Limiting Sobolev inequalities for vector fields and canceling linear differential operators
aa r X i v : . [ m a t h . A P ] A ug LIMITING SOBOLEV INEQUALITIES FOR VECTOR FIELDS ANDCANCELING LINEAR DIFFERENTIAL OPERATORS
JEAN VAN SCHAFTINGENAbstract. The estimate k D k − u k L n/ ( n − ≤ k A ( D ) u k L is shown to hold if and only if A ( D ) is elliptic and canceling. Here A ( D ) is ahomogeneous linear differential operator A ( D ) of order k on R n from a vectorspace V to a vector space E . The operator A ( D ) is de fi ned to be canceling if \ ξ ∈ R n \{ } A ( ξ )[ V ] = { } . This result implies in particular the classical Gagliardo–Nirenberg-Sobolev in-equality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for di ff eren-tial forms due to J. Bourgain and H. Brezis. In the proof, the class of cocancelinghomogeneous linear di ff erential operator L ( D ) of order k on R n from a vectorspace E to a vector space F is introduced. It is proved that L ( D ) is cocancel-ing if and only if for every f ∈ L ( R n ; E ) such that L ( D ) f = 0 , one has f ∈ ˙ W − ,n/ ( n − ( R n ; E ) . The results extend to fractional and Lorentz spacesand can be strengthened using some tools of J. Bourgain and H. Brezis. Contents1. Introduction 22. Estimates on L vector fi elds and cocanceling operators 53. Examples of cocanceling operators 84. Proof of the Sobolev estimate 105. Necessary conditions for the Sobolev estimate 146. Characterization and examples of canceling operators 187. Partially canceling operators 258. Fractional and Lorentz estimates 289. Strong Bourgain–Brezis estimates 37Acknowledgment 38References 38 Date : October 26, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Sobolev embedding; overdetermined elliptic operator; compatibility con-ditions; homogeneous di ff erential operator; canceling operator; cocanceling operator; exterior deriva-tive; symmetric derivative; homogeneous Triebel–Lizorkin space; homogeneous Besov space; Lorentzspace; homogeneous fractional Sobolev–Slobodecki˘ ı space; Korn–Sobolev inequality; Hodge inequal-ity; Saint- Venant compatibility conditions.
1. Introduction1.1.
Norms on vector valued homogeneous Sobolev spaces.
Given n ≥ , k ∈ N , p ≥ and a fi nite-dimensional vector space V , the homogeneous Sobolev space ˙ W k,p ( R n ; V ) can be characterized as the completion of the space of smooth vector fi elds C ∞ c ( R n ; V ) under the norm de fi ned for u ∈ C ∞ c ( R n ; V ) by k D k u k L p = (cid:16)Z R n | D k u | p (cid:17) p . When dim
V > , one can wonder whether the norm can be estimated by aquantity involving only some components of the derivative. More precisely, assumethat A ( D ) is a homogeneous di ff erential operator of order k on R n from V toanother fi nite-dimensional vector space E , that is there exist linear maps A α ∈L ( V ; E ) with α ∈ N n and | α | = k such that for every u ∈ C ∞ ( R n ; V ) , A ( D ) u = X α ∈ N n | α | = k A α ( ∂ α u ) ∈ C ∞ ( R n ; E ) . One can ask the question whether the norms de fi ned for u ∈ C ∞ c ( R n ; V ) by k D k u k L p and k A ( D ) u k L p are equivalent.When p > , the answer is given by the classical result Theorem 1.1 (A. P. Calderón and A. Zygmund, 1952 [13]) . Let < p < ∞ and A ( D ) be a homogeneous differential operator of order k on R n from V to E . Theestimate k D k u k L p ≤ C k A ( D ) u k L p , holds for every u ∈ C ∞ c ( R n ; V ) if and only if A ( D ) is elliptic. Here and in the sequel the constant C is understood to be independent of the vec-tor fi eld u . The ellipticity condition is the classical notion of ellipticity for overdeter-mined di ff erential operators [21, theorem 1; 36, de fi nition 1.7.1] (when dim V = 1 ,see also S. Agmon [2, §7; 3, de fi nition 6.3]): De fi nition 1.1. A homogeneous linear di ff erential operator A ( D ) on R n from V to E is elliptic if for every ξ ∈ R n \ { } , A ( ξ ) is one-to-one.The restriction p > is essential in theorem 1.1. Indeed, D. Ornstein [30] hasshown that there are no nontrivial L -estimates of derivatives . Theorem 1.2 (D. Ornstein, 1962) . Let A ( D ) and B ( D ) be homogeneous linear dif-ferential operators of order k on R n from V to E and from V to R respectively. Iffor every u ∈ C ∞ c ( R n ; V ) , k B ( D ) u k L ≤ C k A ( D ) u k L , then there exists T ∈ L ( E ; R ) such that B ( D ) = T ◦ A ( D ) . Whereas D. Ornstein’s result does not include explicitely vector valued operators, his theoremand his method of proof remain valid in this case. B. Kirchheim and J. Kristensen [22, 23] have givena proof that relies on the convexity of homogeneous rank-one convex functions; their result coversexplicitely the vectorial case.
IMITING SOBOLEV IN
EQUALITIES AND CANCELING OPERATORS 3
Here L ( E ; R ) denotes the set of linear maps from E to R . The derivatives B ( D ) u are then linear combinations of the derivatives A ( D ) u and the estimate istrivial in the sense that it follows immediately from the boundedness of linear mapsde fi ned on fi nite-dimensional vector spaces.1.2. A collection of known Sobolev inequalities and non-inequalities.
Whereastheorem 1.1 fails for p = 1 , one can ask whether in some other estimates the quan-tity k D k u k L can be replaced by some weaker quantity k A ( D ) u k L .Consider the classical Gagliardo–Nirenberg–Sobolev inequality [20; 29, p. 125]which states that for every vector fi eld u ∈ C ∞ c ( R n ; V ) , one has k u k L nn − ≤ C k Du k L . (1.1)One can wonder whether all the components of the derivative Du are necessary inthis estimate when u is a vector- fi eld.A fi rst example of such a possibility is the Korn–Sobolev inequality of M. J. Strauss[38, theorem 1] (see also [9, Corollary 26; 42, theorem 6]): for every u ∈ C ∞ c ( R n ; R n ) ,one has k u k L n/ ( n − ≤ C k∇ s u k L , (1.2)where ∇ s u = (cid:0) Du + ( Du ) ∗ (cid:1) denotes the symmetric part of the derivative Du ∈ C ∞ ( R n ; L ( R n ; R n ) . This inequality does not follow from (1.1), as the norms k∇ s u k L and k Du k L are not equivalent by theorem 1.2 (see also [16, theorem1]). In the three-dimensional space R , one can wonder whether an estimate of thekind k u k L ≤ C (cid:0) k div u k L + k curl u k L (cid:1) (1.3)holds for every u ∈ C ∞ c ( R ; R ) . The answer is known to be negative even in thecase where curl u = 0 ; a contradiction is obtained by taking suitable regularizationsof the gradient of Newton’s kernel x ∈ R − x π | x | . Surprisingly, J. Bourgainand H. Brezis [8, theorem 2; 9, corollary 7] have proved that for every vector fi eld u ∈ C ∞ c ( R ; R ) such that div u = 0 , one has k u k L ≤ C k curl u k L . (1.4)J. Bourgain and H. Brezis [9, Corollary 17] have proved similarly that for every dif-ferential form u ∈ C ∞ c ( R n ; V ℓ R n ) , one has the Hodge–Sobolev inequality k u k L n/ ( n − ≤ C (cid:0) k du k L + k d ∗ u k L (cid:1) . (1.5)(see also L. Lanzani and E. M. Stein [26]).1.3. Limiting Sobolev inequalities and canceling operators.
We would like todetermine whether for a given fi rst order homogeneous di ff erential operator A ( D ) an estimate of the form k u k L nn − ≤ C k A ( D ) u k L (1.6)holds. The answer is given by Theorem 1.3.
Let A ( D ) be a homogeneous linear differential operator of order k on R n from V to E . The estimate k D k − u k L nn − ≤ C k A ( D ) u k L , holds for every u ∈ C ∞ c ( R n ; V ) if and only if A ( D ) is elliptic and canceling. The cancellation is a new condition that we introduce
JEAN VAN SCHAFTINGEN De fi nition 1.2. A homogeneous linear di ff erential operator A ( D ) on R n from V to E is canceling if \ ξ ∈ R n \{ } A ( ξ )[ V ] = { } . In the well-known L p counterpart of theorem 1.3 for < p < n , the ellipticityalone is su ffi cient. One has for every u ∈ C ∞ c ( R n ; R n ) k D k − u k L npn − p ≤ C k A ( D ) u k L p , (1.7)if and only if A ( D ) is elliptic .The cancellation condition for fi rst-order operators is equivalent to a structuralcondition used by J. Bourgain and H. Brezis to prove (1.6) [9, theorem 25] (see propo-sition 6.2 below).The su ffi ciency part of theorem 1.3 will be proved in proposition 4.6; the ne-cessity of the ellipticity in corollary 5.2 and the necessity of the cancellation inproposition 5.5.The estimates (1.1), (1.2) and (1.5) will be derived from theorem 1.3 in section 6as well as the nonestimate (1.3). The case of the Hodge–Sobolev inequality (1.4)will be treated in section 7 in a generalization of theorem 1.3 to partially cancelingoperators.Theorem 1.3 also remains valid for estimates in fractional Sobolev spaces and inLorentz spaces (section 8). Using the tools of J. Bourgain and H. Brezis, a counterpartof theorem 1.3 with a weaker norm is obtained (section 9).1.4.
