Spectral properties of fractional differentiation operators
aa r X i v : . [ m a t h . F A ] M a y Electronic Journal of Differential Equations , Vol. 2018 (2018), No. 29, pp. 1–24.ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
SPECTRAL PROPERTIES OF FRACTIONALDIFFERENTIATION OPERATORS
MAKSIM V. KUKUSHKIN
Communicated by Ludmila S. Pulkina
Abstract.
We consider fractional differentiation operators in various sensesand show that the strictly accretive property is the common property of frac-tional differentiation operators. Also we prove that the sectorial property holdsfor differential operators second order with a fractional derivative in the finalterm, we explore a location of the spectrum and resolvent sets and show thatthe spectrum is discrete. We prove that there exists a two-sided estimate foreigenvalues of the real component of operators second order with the fractionalderivative in the final term. Introduction
It is remarkable that the term accretive, which applicable to a linear operator T acting in Hilbert space H, is introduced by Friedrichs in the paper [5], and meansthat the operator T has the following property: the numerical range Θ( T ) (see [8,p.335]) is a subset of the right half-plane i.e.Re ( T u, u ) H ≥ , u ∈ D ( T ) . Accepting the notation of the paper [9] we assume that Ω is a convex domain of the n - dimensional Euclidean space E n , P is a fixed point of the boundary ∂ Ω , Q ( r, e )is an arbitrary point of Ω; we denote by e a unit vector having the direction from P to Q, denote by r = | P − Q | an Euclidean distance between the points P and Q. Weuse the shorthand notation T := P + e t, t ∈ R . We consider the Lebesgue classes L p (Ω) , ≤ p < ∞ of complex valued functions. For the function f ∈ L p (Ω) , wehave Z Ω | f ( Q ) | p dQ = Z ω dχ d ( e ) Z | f ( Q ) | p r n − dr < ∞ , (1.1)where dχ is the element of the solid angle of the unit sphere surface (the unitsphere belongs to E n ) and ω is a surface of this sphere, d := d ( e ) is the length ofthe segment of the ray going from the point P in the direction e within the domainΩ . Without lose of generality, we consider only those directions of e for which the Mathematics Subject Classification.
Key words and phrases.
Fractional derivative; fractional integral; energetic space;sectorial operator; strong accretive operator;c (cid:13) inner integral on the right side of equality (1.1) exists and is finite. It is the well-known fact that these are almost all directions. We denote by Lip µ, (0 < µ ≤ λ := (cid:8) ρ ( Q ) : | ρ ( Q ) − ρ ( P ) | ≤ M r λ , P, Q ∈ ¯Ω (cid:9) . Consider the Kipriyanov fractional differential operator defined in the paper [10]by the formal expression D α ( Q ) = α Γ(1 − α ) r Z [ f ( Q ) − f ( T )]( r − t ) α +1 (cid:18) tr (cid:19) n − dt + C ( α ) n f ( Q ) r − α , P ∈ ∂ Ω , where C ( α ) n = ( n − / Γ( n − α ) . In accordance with Theorem 2 [10], under theassumptions lp ≤ n, < α < l − np + nq , q > p, (1.2)we have that for sufficiently small δ > k D α f k L q (Ω) ≤ Kδ ν k f k L p (Ω) + δ − ν k f k L lp (Ω) , f ∈ ◦ W lp (Ω) , (1.3)where ν = nl (cid:18) p − q (cid:19) + α + βl . (1.4)The constant K does not depend on δ, f ; the point P ∈ ∂ Ω; β is an arbitrarilysmall fixed positive number. Further, we assume that α ∈ (0 , . Using the notationof the paper [20], we denote by I αa + ( L p ) , I αb − ( L p ) , ≤ p ≤ ∞ the left-side, right-side classes of functions representable by the fractional integral on the segment[ a, b ] respectively. Let d := diam Ω; C, C i = const , i ∈ N . We use a shorthandnotation P · Q = P i Q i = P ni =1 P i Q i for the inner product of the points P =( P , P , ..., P n ) , Q = ( Q , Q , ..., Q n ) which belong to E n . Denote by D i u the weekderivative of the function u with respect to a coordinate variable with index 1 ≤ i ≤ n. We assume that all functions have a zero extension outside of ¯Ω . Denote byD( L ) , R( L ) the domain of definition, range of values of the operator L respectively.Everywhere further, unless otherwise stated, we use the notations of the papers [9],[10], [20]. Let us define the operators( I α g )( Q ) := 1Γ( α ) r Z g ( T )( r − t ) − α (cid:18) tr (cid:19) n − dt, ( I αd − g )( Q ) := 1Γ( α ) d Z r g ( T )( t − r ) − α dt,g ∈ L p (Ω) , ≤ p ≤ ∞ . These operators we call respectively the left-side, right-side directional fractionalintegral. We introduce the classes of functions representable by the directionalfractional integrals. I α ( L p ) := (cid:8) u : u ( Q ) = ( I α g )( Q ) , g ∈ L p (Ω) , ≤ p ≤ ∞ (cid:9) , (1.5) I αd − ( L p ) = (cid:8) u : u ( Q ) = ( I αd − g )( Q ) , g ∈ L p (Ω) , ≤ p ≤ ∞ (cid:9) . (1.6) JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 3
Define the operators ψ + ε , ψ − ε depended on the parameter ε > . In the left-sidecase ( ψ + ε f )( Q ) = r − ε Z f ( Q ) r n − − f ( T ) t n − ( r − t ) α +1 r n − dt, ε ≤ r ≤ d,f ( Q ) α (cid:18) ε α − r α (cid:19) , ≤ r < ε. (1.7)In the right-side case( ψ − ε f )( Q ) = d Z r + ε f ( Q ) − f ( T )( t − r ) α +1 dt, ≤ r ≤ d − ε,f ( Q ) α (cid:18) ε α − d − r ) α (cid:19) , d − ε < r ≤ d, where D( ψ + ε ) , D( ψ − ε ) ⊂ L p (Ω) . Using the definitions of the monograph [20, p.181]we consider the following operators. In the left-side case( D α , ε f )( Q ) = 1Γ(1 − α ) f ( Q ) r − α + α Γ(1 − α ) ( ψ + ε f )( Q ) . (1.8)In the right-side case( D αd − , ε f )( Q ) = 1Γ(1 − α ) f ( Q )( d − r ) − α + α Γ(1 − α ) ( ψ − ε f )( Q ) . The left-side and right-side fractional derivatives are understood respectively as thefollowing limits with respect to the norm L p (Ω) , (1 ≤ p < ∞ ) D α f = lim ε → ( L p ) D α , ε f, D αd − f = lim ε → ( L p ) D αd − , ε f. (1.9)We need auxiliary propositions, which are presented in the next section.2. Some lemmas and theorems
We have the following theorem on boundedness of the directional fractional in-tegral operators.
Theorem 2.1.
The directional fractional integral operators are bounded in L p (Ω) , ≤ p < ∞ , the following estimates holds k I α u k L p (Ω) ≤ C k u k L p (Ω) , k I αd − u k L p (Ω) ≤ C k u k L p (Ω) , C = d α / Γ( α + 1) . (2.1) Proof.
