Characterizations of Fractional Sobolev Spaces From the Perspective of Riemann-Liouville Operators
aa r X i v : . [ m a t h . F A ] S e p CHARACTERIZATIONS OF FRACTIONAL SOBOLEVSPACES FROM THE PERSPECTIVE OFRIEMANN-LIOUVILLE OPERATORS
YULONG LIABSTRACT. Fractional Sobolev spaces b H s ( R ) have beenplaying important roles in analysis of many mathematicalsubjects. In this work, we re-consider fractional Sobolevspaces under the perspective of fractional operators and es-tablish characterizations on the Fourier transform of func-tions of fractional Sobolev spaces, thereby giving anotherequivalent definition.
1. Introduction. b H s ( R ) has been exhibiting important usefulnessin the study of theory of classical integer-order PDEs; and it is wellknown from the standard textbooks that fractional Sobolev spaces b H s ( R ) could be defined in several ways, namely via Fourier transform,Gagliardo norm or interpolation spaces.Our previous work ( [2] , [3] ) have suggested that usual fractionalSobolev spaces have been behaving new features in analysis of fraction-order differential equations due to the simultaneous appearing of left,right and mixed Riemann-Liouville derivatives. In this work we con-tinue to explore usual fractional Sobolev spaces under the perspectiveof fractional calculus theory; and the main result in this work is Theo-rem 2, which characterizes the Fourier transform of elements of b H s ( R )and thus gives anther equivalent definition of b H s ( R ). The material isorganized as follows: • Section 2 introduces the notations and conventions adoptedthroughout the material. • Section 3 introduces the preliminary knowledge on fractionalR-L operators.
Mathematics subject classification.
Keywords and phrases.
Riemann-Liouville fractional operators, weak fractionalderivative, Fourier transform, regularity, decomposition, mixed derivative.Received by the editors September 17, 2018. Y. LI • Section 4 outlines some existing results on the characterizationof b H s ( R ), which are obtained in [2] and will provide the wholeabstract setting for the work. • Section 5 establishes the main results.
2. Notations. • All functions considered in this material are default to be realvalued unless otherwise specified. • ( f, g ) and R R f g shall be used interchangeably. Also, we denoteintegration R A f on set A without pointing out the variableunless it is necessary to specify. • If f, g ∈ L ( R ), f = g means f = g, a.e. , unless statedotherwise. • C ∞ ( R ) denotes the space of all infinitely differentiable func-tions with compact support in R . • F ( u ) denotes the Fourier transform of u with specific expressiondefined in Definition 6, b u denotes the Plancherel transform of u defined in Theorem 3, which is well known that b u is anisometry map from L ( R ) onto L ( R ) and coincides with F ( u )if u ∈ L ( R ) ∩ L ( R ). • u ∨ denotes the inverse of Plancherel transform, and ∗ denotesconvolution.