Estimates for L vector fi elds and cocanceling operators. By the Hölderinequality and classical elliptic estimates, the estimate k D k − u k L nn − ≤ C k A ( D ) u k L for every u ∈∈ C ∞ c ( R n ; V ) is equivalent to Z R n A ( D ) u · ϕ ≤ C ′ k A ( D ) u k L k Dϕ k L n . for every u ∈∈ C ∞ c ( R n ; V ) and ϕ ∈ C ∞ c ( R n ; E ) .This leads us to the related question to determine under which conditions doesone have an estimate Z R n f · ϕ ≤ C k f k L k Dϕ k L n . (1.8)for every ϕ ∈ C ∞ c ( R n ; E ) and every f in some subset of L ( R n ; E ) . Without anyrestriction on f , this estimate fails when n ≥ ; it would be indeed equivalent with k u k L ∞ ≤ C k Du k L n , which is also known to be false. Surprisingly, J. Bourgain and H. Brezis [8, p. 541;9, theorem 1 ′ ] have proved that when E = R n and f is taken in the class ofdivergence-free vector- fi elds, the above estimate holds. We want to determine fora given di ff erential operator L ( D ) on R n from E to F , whether an estimate of thetype (1.8) holds. The answer is given by The su ffi ciency of the ellipticity is a consequence of the classical theorem 1.1 and the Sobolevembedding. The necessity of ellipticity in (1.7) was probably known to the experts; we shall prove inproposition 5.1 that ellipticity is necessary in (1.7) for every p ∈ [1 , n ) . IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 5
Theorem 1.4.
Let n ≥ and L ( D ) be a homogeneous differential operator on R n from E to F . The following conditions are equivalent(i) there exists C > such that for every f ∈ L ( R n ; E ) such that L ( D ) f = 0 and ϕ ∈ C ∞ c ( R n ; E ) , Z R n f · ϕ ≤ C k f k L k Dϕ k L n , (ii) for every f ∈ L ( R n ; E ) such that L ( D ) f = 0 Z R n f = 0 , (iii) L ( D ) is cocanceling. The cocancellation condition is a new condition that we introduce: De fi nition 1.3. Let L ( D ) be a homogeneous linear di ff erential operator on R n from E to F . The operator L ( D ) is cocanceling if \ ξ ∈ R n \{ } ker L ( ξ ) = { } . The equivalence between (ii) and (iii) in theorem 1.4 will be the proved in propo-sition 2.1; (ii) will be deduced from (i) in proposition 2.2; (i) will be proved assuming(iii) in proposition 2.3 relying on results of J. Bourgain and H. Brezis [9] and the au-thor [44].It is possible also to obtain some partial estimate when L ( D ) satis fi es partiallythe cocancellation condition (see section 7) and to obtain fractional estimates (seesection 8). Using the tools of J. Bourgain and H. Brezis, we show that if L ( D ) is acanceling homogenenous di ff erential operator, it allows to characterize vector- fi elds f ∈ L ( R n ; E ) that de fi ne linear functionals on the homogeneous Sobolev space ˙ W ,n ( R n ; E ) (see section 9).2. E stimates on L vector fields and cocanceling operators2.1. Characterization of cocanceling operators.
The following proposition char-acterizes cocanceling operators:
Proposition 2.1.
Let L ( D ) be a homogeneous linear differential operator of order k on R n from E to F . The following are equivalent(i) L ( D ) is cocanceling,(ii) for every e ∈ E , if L ( D ) ( δ e ) = 0 , then e = 0 ,(iii) for every f ∈ L ( R n ; E ) , if L ( D ) f = 0 , then Z R n f = 0 , (iv) for every f ∈ C ∞ c ( R n ; E ) , if L ( D ) f = 0 , then Z R n f = 0 . Here δ denotes Dirac’s measure at . In (ii) and (iii), the di ff erential operator L ( D ) is taken in the sense of distributions. JEAN VAN SCHAFTINGEN
Proof.
Assume that L ( D ) is cocanceling. Fix e ∈ E such that L ( D )( δ e ) = 0 . Forevery ϕ ∈ C ∞ c ( R n ; E ) , by de fi nition of the distributional derivative and propertiesof the Fourier transform b ϕ of ϕ , h L ( D )( δ e ) , ϕ i = ( − k e · (cid:0) L ( D ) ∗ ϕ (cid:1) (0)= Z R n e · (cid:0) ( − πi ) k L ( ξ ) ∗ [ b ϕ ( ξ )] (cid:1) dξ = Z R n (cid:0) (2 πi ) k L ( ξ )[ e ] (cid:1) · b ϕ ( ξ ) dξ. Since by hypothesis L ( D ) ( δ e ) = 0 , we have, for every ϕ ∈ C ∞ c ( R n ; E ) , Z R n (cid:0) (2 πi ) k L ( ξ )[ e ] (cid:1) · b ϕ ( ξ ) dξ = 0; hence for every ξ ∈ R n L ( ξ )[ e ] = 0 . Since L ( D ) is cocanceling, we conclude that e = 0 . We have proved that (i) implies (ii).Now assume that (ii) holds and let f ∈ L ( R n ; E ) . If L ( D ) f = 0 , de fi ne f λ : R n → E for λ > and x ∈ R n by f λ ( x ) = λ n f (cid:0) xλ (cid:1) . One has f λ → δ e in the sense of distributions as λ → , where e = R R n f. Therefore L ( D ) f λ → L ( D )( δ e ) in the sense of distributions as λ → . Since L ( D ) is homogeneous, L ( D ) f λ = 0 and hence L ( D )( δ e ) = 0 . Therefore by assumption, R R n f = e = 0 .We have proved (iii). It is clear that (iii) implies (iv).Finally assume that (iv) holds. Let e ∈ T ξ ∈ R n \{ } ker L ( ξ ) . Choose ψ ∈ C ∞ c ( R n ) such that R R n ψ = 1 . For every x ∈ R n , (cid:0) L ( D )( ψe ) (cid:1) ( x ) = Z R n e πix · ξ (2 πi ) k L ( ξ )[ e ] b ψ ( ξ ) dξ = 0 . By (iv), we conclude that e = R R n ψe = 0 . We have proved that L ( D ) is cocancel-ing. (cid:3) In general, it is not clear whether there exists f ∈ C ∞ c ( R n ; E ) \ { } such that L ( D ) f = 0 . When L ( D ) is not cocanceling , proposition 2.2 shows that there exists f ∈ C ∞ c ( R n ; E ) \ { } such that L ( D ) f = 0 .2.2. Necessity of the cocancellation.
Using a classical construction, we prove that(i) implies (ii) in theorem 1.4
Proposition 2.2.
Let n ≥ and f ∈ L ( R n ; E ) . If for every ϕ ∈ C ∞ c ( R n ; E ) , Z R n f · ϕ ≤ C k f k L k Dϕ k L n , then Z R n f = 0 . Proof.
Let ψ ∈ C ∞ ( R + ) be such that ψ = 1 on [0 , , ψ ∈ [0 , on [1 , and ψ = 0 on [2 , ∞ ) . For λ > de fi ne ϕ λ : R n → R for x ∈ R n by ϕ λ ( x ) = ψ ( | x | λ ) . One has for every x ∈ R n , lim λ → ϕ λ ( x ) = 1 and k Dϕ λ k L n = λ − n k Dϕ k L n . By Lebesgue’s dominated convergence theorem and the estimate, R R n f = 0 . (cid:3) IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 7
Estimates on L vector fi elds. We shall now prove that the cocancellationcondition implies the estimate in theorem 1.4
Proposition 2.3.