Let us prove first estimate (2.1), the proof of the second one is absolutelyanalogous. Using the generalized Minkowski inequality, we have k I α u k L p (Ω) = 1Γ( α ) Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z g ( T )( r − t ) − α (cid:18) tr (cid:19) n − dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dQ /p = 1Γ( α ) Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z g ( Q − τ e ) τ − α (cid:18) r − τr (cid:19) n − dτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dQ /p M. V. KUKUSHKIN EJDE-2018/29 ≤ α ) Z Ω d Z | g ( Q − τ e ) | τ − α dτ p dQ /p ≤ α ) d Z τ α − dτ Z Ω | g ( Q − τ e ) | p dQ /p ≤ d α Γ( α + 1) k u k L p (Ω) . (cid:3) Theorem 2.2.
Suppose f ∈ L p (Ω) , there exists lim ε → ψ + ε f or lim ε → ψ − ε f with respectto the norm L p (Ω) , (1 ≤ p < ∞ ); then f ∈ I α ( L p ) or f ∈ I αd − ( L p ) respectively.Proof. Let f ∈ L p (Ω) and lim ε → ( L p ) ψ + ε f = ψ. Consider the function( ϕ + ε f )( Q ) = 1Γ(1 − α ) (cid:26) f ( Q ) r α + α ( ψ + ε f )( Q ) (cid:27) . Taking into account (1.7), we can easily prove that ϕ + ε f ∈ L p (Ω) . Obviously, thereexists the limit ϕ + ε f → ϕ ∈ L p (Ω) , ε ↓ . Taking into account Theorem 2.1, we cancomplete the proof, if we show that I α ϕ + ε f L p → f, ε ↓ . (2.2)In the case ( ε ≤ r ≤ d ) , we have( I α ϕ + ε f )( Q ) πr n − sin απ = r Z ε f ( P + y e ) y n − − α ( r − y ) − α dy + α r Z ε ( r − y ) α − dy y − ε Z f ( P + y e ) y n − − f ( T ) t n − ( y − t ) α +1 dt + 1 ε α ε Z f ( P + y e )( r − y ) α − y n − dy = I. By direct calculation, we obtain I = 1 ε α r Z f ( P + y e )( r − y ) α − y n − dy − α r Z ε ( r − y ) α − dy y − ε Z f ( T )( y − t ) α +1 t n − dt. (2.3)Changing the variable of integration in the second integral, we have α r Z ε ( r − y ) α − dy y − ε Z f ( T )( y − t ) α +1 t n − dt = α r − ε Z ( r − y − ε ) α − dy y Z f ( T )( y + ε − t ) α +1 t n − dt = α r − ε Z f ( T ) t n − dt r − ε Z t ( r − y − ε ) α − ( y + ε − t ) α +1 dy JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 5 = α r − ε Z f ( T ) t n − dt r Z t + ε ( r − y ) α − ( y − t ) − α − dy. (2.4)Applying formula (13.18) [20, p.184], we get r Z t + ε ( r − y ) α − ( y − t ) − α − dy = 1 αε α · ( r − t − ε ) α r − t . (2.5)Combining relations (2.3),(2),(2.5), using the change of the variable t = r − ετ, weget ( I α ϕ + ε f )( Q ) πr n − sin απ = 1 ε α r Z f ( P + y e )( r − y ) α − y n − dy − r − ε Z f ( T )( r − t − ε ) α r − t t n − dt = 1 ε α r Z f ( T ) (cid:2) ( r − t ) α − ( r − t − ε ) α + (cid:3) r − t t n − dt = r/ε Z τ α − ( τ − α + τ f ( P + [ r − ετ ] e )( r − ετ ) n − dτ, τ + = ( τ, τ ≥ , τ < . . (2.6)Consider the auxiliary function K defined in the paper [20, p.105] K ( t ) = sin αππ · t α + − ( t − α + t ∈ L p ( R ); ∞ Z K ( t ) dt = 1; K ( t ) > . (2.7)Combining (2),(2.7) and taking into account that f has the zero extension outsideof ¯Ω , we obtain( I α ϕ + ε f )( Q ) − f ( Q ) = ∞ Z K ( t ) (cid:8) f ( P + [ r − εt ] e )(1 − εt/r ) n − − f ( P + r e ) (cid:9) dt. (2.8)Consider the case (0 ≤ r < ε ) . Taking into account (1.7), we get( I α ϕ + ε f )( Q ) − f ( Q ) = sin αππε α r Z f ( T )( r − t ) − α (cid:18) tr (cid:19) n − dt − f ( Q )= sin αππε α r Z f ( P + [ r − t ] e ) t − α (cid:18) r − tr (cid:19) n − dt − f ( Q ) . (2.9)Consider the domainsΩ ε := { Q ∈ Ω , d ( e ) ≥ ε } , ˜Ω ε = Ω \ Ω ε . (2.10)In accordance with this definition we can divide the surface ω into two parts ω ε and ˜ ω ε , where ω ε is the subset of ω such that d ( e ) ≥ ε and ˜ ω ε is the subset of ω such that d ( e ) < ε. Using (2.8),(2), we get k ( I α ϕ + ε f ) − f k pL p (Ω) M. V. KUKUSHKIN EJDE-2018/29 = Z ω ε dχ d Z ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Z K ( t )[ f ( Q − εt e )(1 − εt/r ) n − − f ( Q )] dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p r n − dr + Z ω ε dχ ε Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin αππε α r Z f ( P + [ r − t ] e ) t − α (cid:18) r − tr (cid:19) n − dt − f ( Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p r n − dr + Z ˜ ω ε dχ d Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin αππε α r Z f ( P + [ r − t ] e ) t − α (cid:18) r − tr (cid:19) n − dt − f ( Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p r n − dr = I + I + I . (2.11)Consider I , using the generalized Minkovski inequality, we get I p ≤ ∞ Z K ( t ) Z ω ε dχ d Z ε | f ( Q − εt e )(1 − εt/r ) n − − f ( Q ) | p r n − dr p dt. We use the following notation h ( ε, t ) := K ( t ) Z ω ε dχ d Z ε | f ( Q − εt e )(1 − εt/r ) n − − f ( Q ) | p r n − dr p dt. It can easily be checked that | h ( ε, t ) | ≤ K ( t ) k f k L p (Ω) , ∀ ε >
0; (2.12) | h ( ε, t ) | ≤ Z ω ε dχ d Z ε (cid:12)(cid:12) (1 − εt/r ) n − [ f ( Q − εt e ) − f ( Q )] (cid:12)(cid:12) p r n − dr p dt + Z ω ε dχ d Z (cid:12)(cid:12) f ( Q )[1 − (1 − εt/r ) n − ] (cid:12)(cid:12) p r n − dr p dt = I + I . By virtue of the average continuity property in L p (Ω) , we have ∀ t > I → , ε ↓ . Consider I and let us define the function h ( ε, t, r ) := (cid:12)(cid:12) f ( Q )[1 − (1 − εt/r ) n − ] (cid:12)(cid:12) . Obviously, the following relations hold almost everywhere in Ω ∀ t > , h ( ε, t, r ) ≤ | f ( Q ) | , h ( ε, t, r ) → , ε ↓ . Applying the Lebesgue dominated convergence theorem, we get I → , ε ↓ . Itimplies that ∀ t > , lim ε → h ( ε, t ) = 0 . (2.13)Taking into account (2.12), (2.13) and applying the Lebesgue dominated conver-gence theorem again, we obtain I → , ε ↓ . JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 7
Consider I , using the Mincovski inequality, we get I p ≤ sin αππε α Z ω ε dχ ε Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z f ( Q − t e ) t − α (cid:18) r − tr (cid:19) n − dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p r n − dr p + Z ω ε dχ ε Z | f ( Q ) | p r n − dr p = I + I . Applying the generalized Mincovski inequality, we obtain I π sin απ = 1 ε α Z ω ε dχ ε Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z f ( Q − t e ) t − α (cid:18) r − tr (cid:19) n − dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p r n − dr p ≤ ε α Z ω ε ε Z t α − ε Z t | f ( Q − t e ) | p (cid:18) r − tr (cid:19) ( p − n − ( r − t ) n − dr p dt p dχ p ≤ ε α Z ω ε ε Z t α − ε Z t | f ( P + [ r − t ] e ) | p ( r − t ) n − dr p dt p dχ p ≤ ε α Z ω ε ε Z t α − ε Z | f ( P + r e ) | p r n − dr p dt p dχ p = 1 α k f k L p (∆ ε ) , ∆ ε := { Q ∈ Ω ε , r < ε } . Note that mess ∆ ε → , ε ↓ , hence I , I → , ε ↓ . It follows that I → , ε ↓ . In the same way, we obtain I → , ε ↓ . Since we proved that I , I , I → , ε ↓ , then relation (2.2) holds. This completes the proof corresponding to the left-sidecase. The proof corresponding to the right-side case is absolutely analogous. (cid:3) Theorem 2.3.
