Fractional Sobolev Spaces via Riemann-Liouville Derivatives

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.

1. Introduction

Let 1≤𝑝<∞. Classical Sobolev space 𝑊^1,𝑝=𝑊^1,𝑝((𝑎,𝑏),ℝ^𝑚) of order one on an open bounded interval (𝑎,𝑏)⊂ℝ is defined by (cf. [1]) 𝑊^1,𝑝=𝑢∈𝐿^𝑝;∃_{𝑔∈𝐿𝑝}∀_{𝜑∈𝐶∞𝑐}^𝑏_𝑎𝑢(𝑡)𝜑⁽¹⁾(𝑡)𝑑𝑡=−^𝑏_𝑎𝑔(𝑡)𝜑(𝑡)𝑑𝑡, (1) where 𝐿^𝑝=𝐿^𝑝((𝑎,𝑏),ℝ^𝑚) is the space of functions 𝑔∶(𝑎,𝑏)→ℝ^𝑚 that are integrable with power 𝑝, 𝐶^∞_𝑐=𝐶^∞_𝑐((𝑎,𝑏),ℝ^𝑚) is the set of smooth functions 𝜑∶(𝑎,𝑏)→ℝ^𝑚 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 space 𝑊^𝑛,𝑝=𝑊^𝑛,𝑝((𝑎,𝑏),ℝ^𝑚) of order 𝑛>1 is defined by (cf. [1]) 𝑊^𝑛,𝑝=𝑢∈𝑊^{𝑛−1,𝑝};𝑢^′∈𝑊^{𝑛−1,𝑝}. (2) The spaces 𝑊^𝑛,𝑝 can be characterized as follows.

Theorem 1. Let 𝑛∈ℕ. Then, 𝑢∈𝑊^𝑛,𝑝 if and only if there exist functions 𝑔₁,…,𝑔_𝑛∈𝐿^𝑝 such that ^𝑏_𝑎𝑢(𝑡)𝜑^(𝑗)(𝑡)𝑑𝑡=(−1)^𝑗𝑏_𝑎𝑔_𝑗(𝑡)𝜑(𝑡)𝑑𝑡 (3) for 𝜑∈𝐶^∞_𝑐 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]→ℝ^𝑚 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⁾∈𝐿^𝑝, and 𝑔₁=𝑢⁽¹⁾,…,𝑔_𝑛=𝑢^(𝑛). (4)

Remark 2. One shows that 𝑔₁=𝑢^′, 𝑔₂=(𝑢^′)^′,…, 𝑔_𝑛=(⋯(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 functions 𝑔_𝑗, 𝑗=1,…,𝑛, is denoted by 𝐷^𝑗𝑢 and called the weak derivative of 𝑢 of order 𝑗.

The space 𝑊^𝑛,𝑝 endowed with a norm ‖𝑢‖^𝑝_{𝑊𝑛,𝑝}=^𝑛_𝑖=0𝐷^𝑖𝑢^𝑝_𝐿𝑝,𝑢∈𝑊^𝑛,𝑝, (5) 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) order 𝛼, 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⊂ℝ^𝑛, are known (cf. [2]): Gagliardo spaces 𝑊^𝛼,𝑝(Ω), Besov spaces 𝐵^𝛼,𝑝(Ω), and Nikolskii spaces 𝐻^𝛼,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 order 𝛼>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

By 𝐴𝐶^𝑛,1=𝐴𝐶^𝑛,1([𝑎,𝑏],ℝ^𝑚), where 𝑛∈ℕ, we denote the set of all functions 𝑓∶[𝑎,𝑏]→ℝ^𝑚 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,…,𝐷^𝑛𝑓 of 𝑓. It is known (cf. [3, Lemma 2.4]) that 𝑓∈𝐴𝐶^𝑛,1 if and only if there exist 𝑐₀,𝑐₁,…,𝑐_𝑛−1∈ℝ^𝑚 and 𝜑∈𝐿¹ such that 𝑓(𝑡)=^𝑛−1_𝑖=0𝑐_𝑖()𝑖!𝑡−𝑎^𝑖+^𝑡_𝑎^𝑡1_𝑎⋯^𝑡𝑛−1_𝑎𝜑(𝜏)𝑑𝜏𝑑𝑡_𝑛−1⋯𝑑𝑡₁,𝑡∈[𝑎,𝑏]a.e. (6) In such a case, 𝐷^𝑖𝑓(𝑎)=𝑐_𝑖𝐷,𝑖=0,…,𝑛−1,_𝑛𝑓(𝑡)=𝜑(𝑡),𝑡∈[𝑎,𝑏]a.e. (7)

By 𝐴𝐶^𝑛,𝑝=𝐴𝐶^𝑛,𝑝([𝑎,𝑏],ℝ^𝑚) we denote the set of all functions 𝑓∶[𝑎,𝑏]→ℝ^𝑚 that have representation (6) with 𝑐₀,𝑐₁,…,𝑐_𝑛−1∈ℝ^𝑚 and 𝜑∈𝐿^𝑝. It is known that 𝐴𝐶^𝑛,𝑝=𝑊^𝑛,𝑝 for 1≤𝑝<∞.

Let 𝛼>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 𝐼^𝛼_𝑎+ℎ given by 𝐼^𝛼_𝑎+ℎ𝑡)=1(Γ(𝛼)^𝑡_𝑎ℎ(𝜏)(𝑡−𝜏)^1−𝛼𝑑𝜏,𝑡∈[𝑎,𝑏]a.e. (8)

Remark 4. The above integral exists and is finite a.e. on [𝑎,𝑏].

In view of the convergence (cf. [3, Theorem 2.7]) lim_𝛼→0+𝐼^𝛼_𝑎+ℎ(𝑡)=ℎ(𝑡),𝑡∈[𝑎,𝑏]a.e., (9) it is natural to put 𝐼⁰_𝑎+ℎ(𝑡)=ℎ(𝑡),𝑡∈[𝑎,𝑏]a.e. (10)

Remark 5. If 𝛼≥1, the right side of (8) exists (and is finite) everywhere on [𝑎,𝑏]. So, 𝐼^𝛼_𝑎+ℎ 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 if ℎ 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 𝐼^𝛼_𝑎+ℎ can be defined everywhere on (𝑎,𝑏]. From [3, Theorem 3.6] it follows that in such a case 𝐼^𝛼_𝑎+ℎ 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∈𝐿^𝑝 with 1≤𝑝<∞, then 𝐼^𝛼_𝑎+ℎ∈𝐿^𝑝.

Let 𝑛−1<𝛼≤𝑛, where 𝑛∈ℕ. By 𝐴𝐶^𝛼,1_𝑎+=𝐴𝐶^𝛼,1_𝑎+([𝑎,𝑏],ℝ^𝑚) we denote the set of all functions 𝑓∶[𝑎,𝑏]→ℝ^𝑚 that have the representation 𝑓(𝑡)=^𝑛−1_𝑖=0𝑐_𝑖Γ()()𝛼−𝑛+1+𝑖𝑡−𝑎^{𝛼−𝑛+𝑖}+𝐼^𝛼_𝑎+𝜑(𝑡),𝑡∈[𝑎,𝑏]a.e., (11) with 𝑐₀,𝑐₁,…,𝑐_𝑛−1∈ℝ^𝑚 and 𝜑∈𝐿¹. Of course, 𝐴𝐶^𝑛,1_𝑎+=𝐴𝐶^𝑛,1.

We say that 𝑓∈𝐿¹ possesses the left Riemann-Liouville derivative 𝐷^𝛼_𝑎+𝑓 of order 𝛼∈(𝑛−1,𝑛], 𝑛∈ℕ, on the interval [𝑎,𝑏] if 𝐼^𝑛−𝛼_𝑎+𝑓∈𝐴𝐶^𝑛,1. By this derivative we mean the derivative 𝐷^𝑛(𝐼^𝑛−𝛼_𝑎+𝑓). Of course, 𝐷^𝑛_𝑎+𝑓=𝐷^𝑛𝑓.

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 𝐴𝐶^𝑛,1.

Theorem 7. If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, and 𝑓∈𝐿¹, then 𝑓 has the left Riemann-Liouville derivative 𝐷^𝛼_𝑎+𝑓 of order 𝛼 on the interval [𝑎,𝑏] if and only if 𝑓∈𝐴𝐶^𝛼,1_𝑎+; that is, 𝑓 has the representation (11). In such a case, 𝐷^𝑖𝐼^𝑛−𝛼_𝑎+𝑓(𝑎)=𝑐_𝑖𝐷,𝑖−0,…,𝑛−1,_𝛼𝑎+𝑓(𝑡)=𝜑(𝑡),𝑡∈[𝑎,𝑏]𝑎.𝑒. (12)

By 𝐴𝐶^𝛼,𝑝_𝑎+ (1≤𝑝<∞) we denote the set of all functions 𝑓∶[𝑎,𝑏]→ℝ^𝑚 possessing representation (11) with 𝑐₀,𝑐₁,…,𝑐_𝑛−1∈ℝ^𝑚, 𝜑∈𝐿^𝑝.

Remark 8. It is easy to see that the above theorem implies the following one (for any 1≤𝑝<∞): 𝑓 has the left Riemann-Liouville derivative 𝐷^𝛼_𝑎+𝑓∈𝐿^𝑝 if and only if 𝑓∈𝐴𝐶^𝛼,𝑝_𝑎+; that is, 𝑓 has the representation (11) with 𝜑∈𝐿^𝑝.

Let 𝛼>0. By the right Riemann-Liouville fractional integral of ℎ∈𝐿¹ on the interval [𝑎,𝑏] we mean a function 𝐼^𝛼_𝑏−ℎ(𝑡)=1Γ(𝛼)^𝑏_𝑡ℎ(𝜏)(𝜏−𝑡)^1−𝛼𝑑𝜏,𝑡∈[𝑎,𝑏]a.e. (13) Similarly, as in the case of left integral, we put 𝐼⁰_𝑏−ℎ(𝑡)=ℎ(𝑡),𝑡∈[𝑎,𝑏]a.e. (14)

Remark 9. Clearly, 𝐼^𝛼_𝑏−𝜑 has the properties analogous to those described in Remarks 5 and 6.

By 𝐴𝐶^𝛼,1_𝑏−=𝐴𝐶^𝛼,1_𝑏−([𝑎,𝑏],ℝ^𝑚) we denote the set of all functions 𝑓∶[𝑎,𝑏]→ℝ^𝑚 that have the representation 𝑓(𝑡)=^𝑛−1_𝑖=0(−1)^𝑖𝑑_𝑖Γ()()𝛼−𝑛+1+𝑖𝑏−𝑡^{𝛼−𝑛+𝑖}+𝐼^𝛼_𝑏−𝜓(𝜏),𝑡∈[𝑎,𝑏]a.e., (15) with 𝑑₀,𝑑₁,…,𝑑_𝑛−1∈ℝ^𝑚 and 𝜓∈𝐿¹. It is easy to see that 𝐴𝐶^𝑛,1_𝑏−=𝐴𝐶^𝑛,1.

We say that 𝑓∈𝐿¹ possesses the right Riemann-Liouville derivative 𝐷^𝛼_𝑏−𝑓 of order 𝛼∈(𝑛−1,𝑛], 𝑛∈ℕ, on the interval [𝑎,𝑏] if 𝐼^𝑛−𝛼_𝑏−𝑓∈𝐴𝐶^𝑛,1. By this derivative we mean the function (−1)^𝑛𝐷^𝑛(𝐼^1−𝛼_𝑏−𝑓). Of course, 𝐷^𝑛_𝑏−𝑓=(−1)^𝑛𝐷^𝑛𝑓.