Let L ( D ) be a homogeneous differential operator from E to F . If L ( D ) is cocanceling, f ∈ L ( R n ; E ) and L ( D ) f = 0 in the sense of distributions,then for every ϕ ∈ C ∞ c ( R n ; E ) , Z R n f · ϕ ≤ C k f k L k Dϕ k L n . The fi rst ingredient of the proof of proposition 2.3 is a similar result in whichthe vector condition is replaced by a single scalar condition. It will be shown inproposition 3.5 that this is a particular case of proposition 2.3. Proposition 2.4 (Van Schaftingen, 2008 [44, theorem 4]) . Let k ∈ N and f α ∈ L ( R n ) for α ∈ N n with | α | = k . If X α ∈ N n | α | = k ∂ α f α = 0 in the sense of distributions, then for every α ∈ N n with | α | = k and ϕ ∈ C ∞ c ( R n ) Z R n f α ϕ ≤ C k f k L k Dϕ k L n . The proof of proposition 2.4 relies on a slicing argument which is reminiscent ofthat used for the proof of the Gagliardo–Nirenberg embedding [20; 29, pp. 128-129],the Korn–Sobolev inequality [38] and which is a modi fi cation of an argument forestimates of circulation along closed curves [40], divergence-free vector fi elds [41],closed di ff erential forms [26] and vector fi elds that satisfy a second-order condition[42]. This was adapted to fractional spaces [8, remark 1; 9, remark 11; 41, remark 5;43, remark 4.2; 44, remark 2; 45] and noncommutative settings [14, 46]. A strongerversion of Proposition 2.4 can also be obtained by the methods of J. Bourgain andH. Brezis [9] (see theorem 9.2).The second ingredient is an algebraic lemma: Lemma 2.5.
Let L ( D ) = P | α | = k ∂ α L α be a homogeneous differential operator oforder k on R n from E to F . The operator L ( D ) is cocanceling if and only if thereexist K α ∈ L ( F ; E ) for every α ∈ N n with | α | = k such that X α ∈ N n | α | = k K α ◦ L α = id . (2.1)A key consequence of lemma 2.5 is that given f ∈ L ( R n ; E ) such that L ( D ) f =0 , f is the composition of a linear map with a vector fi eld that satis fi es the as-sumptions of proposition 2.4. Indeed by taking g α = L α ( f ) , one can write f = P α ∈ N n , | α | = k K α ( g α ) with P α ∈ N n , | α | = k ∂ α g α = 0 . Proof of lemma 2.5.
Since ( ξ α ) | α | = k is a basis of the vector space of homogeneouspolynomials of degree k , the operator e ∈ E (cid:0) L α ( e ) (cid:1) | α | = k ∈ F ( n + k − k ) is one-to-one if and only if L ( D ) is cocanceling. This is equivalent with this map beinginvertible on the left, which is (2.1). (cid:3) Proposition 2.3 will now be a consequence of proposition 2.4 and lemma 2.5.
JEAN VAN SCHAFTINGEN
Proof of proposition 2.3.
By assumption P | α | = k ∂ α L α ( f ) = 0 . By proposition 2.4,for every α ∈ N n with | α | = k and ψ ∈ C ∞ c ( R n ; V ) , Z R n L α ( f ) · ψ ≤ C k f k L k Dψ k L n . (2.2)For ϕ ∈ C ∞ c ( R n ; E ) , in view of (2.1) and (2.2) Z R n f · ϕ = X α ∈ N n | α | = k Z R n L α ( f ) · K α ∗ ( ϕ ) ≤ C X α ∈ N n | α | = k k L α ( f ) k L k DK α ∗ ( ϕ ) k L n ≤ C ′ k f k L k Dϕ k L n . (cid:3)
3. E xamples of cocanceling operators
Divergence. A fi rst example of cocanceling operator is the divergence opera-tor. Proposition 3.1.
Let L ( D ) be the homogeneous linear differential operator of order on R n from R n to R defined for ξ ∈ R n and e ∈ R n by L ( ξ )[ e ] = ξ · e. The operator L ( D ) is cocanceling.Proof. For every ξ ∈ R n , ker L ( ξ ) = ξ ⊥ . Hence, T ξ ∈ R n \{ } = { } . (cid:3) As a consequence of theorem 1.4, we recover the estimate
Corollary 3.2 (J. Bourgain and H. Brezis, 2004 [8, p. 541; 9, theorem ′ ; 41, theorem1.5]) . For every f ∈ L ( R n ; R n ) such that div f = 0 and every ϕ ∈ C ∞ c ( R n ) , Z R n f · ϕ ≤ C k f k L k Dϕ k L n . Exterior derivative.
The construction for the divergence operator generalizesto di ff erentials forms Proposition 3.3.
Let ℓ ∈ { , . . . , n − } and L ( D ) be the homogeneous lineardifferential operator of order on R n from V ℓ R n to V ℓ +1 R n defined for ξ ∈ R n ≃ V R n and e ∈ V ℓ R n by L ( ξ )[ e ] = ξ ∧ e. The operator L ( D ) is cocanceling.Proof. If e ∈ V ℓ R n with ℓ ≤ n − , one checks that if ξ ∧ e = 0 for every ξ ∈ R n ,then e = 0 . (cid:3) As a consequence we recover from theorem 1.4 the estimate
Corollary 3.4 (J. Bourgain and H. Brezis [9, Corollary 17], 2004 and L. Lanzani andE. Stein, 2005 [26]) . Let ℓ ∈ { , . . . , n − } . For every f ∈ L ( R n ; V ℓ R n ) suchthat df = 0 and every ϕ ∈ C ∞ c ( R n ; V n − ℓ R n ) , Z R n f ∧ ϕ ≤ C k f k L k Dϕ k L n . IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 9
Higher order condition.
One can also replace the divergence with a similarhigher-order condition
Proposition 3.5.
Let k ∈ N ∗ and L ( D ) be the homogeneous linear differential op-erator of order k on R n from R ( n + k − k ) to R defined for ξ ∈ R n and e ∈ R ( n + k − k ) by L ( ξ )[ e ] = X α ∈ N n | α | = k ξ α e α . The operator L ( D ) is cocanceling.Proof. Assume that e ∈ T ξ ∈ R n \{ } ker L ( ξ ) . One has then for every ξ ∈ R n , P α ∈ N n , | α | = k ξ α e α = 0 . By the properties of multivariate polynomials, one con-cludes that e = 0 . (cid:3) As a corollary, one recovers proposition 2.4 from theorem 1.4.3.4.
Saint-Venant compatibility conditions.
The Saint-Venant compatibility con-ditions are an example of cocanceling operator. In order to de fi ne it, denote by S R n the space of symmetric bilinear forms on R n . Proposition 3.6.
Let W ( D ) be the homogeneous linear differential operator of order on R n from S R n to S R n ⊗ S R n defined for ξ ∈ R n , e ∈ S R n , and u, v, w, z ∈ R n by (cid:0) W ( ξ )[ e ] (cid:1) [ u, v, w, z ] = e ( u, v )( ξ · w )( ξ · z ) + e ( w, z )( ξ · u )( ξ · v ) − e ( u, z )( ξ · w )( ξ · v ) − e ( w, v )( ξ · u )( ξ · z ) . The operator W ( D ) is cocanceling if and only if n ≥ .Proof. First note that if n = 1 , L ( D ) = 0 .Assume that n ≥ and let e ∈ S R n be such that for every u, v, w, z ∈ R n and ξ ∈ R n , (cid:0) W ( ξ )[ e ] (cid:1) [ u, v, w, z ] = 0 . (3.1)Let u ∈ R n . Since n ≥ , one can choose w ∈ R n \ { } such that w · u = 0 . Onehas then (cid:0) W ( w )[ e ] (cid:1) [ u, u, w, w ] = e ( u, u ) k w k , from which one deduces by (3.1) that for every u ∈ R n , e ( u, u ) = 0 . Since e issymmetric, e = 0 . (cid:3) Corollary 3.7.
Let n ≥ . For every f ∈ L ( R n ; S R n ) such that W ( D ) f = 0 and every ϕ ∈ C ∞ c ( R n ; S R n ) , Z R n f : ϕ ≤ C k f k L k Dϕ k L n . Here : denotes the scalar product in S R n . Corollary 3.7 is the core of theargument of the proof of the Korn–Sobolev inequality by estimates under secondorder conditions [42, theorem 6].We can also consider higher-order Saint-Venant operators [33, (2.1.9)]. We de-note by S k R n the space of symmetric k -linear forms on R n . Proposition 3.8.
Let W ( D ) be the homogeneous linear differential operator of order k on R n from S k R n to S k R n ⊗ S k R n defined for ξ ∈ R n , e ∈ S k R n , and v , . . . , v k , v , . . . , v k ∈ R n by (cid:0) W ( ξ )[ e ] (cid:1) [ v , . . . , v k , v , . . . , v k ]= X α ∈{ , } n ( − | α | e ( v α , . . . , v α k k )( ξ · v − α ) · · · ( ξ · v − α k k ) . The operator W ( D ) is cocanceling if and only if n ≥ , . The condition W ( D ) f = 0 is satis fi ed by the symmetric derivative of a fi eld ofsymmetric k − -linear forms. Sketch of the proof of proposition 3.8.