3. Preliminary.3.1. Fractional Riemann-Liouville Integrals and Properties.Definition 1.
Let u : R → R and σ >
0. The left and right Riemann-Liouville fractional integrals of order σ are, formally respectively, de-fined as D − σ u ( x ) := 1Γ( σ ) Z x −∞ ( x − s ) σ − u ( s ) d s, (3.1) D − σ ∗ u ( x ) := 1Γ( σ ) Z ∞ x ( s − x ) σ − u ( s ) d s, (3.2)where Γ( σ ) is the usual Gamma function. HARACTERIZATIONS OF FRACTIONAL SOBOLEV SPACES Property 3.1 ( [1] , p. 96) . Given < σ , (3.3) ( φ, D − σ ψ ) = ( D − σ ∗ φ, ψ ) , for φ ∈ L p ( R ) , ψ ∈ L q ( R ) , p > , q > , /p + 1 /q = 1 + σ . Property 3.2 ( [1] , Corollary 2.1) . Let µ, σ > , and w ∈ C ∞ ( R ) ,then (3.4) D − µ D − σ w = D − ( µ + σ ) w and D − µ ∗ D − σ ∗ w = D − ( µ + σ ) ∗ w. Property 3.3 ( [1] , pp. 95, 96) . Let µ > . Given h ∈ R , define thetranslation operator τ h as τ h u ( x ) = u ( x − h ) . Also, given κ > , definethe dilation operator Π κ as Π κ u ( x ) = u ( κx ) . Under the assumptionthat D − µ u and D − µ ∗ u are well-defined, the following is true: (3.5) τ h ( D − µ u ) = D − µ ( τ h u ) , τ h ( D − µ ∗ u ) = D − µ ∗ ( τ h u )Π κ ( D − µ u ) = κ µ D − µ (Π κ u ) , Π κ ( D − µ ∗ u ) = κ µ D − µ ∗ (Π κ u ) . Let u : R → R . Assume µ > n is the smallest integergreater than µ (i.e., n − ≤ µ < n ), and σ = n − µ . The left andright Riemann-Liouville fractional derivatives of order µ are, formallyrespectively, defined as D µ u := 1Γ( σ ) d n d x n Z x −∞ ( x − s ) σ − u ( s ) d s, (3.6) D µ ∗ u := ( − n Γ( σ ) d n d x n Z ∞ x ( s − x ) σ − u ( s ) d s. (3.7) Property 3.4 ( [2] ) . Let < µ and u ∈ C ∞ ( R ) , then D µ u, D µ ∗ u ∈ L p ( R ) for any ≤ p < ∞ . Property 3.5 ( [1] , p. 137) . Let µ > , u ∈ C ∞ ( R ) , then (3.8) F ( D µ u ) = (2 πiξ ) µ F ( u ) and F ( D µ ∗ u ) = ( − πiξ ) µ F ( u ) , ξ = 0 , where F ( · ) is the Fourier Transform as defined in Definition 6 and thecomplex power functions are understood as ( ∓ iξ ) σ = | ξ | σ e ∓ σπi · sign( ξ ) / . Y. LI
Property 3.6 ( [2] ) . Consider τ h and Π κ defined in Property 3.3. Let µ > , n − ≤ µ < n , where n is a positive integer, then (3.9) τ h ( D µ u ) = D µ ( τ h u ) , τ h ( D µ ∗ u ) = D µ ∗ ( τ h u )Π κ ( D µ u ) = κ − µ D µ (Π κ u ) , Π κ ( D µ ∗ u ) = κ − µ D µ ∗ (Π κ u ) .
4. Characterization of Sobolev Space b H s ( R ) . In this section,we will list the necessary concepts and results developed in [2] , whichcharacterize the classical Sobolev space b H s ( R ) defined in 5. Thissection will give us the theoretical framework in which the main resultsin Section 5 will be established. Definition 3 (Weak Fractional Derivatives [2] ) . Let µ >
0, and u, w ∈ L loc ( R ). The function w is called weak µ -order left fractional derivativeof u , written as D µ u = w , provided(4.10) ( u, D µ ∗ ψ ) = ( w, ψ ) ∀ ψ ∈ C ∞ ( R ) . In a similar faschion, w is weak µ -order right fractional derivative of u ,written as D µ ∗ u = w , provided(4.11) ( u, D µ ψ ) = ( w, ψ ) ∀ ψ ∈ C ∞ ( R ) . Definition 4 ( [2] ) . Let s ≥
0. Define spaces(4.12) f W sL ( R ) = { u ∈ L ( R ) , D s u ∈ L ( R ) } , (4.13) f W sR ( R ) = { u ∈ L ( R ) , D s ∗ u ∈ L ( R ) } , where D s u and D s ∗ u are in the weak fractional derivative sense asdefined in Definition 3. A semi-norm(4.14) | u | L := k D s u k L ( R ) for f W sL ( R ) and | u | R := k D s ∗ u k L ( R ) for f W sR ( R ) , is given with the corresponding norm(4.15) k u k ⋆ := ( k u k L ( R ) + | u | ⋆ ) / , ⋆ = L, R.