We also have the following.

Theorem 10. If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, and 𝑓∈𝐿¹, then 𝑓 has the right Riemann-Liouville derivative 𝐷^𝛼_𝑏−𝑓 of order 𝛼 on the interval [𝑎,𝑏] if and only if 𝑓∈𝐴𝐶^𝛼,1_𝑏−𝑓; that is, 𝑓 has the representation (15). In such a case, 𝐷^𝑖𝐼^𝑛−𝛼_𝑏−𝑓(𝑏)=𝑑_𝑖𝐷,𝑖=0,…,𝑛−1,_𝛼𝑏−𝑓(𝑡)=𝜓(𝑡),𝑡∈[𝑎,𝑏]𝑎.𝑒. (16)

By 𝐴𝐶^𝛼,𝑝_𝑏−, 1<𝑝<∞, we denote the set of all functions 𝑓∶[𝑎,𝑏]→ℝ^𝑚 possessing representation (15) with 𝑑₀,𝑑₁,…,𝑑_𝑛−1∈ℝ^𝑚, 𝜓∈𝐿^𝑝.

Remark 11. As in the case of left Riemann-Liouville derivative, the following theorem holds true for any 1≤𝑝<∞: 𝑓 has the right Riemann-Liouville derivative 𝐷^𝛼_𝑏−𝑓∈𝐿^𝑝 if and only if 𝑓∈𝐴𝐶^𝛼,𝑝_𝑏−; that is, 𝑓 has the representation (15) with 𝜓∈𝐿^𝑝.

In the next section, we will use the following two theorems (cf. [6, 7]).

Theorem 12. If 𝛼,𝛽>0, then 𝐴𝐶^𝛼,1_𝑎+⊂𝐴𝐶^𝛽,1_𝑎+ (17) if and only if 𝛽∈(0,1],𝛼∈^∞_𝑖=0𝛽+𝑖,1+𝑖 (18) or 𝛽∈(1,∞),𝛼=𝛽+𝑖,𝑖=0,1,…. (19)

Theorem 13. (a) If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, 0<𝛽<𝛼−𝑛+1, and 𝑥∈𝐴𝐶^𝛼,1_𝑎+ is of the form (11), then 𝑥(𝑡)=𝐼^𝛽_𝑎+𝜓(𝑡),𝑡∈[𝑎,𝑏]𝑎.𝑒., (20) where 𝜓∈𝐿¹ is given by 𝜓(𝑡)=^𝑛−1_𝑖=0𝑐_𝑖Γ()𝛼−𝑛+1−𝛽+𝑖𝑡−𝑎^{𝛼−𝑛−𝛽+𝑖}+𝐼^𝛼−𝛽_𝑎+𝜑(𝑡),𝑡∈[𝑎,𝑏]𝑎.𝑒. (21) (b) If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, 𝑛≥2, 𝛽=𝛼−𝑗, 𝑗∈{1,…,𝑛−1}, and 𝑥∈𝐴𝐶^𝛼,1_𝑎+ are of the form (11), then 𝑥(𝑡)=^{(𝑛−𝑗)−1}_𝑖=0𝑑_𝑖Γ()𝛽−𝑛−𝑗+1+𝑖𝑡−𝑎^{𝛽−(𝑛−𝑗)+𝑖}+𝐼^𝛽_𝑎+𝜓(𝑡),𝑡∈[𝑎,𝑏]𝑎.𝑒., (22) where 𝑑_𝑖=𝑐_𝑖𝜓(𝑡)=,𝑖=0,…,𝑛−𝑗−1,_{𝑗−1𝑖=0}𝐼^𝑖_𝑎+𝑐_{𝑛−𝑗+𝑖}(𝑡)+𝐼^𝑗_𝑎+𝜑(𝑡),𝑡∈[𝑎,𝑏]𝑎.𝑒. (23)(24)

3. Some Special Cases of Integration by Parts

Below, 𝐶=𝐶([𝑎,𝑏],ℝ^𝑚) is the set of continuous functions 𝑓∶[𝑎,𝑏]→ℝ^𝑚 and 𝐶^𝑛=𝐶^𝑛([𝑎,𝑏],ℝ^𝑚) is the set of functions 𝑓∶[𝑎,𝑏]→ℝ^𝑚 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,…,𝐷^𝑛𝑓.

Lemma 14. If 𝑛−1<𝛼<𝑛, 𝑛∈ℕ, 𝑓∈𝐶^𝑛, and 𝐷^𝑖𝑓(𝑏)=0,𝑖=0,…,𝑛−1, (25) then 𝐼^𝑛−𝛼_𝑏−𝑓∈𝐶^𝑛, 𝐷^𝑖𝐼^𝑛−𝛼_𝑏−𝑓(𝑏)𝐷=0,𝑖=0,…,𝑛−1,_𝛼𝑏−𝑓=𝐼^𝑛−𝛼_𝑏−𝐷^𝑛_𝑏−𝑓=𝐼^𝑛−𝛼_𝑏−(−1)^𝑛𝐷^𝑛𝑓∈𝐶. (26) Consequently, 𝑓∈𝐼^𝛼_𝑏−(𝐶).

Proof. If 𝑓∈𝐶^𝑛 and 𝐷^𝑖𝑓(𝑏)=0,𝑖=0,…,𝑛−1, (27)
then (cf. Remark 6 and [3, formulas (2.21) and (2.58)]) 𝐼^𝑛−𝛼_𝑏−𝑓(𝑡)=𝐼^𝑛−𝛼_𝑏−𝐼^𝑛_𝑏−𝐷^𝑛_𝑏−𝑓(𝑡)=𝐼^𝑛_𝑏−𝐼^𝑛−𝛼_𝑏−𝐷^𝑛_𝑏−𝑓(𝑡)=𝐼^𝑛_𝑏−𝐼^𝑛−𝛼_𝑏−(−1)^𝑛𝐷^𝑛𝑓(𝑡) (28) for all 𝑡∈[𝑎,𝑏]. This means (cf. Remark 6) that 𝐼^𝑛−𝛼_𝑏−𝑓∈𝐶^𝑛 and 𝐷^𝛼_𝑏−𝑓=𝐼^𝑛−𝛼_𝑏−((−1)^𝑛𝐷^𝑛𝑓)∈𝐶. Equalities (28) imply also the equalities 𝐷^𝑖𝐼^𝑛−𝛼_𝑏−𝑓(𝑏)=0,𝑖=0,…,𝑛−1. (29) So (cf. Theorem 7), 𝑓∈𝐼^𝛼_𝑏−(𝐶).

Remark 15. Analogous lemma can be obtained for the left integral 𝐼^𝑛−𝛼_𝑎+𝑓 and derivative 𝐷^𝛼_𝑎+𝑓 (without the term (−1)^𝑛).

We have the following special case of the theorem on integration by parts.

Theorem 16. If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, then ^𝑏_𝑎𝑓(𝑡)𝐷^𝛼_𝑏−𝑔(𝑡)𝑑𝑡=^𝑏_𝑎𝐷^𝛼_𝑎+𝑓(𝑡)𝑔(𝑡)𝑑𝑡 (30) for any 𝑓∈𝐴𝐶^𝛼,1_𝑎+ and 𝑔∈𝐶^𝑛 satisfying boundary conditions 𝐷^𝑖𝑔(𝑎)=𝐷^𝑖𝑔(𝑏)=0,𝑖=0,…,𝑛−1. (31)

Proof. Let 𝑓∈𝐴𝐶^𝛼,1_𝑎+ be of the form (11) with 𝑐₀,𝑐₁,…,𝑐_𝑛−1∈ℝ^𝑚 and 𝜑∈𝐿¹. We have (Lemma 14 and Remark 9) ^𝑏_𝑎𝑓(𝑡)𝐷^𝛼_𝑏−𝑔(𝑡)=𝑑𝑡_𝑏𝑎𝑓(𝑡)𝐼^𝑛−𝛼_𝑏−𝐷^𝑛_𝑏−𝑔𝑡)=(𝑑𝑡_{𝑏𝑎𝑛−1𝑖=0}𝑐_𝑖Γ()()𝛼−𝑛+1+𝑖𝑡−𝑎^{𝛼−𝑛+𝑖}+𝐼^𝛼_𝑎+𝜑(𝜏)×𝐼^𝑛−𝛼_𝑏−𝐷^𝑛_𝑏−𝑔(𝑡)=𝑑𝑡_{𝑛−1𝑖=0}𝑐_𝑖𝐼^{𝛼−𝑛+1+𝑖}_𝑏−𝐼^𝑛−𝛼_𝑏−𝐷^𝑛_𝑏−𝑔(𝑎)+^𝑏_𝑎𝐷^𝛼_𝑎+𝑓(𝑡)𝐼^𝑛_𝑏−𝐷^𝑛_𝑏−𝑔(𝑡)=𝑑𝑡_{𝑛−1𝑖=0}𝑐_𝑖𝐼^1+𝑖_𝑏−𝐷^𝑛_𝑏−𝑔(𝑎)+^𝑏_𝑎𝐷^𝛼_𝑎+𝑓(𝑡)𝑔(𝑡)=𝑑𝑡_𝑏𝑎𝐷^𝛼_𝑎+𝑓(𝑡)𝑔(𝑡)𝑑𝑡. (32) 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 𝐷^𝑛𝑔 and in the fifth equality we used the equalities 𝐼¹_𝑏−𝐷^𝑛_𝑏−𝑔(𝑡)=^𝑏_𝑡𝐷^𝑛_𝑏−𝑔(𝑡)=()𝑑𝑡−1^𝑛𝑏_𝑡𝐷^𝑛𝑔𝑡)=()(𝑑𝑡−1^𝑛𝐷^𝑛−1𝑔(𝑏)−𝐷^𝑛−1𝑔(𝑡)=(−1)^𝑛−1𝐷^𝑛−1𝑔(𝑡)=𝐷^𝑛−1_𝑏−𝑔(𝑡). (33) 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. If 0<𝛼<1, 𝛼>1/𝑝, 𝛼>1/𝑞, 1≤𝑝<∞, and 1≤𝑞<∞, then ^𝑏_𝑎𝑓(𝑡)𝐷^𝛼_𝑏−𝑔(𝑡)𝑑𝑡=𝐼^1−𝛼_𝑎+𝑓(𝑎)𝑔(𝑎)−𝐼^1−𝛼_𝑏−𝑔(𝑏)𝑓(𝑏)+^𝑏_𝑎𝐷^𝛼_𝑎+𝑓(𝑡)𝑔(𝑡)𝑑𝑡 (34) for 𝑓∈𝐴𝐶^𝛼,𝑝_𝑎+([𝑎,𝑏],ℝ^𝑛) and 𝑔∈𝐴𝐶^𝛼,𝑞_𝑏−([𝑎,𝑏],ℝ^𝑛).