Assume that e ∈ T ξ ∈ R n \{ } ker L ( ξ ) . Given u ∈ R n , one chooses w ∈ R n \ { } such that w · u = 0 . One has then (cid:0) W ( w )[ e ] (cid:1) [ u, . . . , u, w, . . . , w ] = e ( u, . . . , u ) k w k k , from which one concludes that e = 0 . (cid:3)
4. P roof of the
Sobolev estimateIn this section we prove a Sobolev estimate for elliptic canceling operator. Weproceed in several steps. First we recall in section 4.1 a classical elliptic estimatefor elliptic operators. Next in section 4.2 we recall how the range of a given lineardi ff erential operator can be characterized as the kernel of another linear di ff erentialoperator of compatibility conditions and we study when this operator is cocanceling.Finally, in section 4.3, we prove the estimate by combining the previous ingredientswith theorem 1.4 proved in section 2.4.1. Classical elliptic estimates.
In order to prove theorem 1.3, we shall use aclassical variant of theorem 1.1
Proposition 4.1.
Let A ( D ) be a linear homogeneous differential operator of order k on R n from V to E . If A ( D ) is elliptic and p > , then for every u ∈ C ∞ c ( R n ; V ) , k D k − u k L p ≤ C k A ( D ) u k ˙ W − ,p . Proof.
One has for every α ∈ N n with | α | = k − and for every ξ ∈ R n \ { } , d ∂ α u ( ξ ) = 12 πi ξ α (cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1) − ◦ A ( ξ ) ∗ (cid:0) \ A ( D ) u ( ξ ) (cid:1) . Recall that k A ( D ) u k ˙ W − ,p = k ( − ∆) − A ( D ) u k L p . By the theory of singularintegrals on L p (see for example E. Stein [37, theorem 6 in Chapter 3, § 3.5 togetherwith theorem 3 in Chapter 2, § 4.2]), one has the desired estimate. (cid:3) In general A ( D ) is an overdetermined elliptic operator; as a consequence, thereare many possible choices for a singular integral operator that inverts A ( D ) . In theproof of proposition 4.1, a change of the Euclidean structure on E would result in adi ff erent singular integral operator that would have the same properties. IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 11
Compatibility conditions.
The last tool in the proof of the su ffi ciency part intheorem 1.3 is Proposition 4.2.
Let A ( D ) be a homogeneous differential operator on R n from V to E . If A ( D ) is elliptic, then there exists a finite-dimensional vector space F anda homogeneous differential operator L ( D ) on R n from E to F such that for every ξ ∈ R n \ { } , ker L ( ξ ) = A ( ξ )[ V ] . In the language of homological algebra, for every ξ ∈ R n \ { } , V A ( ξ ) −−−→ E L ( ξ ) −−→ F forms an exact sequence.The proof will be done in two steps. First we will recall the construction dueto L. Ehrenpreis [17; 25, theorem 2; 36, theorem 1.5.5] of compatibility conditionfor an overdetermined linear di ff erential operator that does not need to be elliptic.We then show that under the ellipticity condition, this operator has the requiredproperty.Let P ℓξ ( R n ; V ) be the space of exponential polynomials of degree at most ℓ , thatis the set of functions u : R n → V that can be written for every x ∈ R n as u ( x ) = X α ∈ N n | α |≤ ℓ x α e ξ · x v α . where v α ∈ V for each α ∈ N n with | α | ≤ ℓ . We also set P ξ ( R n ; V ) = S ℓ ∈ N P ℓξ ( R n ; V ) . If we de fi ne for ξ ∈ R n the function e ξ : R n → R by e ξ ( x ) = e ξ · x for every x ∈ R n , one has P ℓξ ( R n ; V ) = e ξ P ℓ ( R n ; V ) .Finally, K ( D ) is a linear di ff erential operator on R n from E to F of order atmost ℓ if it can be written for u ∈ C ∞ as K ( D ) u = P α ∈ N n , | α |≤ ℓ K α ( ∂ α u ) . The next lemma gives a necessary and su ffi cient condition for the solvability ofthe equation A ( D ) u = f in the framework of exponential polynomials. Lemma 4.3.
Let A ( D ) be a linear differential operator of order at most k on R n from V to E and let ξ ∈ R n . For every f ∈ P ℓξ ( R n ; E ) , there exists u ∈P ℓ + kξ ( R n ; V ) such that A ( D ) u = f if and only if for every linear differentialoperator K ( D ) on R n of order at most ℓ from E to R such that K ( D ) ◦ A ( D ) = 0 ,one has K ( D ) f = 0 .Proof. Note that for every linear form φ on P ℓξ ( R n ; E ) there exists a unique di ff er-ential operator K ( D ) of order at most ℓ on R n from E to R such that for every g ∈P ℓξ ( R n ; E ) , h φ, g i = ( K ( D ) g )(0) . If we want to characterize A ( D ) P ℓ + kξ ( R n ; V ) by duality, we are led to study the di ff erential operators K ( D ) of order at most ℓ on R n from E to R such that K ( D ) ◦ A ( D ) u (0) = 0 for every u ∈ P ℓ + kξ ( R n ; V ) .Note that since K ( D ) ◦ A ( D ) is of order at most k + ℓ , this is equivalent with K ( D ) ◦ A ( D ) = 0 , which is the condition appearing in the proposition. (cid:3) The drawback of the previous lemma is that the number of conditions imposedon the data f depends on the degree of f . This can be improved by some commu-tative algebra construction. Lemma 4.4.
Let A ( D ) be a linear differential operator of order k on R n from V to E . There exists a finite dimensional vector space G and a linear differential operator J ( D ) from E to G such that for every f ∈ P ξ ( R n ; E ) , there exists u ∈ P ξ ( R n ; V ) such that A ( D ) u = f if and only if J ( D ) f = 0 . In the language of homological algebra, the sequence P ξ ( R n ; V ) A ( D ) −−−→ P ξ ( R n ; E ) J ( D ) −−−→ P ξ ( R n ; G ) (4.1)is exact. Proof of lemma 4.4.
Let K be the set of linear di ff erential operators K ( D ) on R n from E to R such that K ( D ) ◦ A ( D ) = 0 . The set K is a submodule of the moduleof linear di ff erential operators on R n from V to R on the ring of linear di ff erentialoperators on R n from R to R which is isomorphic to the ring of polynomials on R n . Therefore, K is fi nitely generated (see for example [5, proposition 3.32 andcorollary 4.7]): there exists a fi nite-dimensional space G and a linear di ff erentialoperator J ( D ) on R n from E to G such that for every K ( D ) ∈ K , there exists adi ff erential operator Q ( D ) from G to R such that K ( D ) = Q ( D ) ◦ J ( D ) . Thelemma then follows from the application of lemma 4.3. (cid:3) One can ensure that J ( D ) has minimal order by using tools of computationalcommutative algebra [5, §6.1 and 10.3].In order to complete the proof of proposition 4.2, we need to show that for every ξ ∈ R n \ { } , ker J ( ξ ) = A ( ξ )[ V ] . This is equivalent to the exactness of thesequence P ξ ( R n ; V ) A ( D ) −−−→ P ξ ( R n ; E ) J ( D ) −−−→ P ξ ( R n ; G ) . (4.2)Under the ellipticity condition, the exactness of the sequence (4.1) implies the exact-ness of the sequence (4.2): Lemma 4.5.
Let A ( D ) be a homogeneous linear differential operator of order k on R n from V to E , ξ ∈ R n \ { } , ℓ ∈ N and u ∈ P ξ ( R n ; V ) . If the operator A ( D ) is elliptic and A ( D ) u ∈ P ℓξ ( R n ; E ) , then u ∈ P ℓξ ( R n ; V ) . The lemma implies that if A ( D ) is elliptic, ℓ ∈ N and ξ ∈ R n \{ } , the sequence P ℓξ ( R n ; V ) A ( D ) −−−→ P ℓξ ( R n ; E ) J ( D ) −−−→ P ℓξ ( R n ; G ) is exact. When ℓ = 0 , this is (4.2). Proof of lemma 4.5.
It is su ffi cient to show that if u ∈ P ℓ +1 ξ ( R n ; V ) and A ( D ) u ∈P ℓξ ( R n ; E ) , then u ∈ P ℓξ ( R n ; V ) . Write u = e ξ p , with p ∈ P ℓ +10 ( R n ; V ) . Onehas A ( D )[ e ξ p ] = e ξ ( A ( D + ξ ) p ) = e ξ (cid:0) A ( ξ )[ p ] + (cid:0) A ( ξ + D ) − A ( ξ ) (cid:1) [ p ] (cid:1) . Note that (cid:0) A ( ξ + D ) − A ( ξ ) (cid:1) [ p ] ∈ P ℓξ ( R n ) . Therefore, A ( ξ )[ p ] ∈ P ℓ ( R n ; E ) . Since A ( ξ ) is one-to-one, this implies that p ∈ P ℓ ( R n ; V ) . (cid:3) Proof of proposition 4.2.