Remark . Notice the special case f W L ( R ) = f W R ( R ) = b H ( R ) = L ( R ). HARACTERIZATIONS OF FRACTIONAL SOBOLEV SPACES Property 4.1 (Uniqueness of Weak Fractional R-L Derivatives [2] ) . If v ∈ L loc ( R ) has a weak s -order left (or right) fractional derivative,then it is unique up to a set of zero measure. Now we have the following characterization of Sobolev space b H s ( R ). Theorem 1 ( [2] ) . Given s ≥ , f W sL ( R ) , f W sR ( R ) and b H s ( R ) areidentical spaces with equal norms and semi-norms. Corollary 1 ( [2] ) . u ∈ b H s ( R ) if and only if there exits a sequence { u n } ⊂ C ∞ ( R ) such that { u n } , { D s u n } are Cauchy sequences in L ( R ) , with lim n →∞ u n = u . As a consequence, we have lim n →∞ D s u n = D s u .Likewise, u ∈ b H s ( R ) if and only if there exits a sequence { u n } ⊂ C ∞ ( R ) such that { u n } , { D s ∗ u n } are Cauchy sequences in L ( R ) , with lim n →∞ u n = u .As a consequence, we have lim n →∞ D s ∗ u n = D s ∗ u .
5. Main Results.
Let us keep in mind that throughout the rest ofpaper, fractional derivatives are always understood in the weak sensedefined in Section 4. The main result in this work is the following.
Theorem 2.
Given s ≥ . Let f ( ξ ) be a function of form Q ni =1 (1 + χ ( s i ) · ( ± πξi ) s i ) , where ≤ s i ( i = 1 , · · · , n ) , Σ ni =1 s i ≤ s and χ ( s ) = ( , if s ∈ (0 , ] or (cid:2) + 2 k, + 2 k (cid:3) , k ∈ N , − , if s ∈ (cid:0) + 2 k, + 2 k (cid:1) , k ∈ N . Denote τ = s − Σ ni =1 s i .Then the following is true: v ( x ) ∈ b H s ( R ) if and only if there exists a u ( x ) ∈ b H τ ( R ) such that f ( ξ ) · b v ( ξ ) = b u ( ξ ) .Remark . As convention, the complex power functions are understoodas ( ∓ iξ ) σ = | ξ | σ e ∓ σπi · sign( ξ ) / .Before embarking on the rigorous proof of the theorem above, let usfirst establish several lemmas which are of crucial importance later. Y. LI
Lemma 1.
Fix s ≥ . The following sets are dense in L ( R ) respec-tively: • M = { w : w = ψ + D s ψ, ∀ ψ ∈ C ∞ ( R ) }• M = { w : w = ψ − D s ψ, ∀ ψ ∈ C ∞ ( R ) }• M = { w : w = ψ + D s ∗ ψ, ∀ ψ ∈ C ∞ ( R ) }• M = { w : w = ψ − D s ∗ ψ, ∀ ψ ∈ C ∞ ( R ) } Proof.