4. Fractional Sobolev Spaces

Let 0<𝛼≤1 and let 1≤𝑝<∞. By left Sobolev space of order 𝛼 we will mean the set 𝑊^𝛼,𝑝_𝑎+=𝑊^𝛼,𝑝_𝑎+((𝑎,𝑏),ℝ^𝑚) given by 𝑊^𝛼,𝑝_𝑎+=𝑢∈𝐿^𝑝;∃_{𝑔∈𝐿𝑝}∀_{𝜑∈𝐶∞𝑐}^𝑏_𝑎𝑢(𝑡)𝐷^𝛼_𝑏−𝜑(𝑡)=𝑑𝑡_𝑏𝑎𝑔(𝑡)𝜑(𝑡)𝑑𝑡. (35) 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 𝑢^𝛼_𝑎+. Uniqueness of this weak derivative follows from [1, Lemma IV.2 and Propositions IV.18, IV.21].

Let 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, 𝑛≥2. By the left Sobolev space of order 𝛼 we mean the set 𝑊^𝛼,𝑝_𝑎+=𝑊^𝛼,𝑝_𝑎+([𝑎,𝑏],ℝ^𝑚) given by 𝑊^𝛼,𝑝_𝑎+=𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+;𝑢^{𝛼−(𝑛−1)}_𝑎+∈𝑊^{𝑛−1,𝑝}_𝑎+. (36)

Remark 18. Since 𝐷¹_𝑏−𝜑=−𝐷¹𝜑=−𝜑⁽¹⁾ for 𝜑∈𝐶^∞_𝑐, we see that the weak left fractional derivative 𝑢¹_𝑎+ of 𝑢 coincides with the classical weak derivative 𝑢^′=𝐷¹𝑢 of 𝑢. Consequently, 𝑊^𝑛,𝑝_𝑎+=𝑊^𝑛,𝑝=𝐴𝐶^𝑛,𝑝=𝐴𝐶^𝑛,𝑝_𝑎+ (37) for 𝑛∈ℕ.

We have the following characterization of 𝑊^𝛼,𝑝_𝑎+.

Theorem 19. If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, and 1≤𝑝<∞, then 𝑊^𝛼,𝑝_𝑎+=𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝. (38)

Proof. Case of 𝛼=𝑛 follows from (37) and from the fact that 𝐴𝐶^𝑛,𝑝_𝑎+⊂𝐿^𝑝. So, let us consider the case of 𝑛−1<𝛼<𝑛. We will apply the induction with respect to 𝑛∈ℕ.
Let 𝑛=1. If 𝑢∈𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝, then from Theorem 7 it follows that 𝑢 has the derivative 𝐷^𝛼_𝑎+𝑢∈𝐿^𝑝. Theorem 16 implies that ^𝑏_𝑎𝑢(𝑡)𝐷^𝛼_𝑏−𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝐷^𝛼_𝑎+𝑢(𝑡)𝜑(𝑡)𝑑𝑡 (39) for any 𝜑∈𝐶^∞_𝑐. So, 𝑢∈𝑊^𝛼,𝑝_𝑎+ with 𝑢^𝛼_𝑎+=𝑔=𝐷^𝛼_𝑎+𝑢∈𝐿^𝑝. (40)
Now, let us assume that 𝑢∈𝑊^𝛼,𝑝_𝑎+, that is, 𝑢∈𝐿^𝑝, and there exists a function 𝑔∈𝐿^𝑝 such that ^𝑏_𝑎𝑢(𝑡)𝐷^𝛼_𝑏−𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝑔(𝑡)𝜑(𝑡)𝑑𝑡 (41) for any 𝜑∈𝐶^∞_𝑐. To show that 𝑢∈𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝 it is sufficient to check (cf. Theorem 7 and definition of 𝐴𝐶^𝛼,𝑝_𝑎+) that 𝑢 possesses the left Riemann-Liouville derivative of order 𝛼, belonging to 𝐿^𝑝, that is, that 𝐼^1−𝛼_𝑎+𝑢 is absolutely continuous on [𝑎,𝑏] and its classical derivative of the first order (existing a.e. on [𝑎,𝑏]) belongs to 𝐿^𝑝.
If 𝜑∈𝐶^∞_𝑐, then (cf. Lemma 14) 𝜑∈𝐼^𝛼_𝑏−(𝐶) and 𝐷^𝛼_𝑏−𝜑=𝐼^1−𝛼_𝑏−(−𝐷¹𝜑). From the differential form of the classical fractional theorem on the integration by parts (cf. [3, formula (2.64)]) it follows that ^𝑏_𝑎𝑢(𝑡)𝐷^𝛼_𝑏−𝜑(𝑡)=𝑑𝑡_𝑏𝑎𝑢(𝑡)𝐼^1−𝛼_𝑏−−𝐷¹𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝐷^1−𝛼_𝑎+𝐼^1−𝛼_𝑎+𝑢(𝑡)𝐼^1−𝛼_𝑏−−𝐷¹𝜑(𝑡)=𝑑𝑡_𝑏𝑎𝐼^1−𝛼_𝑎+𝑢(𝑡)−𝐷¹𝜑(𝑡)𝑑𝑡=−^𝑏_𝑎𝐼^1−𝛼_𝑎+𝑢(𝑡)𝐷¹𝜑(𝑡)𝑑𝑡. (42) This means (cf. (41) and (42)) that ^𝑏_𝑎𝐼^1−𝛼_𝑎+𝑢(𝑡)𝐷¹𝜑(𝑡)𝑑𝑡=−^𝑏_𝑎𝑔(𝑡)𝜑(𝑡)𝑑𝑡 (43) for any 𝜑∈𝐶^∞_𝑐. So, 𝐼^1−𝛼_𝑎+𝑢∈𝑊^1,𝑝. Consequently, 𝐼^1−𝛼_𝑎+𝑢 is absolutely continuous and its classical derivative is equal a.e. on [𝑎,𝑏] to 𝑔∈𝐿^𝑝.
Now, let 𝑛∈ℕ and assume that 𝑊^𝛼,𝑝_𝑎+=𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝 (44) for any 𝛼∈(𝑛−1,𝑛). We will show that this equality holds true for any 𝛼∈(𝑛,𝑛+1).
Indeed, let 𝛼∈(𝑛,𝑛+1). If 𝑢∈𝑊^𝛼,𝑝_𝑎+, then 𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+=𝐴𝐶^{𝛼−1,𝑝}_𝑎+∩𝐿^𝑝 and 𝑢^𝛼−𝑛_𝑎+∈𝑊^𝑛,𝑝_𝑎+. So, 𝑢(𝑡)=^𝑛−1_𝑖=0𝑐_𝑖Γ()()𝛼−𝑛+𝑖𝑡−𝑎^{𝛼−1−𝑛+𝑖}+𝐼^𝛼−1_𝑎+𝜑(𝑡),𝑡∈[𝑎,𝑏]a.e., (45) where 𝑐₀,𝑐₁,…,𝑐_𝑛−1∈ℝ^𝑚 and 𝜑∈𝐿^𝑝. If 𝑛=1, from the above formula it follows (cf. (40)) that 𝜑=𝐷^𝛼−1_𝑎+𝑢=𝑢^𝛼−1_𝑎+∈𝑊^1,𝑝_𝑎+=𝑊^1,𝑝. (46) If 𝑛≥2, from Theorem 13 (b) and (40) it follows that 𝑢^𝛼−𝑛_𝑎+=𝐷^𝛼−𝑛_𝑎+𝑢=𝐷^{(𝛼−1)−(𝑛−1)}_𝑎+𝑢=^𝑛−2_𝑖=0𝐼^𝑖_𝑎+𝑐_1+𝑖+𝐼^𝑛−1_𝑎+𝜑. (47) This means that 𝐼^𝑛−1_𝑎+𝜑∈𝑊^𝑛,𝑝_𝑎+ and, consequently, 𝜑=𝐷^𝑛−1_𝑎+𝐼^𝑛−1_𝑎+𝜑=𝐷^𝑛−1𝐼^𝑛−1_𝑎+𝜑∈𝐷^𝑛−1𝑊^𝑛,𝑝_𝑎+=𝐷^𝑛−1𝑊^𝑛,𝑝=𝑊^1,𝑝. (48) So, in both cases (𝑛=1, 𝑛≥2) there exist 𝑐_𝑛∈ℝ^𝑚 and 𝜆∈𝐿^𝑝 such that 𝜑=𝑐_𝑛+𝐼¹_𝑎+𝜆. (49) To show that 𝑢∈𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝 it is sufficient to check that 𝐼^{𝑛+1−𝛼}_𝑎+𝑢∈𝐴𝐶^𝑛+1,𝑝_𝑎+. (50) Indeed, we have (below, we use the following elementary formula 𝐼^𝜈_𝑎+(⋅−𝑎)^𝛿−1(𝑡)=Γ(𝛿)Γ()()𝜈+𝛿𝑡−𝑎^{𝜈+𝛿−1},𝑡∈[𝑎,𝑏]a.e., (51) for 𝜈>0, 𝛿>0) 𝐼^{𝑛+1−𝛼}_𝑎+𝑢(𝑡)=^𝑛−1_𝑖=0𝑐_𝑖Γ()𝐼𝛼−𝑛+𝑖^{𝑛+1−𝛼𝑎+}(⋅−𝑎)^{𝛼−1−𝑛+𝑖}(𝑡)+𝐼^{𝑛+1−𝛼}_𝑎+𝐼^𝛼−1_𝑎+𝜑(𝑡)=^𝑛−1_𝑖=0𝑐_𝑖Γ()Γ()𝛼−𝑛+𝑖𝛼−𝑛+𝑖Γ()()1+𝑖𝑡−𝑎^𝑖+𝐼^𝑛_𝑎+𝜑(𝑡)=^𝑛−1_𝑖=0𝑐_𝑖Γ(1+𝑖)+𝐼^𝑛_𝑎+𝑐_𝑛(𝑡)+𝐼^𝑛+1_𝑎+𝜆(𝑡)=^𝑛_𝑖=0𝑐_𝑖()𝑖!𝑡−𝑎^𝑖+𝐼^𝑛+1_𝑎+𝜆(𝑡) (52) for 𝑡∈[𝑎,𝑏] a.e. Thus, 𝐼^{𝑛+1−𝛼}_𝑎+𝑢∈𝐴𝐶^𝑛+1,𝑝_𝑎+.
Conversely, let 𝑢∈𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝. To show that 𝑢∈𝑊^𝛼,𝑝_𝑎+ we have to check that 𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+,𝑢^𝛼−𝑛_𝑎+∈𝑊^𝑛,𝑝_𝑎+. (53) Theorem 13(b) implies the equality 𝐷^𝛼−1_𝑎+𝑢=𝑐_𝑛+𝐼¹_𝑎+𝜑, (54) where 𝑢(𝑡)=^𝑛_𝑖=0𝑐_𝑖Γ()()𝛼−𝑛+𝑖𝑡−𝑎^{𝛼−(𝑛+1)+𝑖}+𝐼^𝛼_𝑎+𝜑(𝑡),𝑡∈[𝑎,𝑏]a.e., (55) with 𝜑∈𝐿^𝑝. So (cf. Theorem 12), 𝑢∈𝐴𝐶^{𝛼−1,𝑝}_𝑎+∩𝐿^𝑝=𝑊^{𝛼−1,𝑝}_𝑎+. From Theorem 13 (b) it also follows that 𝑢^𝛼−𝑛_𝑎+=𝐷^𝛼−𝑛_𝑎+𝑢=^𝑛−1_𝑖=0𝐼^𝑖_𝑎+𝑐_1+𝑖+𝐼^𝑛_𝑎+𝜑=^𝑛−1_𝑖=0𝑐_1+𝑖()𝑖!𝑡−𝑎^𝑖+𝐼^𝑛_𝑎+𝜑∈𝐴𝐶^𝑛,𝑝_𝑎+=𝑊^𝑛,𝑝_𝑎+ (56) (existence of the weak fractional derivative 𝑢^𝛼−𝑛_𝑎+ and equality 𝑢^𝛼−𝑛_𝑎+=𝐷^𝛼−𝑛_𝑎+𝑢 follow from the relation 𝑢∈𝑊^{𝛼−𝑛,𝑝}_𝑎+ and (40)).