Let J ( D ) be given by lemma 4.4. In view of lemma 4.4 andlemma 4.5, one has for every ξ ∈ R n \ { } , ker J ( ξ ) ≃ (cid:8) f ∈ P ξ ( R n ; E ) : J ( D ) f = 0 (cid:9) = (cid:8) A ( D ) u : u ∈ P ξ ( R n ; V ) (cid:9) ≃ A ( ξ )[ V ] , IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 13 where the isomorphism is given by e ∈ E e ξ e ∈ P ξ ( R n ; E ) .There exist ν ∈ N and for every i ∈ { , . . . , ν } homogeneous di ff erential oper-ators J i ( D ) of order i on R n from E to G such that J ( D ) = ν X i =0 J i ( D ) . Since A ( D ) is homogeneous, one has for every ξ ∈ R n \ { } , ν \ i =0 ker J i ( ξ ) = A ( ξ )[ V ] . Therefore, by taking F = Q νi =0 (cid:0)N ν − i R n (cid:1) ⊗ G and L ( ξ ) = (cid:0) ξ ⊗ ν ⊗ J ( ξ ) , ξ ⊗ ν − ⊗ J ( ξ ) , . . . , ξ ⊗ J ν − ( ξ ) , J ν ( ξ ) (cid:1) , we obtain a homogeneous di ff erential operator that has the required properties. (cid:3) The ellipticity assumption in proposition 4.2 might seen unnatural in the state-ment. It is nevertheless essential as shown by the following example
Example . Consider the homogeneous linear di ff erential operator A ( D ) of order on R from R to R de fi ned by the matrix. A ( ξ ) = (cid:18) ξ − ξ ξ − ξ (cid:19) . The operator A ( D ) is not elliptic, since A ( ξ ) is not one-to-one when | ξ | = | ξ | .(The reader will note that this is a hyperbolic operator.) Assume now that thereexists a homogeneous di ff erential operator L ( D ) from R to a vector space F suchthat for every ξ ∈ R , L ( ξ ) ◦ A ( ξ ) = 0 . Since A ( ξ ) is onto when | ξ | 6 = | ξ | , we have L ( ξ ) = 0 when | ξ | 6 = | ξ | . Fromthis we conclude that L ( ξ ) = 0 for every ξ ∈ R . One has then ker L (1 ,
1) = R = R (1 ,
1) = A (1 , R ] . Also note that since A (1 , R ] = R (1 , and A (1 , − R ] = R (1 , − , A ( D ) is canceling, but L ( D ) = 0 is not cocanceling. Remark . It is also possible to obtain an operator L ( D ) satisfying the conclusionof proposition 4.2 by setting L ( ξ ) = det (cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1) id − A ( ξ ) ◦ adj (cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1) ◦ A ( ξ ) ∗ , (4.3)where adj (cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1) = det (cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1)(cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1) − is the adjugateoperator of A ( ξ ) ∗ ◦ A ( ξ ) . (This construction is up to the multiplicative constant det (cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1) the classical orthogonal projector on A ( ξ )[ V ] used for examplefor least-square solutions of overdetermined systems.) The latter construction of L ( D ) can be much more complicated that necessary. For example, if one is inter-ested in the Hodge–Sobolev inequality (1.5), one takes V = V ℓ R n and for every u ∈ C ∞ c ( R n ; V ) , A ( u ) = ( du, d ∗ u ) . The operator L ( D ) given by (4.3) is L ( g, h ) = (cid:0) ( − ∆) m − d ∗ dg, ( − ∆) m − dd ∗ h (cid:1) , where m = dim V ℓ +1 R n + dim V ℓ − R n = (cid:0) nℓ − (cid:1) + (cid:0) nℓ +1 (cid:1) . It is possible to showthat L ( g, h ) = 0 if and only if dg = 0 and d ∗ h = 0 . Sobolev inequality.
We now have all the ingredients to prove the su ffi ciencypart of theorem 1.3 Proposition 4.6.
Let A ( D ) be a linear differential operator of order k on R n from V to E . If A ( D ) is elliptic and canceling, then for every u ∈ C ∞ c ( R n ; V ) , k D k − u k L nn − ≤ C k A ( D ) u k L . Proof.
Let L ( D ) be given by proposition 4.2. One notes that L ( D ) (cid:0) A ( D ) u (cid:1) = 0 . Since A ( D ) is canceling and for every ξ ∈ R n \ { } , ker L ( ξ ) = A ( ξ )[ V ] , L ( D ) is cocanceling. Therefore, by theorem 1.4, k A ( D ) u k ˙ W − ,n/ ( n − ≤ C k A ( D ) u k L . Finally, we note that by proposition 4.1, one has k D k − u k L nn − ≤ C k A ( D ) u k ˙ W − ,n/ ( n − . (cid:3)
5. N ecessary conditions for the
Sobolev estimateIn the section, we study the necessity of the ellipticity (section 5.1) and cancella-tion (section 5.2) conditions for the Sobolev estimate.5.1.
Necessity of the ellipticity.
We show that the ellipticity condition is neces-sary in Sobolev-type inequalities
Proposition 5.1.
Let A ( D ) be a homogeneous linear differential operator of order k on R n from V to E , B ( D ) be a homogeneous differential operator of order k − on R n from V to F , and p ∈ [1 , n ) . If for every u ∈ C ∞ c ( R n ; E ) , k B ( D ) u k L npn − p ≤ C k A ( D ) u k L p , then for every ξ ∈ R n , ker A ( ξ ) ⊆ ker B ( ξ ) . As a corollary, we have the necessity of the ellipticity in theorem 1.3:
Corollary 5.2.
Let A ( D ) be a homogeneous linear differential operator of order k on R n from V to E . If for every u ∈ C ∞ c ( R n ; V ) , k D k − u k L npn − p ≤ C k A ( D ) u k L p , then A ( D ) is elliptic.Proof. Take B ( D ) = D k − . For every ξ ∈ R n \ { } , one has ker B ( ξ ) = { } . Theconclusion follows from the application of proposition 5.1. (cid:3) Proof of proposition 5.1.
Let ξ ∈ R n \{ } and v ∈ ker A ( ξ ) . Choose ϕ ∈ C ∞ ( R ) \{ } such that supp ϕ ⊂ ( − , and ψ ∈ C ∞ c ( R n ) such that ψ on thehyperplane H = { x ∈ R n : ξ · x = 0 } . For λ > , de fi ne u λ : R n → R for x ∈ R n by u λ ( x ) = ϕ ( ξ · x ) ψ (cid:0) xλ (cid:1) v. Since A ( ξ )[ v ] = 0 , one has for each x ∈ R n and λ > | A ( D ) u λ ( x ) | ≤ C k X i =1 λ − i (cid:12)(cid:12) D i ψ (cid:0) xλ (cid:1)(cid:12)(cid:12) . IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 15
One has therefore, for every λ > , Z R n | A ( D ) u λ | p ≤ C k X i =1 Z H λ λ n − ip | D i ψ | p , where H λ = { x ∈ R n : | ξ · x | ≤ λ − } . Since for every i ∈ { , . . . , k } , lim λ →∞ λ Z H λ | D i ψ | p = Z H | D i ψ | p we conclude that, as λ → ∞ , Z R n | A ( D ) u λ | p = O (cid:0) λ n − − p (cid:1) . On the other hand, for every x ∈ R n and λ > , (cid:12)(cid:12) B ( D ) u λ ( x ) − ψ (cid:0) xλ (cid:1) ϕ ( k − ( ξ · x ) B ( ξ )[ v ] (cid:12)(cid:12) ≤ C k − X i =1 λ − i (cid:12)(cid:12) D i ψ (cid:0) xλ (cid:1)(cid:12)(cid:12) . As previously, we have as λ → ∞ , Z R n (cid:12)(cid:12) B ( D ) u λ ( x ) − ψ (cid:0) xλ (cid:1) ϕ ( k − ( ξ · x ) B ( ξ )[ v ] (cid:12)(cid:12) npn − p dx = O (cid:0) λ n − − npn − p (cid:1) , whence Z R n | B ( D ) u λ | p ∗ = λ n − | B ( ξ )[ v ] | p ∗ Z R | ϕ ( k − | p ∗ Z H | ψ | p ∗ + o ( λ n − ) . Therefore, in view of the assumption, we have, as λ → ∞ , (cid:12)(cid:12) B ( ξ )[ v ] (cid:12)(cid:12) λ n − p − n = O ( λ n − p − ) . This is only possible if v ∈ ker B ( ξ ) . (cid:3) The proof of proposition 5.1 strongly relies on the fact that we are considering k − -th derivatives on the left-hand side of the estimate. For lower derivatives onecan still obtain some inequality without the ellipticity of A ( D ) .Consider the homogeneous linear di ff erential operator A ( D ) of order on R from R to R de fi ned for u ∈ C ∞ ( R ) by A ( D )[ u ] = ( ∂ ∂ u, ∂ ∂ u ) . Since ker A (1 , , ,
0) = R , this operator is not elliptic. By corollary 5.2, thereexists b ∈ R such that the estimate k b · ∇ u k L / ≤ C (cid:0) k ∂ ∂ u k L + k ∂ ∂ u k L (cid:1) does not hold. In fact, the estimate does not hold for any b ∈ R \ { } . Proposition 5.3.