1. The proof is provided for M only and the results for othersets follow analogously without essential differences.2. To use Theorem 5, first notice L ( R ) is a Hilbert space and M ⊂ L ( R ) by Property 3.4. Furthermore, it is effortless to verify M is a subspace of L ( R ). Thus all the hypothesis of Theorem 5 is met.3. Let us assume that g ∈ L ( R ) and ( g, w ) = 0 for any w ∈ M .The proof is done if this implies g = 0 a.e..4. Pick a non-zero function ψ ∈ C ∞ ( R ), then by Plancherel Theoremwe know b ψ ( ξ ) is a non-zero function, namely, b ψ ( ξ ) = 0. On accountof continuity of b ψ ( ξ ), there exists a non-empty open interval ( a, b ) ⊂ R such that b ψ ( ξ ) = 0 on ( a, b ).5. Let v ( x ) = ψ ( ǫx ), where ǫ is any positive fixed number. It’sclear that v ( x ) ∈ C ∞ ( R ) as well. Computing the Fourier transform of w ( x ) = v ( x ) + D s v by Property 3.5 gives(5.16) b w ( ξ ) = (1 + (2 πiξ ) s ) b v ( ξ ) = (1 + (2 πiξ ) s ) · ǫ b ψ ( ξǫ ) . Notice | πiξ ) s | 6 = 0 a.e. and b ψ ( ξǫ ) = 0 on ( aǫ, bǫ ). It follows that b w ( ξ ) = 0 a.e. on ( aǫ, bǫ ).6. Set new function G ( − y ) := R R g ( x ) w ( x − y ) d x = R R g ( x ) τ y w ( x ) d x .Using Property 3.6 gives τ y w ( x ) ∈ M and therefore G ( − y ) = 0 for any y ∈ R by our assumption in Step 3. And thus by Plancherel Theoremwe know b G ( ξ ) = 0 a.e. as well.7. Note that G ( y ) = g ( − x ) ∗ w ( x ) and g, w ∈ L ( R ). Usingconvolution theorem ( [4] , Theorem 1.2, p.12) gives b G = \ g ( − x ) · [ w ( x ) =0, which implies \ g ( − x )( ξ ) = 0 a.e. on ( ǫa, ǫb ) by recalling b w ( ξ ) = 0 a.e.on ( aǫ, bǫ ). HARACTERIZATIONS OF FRACTIONAL SOBOLEV SPACES
78. Because of the arbitrariness of ǫ , we deduce \ g ( − x )( ξ ) = 0 a.e. in R and therefore g ( x ) = 0 a.e in R by another use of inverse PlancherelTheorem. This completes the whole proof. (cid:3) Lemma 2.
Given ≤ s , ψ ∈ C ∞ ( R ) , then k ψ ± D s ψ k L ( R ) = k ψ k L ( R ) ± s π ) k D s/ ψ k L ( R ) + k D s ψ k L ( R ) , (5.17) k ψ ± D s ∗ ψ k L ( R ) = k ψ k L ( R ) ± s π ) k D s/ ψ k L ( R ) + k D s ∗ ψ k L ( R ) . (5.18) Proof.
1. The proof is established for one case of identities aboveonly, namely k ψ + D s ψ k L ( R ) , and the others can be shown analogouslyby repeating the same procedure (even though involve left derivativeand right derivative in different cases).2. Since k ψ + D s ψ k L ( R ) = k ψ k L ( R ) + k D s ψ k L ( R ) + 2( ψ, D s ψ ), weonly need to show ( ψ, D s ψ ) = cos( s π ) k D s/ ψ k L ( R ) .3. Note that if we could show ( ψ, D s ψ ) = ( D s/ ∗ ψ, D s/ ψ ), thenimmediately ( ψ, D s ψ ) = cos( s π ) k D s/ ψ k L ( R ) follows by applicationof the second identity of Theorem 4.1 in [2] and so the proof is done.4. The fact ( ψ, D s ψ ) = ( D s/ ∗ ψ, D s/ ψ ) could be verified by astraightforward calculation as follows. Let us rewrite s = n − δ with0 ≤ δ <