From the first part of the above proof (case of 𝑛=1) and from the uniqueness of the weak fractional derivative the following theorem follows (cf. also [4, fractional fundamental lemma]).

Theorem 20. If 𝛼∈(0,1], then the weak left fractional derivative 𝑢^𝛼_𝑎+ of a function 𝑢∈𝑊^𝛼,𝑝_𝑎+ coincides with its left Riemann-Liouville fractional derivative 𝐷^𝛼_𝑎+𝑢 a.e. on [𝑎,𝑏].

Remark 21. If 𝑛−1<𝛼≤𝑛 and (𝑛−𝛼)𝑝<1, then 𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐿^𝑝 and, consequently, 𝑊^𝛼,𝑝_𝑎+=𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝=𝐴𝐶^𝛼,𝑝_𝑎+. If (𝑛−𝛼)𝑝≥1, then 𝑊^𝛼,𝑝_𝑎+=𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝 is the set of all functions belonging to 𝐴𝐶^𝛼,𝑝_𝑎+ that satisfy the condition (𝐼^𝑛−𝛼_𝑎+𝑓)(𝑎)=0.

5. Weak Fractional Derivatives

Now, we will prove an extension of Theorem 1.

Theorem 22. Let 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, 1≤𝑝<∞, and 𝑢∈𝐿^𝑝. Then, 𝑢∈𝑊^𝛼,𝑝_𝑎+ if and only if there exist functions 𝑔₁,…,𝑔_𝑛∈𝐿^𝑝 such that ^𝑏_𝑎𝑢(𝑡)𝐷^{𝛼−(𝑛−𝑖)}_𝑏−𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝑔_𝑖(𝑡)𝜑(𝑡)𝑑𝑡,𝜑∈𝐶^∞_𝑐, (57) for any 𝑖∈{1,…,𝑛}. In such a case there exist the left Riemann-Liouville derivatives 𝐷^{𝛼−(𝑛−1)}_𝑎+𝑢,𝐷^{𝛼−(𝑛−2)}_𝑎+𝑢,…,𝐷^𝛼_𝑎+𝑢 of 𝑢 and 𝑔₁=𝐷^{𝛼−(𝑛−1)}_𝑎+𝑢,𝑔₂=𝐷^{𝛼−(𝑛−2)}_𝑎+𝑢,…,𝑔_𝑛=𝐷^𝛼_𝑎+𝑢. (58)

Proof. Case of 𝛼=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.
When 𝑛=1, it is sufficient to use the definition of 𝑊^𝛼,𝑝_𝑎+ with 𝛼∈(0,1) and Theorem 20.
So, let 𝑛∈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, if 𝑢∈𝑊^𝛼,𝑝_𝑎+ with a fixed 𝛼∈(𝑛,𝑛+1), then 𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+,𝐷^𝛼−𝑛_𝑎+𝑢=𝑢^𝛼−𝑛_𝑎+∈𝑊^𝑛,𝑝_𝑎+. (59)(60) Relation (59) implies (by the induction assumption) the existence of functions 𝑔₁,…,𝑔_𝑛∈𝐿^𝑝 such that (of course, 𝛼−1∈(𝑛−1,𝑛)) ^𝑏_𝑎𝑢(𝑡)𝐷^{𝛼−((𝑛+1)−𝑖)}_𝑏−𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝑢(𝑡)𝐷^{(𝛼−1)−(𝑛−𝑖)}_𝑏−𝜑(𝑡)=𝑑𝑡_𝑏𝑎𝑔_𝑖(𝑡)𝜑(𝑡)𝑑𝑡 (61) for any 𝜑∈𝐶^∞_𝑐 and 𝑖∈{1,…,𝑛}. Moreover, 𝑔₁=𝐷^{(𝛼−1)−(𝑛−1)}_𝑎+𝑢=𝐷^{𝛼−((𝑛+1)−1)}_𝑎+𝑔𝑢,₂=𝐷^{(𝛼−1)−(𝑛−2)}_𝑎+𝑢=𝐷^{𝛼−((𝑛+1)−2)}_𝑎+⋮𝑔𝑢,_𝑛=𝐷^{(𝛼−1)−(𝑛−𝑛)}_𝑎+𝑢=𝐷^{𝛼−((𝑛+1)−𝑛)}_𝑎+𝑢. (62) From the existence of 𝐷^𝛼−𝑛_𝑎+𝑢 and from (60) it follows that 𝐼^{1−(𝛼−𝑛)}_𝑎+𝑢∈𝑊^𝑛+1,𝑝. In particular, there exists derivative 𝐷^𝛼_𝑎+𝑢∈𝐿^𝑝. If we put 𝑔_𝑛+1=𝐷^𝛼_𝑎+𝑢 we see that 𝑔_𝑛+1=𝐷^{𝛼−((𝑛+1)−(𝑛+1))}_𝑎+𝑢 and ^𝑏_𝑎𝑢(𝑡)𝐷^{𝛼−((𝑛+1)−(𝑛+1))}_𝑏−𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝑢(𝑡)𝐷^𝛼_𝑏−𝜑(𝑡)=𝑑𝑡_𝑏𝑎𝐷^𝛼_𝑎+𝑢(𝑡)𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝑔_𝑛+1(𝑡)𝜑(𝑡)𝑑𝑡 (63) for any 𝜑∈𝐶^∞_𝑐 (the second equality follows from Theorem 16).
Now, let us assume that there exist functions 𝑔₁,…,𝑔_𝑛+1∈𝐿^𝑝 such that ^𝑏_𝑎𝑢(𝑡)𝐷^{𝛼−((𝑛+1)−𝑖)}_𝑏−𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝑔_𝑖(𝑡)𝜑(𝑡)𝑑𝑡,𝜑∈𝐶^∞_𝑐, (64) 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 𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+. (65)
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 ^𝑏_𝑎𝑔₁(𝑡)𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝑢(𝑡)𝐷^𝛼−𝑛_𝑏−𝜑(𝑡)=𝑑𝑡_𝑏𝑎𝑢(𝑡)𝐼^{𝑛−𝛼+1}_𝑏−𝐷¹_𝑏−𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝐼^{𝑛−𝛼+1}_𝑎+𝑢(𝑡)𝐷¹_𝑏−𝜑(𝑡)𝑑𝑡=−^𝑏_𝑎𝐼^{𝑛−𝛼+1}_𝑎+𝑢(𝑡)𝐷¹𝜑(𝑡)𝑑𝑡 (66) for any 𝜑∈𝐶^∞_𝑐. This means that 𝐼^{𝑛−𝛼+1}_𝑎+𝑢∈𝑊^1,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 ^𝑏_𝑎𝑔_𝑖(𝑡)𝜑(𝑡)=𝑑𝑡_𝑏𝑎𝑢(𝑡)𝐷^{𝛼−((𝑛+1)−𝑖)}_𝑏−𝜑(𝑡)𝑑𝑡=^𝑏_𝑎𝑢(𝑡)𝐼^{(𝑛+1)−𝛼}_𝑏−𝐷^𝑖_𝑏−𝜑(𝑡)=𝑑𝑡_𝑏𝑎𝐼^{(𝑛+1)−𝛼}_𝑎+𝑢(𝑡)𝐷^𝑖_𝑏−𝜑(𝑡)=()𝑑𝑡−1^𝑖𝑏_𝑎𝐼^{(𝑛+1)−𝛼}_𝑎+𝑢(𝑡)𝐷^𝑖𝜑(𝑡)=()𝑑𝑡−1^𝑖𝑏_𝑎𝐼^{(𝑛+1)−𝛼}_𝑎+𝑢(𝑡)𝐷¹𝐷^𝑖−1𝜑(𝑡)=()𝑑𝑡−1^𝑖−1𝑏_𝑎𝐷¹𝐼^{1−(𝛼−𝑛)}_𝑎+𝑢(𝑡)𝐷^𝑖−1𝜑(𝑡)=()𝑑𝑡−1^𝑖−1𝑏_𝑎𝐷^𝛼−𝑛_𝑎+𝑢(𝑡)𝐷^𝑖−1𝜑(𝑡)𝑑𝑡. (67) Thus, from Theorem 1 it follows that 𝑢^𝛼−𝑛_𝑎+=𝐷^𝛼−𝑛_𝑎+𝑢∈𝑊^𝑛,𝑝. (68) Using (65) and (68) we assert that 𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+.

Functions 𝑔₁,…,𝑔_𝑛 will be called the weak left fractional derivatives of 𝑢∈𝑊^𝛼,𝑝_𝑎+ 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 Remark 2.

Theorem 23. If 𝑢∈𝑊^𝛼,𝑝_𝑎+, 𝑛−1<𝛼≤𝑛, and 𝑛∈ℕ, then 𝑔_𝑖=𝐷^𝑖−1𝐷^{𝛼−(𝑛−1)}_𝑎+𝑢,𝑖=1,…,𝑛. (69)

Proof. If 𝑢∈𝑊^𝛼,𝑝_𝑎+, then 𝐼^𝑛−𝛼_𝑎+𝑢∈𝑊^𝑛,𝑝 and from the definition of the Riemann-Liouville derivative and Theorem 22 it follows that 𝑔₁=𝐷^{𝛼−𝑛+1}_𝑎+𝑢=𝐷¹𝐼^{1−(𝛼−𝑛+1)}_𝑎+𝑢=𝐷¹𝐼^𝑛−𝛼_𝑎+𝑢,𝑔₂=𝐷^{𝛼−𝑛+2}_𝑎+𝑢=𝐷²𝐼^{2−(𝛼−𝑛+2)}_𝑎+𝑢=𝐷²𝐼^𝑛−𝛼_𝑎+𝑢=𝐷¹𝐷¹𝐼^𝑛−𝛼_𝑎+𝑢=𝐷¹𝐷^{𝛼−𝑛+1}_𝑎+𝑢,⋮𝑔_𝑛=𝐷^{𝛼−𝑛+𝑛}_𝑎+𝑢=𝐷^𝑛𝐼^{𝑛−(𝛼−𝑛+𝑛)}_𝑎+𝑢=𝐷^𝑛−1𝐷¹𝐼^𝑛−𝛼_𝑎+𝑢=𝐷^𝑛−1𝐷^{𝛼−𝑛+1}_𝑎+𝑢. (70) The proof is completed.

6. Norms in 𝑊^𝛼,𝑝_𝑎+

Let us fix 𝛼∈(𝑛−1,𝑛] where 𝑛∈ℕ and consider in the space 𝑊^𝛼,𝑝_𝑎+ a norm ‖⋅‖_𝑊^𝛼,𝑝_𝑎+ given by ‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+}=‖𝑢‖^𝑝_𝐿𝑝+^𝑛−1_𝑖=0𝐷^{𝛼−(𝑛−1)+𝑖}_𝑎+𝑢^𝑝_𝐿𝑝,𝑢∈𝑊^𝛼,𝑝_𝑎+ (71) (here ‖⋅‖_𝐿^𝑝 denotes the classical norm in 𝐿^𝑝).