Suppose f = I α ψ or f = I αd − ψ, ψ ∈ L p (Ω) , ≤ p < ∞ ; then D α f = ψ or D αd − f = ψ respectively.Proof. Consider r n − f ( Q ) − ( r − τ ) n − f ( Q − τ e )= 1Γ( α ) r Z ψ ( Q − t e ) t − α ( r − t ) n − dt − α ) r Z τ ψ ( Q − t e )( t − τ ) − α ( r − t ) n − dt = τ α − r Z ψ ( Q − t e ) k (cid:18) tτ (cid:19) ( r − t ) n − dt, k ( t ) = 1Γ( α ) ( t α − , < t < t α − − ( t − α − , t > . Hence in the case ( ε ≤ r ≤ d ) , we have( ψ + ε f )( Q ) = r Z ε r n − f ( Q ) − ( r − τ ) n − f ( Q − τ e ) r n − τ α +1 dτ M. V. KUKUSHKIN EJDE-2018/29 = r Z ε τ − dτ r Z ψ ( Q − t e ) k (cid:18) tτ (cid:19) (1 − t/r ) n − dt = r Z ψ ( Q − t e ) (1 − t/r ) n − dt r Z ε k (cid:18) tτ (cid:19) τ − dτ = r Z ψ ( Q − t e ) (1 − t/r ) n − t − dt t/ε Z t/r k ( s ) ds. Applying formula (6.12) [20, p.106], we get( ψ + ε f )( Q ) · α Γ(1 − α ) = r Z ψ ( Q − t e ) (1 − t/r ) n − (cid:20) ε K (cid:18) tε (cid:19) − r K (cid:18) tr (cid:19)(cid:21) dt. Since in accordance with (2.7), we have K (cid:18) tr (cid:19) = [Γ(1 − α )Γ( α )] − (cid:18) tr (cid:19) α − , then( ψ + ε f )( Q ) · α Γ(1 − α ) = r/ε Z K ( t ) ψ ( Q − εt e ) (1 − εt/r ) n − dt − f ( Q )Γ(1 − α ) r α . Taking into account (1.8),(2.7), and that the function ψ ( Q ) has the zero extensionoutside of ¯Ω , we obtain( D α ,ε f )( Q ) − ψ ( Q ) = ∞ Z K ( t ) (cid:2) ψ ( Q − εt e )(1 − εt/r ) n − − ψ ( Q ) (cid:3) dt, ε ≤ r ≤ d. Consider the case (0 ≤ r < ε ) . In accordance with (1.7), we have( D α ,ε f )( Q ) − ψ ( Q ) = f ( Q ) ε α Γ(1 − α ) − ψ ( Q ) . Using the generalized Mincovski inequality, we get k ( D α ,ε f )( Q ) − ψ ( Q ) k L p (Ω) ≤ ∞ Z K ( t ) k ψ ( Q − εt e )(1 − εt/r ) n − − ψ ( Q ) k L p (Ω) dt + 1Γ(1 − α ) ε α k f k L p (∆ ′ ε ) + k ψ k L p (∆ ′ ε ) , ∆ ′ ε = ∆ ε ∪ ˜Ω ε , here we use the denotations that were used in Theorem 2.2. Arguing as above (seeTheorem 2.2), we see that all three summands of the right side of the previousinequality tend to zero, when ε ↓ . (cid:3) Theorem 2.4.
Suppose ρ ∈ Lip λ, α < λ ≤ , f ∈ H (Ω); then ρf ∈ I α ( L ) ∩ I αd − ( L ) . JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 9
Proof.
We provide a proof only for the left-side case, the proof corresponding tothe right-side case is absolutely analogous. First, assume that f ∈ C ∞ (Ω) . Usingthe denotations that were used in Theorem 2.2, we have k ψ + ε f − ψ + ε f k L (Ω) ≤k ψ + ε f − ψ + ε f k L (Ω ε ) + k ψ + ε f − ψ + ε f k L (˜Ω ε ) , (2.14)where ε > ε > . We have the following reasoning k ψ + ε f − ψ + ε f k L (Ω ε ) ≤ Z ω ε dχ d Z ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − ε Z r − ε ( ρf )( Q ) r n − − ( ρf )( T ) t n − r n − ( r − t ) α +1 dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r n − dr + Z ω ε dχ ε Z ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − ε Z ( ρf )( Q ) r n − r n − ( r − t ) α +1 dt − r − ε Z ( ρf )( Q ) r n − − ( ρf )( T ) t n − r n − ( r − t ) α +1 dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r n − dr + Z ω ε dχ ε Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − ε Z ( ρf )( Q ) r n − r n − ( r − t ) α +1 dt − r − ε Z ( ρf )( Q ) r n − r n − ( r − t ) α +1 dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r n − dr = I + I + I . Since f ∈ C ∞ (Ω) , then for sufficiently small ε > f ( Q ) = 0 , r < ε . This implies that I = I = 0 and that the second summand of the right side ofinequality (2.14) equals zero. Making the change the variable in I , then using thegeneralized Minkowski inequality, we get I = Z ω ε dχ d Z ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε Z ε ( ρf )( Q ) r n − − ( ρf )( Q − e t )( r − t ) n − r n − t α +1 dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r n − dr ≤ ε Z ε t − α − Z ω ε dχ d Z ε (cid:12)(cid:12) ( ρf )( Q ) − (1 − t/r ) n − ( ρf )( Q − e t ) (cid:12)(cid:12) r n − dr dt ≤ ε Z ε t − α − Z ω ε dχ d Z ε | ( ρf )( Q ) − ( ρf )( Q − e t ) | r n − dr dt + ε Z ε t − α − Z ω ε dχ d Z ε (cid:2) − (1 − t/r ) n − (cid:3) | ( ρf )( Q − e t ) | r n − dr dt ≤ C ε Z ε t λ − α − dt + ε Z ε t − α Z ω ε dχ d Z ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r n − X i =0 (cid:18) tr (cid:19) i ( ρf )( Q − e t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r n − dr dt. Using the function f property, we see that there exists a constant δ such that f ( Q − e t ) = 0 , r < δ. In accordance with the above reasoning, we have I ≤ C ε λ − α − ε λ − α λ − α + k f k L (Ω) ε − α − ε − α δ (1 − α ) ( n − . Applying Theorem 2.1, we complete the proof for the case ( f ∈ C ∞ (Ω)) . Nowassume that f ∈ H (Ω) , then there exists the sequence { f n } ⊂ C ∞ (Ω) , f n H −→ f. It is easy to prove that ρf n L −→ ρf. In accordance with the proven above fact, wehave ρf n = I α ϕ n , { ϕ n } ∈ L (Ω) , therefore I α ϕ n L −→ ρf. (2.15)To conclude the proof, it is sufficient to show that ϕ n L −→ ϕ ∈ L (Ω) . Note that byvirtue of Theorem 2.2 we have D α ρf n = ϕ n . Let c n,m := f n + m − f n , we have k ϕ n + m − ϕ n k L (Ω) ≤ α Γ(1 − α ) Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z ( ρc n,m )( Q ) r n − − ( ρc n,m )( T ) t n − r n − ( t − r ) α +1 dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ + 1Γ(1 − α ) Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ( ρc n,m )( Q ) r α (cid:12)(cid:12)(cid:12)(cid:12) dQ = I + I . Consider I . It can be shown in the usual way thatΓ(1 − α ) α I ≤ Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z ( ρc n,m )( Q ) − ( ρc n,m )( Q − e t ) t α +1 dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ + Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z ( ρc n,m )( Q − e t )[1 − (1 − t/r ) n − ] t α dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ = I + I ; I ≤ sup Q ∈ Ω | ρ ( Q ) | Z Ω r Z | c n,m ( Q ) − c n,m ( Q − e t ) | t α +1 dt dQ + Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z c n,m ( Q − e t )[ ρ ( Q ) − ρ ( Q − e t )] t α +1 dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ = I + I . Applying the generalized Minkowski inequality, then representing the function un-der the inner integral by the directional derivative, we get I ≤ C d Z t − α − Z Ω | c n,m ( Q ) − c n,m ( Q − e t ) | dQ dt = C d Z t − α − Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Z c ′ n,m ( Q − e τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ dt. JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 11
Using the Cauchy-Schwarz inequality, the Fubini theorem, we have I ≤ C d Z t − α − Z Ω dQ t Z (cid:12)(cid:12) c ′ n,m ( Q − e τ ) (cid:12)(cid:12) dτ t Z dτ dt = C d Z t − α − / t Z dτ Z Ω (cid:12)(cid:12) c ′ n,m ( Q − e τ ) (cid:12)(cid:12) dQ dt ≤ C d − α − α k c ′ n,m k L (Ω) . Arguing as above, using the Holder property of the function ρ, we see that I ≤ M d Z t λ − α − Z Ω | c n,m ( Q − e t ) | dQ dt ≤ M d λ − α λ − α k c n,m k L (Ω) . It can be shown in the usual way that I ≤ C Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z | c n,m ( Q − e t ) | n − X i =0 (cid:18) tr (cid:19) i r − t − α dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ ≤ C Z Ω r Z t − α dt r Z t (cid:12)(cid:12) c ′ n,m ( Q − e τ ) (cid:12)(cid:12) dτ r − dQ = C Z Ω r Z (cid:12)(cid:12) c ′ n,m ( Q − e τ ) (cid:12)(cid:12) dτ τ Z t − α dt r − dQ ≤ C − α Z Ω r Z (cid:12)(cid:12) c ′ n,m ( Q − e τ ) (cid:12)(cid:12) τ − α dτ dQ . Applying the generalized Minkowski inequality, we have I ≤ C d Z τ − α dτ Z Ω (cid:12)(cid:12) c ′ n,m ( Q − e τ ) (cid:12)(cid:12) dQ ≤ C d − α − α k c ′ n,m k L (Ω) . Consider I , we have I ≤ C Γ(1 − α ) Z Ω | c n,m ( Q ) | r − α dQ = C Γ(1 − α ) Z Ω r − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z c ′ n,m ( Q − e t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ ≤ C Γ(1 − α ) Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z c ′ n,m ( Q − e t ) t − α dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ . Using the generalized Minkowski inequality, then applying the trivial estimates, weget I ≤ C Z ω d Z t − α dt d Z t | c ′ n,m ( Q − e t ) | r n − dr dχ ≤ C Z ω d Z t − α dt d Z | c ′ n,m ( Q − e t ) | r n − dr dχ = C d Z t − α dt Z ω dχ d Z | c ′ n,m ( Q − e t ) | r n − dr ≤ C d − α − α k c ′ n,m k L (Ω) . Taking into account that the sequences { f n } , { f ′ n } are fundamental, we obtain I , I → . Hence the sequence { ϕ n } is fundamental and ϕ n L −→ ϕ ∈ L (Ω) . Notethat by virtue of Theorem 2.1 the directional fractional integral operator is boundedon the space L (Ω) . Hence I α ϕ n L −→ I α ϕ. Combining this fact with (2.15), we have ρf = I α ϕ. (cid:3) Lemma 2.5.
The operator D α is a restriction of the operator D α . Proof.
We need to show that the next equality holds( D α f )( Q ) = (cid:0) D α f (cid:1) ( Q ) , f ∈ ◦ W lp (Ω) . (2.16)It can be shown in the usual way that D α v = α Γ(1 − α ) r Z v ( Q ) − v ( T )( r − t ) α +1 (cid:18) tr (cid:19) n − dt + C ( α ) n v ( Q ) r − α = α Γ(1 − α ) r Z r n − v ( Q ) − t n − v ( T ) r n − ( r − t ) α +1 dt − α v ( Q )Γ(1 − α ) r Z r n − − t n − r n − ( r − t ) α +1 dt + C ( α ) n v ( Q ) r − α = ( D α v )( Q ) − α v ( Q )Γ(1 − α ) n − X i =0 r − − i r Z t i ( r − t ) α dt + C ( α ) n v ( Q ) r − α − v ( Q ) r − α Γ(1 − α ) = ( D α v )( Q ) − I + I − I . (2.17) JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 13
Using the formula of the fractional integral of a power function (2.44) [20, p.47],we have I = α v ( Q ) r − Γ(1 − α ) r Z dt ( r − t ) α + α v ( Q )Γ(1 − α ) n − X i =1 r − − i r Z t i ( r − t ) α dt = v ( Q ) α Γ(2 − α ) r − α + v ( Q ) α n − X i =1 r − − i ( I − α t i )( r )= v ( Q ) α Γ(2 − α ) r − α + v ( Q ) α n − X i =1 r − α i !Γ(2 − α + i ) . Hence I + I = v ( Q ) r − α Γ(2 − α ) + v ( Q ) r − α α n − X i =1 i !Γ(2 − α + i ) = 2 v ( Q ) r − α Γ(3 − α )+ v ( Q ) r − α α n − X i =2 i !Γ(2 − α + i ) = 3! v ( Q ) r − α Γ(4 − α ) + v ( Q ) r − α α n − X i =3 i !Γ(2 − α + i )= ( n − v ( Q ) r − α Γ( n − − α ) + v ( Q ) r − α α ( n − n − α ) = C ( α ) n v ( Q ) r − α . (2.18)Therefore I − I − I = 0 and obtain equality (2.16). Let us prove that theconsidered operators do not coincide with each other. For this purpose considerthe function f = I α ϕ, ϕ ∈ L p (Ω) , then in accordance with Theorem 2.2, we have D α I α ϕ = ϕ. Hence I α ( L p ) ⊂ D (cid:0) D α (cid:1) . Now it is sufficient to notice that ∃ f ∈ I α ( L p ) , f (Λ) = 0 , where Λ ⊂ ∂ Ω , mess Λ = 0 . On the other hand, we know that f ( ∂ Ω) = 0 a.e., ∀ f ∈ D ( D α ) . (cid:3) Lemma 2.6.