Let b ∈ R . If for every u ∈ C ∞ c ( R ; R ) , k b · ∇ u k L / ≤ C (cid:0) k ∂ ∂ u k L + k ∂ ∂ u k L (cid:1) , then b = 0 .Proof. By proposition 5.1, if ξ ∈ R satis fi es ξ ξ = 0 and ξ ξ = 0 , then b · ξ = 0 .By taking for ξ elements of the canonical basis of R , one concludes that b = 0 . (cid:3) On the other hand
Proposition 5.4.
For every u ∈ C ∞ c ( R ; R ) , k u k L ≤ C (cid:0) k ∂ ∂ u k L + k ∂ ∂ u k L (cid:1) . (5.1) Proof.
The proof is a direct adaptation of a proof of E. Gagliardo [20, teorema5.I] and L. Nirenberg [29, 128–129]. The proof goes as follows: for every x =( x , x , x , x ) ∈ R u ( x ) = Z x −∞ Z x ∞ ∂ ∂ u ( s, t, x , x ) dt ds. Hence, for every x ∈ R , | u ( x ) | ≤ Z R | ∂ ∂ u ( s, t, x , x ) | ds dt. Similarly, one has for every x ∈ R , | u ( x ) | ≤ Z R | ∂ ∂ u ( x , x , s, t ) | ds dt. Therefore, for every x ∈ R | u ( x ) | ≤ Z R | ∂ ∂ u ( s, t, x , x ) | ds dt Z R | ∂ ∂ u ( x , x , s, t ) | ds dt. The integration of this inequality with respect to x on R and the application ofYoung’s inequality yields (5.1). (cid:3) We have thus an operator which is not elliptic. By proposition 5.3, there is no fi rst-order Sobolev inequality, but there is a second-order Sobolev inequality ofproposition 5.4.5.2. Necessity of the cancellation.
The necessity of the cancellation property forSobolev-type estimates is given by the following
Proposition 5.5.
Assume that A ( D ) is an elliptic homogeneous linear differentialoperator of order k on R n from V to E . Let ℓ ∈ { , . . . , k − } be such that ℓ > k − n . If for every u ∈ C ∞ c ( R n ; V ) k D ℓ u k L nn − ( k − ℓ ) ≤ C k A ( D ) u k L , then A ( D ) is canceling. In this statement the operator is assumed to be elliptic, which is not necessaryfor the estimate when ℓ < k − . We do not have any examples that show that thisassumption is necessary: Open Problem 5.1.
Does proposition 5.5 remain true without the ellipticity assump-tion?
Remark . Proposition 5.5 does not cover the case ℓ = n − k . In the case n = 1 ,for every k ∈ N ∗ the homogeneous linear di ff erential operator A ( D ) de fi ned for ξ ∈ R by A ( ξ ) = ξ k is elliptic but not canceling. Nonetheless, for every u ∈ C ∞ c ( R ) , k u ( k − k L ∞ ≤ k u ( k ) k L . We did not fi nd higher-dimensional examples. IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 17
Proof of proposition 5.5.
Let e ∈ T ξ ∈ R n \{ } A ( ξ )[ V ] . Since for every ξ ∈ R n \{ } , A ( ξ ) is one-to-one, the function U : R n \ { } → V de fi ned for each ξ ∈ R n \ { } A ( ξ ) (cid:2) U ( ξ ) (cid:3) = e is smooth. This can be seen by the implicit function theorem or by the formula U ( ξ ) = (cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1) − ◦ A ( ξ ) ∗ [ e ] . Since A ( ξ ) is homogeneous of degree k , forevery ξ ∈ R n \ { } and t ∈ R \ { } , U ( tξ ) = t − k U ( ξ ) . Choose now a function ψ ∈ C ∞ ( R n ) such that supp b ψ ⊂ B (0) and b ψ = 1 on B / (0) . For λ > , de fi ne ψ λ : R n → R for x ∈ R n by ψ λ ( x ) = λ n ψ ( λx ) , andde fi ne u λ : R n → V such that for each ξ ∈ R n , c u λ ( ξ ) = (2 πi ) − k (cid:0)c ψ λ ( ξ ) − d ψ /λ ( ξ ) (cid:1) U ( ξ ) . If λ > , supp( c ψ λ − d ψ /λ ) ⊂ B λ (0) \ B / (2 λ ) (0) . Hence, u λ is well-de fi ned andbelongs to the Schwartz class of fast decaying smooth functions.We now claim that for every λ > , k D ℓ u λ k L nn − ( k − ℓ ) ≤ C k A ( D ) u λ k L . (5.2)To see this, consider a function ϕ ∈ C ∞ c ( R n ) such that ϕ = 1 on B (0) . For R > , de fi ne ϕ R : R n → R for x ∈ R n by ϕ R ( x ) = ϕ ( x/R ) . By hypothesis, forevery R > , k D ℓ ( ϕ R u λ ) k L nn − ( k − ℓ ) ≤ C k A ( D )( ϕ R u λ ) k L . By letting R → ∞ , we obtain (5.2).Now, by de fi nition of u λ and the choice of e , one has A ( D ) u λ = ( ψ λ − ψ /λ ) e, (5.3)and therefore, k A ( D ) u λ k L ≤ k ψ k L . (5.4)On the other hand, for every α ∈ N n , λ > and x ∈ R n ∂ α u λ ( x ) = Z R n e πiξ · x (cid:0) b ψ ( ξ/λ ) − b ψ ( λξ ) (cid:1) (2 πi ) | α |− k ξ α U ( ξ ) dξ. By writing for every ξ ∈ R n b ψ ( ξ/λ ) − b ψ ( λξ ) = − Z λ /λ ξt · ∇ b ψ (cid:16) ξt (cid:17) dtt , we have, by Fubini’s theorem, ∂ α u λ ( x ) = Z λ /λ w α ( tx ) t n − ( k −| α | ) dtt . where w α : R n → V is de fi ned for x ∈ R n by w α ( x ) = − (2 πi ) | α |− k Z R n e πiξ · x ξ · ∇ b ψ ( ξ ) ξ α U ( ξ ) dξ, Since w α decays fast at in fi nity, if | α | > k − n and x ∈ R n \ { } , the limit u α ( x ) = lim λ →∞ ∂ α u λ ( x ) = Z ∞ w α ( tx ) t n − ( k −| α | ) dtt (5.5) is well-de fi ned.Assume by contradiction that there exists α ∈ N n such that | α | = ℓ and u α .For every x ∈ R n and t > , one has by (5.5) u α ( tx ) = u α ( x ) t n − ( k − ℓ ) . (5.6)Since u α , this implies that Z R n | u α | nn − ( k − ℓ ) = ∞ . By Fatou’s lemma we have lim inf λ →∞ Z R n | ∂ α u λ | nn − ( k − ℓ ) ≥ Z R n | u α | nn − ( k − ℓ ) = ∞ , in contradiction with (5.2) and (5.4).We have thus u α ≡ for every α ∈ N n with | α | = ℓ . For each x ∈ R n \ { } , λ > and α ∈ N n with | α | = ℓ , we have by (5.5) | ∂ α u λ ( x ) | ≤ Z ∞ | w α ( tx ) | t n − ( k − ℓ ) dtt = 1 | x | n − ( k − ℓ ) Z ∞ (cid:12)(cid:12)(cid:12) w α (cid:16) t x | x | (cid:17)(cid:12)(cid:12)(cid:12) t n − ( k − ℓ ) dtt , and therefore | ∂ α u λ ( x ) | ≤ C | x | n − ( k − ℓ ) . By Lebesgue’s dominated convergence theorem, D ℓ u λ → in L ( R n ) . Takingnow ζ ∈ C ∞ c ( R n ) , we obtain by a suitable integration by parts that lim λ →∞ Z R n ζA ( D ) u λ = 0 . (5.7)On the other hand, in view of (5.3), one has Z R n ζA ( D ) u λ = Z R n ( ψ λ − ψ /λ ) ζe, whence lim λ →∞ Z R n ζA ( D ) u λ = ζ (0) e. Since this should hold for every ζ ∈ C ∞ c ( R n ) , this implies in view of (5.7) that e = 0 . (cid:3)
6. C haracterization and examples of canceling operators6.1.
Analytic characterization of elliptic canceling operators.
We have seen inproposition 2.1 that the cocanceling condition is equivalent with a property of thevector fi elds that are in its kernel. For elliptic canceling operators, the same methodsallow to characterize canceling operators by properties of the image of vector fi elds. Proposition 6.1.
Let A ( D ) be a homogeneous differential operator of order k on R n from V to E . If A ( D ) is elliptic, the following are equivalent(i) A ( D ) is canceling, IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 19 (ii) for every u ∈ L ( R n ; V ) , if A ( D ) u ∈ L ( R n ; E ) , then Z R n A ( D ) u = 0 , (iii) for every u ∈ C ∞ ( R n ; V ) , if supp A ( D ) u is compact, then Z R n A ( D ) u = 0 , If for every j ∈ { , . . . , k − } , lim | x |→∞ | D j u ( x ) || x | n − j = 0 then Z R n A ( D ) u = 0 , for any operator A ( D ) . It is thus crucial that no decay assumption is imposed on u in (iii). Proof.