We have the following theorem.

Theorem 24. If 𝑛∈ℕ and 𝛼∈(𝑛−1,𝑛], then the norm ‖⋅‖_𝑊^𝛼,𝑝_𝑎+ is equivalent to a norm ‖𝑢‖_𝑎,𝑊^𝛼,𝑝_𝑎+ given by ‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+}=^𝑛−1_𝑖=0𝐷^𝑖𝐼^𝑛−𝛼_𝑎+𝑢(𝑎)^𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝,𝑢∈𝑊^𝛼,𝑝_𝑎+. (72)

Proof. We will use the induction with respect to 𝑛∈ℕ.
Assume that 𝑛=1 and (1−𝛼)𝑝<1. Then, for 𝑢∈𝑊^𝛼,𝑝_𝑎+ given by 𝑢(𝑡)=𝑐Γ(𝛼)1(𝑡−𝑎)^1−𝛼+𝐼^𝛼_𝑎+𝜑(𝑡) (73) with 𝑐∈ℝ^𝑛 and 𝜑∈𝐿^𝑝, we have ‖𝑢‖^𝑝_𝐿𝑝=^𝑏_𝑎𝑐1Γ(𝛼)(𝑡−𝑎)^1−𝛼+𝐼^𝛼_𝑎+𝜑(𝑡)^𝑝𝑑𝑡≤2^𝑝−1|𝑐|^𝑝Γ(𝛼)^𝑝𝑏_𝑎(𝑡−𝑎)^{(𝛼−1)𝑝}𝑑𝑡+𝐼^𝛼_𝑎+𝜑^𝑝_𝐿𝑝≤2^𝑝−1|𝑐|^𝑝Γ(𝛼)^𝑝1()()𝛼−1𝑝+1𝑏−𝑎^{(𝛼−1)𝑝+1}+𝐾^𝑝𝜑^𝑝_𝐿𝑝, (74) where 𝐾=(𝑏−𝑎)^𝛼/Γ(𝛼+1) (cf. [3, formula (2.72)]). Thus (of course, 𝑐=𝐼^1−𝛼_𝑎+𝑢(𝑎) and 𝜑=𝐷^𝛼_𝑎+𝑢), ‖𝑢‖^𝑝_𝐿𝑝≤𝐿_𝛼,0|𝑐|^𝑝+𝜑^𝑝_𝐿𝑝=𝐿_𝛼,0𝐼^1−𝛼_𝑎+𝑢(𝑎)^𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝=𝐿_𝛼,0‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+}, (75) where 𝐿_𝛼,0=2^𝑝−1(𝑏−𝑎)^{1−(1−𝛼)𝑝}Γ(𝛼)^𝑝1−(1−𝛼)𝑝+𝐾^𝑝. (76)
Consequently, ‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+}=‖𝑢‖^𝑝_𝐿𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝≤𝐿_𝛼,1‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+}, (77) where 𝐿_𝛼,1=𝐿_𝛼,0+1.
Now, we will prove that there exists a constant 𝑀_𝛼,1>0 such that ‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+}≤𝑀_𝛼,1‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+},𝑢∈𝑊^𝛼,𝑝_𝑎+. (78) Indeed, let 𝑢∈𝑊^𝛼,𝑝_𝑎+ and consider a coordinate function (𝐼^1−𝛼_𝑎+𝑢)^𝑖 of 𝐼^1−𝛼_𝑎+𝑢 with a fixed 𝑖∈{1,…,𝑚}. The mean value theorem implies the existence of 𝑡₀∈(𝑎,𝑏) such that 𝐼^1−𝛼_𝑎+𝑢^𝑖𝑡₀=1𝑏−𝑎^𝑏_𝑎𝐼^1−𝛼_𝑎+𝑢^𝑖(𝑠)𝑑𝑠. (79) From the absolute continuity of (𝐼^1−𝛼_𝑎+𝑢)^𝑖 it follows that 𝐼^1−𝛼_𝑎+𝑢^𝑖(𝑡)=𝐼^1−𝛼_𝑎+𝑢^𝑖𝑡₀+^𝑡_𝑡0𝐷¹𝐼^1−𝛼_𝑎+𝑢^𝑖(𝑠)𝑑𝑠 (80) for any 𝑡∈[𝑎,𝑏]. Consequently, 𝐼^1−𝛼_𝑎+𝑢^𝑖(𝑡≤1)𝐼𝑏−𝑎^{1−𝛼𝑎+}𝑢_𝐿1+^𝑡_𝑡0𝐷^𝛼_𝑎+𝑢(𝑠≤1)𝑑𝑠𝐼𝑏−𝑎^{1−𝛼𝑎+}𝑢_𝐿1+𝐷^𝛼_𝑎+𝑢_𝐿1≤1()𝑏−𝑎𝑏−𝑎^1−𝛼Γ()‖𝑢‖2−𝛼^𝐿1+𝐷^𝛼_𝑎+𝑢_𝐿1 (81) for 𝑡∈[𝑎,𝑏]. In particular, 𝐼^1−𝛼_𝑎+𝑢^𝑖(𝑎≤1)()𝑏−𝑎𝑏−𝑎^1−𝛼Γ()‖𝑢‖2−𝛼^𝐿1+𝐷^𝛼_𝑎+𝑢_𝐿1. (82) So, 𝐼^1−𝛼_𝑎+𝑢(𝑎)≤𝑚𝑀_𝛼,0‖𝑢‖_𝐿¹+𝐷^𝛼_𝑎+𝑢_𝐿1≤𝑚𝑀_𝛼,0(𝑏−𝑎)^{(𝑝−1)/𝑝}‖𝑢‖_𝐿^𝑝+𝐷^𝛼_𝑎+𝑢_𝐿𝑝, (83) where 𝑀_𝛼,0=(𝑏−𝑎)^−𝛼/Γ(2−𝛼)+1. Thus, 𝐼^1−𝛼_𝑎+𝑢(𝑎)^𝑝≤𝑚^𝑝𝑀^𝑝_𝛼,0(𝑏−𝑎)^𝑝−12^𝑝−1×‖𝑢‖^𝑝_𝐿𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝 (84) and, consequently, ‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+}=𝐼^1−𝛼_𝑎+𝑢(𝑎)^𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝≤𝑚^𝑝𝑀^𝑝_𝛼,0(𝑏−𝑎)^𝑝−12^𝑝−1×‖𝑢‖+1_𝑝𝐿𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝=𝑀_𝛼,1‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+}, (85) where 𝑀_𝛼,1=𝑚^𝑝𝑀^𝑝_𝛼,0(𝑏−𝑎)^𝑝−12^𝑝−1+1.
When (1−𝛼)𝑝≥1, then (cf. Remark 21) 𝑊^𝛼,𝑝_𝑎+=𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝 is the set of all functions 𝑢 belonging to 𝐴𝐶^𝛼,𝑝_𝑎+ that satisfy the condition (𝐼^1−𝛼_𝑎+𝑢)(𝑎)=0. Consequently, in the same way as in the case of (1−𝛼)𝑝<1 (putting 𝑐=0) we obtain the inequality ‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+}≤𝐿_𝛼,1‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+} with some 𝐿_𝛼,1>0. The inequality ‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+}≤𝑀_𝛼,1‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+} with some 𝑀_𝛼,1>0 is obvious (it is sufficient to put 𝑀_𝛼,1=1 and use the fact that (𝐼^1−𝛼_𝑎+𝑢)(𝑎)=0).
Now, let us assume that the assertion holds true for some 𝑛∈ℕ. We will prove that it is true for 𝑛+1.
Let 𝛼∈(𝑛,𝑛+1]. We know that there exist constants 𝐿_𝛼,𝑛 and 𝑀_𝛼,𝑛>0 such that 1𝑀_𝛼,𝑛‖𝑢‖^𝑝_{𝑎,𝑊𝛼−1,𝑝𝑎+}≤‖𝑢‖^𝑝_{𝑊𝛼−1,𝑝𝑎+}≤𝐿_𝛼,𝑛‖𝑢‖^𝑝_{𝑎,𝑊𝛼−1,𝑝𝑎+} (86) for any 𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+. If 𝑢∈𝑊^𝛼,𝑝_𝑎+, then 𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+ and therefore (from the induction assumption) ‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+}=‖𝑢‖^𝑝_𝐿𝑝+^𝑛_𝑖=0𝐷^{𝛼−𝑛+𝑖}_𝑎+𝑢^𝑝_𝐿𝑝=‖𝑢‖^𝑝_𝐿𝑝+^𝑛−1_𝑖=0𝐷^{(𝛼−1)−(𝑛−1)+𝑖}_𝑎+𝑢^𝑝_𝐿𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝=‖𝑢‖^𝑝_{𝑊𝛼−1,𝑝𝑎+}+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝≤𝐿_{𝛼−1,𝑛}‖𝑢‖^𝑝_{𝑎,𝑊𝛼−1,𝑝𝑎+}+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝=𝐿_{𝛼−1,𝑛}^𝑛−1_𝑖=0𝐷^𝑖𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝+𝐷^𝛼−1_𝑎+𝑢^𝑝_𝐿𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝. (87) Since 𝐷^𝛼−1_𝑎+𝑢=𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢=𝐷^𝑛𝐼^{𝑛+1−𝛼}_𝑎+𝑢∈𝑊^1,𝑝_𝑎+ (because of 𝐼^{𝑛+1−𝛼}_𝑎+𝑢∈𝑊^𝑛+1,𝑝_𝑎+), therefore (cf. case of 𝑛=1) 𝐷^𝛼−1_𝑎+𝑢^𝑝_𝐿𝑝=𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢^𝑝_𝐿𝑝≤𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢^𝑝_{𝑊1,𝑝𝑎+}≤𝐿_1,1𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢^𝑝_{𝑎,𝑊1,𝑝𝑎+}=𝐿_1,1𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝+𝐷¹𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢_𝐿𝑝=𝐿_1,1𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝+𝐷^𝑛+1𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢_𝐿𝑝, (88) where 𝐿_1,1 is such that ‖𝑣‖^𝑝_{𝑊1,𝑝𝑎+}≤𝐿_1,1‖𝑣‖^𝑝_{𝑎,𝑊1,𝑝𝑎+} (89) for 𝑣∈𝑊^1,𝑝_𝑎+. Thus, ‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+}≤𝐿_{𝛼−1,𝑛}^𝑛−1_𝑖=0𝐷^𝑖𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝+𝐿_{𝛼−1,𝑛}𝐿_1,1×𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝+𝐷^𝑛+1𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢_𝐿𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝≤𝐿_𝛼,𝑛+1^𝑛_𝑖=0𝐷^𝑖𝐼^{𝑛+1−𝛼}_𝑎+𝑢(𝑎)^𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝=𝐿_𝛼,𝑛+1‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+}, (90) where 𝐿_𝛼,𝑛+1=𝐿_{𝛼−1,𝑛}+𝐿_{𝛼−1,𝑛}𝐿_1,1+1 (we used the equality 𝐷^𝑛+1𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢=𝐷^𝑛+1𝐼^{𝑛+1−𝛼}_𝑎+𝑢=𝐷^𝛼_𝑎+𝑢).
On the other hand, since 𝑢∈𝑊^{𝛼−1,𝑝}_𝑎+, therefore (from the induction assumption) ^𝑛−1_𝑖=0𝐷^𝑖𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝+𝐷^𝛼−1_𝑎+𝑢^𝑝_𝐿𝑝≤𝑀_{𝛼−1,𝑛}‖𝑢‖^𝑝_𝐿𝑝+^𝑛−1_𝑖=0𝐷^{𝛼−1−(𝑛−1)+𝑖}_𝑎+𝑢^𝑝_𝐿𝑝. (91) So, ^𝑛−1_𝑖=0𝐷^𝑖𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝≤𝑀_{𝛼−1,𝑛}‖𝑢‖^𝑝_𝐿𝑝+^𝑛−2_𝑖=0𝐷^{𝛼−1−(𝑛−1)+𝑖}_𝑎+𝑢^𝑝_𝐿𝑝+𝐷^𝛼−1_𝑎+𝑢^𝑝_𝐿𝑝. (92) Let us recall that 𝐼^{𝑛+1−𝛼}_𝑎+𝑢∈𝑊^𝑛+1,𝑝, so 𝐼^{𝑛+1−𝛼}_𝑎+𝑢∈𝑊^𝑛,𝑝, 𝐷^𝑛𝐼^{𝑛+1−𝛼}_𝑎+𝑢∈𝑊^1,𝑝, and 𝐷^𝛼_𝑎+𝑢=𝐷^𝑛+1𝐼^{𝑛+1−𝛼}_𝑎+𝑢=𝐷¹𝐷^𝑛𝐼^{𝑛+1−𝛼}_𝑎+𝑢. Thus, 𝐷^𝑛𝐼^{𝑛+1−𝛼}_𝑎+𝑢(𝑎)^𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝=𝐷^𝑛𝐼^{𝑛+1−𝛼}_𝑎+𝑢^𝑝_{𝑎,𝑊1,𝑝𝑎+}≤𝑀_1,1𝐷^𝑛𝐼^{𝑛+1−𝛼}_𝑎+𝑢^𝑝_{𝑊1,𝑝𝑎+}=𝑀_1,1𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢^𝑝_𝐿𝑝+𝐷^𝛼_𝑎+𝑢)^𝑝_𝐿𝑝, (93) where 𝑀_1,1 is such that ‖𝑣‖^𝑝_{𝑎,𝑊1,𝑝𝑎+}≤𝑀_1,1‖𝑣‖^𝑝_{𝑊1,𝑝𝑎+}. (94) Consequently, ‖𝑢‖^𝑝_{𝑎,𝑊𝛼,𝑝𝑎+}=^𝑛−1_𝑖=0𝐷^𝑖𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝+𝐷^𝑛𝐼^{𝑛−(𝛼−1)}_𝑎+𝑢(𝑎)^𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝≤𝑀_{𝛼−1,𝑛}‖𝑢‖^𝑝_𝐿𝑝+^𝑛−2_𝑖=0𝐷^{𝛼−1−(𝑛−1)+𝑖}_𝑎+𝑢^𝑝_𝐿𝑝+𝐷^𝛼−1_𝑎+𝑢^𝑝_𝐿𝑝+𝑀_1,1𝐷^𝛼−1_𝑎+𝑢^𝑝_𝐿𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝≤𝑀_𝛼,𝑛+1‖𝑢‖^𝑝_𝐿𝑝+^𝑛−1_𝑖=0𝐷^{𝛼−1−(𝑛−1)+𝑖}_𝑎+𝑢^𝑝_𝐿𝑝+𝐷^𝛼_𝑎+𝑢^𝑝_𝐿𝑝=𝑀_𝛼,𝑛+1‖𝑢‖^𝑝_𝐿𝑝+^𝑛_𝑖=0𝐷^{𝛼−𝑛+𝑖}_𝑎+𝑢^𝑝_𝐿𝑝=𝑀_𝛼,𝑛+1‖𝑢‖^𝑝_{𝑊𝛼,𝑝𝑎+}, (95) where 𝑀_𝛼,𝑛+1=𝑀_{𝛼−1,𝑛}+𝑀_1,1.