The following identity holds ( D α ) ∗ = D αd − , where limits (1.9) are understood as the limits with respect to the L (Ω) norm.Proof. Let us show that the next relation is true( D α f, g ) L (Ω) = ( f, D αd − g ) L (Ω) , (2.19) f ∈ I α ( L ) , g ∈ I αd − ( L ) . Note that by virtue of Theorem 2.3, we have D α I α ϕ = ϕ, D αd − I αd − ψ = ψ, where ψ, ψ ∈ L (Ω) . Hence, using Theorem 2.1, we have that the left and right side of(2.19) are finite. Therefore using the Fubini theorem, we have( D α f, g ) L (Ω) = Z ω dχ d Z ϕ ( Q ) (cid:0) I αd − ψ (cid:1) ( Q ) r n − dr = 1Γ( α ) Z ω dχ d Z ϕ ( Q ) r n − dr d Z r ψ ( T )( t − r ) − α dt = 1Γ( α ) Z ω dχ d Z ψ ( T ) t n − dt t Z ϕ ( Q )( t − r ) − α (cid:16) rt (cid:17) n − dr = Z Ω (cid:0) I α ϕ (cid:1) ( Q ) ψ ( Q ) dQ = ( f, D αd − g ) L (Ω) . (2.20)Thus inequality (2.19) is proved. It follows that D( D αd − ) ⊂ D (cid:0) [ D α ] ∗ (cid:1) . Let us provethat D (cid:0) [ D α ] ∗ (cid:1) ⊂ D( D αd − ) . In accordance with the definition of adjoint operator,we have (cid:0) D α f, g (cid:1) L (Ω) = (cid:0) f, [ D α g ] ∗ (cid:1) L (Ω) , f ∈ D( D α ) , g ∈ D (cid:0) [ D α ] ∗ (cid:1) . Note that since R( D αd − ) = L (Ω) , then R (cid:0) [ D α ] ∗ (cid:1) = L (Ω) . Using the Fubinitheorem, it can be easily shown that (cid:0) D α f, g − I αd − [ D α ] ∗ g (cid:1) L (Ω) = 0 . By virtue of Theorem 2.3, we have R( D α ) = L (Ω) . Hence g = I αd − [ D α ] ∗ g a.e.It implies that D (cid:0) [ D α ] ∗ (cid:1) ⊂ D (cid:0) D αd − (cid:1) . (cid:3) Strictly accretive property
The following theorem establishes the strictly accretive property (see [8, p. 352])of the Kipriyanov fractional differential operator.
Theorem 3.1.
Suppose ρ ( Q ) is a real non-negative function, ρ ∈ Lip λ, λ > α ; then the following inequality holds Re( f, D α f ) L (Ω ,ρ ) ≥ µ k f k L (Ω ,ρ ) , f ∈ H (Ω) , (3.1) where µ = 12 d − α (cid:16) Γ − (1 − α ) + C ( α ) n (cid:17) − αM d λ − α − α )( λ − α ) inf ρ . Moreover, if we have in additional that for any fixed direction e the function ρ ismonotonically non-increasing, then µ = 12 d − α (cid:16) Γ − (1 − α ) + C ( α ) n (cid:17) . Proof.
Consider a real case and let f ∈ C ∞ (Ω) , we have ρ ( Q ) f ( Q )( D α f )( Q ) = 12 ( D α ρf )( Q )+ α − α ) r Z ρ ( Q )[ f ( Q ) − f ( T )] ( r − t ) α +1 (cid:18) tr (cid:19) n − dt + α − α ) r Z f ( T )[ ρ ( T ) − ρ ( Q )]( r − t ) α +1 (cid:18) tr (cid:19) n − dt + C αn ρf )( Q ) r − α = I ( Q ) + I ( Q ) + I ( Q ) + I ( Q ) . (3.2) JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 15
Applying Theorem 1.4, we have Z Ω I ( Q ) dQ = 12 Z Ω ( D αd − Q )( ρf )( Q ) dQ = 12Γ(1 − α ) Z Ω ( d ( e ) − r ) − α ( ρf )( Q ) dQ ≥ d − α − α ) k f k L (Ω ,ρ ) . (3.3)Using the Fubini theorem, it can be shown in the usual way that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω I ( Q ) dQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α − α ) Z ω dχ d ( e ) Z r n − dr r Z f ( T ) | ρ ( T ) − ρ ( Q ) | ( r − t ) α +1 (cid:18) tr (cid:19) n − dt = α − α ) Z ω dχ d ( e ) Z f ( T ) t n − dt d ( e ) Z t | ρ ( T ) − ρ ( Q ) | ( r − t ) α +1 dr = α − α ) Z ω dχ d ( e ) Z f ( T ) t n − dt d ( e ) − t Z | ρ ( Q − τ e ) − ρ ( Q ) | τ α +1 dτ ≤ αM − α ) Z ω dχ d ( e ) Z f ( T ) t n − dt d ( e ) − t Z τ λ − α +1 dτ ≤ αM d λ − α − α )( λ − α ) k f k L (Ω) . (3.4)Consider Z Ω I ( Q ) dQ = C ( α ) n Z Ω ( ρf )( Q ) r − α dQ ≥ C ( α ) n d − α k f k L (Ω ,ρ ) . (3.5)Combining (3),(3),(3),(3.5), and the fact that I is non-negative, we obtain( f, D α f ) L (Ω ,ρ ) ≥ µ k f k L (Ω ,ρ ) , f ∈ C ∞ (Ω) . (3.6)In the case when for any fixed direction e the function ρ is monotonically non-increasing, we have I ≥ . Hence (3.6) is fulfilled. Now assume that f ∈ H (Ω) , then there exists a sequence { f k } ∈ C ∞ (Ω) , f k H −→ f. Using this fact, it is not hardto prove that f k L (Ω ,ρ ) −→ f. Using inequality (1.3), we prove that k D α f k L (Ω ,ρ ) ≤ C k f k H (Ω) . Therefore D α f k L (Ω ,ρ ) −→ D α f. Hence using the continuity property ofthe inner product, we get( f k , D α f k ) L (Ω ,ρ ) → ( f, D α f ) L (Ω ,ρ ) . Passing to the limit on the left and right side of inequality (3.6), we obtain( f, D α f ) L (Ω ,ρ ) ≥ µ k f k L (Ω ,ρ ) , f ∈ H (Ω) . (3.7)Now let us consider the complex case. Note that the following equality is trueRe( f, D α f ) L (Ω ,ρ ) = ( u, D α u ) L (Ω ,ρ ) + ( v, D α v ) L (Ω ,ρ ) , u = Re f, v = Im f. (3.8)Combining (3.8), (3.7), we obtain (3.1). (cid:3) Sectorial property
Consider a uniformly elliptic operator with real coefficients and the Kipriyanovfractional derivative in the final term Lu := − D j ( a ij D i u ) + ρ D α u, ( i, j = 1 , n ) , D( L ) = H (Ω) ∩ H (Ω) ,a ij ( Q ) ∈ C ( ¯Ω) , a ij ξ i ξ j ≥ a | ξ | , a > , (4.1) ρ ( Q ) > , ρ ( Q ) ∈ Lip λ, α < λ ≤ . (4.2)We assume in additional that µ > , here we use the denotation that is used inTheorem (3.1). We also consider the formal adjoint operator L + u := − D i ( a ij D j u ) + D αd − ρu, D( L + ) = D( L ) , and the operator H = 12 ( L + L + ) . We use a special case of the Green formula − Z Ω D j ( a ij D i u ) ¯ v dQ = Z Ω a ij D i u D j v dQ , u ∈ H (Ω) , v ∈ H (Ω) . (4.3) Remark 4.1.