First note that since A ( D ) is elliptic, proposition 4.2 applies and yields ahomogeneous di ff erential operator L ( D ) on R n from E to F . This operator L ( D ) is cocanceling if and only if A ( D ) is canceling.Let us now prove that (i) implies (ii). Let u ∈ L ( R n ; E ) be such that A ( D ) u ∈ L ( R n ; E ) . By construction of L ( D ) L ( D ) (cid:0) A ( D ) u (cid:1) = 0 . Since by assumption L ( D ) is cocanceling, in view of proposition 2.1 (iii), Z R n A ( D ) u = 0 . It is clear that (ii) implies (iii). Assume now that (iii) holds. Let f ∈ C ∞ c ( R n ; E ) be such that L ( D ) f = 0 .This latter condition allows to de fi ne w : R n → L k ( R n ; V ) such that its Fouriertransform b w satis fi es for every ξ ∈ R n A ( ξ ) (cid:2) b w ( ξ )[ v , . . . , v k ] (cid:3) = ( ξ · v ) · · · ( ξ · v k ) b f ( ξ ) . Since A ( D ) is elliptic and f is smooth, w is smooth. Write now u ( x ) = Z w ( tx )[ x, . . . , x ] (1 − t ) k − ( k − dt, so that D k u = w and hence A ( D ) u = f . By assumption we have that Z R n f = Z R n A ( D ) u = 0 . In view of proposition 2.1, we have proved that L ( D ) is cocanceling. This allows toconclude that A ( D ) is canceling. (cid:3) Equivalence between cancellation and the Bourgain–Brezis algebraic con-dition.
J. Bourgain and H. Brezis [9, theorem 25] have proved the estimate k u k L nn − ≤ C k A ( D ) u k L for an elliptic operator A ( D ) under the structural condition that there exist a basis e , . . . , e ℓ of E and vectors ξ , . . . , ξ ℓ ∈ R n \{ } such that for every i ∈ { , . . . , ℓ } , e i ⊥ A ( ξ i )[ V ] . This condition is in fact equivalent with the cancellation condition Proposition 6.2.
Let A ( D ) be a homogeneous differential operator on R n from V to E . The operator A ( D ) is canceling if and only if span [ ξ ∈ R n \{ } (cid:0) A ( ξ )[ V ] ⊥ (cid:1) = E. Proof.
For every ξ ∈ R n , since A ( ξ ) is a linear operator, one has e ∈ A ( ξ )[ V ] ifand only if for every f ∈ A ( ξ )[ V ] ⊥ , f · e = 0 . Therefore, e ∈ T ξ ∈ R n \{ } A ( ξ )[ V ] if and only if for every ξ ∈ R n \ { } and for every f ∈ A ( ξ )[ V ] ⊥ , f · e = 0 . Wehave thus \ ξ ∈ R n \{ } A ( ξ )[ V ] = (cid:16) [ ξ ∈ R n \{ } (cid:0) A ( ξ )[ V ] ⊥ (cid:1)(cid:17) ⊥ . Hence, one has that \ ξ ∈ R n \{ } A ( ξ )[ V ] = { } if and only if span (cid:18) [ ξ ∈ R n \{ } (cid:0) A ( ξ )[ V ] ⊥ (cid:1)(cid:19) = E, which is the statement that we wanted to prove. (cid:3) Remark . The same argument shows that a linear homogeneous di ff erential op-erator L ( D ) on R n from V to E is cocanceling if and only if span (cid:18) [ ξ ∈ R n \{ } (cid:0) ker L ( ξ ) ⊥ (cid:1)(cid:19) = E, or equivalently span (cid:18) [ ξ ∈ R n \{ } L ( ξ ) ∗ [ V ] (cid:19) = E. First-order canceling operators.
We shall now give explicit examples of can-celing operators.6.3.1.
Gradient operator.
The simplest example is the gradient operator:
Proposition 6.3.
Let A ( D ) be the homogeneous linear differential operator of order on R n from R to R n defined for ξ ∈ R n by A ( ξ ) = ξ. The operator A ( D ) is elliptic.The operator A ( D ) is canceling if and only if n ≥ . The statement and the proof [9, theorem 25] are written for dim V = n ; the arguments adaptstraightforwardly when dim V = n . IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 21
Proof.
For every ξ ∈ R n A ( ξ )[ R ] = R ξ , therefore T ξ ∈ R n \{ } A ( ξ )[ R ] = { } if n ≥ and T ξ ∈ R n \{ } A ( ξ )[ R ] = R if n = 1 . (cid:3) Symmetric derivative.
The symmetric derivative operator appearing in theKorn–Sobolev inequality (1.2) is also an elliptic canceling operator. Recall that S R n is the space of symmetric bilinear forms on R n . Proposition 6.4.
Let A ( D ) be the homogeneous linear differential operator of order on R n from R n to S R n defined for ξ ∈ R n , v ∈ R n and w, z ∈ R n by A ( ξ )[ v ]( w, z ) = 12 (cid:0) ( ξ · w )( v · z ) + ( ξ · z )( v · w ) (cid:1) . The operator A ( D ) is elliptic.The operator A ( D ) is canceling if and only if n ≥ .Proof. The operator A ( D ) is elliptic: assume that v ∈ R n and ξ ∈ R n \ { } aresuch that for every w, z ∈ R n , A ( ξ )[ v ]( w, z ) = 0 . In particular, for every w ∈ R n , A ( ξ )[ v ]( w, w ) = ( ξ · w )( v · w ) . We have thus for every w ∈ R n such that ξ · w = 0 , v · w = 0 . Since such w span R n , we have proved that A ( D ) is elliptic.Now we prove that A ( D ) is canceling when n ≥ . Let e ∈ T ξ ∈ R n \{ } A ( ξ )[ R n ] .For every w ∈ R n , choosing ξ ∈ R n \ { } such that ξ · w = 0 , one has for every v ∈ R n , A ( ξ )[ v ]( w, w ) = 0 and therefore, e ( w, w ) = 0 . Since w ∈ R n is arbitrary and e is symmetric, weconclude that e = 0 . (cid:3) The application of theorem 1.3 yields the Korn–Sobolev inequality (1.2). Theapplication of theorem 8.1 would yield fractional Korn–Sobolev inequalities.This example has a counterpart for the symmetric (or inner) derivative of a sym-metric multilinear forms [33, p. 25]
Proposition 6.5.
Let A ( D ) be the homogeneous linear differential operator of order on R n from S k R n to S k +1 R n defined for v ∈ S k R n , ξ ∈ R n and w , . . . , w k +1 ∈ R n by A ( ξ )[ v ]( w , . . . , w k +1 )= k +1 (cid:0) ( ξ · w ) v ( w , . . . , w k +1 ) + ( ξ · w ) v ( w , w , . . . , w k +1 )+ · · · + ( ξ · w k +1 ) v ( w , . . . , w k ) (cid:1) The operator A ( D ) is elliptic.The operator A ( D ) is canceling if and only if n ≥ .Proof. For the ellipticity, assume that v ∈ S k R n and ξ ∈ R n \ { } are such thatfor every w , . . . , w k +1 ∈ R n , L ( ξ )[ v ]( w , . . . , w k +1 ) = 0 . In particular, for every ξ ∈ R n , A ( ξ )[ v ]( w, . . . , w ) = ( ξ · w ) v ( w, . . . , w ) = 0 . Therefore, for every w ∈ R n such that ξ · w = 0 , v ( w, . . . , w ) = 0 . This impliesthat v = 0 . Now we prove that A ( D ) is canceling when n ≥ . Let e ∈ T ξ ∈ R n \{ } A ( ξ )[ S k R n ] .For every w ∈ R n , choosing ξ ∈ R n \ { } such that ξ · w = 0 , one has for every v ∈ S k R n , A ( ξ )[ v ]( w, w, . . . , w ) = 0 and therefore, e ( w, . . . , w ) = 0 . Since w ∈ R n is arbitrary and e is symmetric, weconclude that e = 0 . (cid:3) Exterior derivative.
We now turn to the study of canceling operators appear-ing in the framework of exterior di ff erential calculus. Proposition 6.6.
Let ℓ ∈ { , . . . , n − } and let A ( D ) = ( d, d ∗ ) be the homo-geneous linear differential operator of order on R n from V ℓ R n to V ℓ +1 R n × V ℓ − R n such that for every ξ ∈ R n and v ∈ V ℓ R n A ( ξ )[ v ] = (cid:0) ξ ∧ v, ∗ ( ξ ∧ ∗ v ) (cid:1) . The operator A ( D ) is elliptic.The operator A ( D ) is canceling if and only if ℓ ∈ { , . . . , n − } .Proof. The ellipticity follows from the Lagrange identity | v | | ξ | = | ξ ∧ v | + |∗ ( ξ ∧∗ v ) | .For the cancellation, if ( f, g ) ∈ T ξ ∈ R n \{ } A ( ξ )[ V ℓ R n ] , one should have forevery ξ ∈ R n , ξ ∧ f = 0 and ξ ∧ ∗ g = 0 . Since ≤ ℓ ≤ n − , this implies that f = 0 and g = 0 . (cid:3) As a consequence of proposition 6.6, one gets the Hodge–Sobolev inequality (1.5).6.3.4.