7. Basic Properties of 𝑊^𝛼,𝑝_𝑎+

Now, we are in a position to prove some basic properties of the introduced spaces.

Theorem 25. The space 𝑊^𝛼,𝑝_𝑎+ is complete with respect to each of the norms ‖⋅‖_𝑊^𝛼,𝑝_𝑎+ and ‖⋅‖_𝑎,𝑊^𝛼,𝑝_𝑎+, for any 𝛼>0 and 1≤𝑝<∞.

Proof. Let 𝑛∈ℕ be such that 𝛼∈(𝑛−1,𝑛]. Of course, it is sufficient to show that 𝑊^𝛼,𝑝_𝑎+ with the norm ‖⋅‖_𝑎,𝑊^𝛼,𝑝_𝑎+ is complete. Let (𝑢_𝑘)⊂𝑊^𝛼,𝑝_𝑎+ be a Cauchy sequence with respect to this norm. So, the sequences (𝐷^𝑖𝐼^𝑛−𝛼_𝑎+𝑢_𝑘(𝑎)), 𝑖=0,…,𝑛−1, are Cauchy sequences in ℝ^𝑚 and (𝐷^𝛼_𝑎+𝑢_𝑘) is the Cauchy sequence in 𝐿^𝑝. Let 𝑐₀,𝑐₁,…,𝑐_𝑛−1∈ℝ^𝑚 and 𝜑∈𝐿^𝑝 be limits of the above sequences in ℝ^𝑚 and 𝐿^𝑝, respectively. Then the function 𝑢(𝑡)=^𝑛−1_𝑖=0𝑐_𝑖Γ()()𝛼−𝑛+1+𝑖𝑡−𝑎^{𝛼−𝑛+𝑖}+𝐼^𝛼_𝑎+𝜑(𝑡),𝑡∈[𝑎,𝑏]a.e., (96) belongs to 𝑊^𝛼,𝑝_𝑎+ and is the limit of (𝑢_𝑘) in 𝑊^𝛼,𝑝_𝑎+ with respect to ‖⋅‖_𝑎,𝑊^𝛼,𝑝_𝑎+ (to assert that 𝑢∈𝐿^𝑝 it is sufficient to consider the cases (𝑛−𝛼)𝑝<1 and (𝑛−𝛼)𝑝≥1—in the second case 𝐼^𝑛−𝛼_𝑎+𝑢_𝑘(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 space 𝑊^𝛼,𝑝_𝑎+ is reflexive with respect to each of the norms ‖⋅‖_𝑊^𝛼,𝑝_𝑎+ and ‖⋅‖_𝑎,𝑊^𝛼,𝑝_𝑎+, for any 𝛼>0 and 1<𝑝<∞.

Proof. Let us consider 𝑊^𝛼,𝑝_𝑎+ with the norm ‖⋅‖_𝑊^𝛼,𝑝_𝑎+ and define a mapping 𝜆∶𝑊^𝛼,𝑝_𝑎+∋𝑢⟼𝑢,𝐷^{𝛼−(𝑛−1)}_𝑎+𝑢,…,𝐷^𝛼−1_𝑎+𝑢,𝐷^𝛼_𝑎+𝑢∈𝐿^𝑝×𝐿^𝑝×⋯×𝐿^𝑝×𝐿^𝑝. (97) Since 𝜆 is the isometry, 𝜆(𝑊^𝛼,𝑝_𝑎+) is the closed linear subspace of the reflexive space 𝐿^𝑝×⋯×𝐿^𝑝. So (cf. [8, Corollary 1 in Part V.73]), it is reflexive and, consequently, 𝑊^𝛼,𝑝_𝑎+ is reflexive with respect to ‖⋅‖_𝑊^𝛼,𝑝_𝑎+.
From the equivalence of the norms it follows that 𝑊^𝛼,𝑝_𝑎+ with the norm ‖⋅‖_𝑎,𝑊^𝛼,𝑝_𝑎+ is also reflexive (it is sufficient to consider the identity mapping 𝑖∶(𝑊^𝛼,𝑝_𝑎+,‖⋅‖_𝑊^𝛼,𝑝_𝑎+)→(𝑊^𝛼,𝑝_𝑎+,‖⋅‖_𝑊^𝛼,𝑝_𝑎+) being the linear homeomorphism and use [8, Remark in Part V.7.3]).

Theorem 27. The space 𝑊^𝛼,𝑝_𝑎+ is separable with respect to each of the norms ‖⋅‖_𝑊^𝛼,𝑝_𝑎+ and ‖⋅‖_𝑎,𝑊^𝛼,𝑝_𝑎+, for any 𝛼>0 and 1≤𝑝<∞.

Proof. Let us consider 𝑊^𝛼,𝑝_𝑎+ with the norm ‖⋅‖_𝑊^𝛼,𝑝_𝑎+ and mapping 𝜆 defined in the proof of Theorem 26. Of course, 𝜆(𝑊^𝛼,𝑝_𝑎+) is separable as a subset of separable space 𝐿^𝑝×⋯×𝐿^𝑝. Since 𝜆 is the isometry, 𝑊^𝛼,𝑝_𝑎+ is also separable with respect to the norm ‖⋅‖_𝑊^𝛼,𝑝_𝑎+. Equivalence of the norms ‖⋅‖_𝑎,𝑊^𝛼,𝑝_𝑎+ and ‖⋅‖_𝑊^𝛼,𝑝_𝑎+ implies separability of 𝑊^𝛼,𝑝_𝑎+ with respect to ‖⋅‖_𝑊^𝛼,𝑝_𝑎+.

8. Imbeddings

We have the following extension of Theorem 12.

Theorem 28. (a) If 0<𝛽<𝛼≤1, then 𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐼^𝛽_𝑎+𝐿^𝑞⊂𝐴𝐶^𝛽,𝑞_𝑎+ (98) for 1≤𝑞≤𝑝<∞ and 1≤𝑞<1/(1−𝛼+𝛽); consequently, 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛽,𝑞_𝑎+ (99) for 1≤𝑞≤𝑝<∞ and 1≤𝑞<1/(1−𝛼+𝛽).
(b) If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, 𝑛≥2, and 0<𝛽<𝛼−(𝑛−1), then 𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐼^𝛽_𝑎+𝐿^𝑞⊂𝐴𝐶^𝛽,𝑞_𝑎+ (100) for 1≤𝑝<∞ and 1≤𝑞<1/(𝑛−𝛼+𝛽); consequently, 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛽,𝑞_𝑎+ (101) for 1≤𝑝<∞ and 1≤𝑞<(1/(𝑛−𝛼+𝛽)).
(c) If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, 𝑛≥2, 𝛽=𝛼−𝑖, and 𝑖∈{1,…,𝑛−1}, then 𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐴𝐶^𝛽,𝑞_𝑎+ (102) for 1≤𝑝<∞ and 1≤𝑞<∞; consequently, 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛽,𝑞_𝑎+ (103) 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) If 0<𝛼=𝛽, then 𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐴𝐶^𝛽,𝑞_𝑎+ (104) for 1≤𝑞≤𝑝<∞; consequently, 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛼,𝑞_𝑎+ (105) for 1≤𝑞≤𝑝<∞.