The operators
L, L + , H are closeable. We can easily check this fact,if we apply Theorem 3.4 [8, p.337] . We have the following lemma.
Theorem 4.2.
The operators ˜ L, ˜ L + are strictly accretive, their numerical rangebelongs to the sector S := { ζ ∈ C : | arg ( ζ − γ ) | ≤ θ } , where θ and γ are defined by the coefficients of the operator L. Proof.
Consider the operator L. It is not hard to prove that − Re (cid:0) D j [ a ij D i f ] , f (cid:1) L (Ω) ≥ a k f k L (Ω) , f ∈ D( L ) . (4.4)HenceRe( f n , Lf n ) L (Ω) ≥ a k f n k L (Ω) + Re( f n , D α f n ) L (Ω ,ρ ) , { f n } ⊂ D( L ) . (4.5)Assume that f ∈ D( ˜ L ) . In accordance with the definition, there exists a sequence { f n } ⊂ D( L ) , f n −→ L f. By virtue of (4.5), we easily prove that f ∈ H (Ω) . Usingthe continuity property of the inner product, we pass to the limit on the left andright side of inequality (4.5). Thus, we haveRe( f, ˜ Lf ) L (Ω) ≥ a k f k L (Ω) + Re( f, D α f ) L (Ω ,ρ ) , f ∈ D( ˜ L ) . (4.6)By virtue of Theorem 3.1, we can rewrite the previous inequality as followsRe( f, ˜ Lf ) L (Ω) ≥ a k f k L (Ω) + µ k f k L (Ω ,ρ ) , f ∈ D( ˜ L ) . (4.7)Applying the Friedrichs inequality to the first summand of the right side, we getRe( f, ˜ Lf ) L (Ω) ≥ µ k f k L (Ω) , f ∈ D( ˜ L ) , µ = a + µ inf ρ ( Q ) . (4.8) JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 17
Consider the imaginary component of the form, generated by the operator L (cid:12)(cid:12) Im( f, Lf ) L (Ω) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω (cid:0) a ij D i uD j v − a ij D i vD j u (cid:1) dQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) ( u, D α v ) L (Ω ,ρ ) − ( v, D α u ) L (Ω ,ρ ) (cid:12)(cid:12) = I + I . (4.9)Using the Cauchy Schwarz inequality for sums, the Young inequality, we have a ij D i uD j v ≤ a | Du || Dv | ≤ a (cid:0) | Du | + | Dv | (cid:1) , a ( Q ) = n X i,j =1 | a ij ( Q ) | / . (4.10)Hence I ≤ a k f k L (Ω) , a = sup a ( Q ) . (4.11)Applying inequality (1.3), the Young inequality, we get (cid:12)(cid:12) ( u, D α v ) L (Ω ,p ) (cid:12)(cid:12) ≤ C k u k L (Ω) k D α v k L q (Ω) ≤ C k u k L (Ω) (cid:26) Kδ ν k v k L (Ω) + δ − ν k v k L (Ω) (cid:27) ≤ ε k u k L (Ω) + ε (cid:18) KC √ δ ν (cid:19) k v k L (Ω) + ε (cid:0) C δ − ν (cid:1) k v k L (Ω) , (4.12)where 2 < q < n/ (2 α − n ) . Hence I ≤ (cid:12)(cid:12) ( u, D α v ) L (Ω ,ρ ) (cid:12)(cid:12) + (cid:12)(cid:12) ( v, D α u ) L (Ω ,ρ ) (cid:12)(cid:12) ≤ ε (cid:16) k u k L (Ω) + k v k L (Ω) (cid:17) + ε (cid:18) KC √ δ ν (cid:19) (cid:16) k u k L (Ω) + k v k L (Ω) (cid:17) + ε (cid:0) C δ − ν (cid:1) (cid:16) k u k L (Ω) + k v k L (Ω) (cid:17) = (cid:18) εδ − ν C + 1 ε (cid:19) k f k L (Ω) + εδ − ν C k f k L (Ω) . (4.13)Taking into account (4) and combining (4.11), (4.13), we easily prove that (cid:12)(cid:12)(cid:12) Im( f, ˜ Lf ) L (Ω) (cid:12)(cid:12)(cid:12) ≤ (cid:18) εδ − ν C + 1 ε (cid:19) k f k L (Ω) + (cid:0) εδ − ν C + a (cid:1) k f k L (Ω) , f ∈ D( ˜ L ) . Thus by virtue of (4.8) for an arbitrary number k > , the next inequality holdsRe( f, ˜ Lf ) L (Ω) − k (cid:12)(cid:12)(cid:12) Im( f, ˜ Lf ) L (Ω) (cid:12)(cid:12)(cid:12) ≥ (cid:0) a − k [ εδ − ν C + a ] (cid:1) k f k L (Ω) + (cid:18) µ inf ρ ( Q ) − k (cid:20) εδ − ν C + 1 ε (cid:21)(cid:19) k f k L (Ω) . Choose k = a (cid:0) εδ − ν C + a (cid:1) − , we get (cid:12)(cid:12)(cid:12) Im( f, ( ˜ L − γ ) f ) L (Ω) (cid:12)(cid:12)(cid:12) ≤ k Re( f, ( ˜ L − γ ) f ) L (Ω) ,γ = µ inf ρ ( Q ) − k (cid:20) εδ − ν C + 1 ε (cid:21) . (4.14)This inequality shows that the numerical range Θ( ˜ L ) belongs to the sector with thetop γ and the semi-angle θ = arctan(1 /k ) . The prove corresponding to the operator˜ L + is analogous. (cid:3) We do not study in detail the conditions under which γ > , but we just notethat relation (4) gives us an opportunity to formulate them in an easy way. Further,we assume that the coefficients of the operator L such that γ > . Theorem 4.3.