1, where n is a positive integer. Notice ψ ∈ C ∞ ( R ), if integer n is even, using definition of R-L derivative and Lemma 2.2 ( [4] , p.73)gives(5.19) ( ψ, D s ψ ) = ( ψ, ddx (n / D n/ − δ ψ ) = ( ψ, ddx (n / D − δ ψ ( n/ ) . Using integration by parts and semigroup property 3.2 the last termbecomes( ψ, ddx (n / D − δ ψ ( n/ ) = ( D n/ ∗ ψ, D − δ/ D − δ/ ψ ( n/ ) . To simplify the right-hand side, notice the fact that D − δ/ ψ ( n/ = D n/ − δ/ ψ by applying Lemma 2.2 ( [4] , p.73), and this gives D − δ/ ψ ( n/ Y. LI ∈ L p ( R ) for p ≥ D n/ ∗ ψ, D − δ/ D − δ/ ψ ( n/ ) = ( D s/ ∗ ψ, D s/ ψ )follows immediately from Property 3.1 and another use of Lemma 2.2( [4] , p.73). Therefore ( ψ, D s ψ ) = ( D s/ ∗ ψ, D s/ ψ ).5. If integer n is odd, using definition of R-L derivative and Lemma2.2 ( [4] , p.73), it is not difficult to verify( ψ, D s ψ ) = ( ψ, ddx ((n+1) / D − ( δ +1) ψ (( n +1) / ) . Now n + 1 is even. Repeating above procedure starting from Equa-tion (5.19) gives us the desired result, which is omitted here. Thus weare done. (cid:3) Lemma 3. (a). Let χ ( s ) = ( , if s ∈ (0 , ] or (cid:2) + 2 k, + 2 k (cid:3) , k ∈ N , − , if s ∈ (cid:0) + 2 k, + 2 k (cid:1) , k ∈ N . Given ≤ s , there exists a one to one and onto map T : v u from b H s ( R ) to L ( R ) such that (5.20) v + χ ( s ) · D s v = u. Analogously, there exists a one to one and onto map T ∗ : v u from b H s ( R ) to L ( R ) such that (5.21) v + χ ( s ) · D s ∗ v = u. (b). Furthermore, for each case, v ∈ b H s + t ( R ) if and only if u ∈ b H t ( R ) for any t > . Proof.
1. For part (a), the proof is shown only for Equality (5.20)since the second one could be shown similarly.2. First we show the map is onto. Fix a u ∈ L ( R ), and withoutloss of generality, we assume χ ( s ) = 1. Invoking Lemma 1, thereexists a Cauchy sequence { v n − D s v n } converging to u in L ( R ), where { v n } ⊂ C ∞ ( R ). This implies that k ( v m − v n ) − ( D s v m − D s v n ) k L ( R ) → m, n → ∞ . HARACTERIZATIONS OF FRACTIONAL SOBOLEV SPACES m, n → ∞ , k v m − v n k L ( R ) +2 cos( s π ) k D s/ v m − D s/ v n k L ( R ) + k D s v m − D s v n k L ( R ) → . Now notice that 2 cos( s π ) ≥ χ ( s ) = 1. Therefore, wededuce that { v n } and { D s v n } are Cauchy sequences respectively. Thisconcludes that , if we denote v = lim n →∞ v n , v + χ ( s ) · D s v = u and v ∈ b H s ( R ) by Corollary 1 .3. We now show the map is one to one. First notice the factthat D s v ∈ L ( R ) if v ∈ b H s ( R ) by the definition of weak fractionalderivative, therefore v + χ ( s ) · D s v ∈ L ( R ), which means T : v u is a map by the uniqueness of weak fractional derivative 4.1. Second,assume there exists another ˜ v such that ˜ v + D s ˜ v = v + D s v , then(˜ v − v ) + D s (˜ v − v ) = 0 immediately follows. By using a standard normestimate argument and application of Lemma 2, it is easy to obtain k ˜ v − v k L ( R ) = 0, which implies ˜ v = v . Thus T : v u is a one to onemap.4. We remain to show part (b). First suppose v ∈ b H s + t ( R ),obviously v ∈ b H t ( R ) since b H s + t ( R ) ⊂ b H t ( R ). Now we claim D s v ∈ b H t ( R ). To see this, using Corollary 1, there exist Cauchy sequences { v n } ⊂ C ∞ ( R ) and { D s + t v n } in L ( R ) such that v n → v, D s + t v n → D s + t v as n → ∞ . Thus for any ψ ∈ C ∞ ( R ), we have( D s v, D t ∗ ψ ) = lim m →∞ ( D s v n , D t ∗ ψ ) = lim m →∞ ( D s + t v n , ψ ) = ( D s + t v, ψ ) . By definition of weak fractional derivative we conclude D s v ∈ b H t ( R )and thus u ( x ) ∈ b H t ( R ).5. The last step is to show u ( x ) ∈ b H t ( R ) implies v ∈ b H s + t ( R ).First notice that it is always possible to rewrite t = t + · · · + t n suchthat t i ≥ , and t i ≤ s ( i = 1 , · · · , n ). Thus, b H s ( R ) ⊂ b H t and b H t ( R ) ⊂ b H t , and this deduces D s v = u − v ∈ b H t ( R ). Accordingto Theorem 1, D s v ∈ f W t L ( R ), which by definition means that thereexists a Q ( x ) ∈ L ( R ) such that(5.22) ( D s v, D t ∗ ψ ) = ( Q, ψ ) ∀ ψ ∈ C ∞ ( R ) . Again, by invoking Corollary 1, the left-hand side of Equation (5.22)0
Y. LI becomes(5.23) ( D s v, D t ∗ ψ ) = lim m →∞ ( D s v n , D t ∗ ψ )= lim m →∞ ( v n , D ( s + t ) ∗ ψ )= ( v, D ( s + t ) ∗ ψ ) . This implies that v ∈ b H s + t by another utilization of definition of f W sL ( R ) and Theorem 1. By repeating this procedure we could in-crease the regularity of v gradually, namely, we could show that v ∈ b H s + t + t ( R ) , · · · , v ∈ b H s + t + ··· + t n ( R ) = b H s + t ( R ), as desired.Thus the whole proof is complete. (cid:3) Lemma 4. (a). Let χ ( s ) = ( , if s ∈ (0 , ] or (cid:2) + 2 k, + 2 k (cid:3) , k ∈ N , − , if s ∈ (cid:0) + 2 k, + 2 k (cid:1) , k ∈ N . Given ≤ s , there exists a one to one and onto map T : v u from b H s ( R ) to L ( R ) such that (5.24) (1 + χ ( s ) · (2 πξi ) s ) · b v ( ξ ) = b u ( ξ ) . Analogously, there exists a one to one and onto map T ∗ : v u from b H s ( R ) to L ( R ) such that (5.25) (1 + χ ( s ) · ( − πξi ) s ) · b v ( ξ ) = b u ( ξ ) . (b). Furthermore, for each case, v ∈ b H s + t ( R ) if and only if u ∈ b H t ( R ) for any t > . Proof.
1. For part (a), the proof is shown only for first Equa-tion (5.24), the other one follows by repeating the same procedure.2. From Lemma 3, there exists a one to one and onto map T : v u from b H s ( R ) to L ( R ) such that(5.26) v + χ ( s ) · D s v = u. Taking the Plancherel transform at both sides gives b v + χ ( s ) · d D s v = b u. So far, we could not yet directly apply Property 3.5 to obtain d D s v ( ξ ) =(2 πξi ) s · b v ( ξ ) since the condition is not met, namely v does not nec- HARACTERIZATIONS OF FRACTIONAL SOBOLEV SPACES C ∞ ( R ). The argument for d D s v ( ξ ) = (2 πξi ) s · b v ( ξ )is justified in the proof of Theorem 3.3 ( [2] ) and will not be repeatedhere. Thus, (1 + χ ( s ) · (2 πξi ) s ) · b v ( ξ ) = b u ( ξ ) .
3. The proof of part (b) directly follows from the part (b) ofLemma 3, which completes the whole proof. (cid:3)
Now we are in the position to prove Theorem 2.
Proof.