Proof. (a) Let us fix 𝑝∈[1,∞). Theorems 12 and 13 (a) imply that 𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐴𝐶^𝛼,1_𝑎+⊂𝐼^𝛽_𝑎+𝐿^𝑞⊂𝐴𝐶^𝛽,𝑞_𝑎+ (106) 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, 𝑊^𝛼,𝑝_𝑎+=𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝⊂𝐴𝐶^𝛽,𝑞_𝑎+∩𝐿^𝑞=𝑊^𝛽,𝑞_𝑎+. (107)
(b) Let us fix 𝑝∈[1,∞). In this case 𝛼−𝛽>1 and, consequently, 𝐼^𝛼−𝛽_𝑎+𝜑=𝐼¹_𝑎+𝐼^{𝛼−𝛽−1}_𝑎+𝜑∈𝐶 for any 𝜑∈𝐿¹. So (cf. Theorems 12 and 13 (a)) 𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐴𝐶^𝛼,1_𝑎+⊂𝐼^𝛽_𝑎+𝐿^𝑞⊂𝐴𝐶^𝛽,𝑞_𝑎+ (108) 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<∞, 𝑊^𝛼,𝑝_𝑎+=𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝⊂𝐼^𝛽_𝑎+𝐿^𝑞=𝐼^𝛽_𝑎+𝐿^𝑞∩𝐿^𝑞⊂𝐴𝐶^𝛽,𝑞_𝑎+∩𝐿^𝑞=𝑊^𝛽,𝑞_𝑎+ (109) (we used here the inclusion 𝐼^𝛽_𝑎+(𝐿^𝑞)⊂𝐿^𝑞).
(c) Let us fix 𝑝∈[1,∞). Theorems 12 and 13 (b) imply that 𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐴𝐶^𝛼,1_𝑎+⊂𝐴𝐶^𝛽,𝑞_𝑎+ (110) for any 𝑞∈[1,∞) (cf. (22), (24)).
If (𝑛−𝛼)𝑝<1, then (cf. Remark 21) 𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝=𝐴𝐶^𝛼,𝑝_𝑎+. So, 𝑊^𝛼,𝑝_𝑎+=𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝=𝐴𝐶^𝛼,𝑝_𝑎+⊂𝐴𝐶^𝛽,𝑞_𝑎+⊂𝐴𝐶^𝛽,𝑞_𝑎+∩𝐿^𝑞=𝑊^𝛽,𝑞_𝑎+ (111) provided that (cf. Theorem 13 (b)) (𝑛−𝛼)𝑞<1. If (𝑛−𝛼)𝑝≥1, then (cf. Remark 21) 𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝 is the set of all functions belonging to 𝐴𝐶^𝛼,𝑝_𝑎+ that satisfy the condition (𝐼^𝑛−𝛼_𝑎+𝑓)(𝑎)=0. Consequently (cf. Theorem 13 (b)), 𝐴𝐶^𝛼,𝑝_𝑎+∩𝐿^𝑝⊂𝐴𝐶^𝛽,𝑞_𝑎+∩𝐿^𝑞, that is, 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛽,𝑞_𝑎+ 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. If 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, 𝑛≥2, 𝛽=𝛼−𝑖, and 𝑖∈{1,…,𝑛−1}, then 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛽,𝑞_𝑎+ (112) for 1≤𝑞≤𝑝<∞.

In the next section we will use the following important result obtained in [9, Lemma 1.1].

Theorem 30. If 𝛼>0 and 1≤𝑝<∞, then the operator 𝐼^𝛼_𝑎+∶𝐿^𝑝→𝐿^𝑝 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 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛽,𝑞_𝑎+ (113) for 0<𝛽<𝛼≤1, 1≤𝑞≤𝑝<∞, and 1≤𝑞<1/(1−𝛼+𝛽), given in Theorem 28 (a), is compact.

Proof. Let 0<𝛽<𝛼≤1, 1≤𝑞≤𝑝<∞, 1≤𝑞<1/(1−𝛼+𝛽), and (𝑢_𝑘), where 𝑢_𝑘(𝑡)=𝑐_𝑘Γ(𝛼)1(𝑡−𝑎)^1−𝛼+𝐼^𝛼_𝑎+𝜑_𝑘(𝑡),𝑡∈[𝑎,𝑏]a.e.,𝑘∈ℕ, (114) be a bounded sequence in 𝑊^𝛼,𝑝_𝑎+. We will show that it contains a subsequence which is convergent in 𝑊^𝛽,𝑞_𝑎+.
Since (𝑢_𝑘) is bounded (cf. Theorem 24), the sequences (𝑐_𝑘) and (𝜑_𝑘) are bounded in ℝ^𝑚 and 𝐿^𝑝, respectively. So, one can choose a subsequence (𝑘_𝑗)_𝑗∈ℕ of positive integers such that (𝑐_𝑘𝑗) is convergent to some 𝑐₀ in ℝ^𝑚 and (cf. Theorem 30) such that (𝐼^𝛼−𝛽_𝑎+𝜑_𝑘𝑗) is convergent to some 𝜓₀ in 𝐿^𝑝. Of course, (𝐼^𝛼−𝛽_𝑎+𝜑_𝑘𝑗) is convergent to 𝜓₀ in 𝐿^𝑞. Moreover, the sequence ((𝑐_𝑘𝑗/Γ(𝛼−𝛽))(⋅−𝑎)^{𝛼−1−𝛽}) converges in 𝐿^𝑞 to (𝑐₀/Γ(𝛼−𝛽))(⋅−𝑎)^{𝛼−1−𝛽}. This means (cf. (20), (21)) that the sequence (𝑢_𝑘𝑗) converges in 𝑊^𝛽,𝑞_𝑎+ to 𝑢₀ given by 𝑢₀(𝑡)=𝐼^𝛽_𝑎+𝜓₀(𝑡),𝑡∈[𝑎,𝑏]a.e., (115) where 𝜓₀(𝑡)=(𝑐₀/Γ(𝛼−𝛽))(𝑡−𝑎)^{𝛼−1−𝛽}+𝜓₀(𝑡). The proof is completed.

The above theorem implies the following.

Corollary 32. The imbedding 𝑊^𝛼,𝑝_𝑎+⊂𝐿^𝑞 (116) 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 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛽,𝑞_𝑎+, (117) 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 (𝑢_𝑘) in 𝑊^𝛼,𝑝_𝑎+ where 𝑢_𝑘(𝑡)=𝑐^𝑘₀Γ()()𝛼−𝑛+1𝑡−𝑎^𝛼−𝑛+𝑐^𝑘₁Γ()()𝛼−𝑛+2𝑡−𝑎^{𝛼−𝑛+1}𝑐+⋯+^𝑘𝑛−1Γ(𝛼)(𝑡−𝑎)^𝛼−1+𝐼^𝛼_𝑎+𝜑_𝑘(𝑡),𝑡∈[𝑎,𝑏]a.e., (118) with 𝑐^𝑘₀,𝑐^𝑘₁,…,𝑐^𝑘_𝑛−1∈ℝ^𝑚 and 𝜑_𝑘∈𝐿^𝑝. We will show that it contains a subsequence which is convergent in 𝑊^𝛽,𝑞_𝑎+.
Since (𝑢_𝑘) is bounded, the sequences (𝑐^𝑘₀),(𝑐^𝑘₁),…,(𝑐^𝑘_𝑛−1)∈ℝ^𝑚 and (𝜑_𝑘) are bounded in ℝ^𝑚 and 𝐿^𝑝, respectively. So, one can choose a subsequence (𝑘_𝑗)_𝑗∈ℕ of positive integers such that (𝑐^𝑘𝑗₀),(𝑐^𝑘𝑗₁),…,(𝑐^𝑘𝑗_𝑛−1) are convergent to some 𝑐₀,𝑐₁,…,𝑐_𝑛−1 in ℝ^𝑚 and (cf. Theorem 30) such that the sequence (𝐼^𝛾_𝑎+𝜑_𝑘𝑗) is convergent in 𝐿^𝑝 to some 𝜔₀ where 𝛾=(𝛼−𝛽)−1>0. Consequently, (𝐼^𝛾_𝑎+𝜑_𝑘𝑗) is convergent in 𝐿¹ to 𝜔₀ and 𝐼^𝛼−𝛽_𝑎+𝜑_𝑘𝑗=𝐼¹_𝑎+𝐼^𝛾_𝑎+𝜑_𝑘𝑗⟶𝐼¹_𝑎+𝜔₀ (119) in 𝐿^𝑞, because ^𝑏_𝑎𝐼¹_𝑎+𝐼^𝛾_𝑎+𝜑_𝑘𝑗(𝑡)−𝐼¹_𝑎+𝜔₀(𝑡)^𝑞≤𝑑𝑡_{𝑏𝑎𝑡𝑎}𝐼^𝛾_𝑎+𝜑_𝑘𝑗(𝑠)−𝜔₀(𝑠)𝑑𝑠^𝑞≤𝐼𝑑𝑡_𝛾𝑎+𝜑_𝑘𝑗−𝜔₀^𝑞_𝐿1(𝑏−𝑎). (120) So, the sequence (𝐷^𝛽_𝑎+𝑢_𝑘𝑗), where 𝐷^𝛽_𝑎+𝑢_𝑘𝑗(𝑡)=𝑐^𝑘𝑗₀Γ()𝛼−𝑛+1−𝛽𝑡−𝑎^{𝛼−𝑛−𝛽}+𝑐^𝑘𝑗₁Γ()𝛼−𝑛+2−𝛽𝑡−𝑎^{𝛼−𝑛−𝛽+1}𝑐+⋯+^{𝑘𝑗𝑛−1}Γ()𝛼−𝛽𝑡−𝑎^{𝛼−𝛽−1}+𝐼^𝛼−𝛽_𝑎+𝜑_𝑘𝑗(𝑡), (121) converges to the function 𝜓₀(𝑡)=𝑐₀Γ()𝛼−𝑛+1−𝛽𝑡−𝑎^{𝛼−𝑛−𝛽}+𝑐₁Γ()𝛼−𝑛+2−𝛽𝑡−𝑎^{𝛼−𝑛−𝛽+1}𝑐+⋯+^𝑛−1Γ()𝛼−𝛽𝑡−𝑎^{𝛼−𝛽−1}+𝐼¹_𝑎+𝜔₀(𝑡) (122)
in 𝐿^𝑞. Thus, the sequence (𝑢_𝑘𝑗) converges in 𝑊^𝛽,𝑞_𝑎+ to 𝐼^𝛽_𝑎+𝜓₀.
The proof is completed.

The above theorem implies the following.