The operators ˜ L, ˜ L + , ˜ H is m-sectorial, the operator ˜ H is selfadjoint.Proof. By virtue of Theorem 4.2 we have that the operator ˜ L is sectorial i.e. Θ( L ) ⊂ S . Applying Theorem 3.2 [8, p. 336] we conclude that R( ˜ L − ζ ) is a closed spacefor any ζ ∈ C \ S and that the next relation holdsdef( ˜ L − ζ ) = η, η = const . (4.15)Using (4.8), it is not hard to prove that k ˜ Lf k L (Ω) ≥ √ µ k f k L (Ω) , f ∈ D( ˜ L ) . Hence the inverse operator ( ˜ L + ζ ) − is defined on the subspace R( ˜ L + ζ ) , Re ζ > . In accordance with condition (3.38) [8, p.350], we need to show thatdef( ˜ L + ζ ) = 0 , k ( ˜ L + ζ ) − k ≤ (Re ζ ) − , Re ζ > . (4.16)Since γ > , then the left half-plane is included in the the set C \ S . Note that byvirtue of inequality (4.8), we haveRe( f, ( ˜ L − ζ ) f ) L (Ω) ≥ ( µ − Re ζ ) k f k L (Ω) . (4.17)Let ζ ∈ C \ S , Re ζ < . Since the operator ˜ L − ζ has a closed range R( ˜ L − ζ ) , then we have L (Ω) = R( ˜ L − ζ ) ⊕ R( ˜ L − ζ ) ⊥ . Note that C ∞ (Ω) ∩ R( ˜ L − ζ ) ⊥ = 0 , because if we assume the contrary, then applyinginequality (4.17) for any element u ∈ C ∞ (Ω) ∩ R( ˜ L − ζ ) ⊥ , we get( µ − Re ζ ) k u k L (Ω) ≤ Re( u, ( ˜ L − ζ ) u ) L (Ω) = 0 , hence u = 0 . Thus this fact implies that( g, v ) L (Ω) = 0 , g ∈ R( ˜ L − ζ ) ⊥ , ∈ C ∞ (Ω) . Since C ∞ (Ω) is a dense set in L (Ω) , then R( ˜ L − ζ ) ⊥ = 0 . It follows that def( ˜ L − ζ ) = 0 . Now if we note (4.15) then we came to the conclusion that def( ˜ L − ζ ) =0 , ζ ∈ C \ S . Hence def( ˜ L + ζ ) = 0 , Re ζ > . Thus the proof of the first relationof (4.16) is complete. To prove the second relation (4.16) we should note that( µ + Re ζ ) k f k L (Ω) ≤ Re( f, ( ˜ L + ζ ) f ) L (Ω) ≤ k f k L (Ω) k ( ˜ L + ζ ) k L (Ω) ,f ∈ D( ˜ L ) , Re ζ > . Using first relation (4.16), we have k ( ˜ L + ζ ) − g k L (Ω) ≤ ( µ + Re ζ ) − k g k L (Ω) ≤ (Re ζ ) − k g k L (Ω) , g ∈ L (Ω) . This implies that k ( ˜ L + ζ ) − k ≤ (Re ζ ) − , Re ζ > . This concludes the proof corresponding to the operator ˜ L. The proof correspondingto the operator ˜ L + is analogous. Consider the operator ˜ H. It is obvious that ˜ H isa symmetric operator. Hence Θ( ˜ H ) ⊂ R . Using (4.5) and arguing as above, we seethat ( f, ˜ Hf ) L (Ω) ≥ µ k f k L (Ω) . (4.18) JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 19
Continuing the used above line of reasoning and applying Theorem 3.2 [8, p.336],we see that def( ˜ H − ζ ) = 0 , Im ζ = 0; (4.19)def( ˜ H + ζ ) = 0 , k ( ˜ H + ζ ) − k ≤ (Re ζ ) − , Re ζ > . (4.20)Combining (4.19) with Theorem 3.16 [8, p.340], we conclude that the operator˜ H is selfadjoint. Finally, note that in accordance with the definition, relation(4.20) implies that the operator ˜ H is m-accretive. Since we already know thatthe operators ˜ L, ˜ L + , ˜ H are sectorial and m-accretive, then in accordance with thedefinition they are m-sectorial. (cid:3) Main theorems
In this section we need using the theory of sesquilinear forms. If it is not statedotherwise, we use the definitions and the notation of the monograph [8]. Considerthe forms t [ u, v ] = Z Ω a ij D i u D j vdQ + Z Ω ρ D α u ¯ vdQ, u, v ∈ H (Ω) , (5.1) t ∗ [ u, v ] := t [ v, u ] = Z Ω a ij D j u D i vdQ + Z Ω uρ D α vdQ, Re t := 12 ( t + t ∗ ) . For convenience, we use the shorthand notation h := Re t. Lemma 5.1.
The form t is a closed sectorial form, moreover t = ˜ f , where f [ u, v ] = ( ˜ Lu, v ) L (Ω) , u, v ∈ D( ˜ L ) . Proof.
First we shall show that the following inequality holds C k f k H (Ω) ≤ | t [ f ] | ≤ C k f k H (Ω) , f ∈ H (Ω) . (5.2)Using (4.6), Theorem 3.1, we obtain C k f k H (Ω) ≤ Re t [ f ] ≤ | t [ f ] | , f ∈ H (Ω) . (5.3)Applying (4.10),(4), we get | t [ f ] | ≤ (cid:12)(cid:12)(cid:12)(cid:0) a ij D i f, D j f (cid:1) L (Ω) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( ρ D α f, f ) L (Ω) (cid:12)(cid:12)(cid:12) ≤ C k f k H (Ω) , f ∈ H (Ω) . (5.4)Note that H (Ω) ⊂ D(˜ t ) . If f ∈ D(˜ t ) , then in accordance with the definition, thereexists a sequence { f n } ⊂ D( t ) , f n −→ t f. Applying (5.2), we get f n H −−→ f. Since thespace H (Ω) is complete, then D(˜ t ) ⊂ H (Ω) . It implies that D(˜ t ) = D( t ) . Hence t is a closed form. The proof of the sectorial property contains in the proof ofTheorem 4.2. Let us prove that t = ˜ f . First, we shall show that f [ u, v ] = t [ u, v ] , u, v ∈ D( f ) . (5.5)Using formula (4.3), we have( Lu, v ) L (Ω) = t [ u, v ] , u, v ∈ D( L ) . (5.6)Hence we can rewrite relation (5.2) in the following form C k f k H (Ω) ≤ (cid:12)(cid:12) ( Lf, f ) L (Ω) (cid:12)(cid:12) ≤ C k f k H (Ω) , f ∈ D( L ) . (5.7) Assume that f ∈ D( ˜ L ) , then there exists a sequence { f n } ∈ D( L ) , f n −→ L f. Com-bining (5.7),(5.2), we obtain f n −→ t f. These facts give us an opportunity to passto the limit on the left and right side of (5.6). Thus, we obtain (5.5). Combining(5.5),(5.2), we get C k f k H (Ω) ≤ | f [ f ] | ≤ C k f k H (Ω) , f ∈ D( f ) . (5.8)Note that by virtue of Theorem 4.2 the operator ˜ L is sectorial, hence due to Theo-rem 1.27 [8, p.399] the form f is closable. Using the facts established above, Theorem1.17 [8, p.395], passing to the limit on the left and right side of inequality (5.5), weget ˜ f [ u, v ] = t [ u, v ] , u, v ∈ H (Ω) . This concludes the proof. (cid:3)
Lemma 5.2.
The form h is a closed symmetric sectorial form, moreover h = ˜ k , where k [ u, v ] = ( ˜ Hu, v ) L (Ω) , u, v ∈ D( ˜ H ) . Proof.
