWa+α,p⊂Wa+α,q for α>0 and 1≤q≤p<∞, given in Theorem 28 (d), are not compact.Remark 38.
From Corollary32 it follows that the imbedding W1,p⊂Lp is compact for any 1≤p<∞. Corollary 34 implies the compactness of the imbedding Wn,1⊂Lq for any n∈ℕ, n≥2, and 1≤q<∞. The following problem is open: is it possible to strengthen Theorem 31 or Theorem 33 to deduce the compactness of the imbedding W1,1⊂Lq for any 1≤q<∞?


## 9. Application to Boundary Value Problems

In this section, we will demonstrate an application of the obtained results to fractional boundary value problems.Namely, let us fixα∈(1/2,1) and consider the following problem:
(132)Db-αDa+αx(t)+x(t)=f(t),t∈[a,b]a.e.Da+αx(a)=0,x(b)=0,
where f∈L2. By a solution to this problem we mean a function x∈Wa+α,2 such that Da+αx and Db-αDa+αx exist, and satisfying the above equation and boundary conditions. Under the assumption on α the boundary conditions make sense ([10, Property 4]).Leta:H×H→ℝ, where
(133)H={x∈Wa+α,2;x(b)=0},
to be in the following bilinear form
(134)a(x,y)=∫ab‍Da+αx(t)Da+αy(t)dt+∫ab‍x(t)y(t)dt.
It is easy to see that H is the closed subspace of the Hilbert space (Wa+α,2, ∥·∥Wa+α,p); it is sufficient to observe that it is closed in Wa+α,2 with respect to ∥·∥a,Wa+α,p and to use Theorem 24. Of course, a is a scalar product in H and the norm generated by a is simply the norm ∥·∥Wa+α,2 restricted to H. Clearly, a is continuous and coercive; that is, there exists a constant c>0 such that a(x,x)≥c∥x∥Wa+α,22 for x∈Wa+α,2; in fact a(x,x)=∥x∥Wa+α,22 for x∈H. So, Lax-Milgram theorem [1] or simply Riesz-Frechet theorem implies that there exists x∈Wa+α,2 such that
(135)a(x,h)=∫ab‍f(t)h(t)dt,h∈H,(136)12∥x∥Wa+α,22-∫ab‍f(t)x(t)dt=minh∈H{12∥h∥Wa+α,22-∫ab‍f(t)h(t)dt}.
Condition (135) means that
(137)∫ab‍Da+αx(t)Da+αh(t)dt=∫ab‍(f(t)-x(t))h(t)dt,h∈H.
From a counterpart of Theorem 20 for the weak right fractional derivative it follows that Da+αx∈Wb-α,2 and
(138)Db-αDa+αx(t)+x(t)=f(t),t∈[a,b]a.e.
Since x∈H, x(b)=0. Moreover, applying Theorem 17 to the left side of (137) we obtain
(139)∫ab‍(Db-αDa+αx(t)+x(t)-f(t))h(t)dt-Ia+1-αh(a)Da+αx(a)+Ib-1-αDa+αx(b)h(b)=0
for h∈H. So,
(140)Ia+1-αh(a)Da+αx(a)=0,h∈H.
This means that Da+αx(a)=0. Indeed, it is sufficient to consider functions hi, i=1,…,m, of the form
(141)hi(t)=(0,…,0,1Γ(α)c(t-a)1-α+1Γ(α)∫at‍1(t-τ)1-α,0,…,0),
where c=-(b-a)1-α∫ab‍(1/(b-τ)1-α)dt, with nonzero ith coordinate function (of course, hi∈H and Ia+1-αhi(a)=(0,…,0,c,0…,0)≠0).Remark 39.
In the same way one can prove the existence of a solution to system(142)Db-αDa+αx(t)+x(t)=f(t),t∈[a,b]a.e.
with boundary conditions
(143)Ia+1-αx(a)=0,Ib-1-αDa+αx(b)=0
or
(144)Ia+1-αx(a)=0,x(b)=0,
in the space Wa+α,2. In [4], system (142) with nonhomogeneous boundary conditions
(145)Ia+1-αx(a)=C,x(b)=D,
where C,D∈ℝm, is investigated using the Stampacchia theorem (cf. [1]). Nonlinear system of the form
(146)Db-αDa+αx(t)=-Fx(t,x(t)),t∈[a,b]a.e.,
where Fx is the gradient (in x) of a potential F=F(t,x), with the above boundary conditions can be studied using the direct method of calculus of variations (cf. [11] for the case of (144)).
It is worth noting that ifα∈(1/2,1), x∈Wa+α,2, and Ia+1-αx(a)=0, then x(a)=0. It follows from the integral representation of x and from the fact that Ia+αf is Hölder continuous on (a,b] and limt→a+Ia+αf(t)=0 when f∈L2 and α∈(1/2,1) (cf. [10, Property 4]). So, it is natural to put in such a case x(a)=0. It seems that the most accurate functions for investigating the above systems with boundary conditions involving condition x(a)=0 (or more general x(a)=C), in general case of α∈(0,1), are the functions possessing the fractional derivatives in Caputo sense; on such functions one assumes that they are absolutely continuous on [a,b] and, consequently, condition x(a)=0 makes sense. To our best knowledge, fractional Sobolev spaces via Caputo derivatives have not been investigated up to now and are an open problem.Remark 40.
From the condition (136) it follows that one can search approximate solutions to (132) using numerical methods, for example, the gradient or projection of gradient methods applied to the functional
(147)h⟼12∥h∥Wa+α,22-∫ab‍f(t)h(t)dt
defined on H and Wa+α,2, respectively.



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