Corollary 34. The imbedding 𝑊^𝛼,𝑝_𝑎+⊂𝐿^𝑞 (123) 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 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛽,𝑞_𝑎+, (124) 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 (𝑢_𝑘) in 𝑊^𝛼,𝑝_𝑎+ where 𝑢_𝑘(𝑡)=𝑐^𝑘₀Γ()()𝛼−𝑛+1𝑡−𝑎^𝛼−𝑛+𝑐^𝑘₁Γ()()𝛼−𝑛+2𝑡−𝑎^{𝛼−𝑛+1}𝑐+⋯+^𝑘𝑛−1Γ(𝛼)(𝑡−𝑎)^𝛼−1+𝐼^𝛼_𝑎+𝜑_𝑘(𝑡),𝑡∈[𝑎,𝑏]a.e., (125) with 𝑐^𝑘₀,𝑐^𝑘₁,…,𝑐^𝑘_𝑛−1∈ℝ^𝑚 and 𝜑_𝑘∈𝐿^𝑝. We will show that it contains a subsequence which is convergent in 𝑊^𝛽,𝑞_𝑎+.
Since (𝑢_𝑘) is bounded, the sequences (𝑐^𝑘₀),(𝑐^𝑘₁),…,(𝑐^𝑘_𝑛−1)∈ℝ^𝑚 and (𝜑_𝑘) are bounded in ℝ^𝑚 and 𝐿^𝑝, respectively. So, one can choose a subsequence (𝑘_𝑗)_𝑗∈ℕ of positive integers such that (𝑐^𝑘𝑗₀),(𝑐^𝑘𝑗₁),…,(𝑐^𝑘𝑗_𝑛−1) are convergent to some 𝑐₀,𝑐₁,…,𝑐_𝑛−1 in ℝ^𝑚 and (cf. Theorem 30) such that 𝐼¹_𝑎+𝑐^𝑘𝑗_{𝑛−𝑖+1},…,𝐼^𝑖−1_𝑎+𝑐^𝑘𝑗_𝑛−1,𝐼^𝑖_𝑎+𝜑_𝑘𝑗 (126) are convergent to some 𝜓_1,0,…,𝜓_𝑖−1,0,𝜓₀ in 𝐿^𝑝 and, consequently, in 𝐿^𝑞. So (cf. Theorem 13 (b)), 𝐷^𝛽_𝑎+𝑢_𝑘𝑗=𝑐^𝑘𝑗_𝑛−𝑖+𝐼¹_𝑎+𝑐^𝑘𝑗_{𝑛−𝑖+1}+⋯+𝐼^𝑖−1_𝑎+𝑐^𝑘𝑗_𝑛−1+𝐼^𝑖_𝑎+𝜑_𝑘𝑗⟶𝜓₀ (127) in 𝐿^𝑞, where 𝜓₀∈𝐿^𝑞 is given by 𝜓₀(𝑡)=𝑐_𝑛−𝑖+𝜓_1,0(𝑡)+⋯+𝜓_𝑖−1,0(𝑡)+𝜓₀(𝑡),𝑡∈[𝑎,𝑏]a.e. (128) This means (cf. (22), (23), and (24)) that the sequence (𝑢_𝑘𝑗) converges in 𝑊^𝛽,𝑞_𝑎+ to 𝑢₀ given by 𝑢₀(𝑡)=𝑑₀Γ()()𝛽−𝑛−𝑖+1𝑡−𝑎^{𝛽−(𝑛−𝑖)}+𝑑₁Γ()()𝛽−𝑛−𝑖+2𝑡−𝑎^{𝛽−(𝑛−𝑖)+1}𝑑+⋯+^{𝑛−(𝑖+1)}Γ𝛽(𝑡−𝑎)^𝛽−1+𝐼^𝛽_𝑎+𝜓₀(𝑡),𝑡∈[𝑎,𝑏]a.e., (129)
where 𝑑₀=𝑐₀,𝑑₁=𝑐₁,…,𝑑_{𝑛−(𝑖+1)}=𝑐_{𝑛−(𝑖+1)}. (130)
The proof is completed.

The above theorem implies the following.

Corollary 36. The imbedding 𝑊^𝛼,𝑝_𝑎+⊂𝐿^𝑞 (131) for 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, 𝑛≥2, and 1≤𝑞≤𝑝<∞ is compact.

Remark 37. It is easy to see that imbeddings 𝑊^𝛼,𝑝_𝑎+⊂𝑊^𝛼,𝑞_𝑎+ for 𝛼>0 and 1≤𝑞≤𝑝<∞, given in Theorem 28 (d), are not compact.

Remark 38. From Corollary 32 it follows that the imbedding 𝑊^1,𝑝⊂𝐿^𝑝 is compact for any 1≤𝑝<∞. Corollary 34 implies the compactness of the imbedding 𝑊^𝑛,1⊂𝐿^𝑞 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 𝑊^1,1⊂𝐿^𝑞 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 fix 𝛼∈(1/2,1) and consider the following problem: 𝐷^𝛼_𝑏−𝐷^𝛼_𝑎+𝑥(𝑡)(𝑡)(𝑡)[]𝐷+𝑥=𝑓,𝑡∈𝑎,𝑏a.e._𝛼𝑎+𝑥(𝑎)=0,𝑥(𝑏)=0, (132) where 𝑓∈𝐿². By a solution to this problem we mean a function 𝑥∈𝑊^𝛼,2_𝑎+ such that 𝐷^𝛼_𝑎+𝑥 and 𝐷^𝛼_𝑏−𝐷^𝛼_𝑎+𝑥 exist, and satisfying the above equation and boundary conditions. Under the assumption on 𝛼 the boundary conditions make sense ([10, Property 4]).

Let 𝑎∶𝐻×𝐻→ℝ, where 𝐻=𝑥∈𝑊^𝛼,2_𝑎+;𝑥(𝑏)=0, (133) to be in the following bilinear form 𝑎𝑥,𝑦=^𝑏_𝑎𝐷^𝛼_𝑎+𝑥(𝑡)𝐷^𝛼_𝑎+𝑦(𝑡)𝑑𝑡+^𝑏_𝑎𝑥(𝑡)𝑦(𝑡)𝑑𝑡. (134) It is easy to see that 𝐻 is the closed subspace of the Hilbert space (𝑊^𝛼,2_𝑎+, ‖⋅‖_𝑊^𝛼,𝑝_𝑎+); it is sufficient to observe that it is closed in 𝑊^𝛼,2_𝑎+ with respect to ‖⋅‖_𝑎,𝑊^𝛼,𝑝_𝑎+ and to use Theorem 24. Of course, 𝑎 is a scalar product in 𝐻 and the norm generated by 𝑎 is simply the norm ‖⋅‖_𝑊^𝛼,2_𝑎+ restricted to 𝐻. Clearly, 𝑎 is continuous and coercive; that is, there exists a constant 𝑐>0 such that 𝑎(𝑥,𝑥)≥𝑐‖𝑥‖²_{𝑊𝛼,2𝑎+} for 𝑥∈𝑊^𝛼,2_𝑎+; in fact 𝑎(𝑥,𝑥)=‖𝑥‖²_{𝑊𝛼,2𝑎+} for 𝑥∈𝐻. So, Lax-Milgram theorem [1] or simply Riesz-Frechet theorem implies that there exists 𝑥∈𝑊^𝛼,2_𝑎+ such that 𝑎(𝑥,ℎ)=^𝑏_𝑎𝑓(𝑡)ℎ(𝑡)1𝑑𝑡,ℎ∈𝐻,2‖𝑥‖²_{𝑊𝛼,2𝑎+}−^𝑏_𝑎𝑓(𝑡)𝑥(𝑡)𝑑𝑡=min_ℎ∈𝐻12‖ℎ‖²_{𝑊𝛼,2𝑎+}−^𝑏_𝑎𝑓(𝑡)ℎ(𝑡)𝑑𝑡. (135)(136) Condition (135) means that ^𝑏_𝑎𝐷^𝛼_𝑎+𝑥(𝑡)𝐷^𝛼_𝑎+ℎ(𝑡)𝑑𝑡=^𝑏_𝑎𝑓(𝑡)−𝑥(𝑡)ℎ(𝑡)𝑑𝑡,ℎ∈𝐻. (137) From a counterpart of Theorem 20 for the weak right fractional derivative it follows that 𝐷^𝛼_𝑎+𝑥∈𝑊^𝛼,2_𝑏− and 𝐷^𝛼_𝑏−𝐷^𝛼_𝑎+𝑥(𝑡)+𝑥(𝑡)=𝑓(𝑡),𝑡∈[𝑎,𝑏]a.e. (138) Since 𝑥∈𝐻, 𝑥(𝑏)=0. Moreover, applying Theorem 17 to the left side of (137) we obtain ^𝑏_𝑎𝐷^𝛼_𝑏−𝐷^𝛼_𝑎+𝑥(𝑡)+𝑥(𝑡)−𝑓(𝑡)ℎ(𝑡)𝑑𝑡−𝐼^1−𝛼_𝑎+ℎ(𝑎)𝐷^𝛼_𝑎+𝑥(𝑎)+𝐼^1−𝛼_𝑏−𝐷^𝛼_𝑎+𝑥(𝑏)ℎ(𝑏)=0 (139) for ℎ∈𝐻. So, 𝐼^1−𝛼_𝑎+ℎ(𝑎)𝐷^𝛼_𝑎+𝑥(𝑎)=0,ℎ∈𝐻. (140) This means that 𝐷^𝛼_𝑎+𝑥(𝑎)=0. Indeed, it is sufficient to consider functions ℎ_𝑖, 𝑖=1,…,𝑚, of the form ℎ_𝑖(𝑡)=10,…,0,Γ(𝛼)𝑐(𝑡−𝑎)^1−𝛼+1Γ(𝛼)^𝑡_𝑎1(𝑡−𝜏)^1−𝛼,0,…,0, (141) where 𝑐=−(𝑏−𝑎)^1−𝛼∫^𝑏_𝑎(1/(𝑏−𝜏)^1−𝛼)𝑑𝑡, with nonzero 𝑖th coordinate function (of course, ℎ_𝑖∈𝐻 and 𝐼^1−𝛼_𝑎+ℎ_𝑖(𝑎)=(0,…,0,𝑐,0…,0)≠0).

Remark 39. In the same way one can prove the existence of a solution to system 𝐷^𝛼_𝑏−𝐷^𝛼_𝑎+𝑥(𝑡)+𝑥(𝑡)=𝑓(𝑡),𝑡∈[𝑎,𝑏]a.e. (142) with boundary conditions 𝐼^1−𝛼_𝑎+𝑥(𝑎)=0,𝐼^1−𝛼_𝑏−𝐷^𝛼_𝑎+𝑥(𝑏)=0 (143) or 𝐼^1−𝛼_𝑎+𝑥(𝑎)=0,𝑥(𝑏)=0, (144) in the space 𝑊^𝛼,2_𝑎+. In [4], system (142) with nonhomogeneous boundary conditions 𝐼^1−𝛼_𝑎+𝑥(𝑎)=𝐶,𝑥(𝑏)=𝐷, (145) where 𝐶,𝐷∈ℝ^𝑚, is investigated using the Stampacchia theorem (cf. [1]). Nonlinear system of the form 𝐷^𝛼_𝑏−𝐷^𝛼_𝑎+𝑥(𝑡)=−𝐹_𝑥(𝑡,𝑥(𝑡)),𝑡∈[𝑎,𝑏]a.e., (146) where 𝐹_𝑥 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 if 𝛼∈(1/2,1), 𝑥∈𝑊^𝛼,2_𝑎+, and 𝐼^1−𝛼_𝑎+𝑥(𝑎)=0, then 𝑥(𝑎)=0. It follows from the integral representation of 𝑥 and from the fact that 𝐼^𝛼_𝑎+𝑓 is Hölder continuous on (𝑎,𝑏] and lim_𝑡→𝑎+𝐼^𝛼_𝑎+𝑓(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 1ℎ⟼2‖ℎ‖²_{𝑊𝛼,2𝑎+}−^𝑏_𝑎𝑓(𝑡)ℎ(𝑡)𝑑𝑡 (147) defined on 𝐻 and 𝑊^𝛼,2_𝑎+, respectively.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The project was financed with funds of National Science Centre, granted on the basis of decision DEC-2011/01/B/ST7/03426.