To prove the symmetric property (see(1.5) [8, p.387]) of the form h, it issufficient to note that h [ u, v ] = 12 (cid:16) t [ u, v ] + t [ v, u ] (cid:17) = 12 (cid:16) t [ v, u ] + t [ u, v ] (cid:17) = h [ v, u ] , u, v ∈ D( h ) . Obviously, we have h [ f ] = Re t [ f ] . Hence applying (5.3), (5.4), we have C k f k H (Ω) ≤ h [ f ] ≤ C k f k H (Ω) , f ∈ H (Ω) . (5.9)Arguing as above, using (5.9), it is easy to prove that D(˜ h ) = H (Ω) . Hence theform h is a closed form. The proof of the sectorial property of the form h containsin the proof of Theorem 4.2. Let us prove that h = ˜ k . We shall show that k [ u, v ] = h [ u, v ] , u, v ∈ D( k ) . (5.10)Applying 2.5, Lemma 2.6, we have( ρ D α f, g ) L (Ω) = ( f, D αd − ρg ) L (Ω) , f, g ∈ H (Ω) . Combining this fact with formula (4.3), it is not hard to prove that(
Hu, v ) L (Ω) = h [ u, v ] , u, v ∈ D( H ) . (5.11)Using (5.11), we can rewrite estimate (5.9) as follows C k f k H (Ω) ≤ ( Hf, f ) L (Ω) ≤ C k f k H (Ω) , f ∈ D( H ) . (5.12)Note that in consequence of Remark 4.1 the operator H is closeable. Assumethat f ∈ D( ˜ H ) , then there exists a sequence { f n } ⊂ D( H ) , f n −→ H f. Combining(5.12),(5.9), we obtain f n −→ h f. Passing to the limit on the left and right side of(5.11), we get (5.10). Combining (5.10),(5.9), we obtain C k f k H (Ω) ≤ k [ f ] ≤ C k f k H (Ω) , f ∈ D( k ) . (5.13)Note that in consequence of Theorem 4.2 the operator ˜ H is sectorial. Hence byvirtue of Theorem 1.27 [8, p.399] the form k is closable. Using the proven abovefacts, Theorem 1.17 [8, p.395], passing to the limits on the left and right side ofinequality (5.10), we get ˜ k [ u, v ] = h [ u, v ] , u, v ∈ H (Ω) . JDE-2018/29 PROPERTIES OF FRACTIONAL DIFFERENTIATION OPERATORS 21
This completes the proof. (cid:3)
Theorem 5.3.
The operator ˜ H has a compact resolvent, the following estimateholds λ n ( L ) ≤ λ n ( ˜ H ) ≤ λ n ( L ) , n ∈ N , (5.14) where λ n ( L k ) , k = 0 , are respectively the eigenvalues of the following operatorswith real constant coefficients L k f = − a ijk D j D i f + ρ k f, D( L k ) = D( L ) ,a ijk ξ i ξ j > , ρ k > . (5.15) Proof.
First, we shall prove the following propositionsi)
The operators ˜ H, L k are positive-definite. Using the fact that the operator H is selfadjoint, relation (4.18), we conclude that the operator ˜ H is positive-definite.Using the definition, we can easily prove that the operators L k are positive-definite.ii) The space H (Ω) coincides with the energetic spaces H ˜ H , H L k as a set of ele-ments. Using Lemma 5.2, we have k f k H ˜ H = ˜ k [ f ] = h [ f ] , f ∈ H (Ω) . (5.16)Hence the space H ˜ H coincides with H (Ω) as a set of elements. Using this fact, weobtain the coincidence of the spaces H (Ω) and H L k as the particular case.iii) We have the following estimates k f k H L ≤ k f k H ˜ H ≤ k f k H L , f ∈ H (Ω) . (5.17)We obtain the equivalence of the norms k · k H and k · k H Lk as the particular caseof relation (5.2). It is obvious that there exist such operators L k that the nextinequalities hold k f k H L ≤ C k f k H (Ω) , C k f k H (Ω) ≤ k f k H L , f ∈ H (Ω) . (5.18)Combining (5.9),(5.16),(5.18), we get (5.17).Now we can prove the proposal of this theorem. Note that the operators ˜ H, L k are positive-definite, the norms k · k H , k · k H Lk , k · k H ˜ H are equivalent. Applying theRellich-Kondrashov theorem, we have that the energetic spaces H ˜ H , H L k are com-pactly embedded into L (Ω) . Using Theorem 3 [17, p.216], we obtain the fact thatthe operators L , L , ˜ H have a discrete spectrum. Taking into account (i),(ii),(iii),in accordance with the definition [17, p.225], we have L ≤ ˜ H ≤ L . Applying Theorem 1 [17, p.225], we obtain (5.14). Note that by virtue of Theorem4.3 the operator ˜ H is m-accretive. Hence 0 ∈ P ( ˜ H ) . Due to Theorem 5 [17, p.222]the operator ˜ H has a compact resolvent at the point zero. Applying Theorem 6.29[8, p.237], we conclude that the operator ˜ H has a compact resolvent. (cid:3) Theorem 5.4.
Operator ˜ L has a compact resolvent, discrete spectrum.Proof. Note that in accordance with Theorem 4.3 the operators ˜ L, ˜ H are m-sectorial,the operator ˜ H is self-adjoint. Applying Lemma 5.1, Lemma 5.2, Theorem 2.9 [8,p.409], we get T t = ˜ L, T h = ˜ H, where T t , T h are the Fridrichs extensions of theoperators ˜ L, ˜ H (see [8, p.409]) respectively. Since in accordance with the definition[8, p.424] the operator ˜ H is a real part of the operator ˜ L, then due to Theorem 5.3, Theorem 3.3 [8, p.424] the operator ˜ L has a compact resolvent. Applying Theorem6.29 [8, p.237], we conclude that the operator ˜ L has a discrete spectrum. (cid:3) Remark 5.5.
It can easily be checked that the Kypriaynov operator is reduced tothe Marchaud operator in the one-dimensional case. At the same time, the resultsof this work are only true for the dimensions n ≥ . However, using Corollary 1 [14] , which establishes the strictly accretive property of the Marchaud operator, wecan apply the obtained technique to the one-dimensional case. Conclusions
The paper presents the results obtained in the spectral theory of fractional differ-ential operators. A number of propositions of independent interest in the fractionalcalculus theory are proved, the new concept of a multidimensional directional frac-tional integral is introduced. The sufficient conditions of the representability bythe directional fractional integral are formulated. In particular, the inclusion ofthe Sobolev space to the class of functions that are representable by the directionalfractional integral is established. Note that the technique of the proofs, which isanalogous to the one-dimensional case, is of particular interest. It should be notedthat the extension of the Kipriyanov fractional differential operator is obtained,the adjoint operator is found, and the strictly accretive property is proved. Theseall create a complete description reflecting qualitative properties of fractional dif-ferential operators. As the main results, the following theorems establishing theproperties of an uniformly elliptic operator with the Kipriyanov fractional deriva-tive in the final term are proved: the theorem on the strictly accretive property,the theorem on the sectorial property, the theorem on the m-accretive property,the theorem establishing a two-sided estimate for the eigenvalues and discretenessof the spectrum of the real component. Using the sesquilinear forms theory, we ob-tained the major theoretical results. We consider the proofs corresponding to themultidimensional case, however the reduction to the one-dimensional case is possi-ble. For instance, the one-dimensional case is described in the paper [13]. We alsonote that the results in this direction can be obtained for the real axis. It is worthnoticing that the application of the sesquilinear forms theory, as a tool to studysecond order differential operators with a fractional derivative in the final term,gives an opportunity to analyze the major role of the senior term in the functionalproperties of the operator. This technique is novel and can be used for studyingthe spectrum of perturbed fractional differential operators. Therefore, the idea ofthe proof may be of interest regardless of the results.
Acknowledgments.
The author thanks Professor Alexander L. Skubachevskiifor valuable remarks and comments made during the report, which took place31.10.2017 at Peoples’ Friendship University of Russia, Moscow.
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Maksim V. KukushkinInternational Committee ”Continental”, Geleznovodsk 357401, Russia
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