1. We intend to construct a one to one and onto map v u from b H s ( R ) to b H τ ( R ) such that f ( ξ ) · b v ( ξ ) = b u ( ξ ) . Then the proof is done.2. Utilizing both part (a) and (b) of Lemma 4, we know there existsa one to one and onto map T : v u from b H s ( R ) to b H s − s ( R ) suchthat(5.27) (1 + χ ( s ) · (2 πξi ) s ) · b v ( ξ ) = c u ( ξ ) . Repeating the same application of Lemma 4 for u , we know thereexists a one to one and onto map T : u u from b H s − s ( R ) to b H s − s − s ( R ) such that(5.28) (1 + χ ( s ) · (2 πξi ) s ) · c u ( ξ ) = c u ( ξ ) . By repeating the same procedure, we obtain one to one and ontomaps T , · · · , T n , where T n : u n − → u n from b H s − s −···− s n − ( R ) to b H s − s −···− s n ( R ) such that(5.29) (1 + χ ( s ) · (2 πξi ) s n ) · [ u n − ( ξ ) = c u n ( ξ ) . Recall τ = s − Σ ni =1 s i , therefore, T = T ◦ T · · · ◦ T n : v u n is aone to one and onto map v u n from b H s ( R ) to b H τ ( R ), satisfying f ( ξ ) · b v ( ξ ) = c u n ( ξ ). This completes Step 1 above by regarding u n as u ,and thus completes the whole proof for Theorem 2. (cid:3) Y. LI
Appendices
A. Several Definitions and Theorems.Definition 5 (Sobolev Spaces Via Fourier Transform) . Let µ ≥ b H µ ( R ) = (cid:26) w ∈ L ( R ) : Z R (1 + | πξ | µ ) | b w ( ξ ) | d ξ < ∞ (cid:27) , where b w is Plancherel transform defined in Theorem 3. The space isendowed with semi-morn(A.2) | u | b H µ ( R ) := k| πξ | µ b u k L ( R ) , and norm(A.3) k u k b H µ ( R ) := (cid:16) k u k L ( R ) + | u | b H µ ( R ) (cid:17) / . And it is well-known that b H µ ( R ) is a Hilbert space. Definition 6 (Fourier Transform) . Given a function f : R → R , theFourier Transform of f is defined as F ( f )( ξ ) := Z ∞−∞ e − πixξ f ( x ) d x ∀ ξ ∈ R . Theorem 3 (Plancherel Theorem ( [5] p. 187)) . One can associate toeach f ∈ L ( R ) a function b f ∈ L ( R ) so that the following propertieshold: • If f ∈ L ( R ) ∩ L ( R ) , then b f is the defined Fourier transformof f in Definition 6. • For every f ∈ L ( R ) , k f k = k b f k . • The mapping f → b f is a Hilbert space isomorphism of L ( R ) onto L ( R ) . Theorem 4. ( [6] , p. 189 ) Assume u, v ∈ L ( R n ) . Then • Z R n uv = Z R n b u b v. • u = ( b u ) ∨ . Theorem 5 ( [7] , Theorem 4.3-2, p. 191) . Let ( X, ( · , · )) be a Hilbertspace and let Y be a subspace of X , then Y = X if and only if element x ∈ X that satisfy ( x, y ) = 0 for all y ∈ Y is x = 0 . REFERENCES . S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives,Gordon and Breach Science Publishers, Yverdon, 1993. . V. Ginting, Y. Li, On fractional diffusion-advection-reaction equation in R arXiv:1805.09398v1 (submitted), 2018. . Y. Li, Symmetric Decompositions of f ∈ L ( R ) Via Fractional Riemann-Liouville Operators arXiv:1807.01847 (submitted), 2018. . A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications offractional differential equations, Vol. 204 of North-Holland Mathematics Studies,Elsevier Science B.V., Amsterdam, 2006. . W. Rudin, Real and complex analysis, 3rd Edition, McGraw-Hill Book Co., NewYork, 1987. . L. C. Evans, Partial differential equations, 2nd Edition, Vol. 19 of GraduateStudies in Mathematics, American Mathematical Society, Providence, RI, 2010. . L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Vol. 3 ofLecture Notes of the Unione Matematica Italiana, Springer, Berlin; UMI, Bologna,2007. . W.Rudin, Functional analysis, 2nd Edition, International Series in Pure andApplied Mathematics, McGraw-Hill, Inc., New York, 1991. Department of Mathematics and Statistics, University of Wyoming, Laramie,Wyoming, USA
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