Directional derivatives of vector fields.
One has also a general constructionto control a vector fi eld by directional derivatives of some components Proposition 6.7.
Let m = dim V . Consider a family of n + m − n –wise lin-early independent vectors ( η i ) ≤ i ≤ n + m − of R n and m –wise linearly independentvectors ( w i ) ≤ i ≤ n + m − of V and define for ξ ∈ R n and v ∈ V , A ( ξ )[ v ] = (cid:0) ( η · ξ )( w · v ) , . . . , ( η m + n − · ξ )( w m + n − · v ) (cid:1) . The operator A ( D ) is elliptic.The operator A ( D ) is canceling if and only if n ≥ . This construction is due to D. G. de Figueiredo [18, inequality (K)] in the frame-work of L estimates. It was introduced by the author in the context of generalizedKorn–Sobolev inequalities [9, remark 16]. Proof of proposition 6.7.
Let us fi rst show that v is elliptic. Let ξ ∈ R n \ { } and v ∈ V be such that A ( ξ )[ v ] = 0 . Since the vectors ( η i ) ≤ i ≤ n + m − are n –wiselinearly independent, there is an increasing sequence of indices i , . . . , i m suchthat for every j ∈ { , . . . , m } , ( η i j · ξ ) = 0 . Therefore, for every j ∈ { , . . . , m } , ( w i j · v ) = 0 . Since the vectors w j , . . . , w j m form a basis of V , we conclude that v = 0 .For the cancellation, assume that e ∈ T ξ ∈ R n \{ } A ( ξ )[ V ] . By taking ξ i ∈ R n \{ } such that ξ i · η i = 0 , we have that for every e ∈ A ( ξ i )[ V ] , e i = 0 . Since e ∈ T n + m − i =1 A ( ξ i )[ V ] , we conclude that e = 0 . We have thus proved that A ( D ) is canceling. (cid:3) By theorem 1.3, this yields
IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 23
Proposition 6.8.
Let m = dim V . Consider a family of n + m − n –wise lin-early independent vectors ( η i ) ≤ i ≤ n + m − of R n and m –wise linearly independentvectors ( w i ) ≤ i ≤ n + m − of V . For every u ∈ C ∞ c ( R n ; V ) k u k L n/ ( n − ≤ C m + n − X i =1 k w i · Du [ η i ] k L . Minimizing the number of components of the derivative.
The previous exam-ple shows that a vector fi eld u ∈ C ∞ c ( R n ; R m ) can be estimated by n + m − directional derivatives of components. One may wonder whether it is possible touse less derivatives [9, open problem 3].For a lower bound we have Proposition 6.9.
Assume that A ( D ) is a differential operator of order on R n from V to E that is canceling and elliptic. Then dim E > dim V and dim E ≥ n .Proof. Since A ( D ) is canceling, there exists ξ ∈ R n such that A ( ξ )[ V ] = E . Since A ( D ) is elliptic, this implies that dim E > dim V .Next fi x v ∈ V and consider the linear map T : R n → E de fi ned by T ( ξ ) = A [ ξ ]( v ) . Since A ( D ) is elliptic, ker T = { } . Therefore, n = dim T ( R n ) ≤ dim E . (cid:3) If we de fi ne l ∗ ( n, m ) to be the minimal dimension l such that there is a cancel-ing elliptic linear di ff erential operator on R n from R m to R l , we have by proposi-tions 6.7 and 6.9 max( n, m + 1) ≤ l ∗ ( n, m ) ≤ m + n − . (6.1)In particular, the construction of proposition 6.7 is optimal if m = 1 (the scalar case)or n = 2 .The Hodge–Sobolev estimate for n = 4 and ℓ = 2 uses less components: onehas V = V R , and thus m = dim V = 6 whereas E = V R × V R , sothat dim E = 8 < n + m − . We have thus ≤ l ∗ (4 , ≤ . In allthe other cases the Hodge–Sobolev inequality does not allow to estimate with lesscomponents than n + dim V − . Indeed, one has dim (cid:16)V ℓ − R n × V ℓ +1 R n (cid:17) =dim V ℓ R n + (cid:0) n − ℓ − (cid:1) + (cid:0) n − ℓ +1 (cid:1) . The condition to have the Hodge–Sobolev inequalityis ≤ ℓ ≤ n − . If we want to use less components than n + m − , we need tohave (cid:0) n − ℓ − (cid:1) + (cid:0) n − ℓ +1 (cid:1) < n − . This is only possible if n = 4 and ℓ = 2 .The Korn–Sobolev uses dim E = n ( n +1)2 components, which is always larger orequal to n − .There are now speci fi c constructions that work in some cases. Let H ≃ R bethe algebra of quaternions Proposition 6.10.
Let A ( D ) the homogeneous linear differential operator of order on R from V = { x ∈ H : Re x = 0 } to H , defined for every v ∈ V and ξ ∈ R by A ( ξ )[ v ] = ξv. The operator A ( D ) is canceling and elliptic. Alternatively, writing ξ = ( ξ , ξ ′′ ) ∈ R × R , one has A ( ξ )[ v ] = ( − ξ ′′ · v, ξ v + ξ ′′ × v ) . Proof.
Since the multiplication of quaternions is invertible, A ( D ) is elliptic.For the cancellation property, for every v ∈ V and ξ ∈ R \ { } , one has Re (cid:0) ξ − A ( ξ )[ v ] (cid:1) = Re v = 0 . Hence, if e ∈ A ( ξ )[ V ] for every ξ ∈ R \ { } , onehas for every ξ ∈ R \ { } , Re ξ − e = 0 , whence e = 0 . (cid:3) This gives the estimate for every u ∈ C ∞ c ( R ; R ) , k u k L / ≤ C (cid:0) k div ′′ u k L + k ∂ u + curl ′′ u k L (cid:1) , where div ′′ u and curl ′′ u denote respectively the divergence and the curl with re-spect to the last three variables.The previous example shows that l ∗ (4 ,
3) = 4 . The same construction can bemade with the octonions and allows to control a vector fi eld from R to R , show-ing that l ∗ (8 ,
7) = 8 . If the same construction is made with complex numbersinstead of the octonions, one recovers the limiting Sobolev inequality for scalarfunctions on R .The previous construction also allows to show again that l ∗ (4 , ≤ and toshow that that l ∗ (8 , j ) ≤ j ; which is an improvement of the previous bound (6.1)when j ≤ .6.4. Second-order estimates.
We now give example of second-order canceling el-liptic operators and of application of theorem 1.3.6.4.1.
Splitting the Laplace–Beltrami operator.
The Laplacian is never a cancelingoperator. However, when split into two parts, it might become canceling
Proposition 6.11.
Let n ≥ , ℓ ∈ { , . . . , n − } and let A ( D ) be the homogeneouslinear differential operator of order from V ℓ R n to V ℓ R n × V ℓ R n defined for u ∈ C ∞ ( R n ; V ℓ R n ) by A ( D )[ u ] = ( dd ∗ u, d ∗ du ) . The operator A ( D ) is elliptic and canceling.Proof. Since dd ∗ + d ∗ d = ∆ is elliptic, A ( D ) is clearly elliptic. For the cancellation,let f, g ∈ T ξ ∈ R n \{ } A ( ξ )[ V ] . One has for every ξ ∈ R n , ξ ∧ f = 0 and ξ ∧∗ g = 0 .Since f, g ∈ V ℓ R n with ℓ ∈ { , . . . , n − } , this implies that f = g = 0 . (cid:3) Corollary 6.12.
Let n ≥ , ℓ ∈ { , . . . , n − } . For every u ∈ C ∞ c ( R n ; V ℓ R n ) , k Du k L nn − ≤ C (cid:0) k dd ∗ u k L + k d ∗ du k L (cid:1) . Linearly independent collections of operators.
A similar situation can be ob-served for a collection of scalar operators
Proposition 6.13.
Let ( w i ) ≤ i ≤ m +1 be m –wise linearly independent vectors of V and ( a i ) ≤ i ≤ m +1 be quadratic forms on R n such that if for every i, j ∈ { , . . . , m +1 } with i < j , then (cid:8) ξ ∈ R n : a i ( ξ ) = 0 (cid:9) ∩ (cid:8) ξ ∈ R n : a j ( ξ ) = 0 (cid:9) = { } and for every i ∈ { , . . . , m + 1 } (cid:8) ξ ∈ R n : a i ( ξ ) = 0 (cid:9) = { } . Define A ( ξ )[ v ] = (cid:0) a ( ξ )( w · v ) , . . . , a m +1 ( ξ )( w m +1 · v ) (cid:1) . The operator A ( D ) is elliptic and canceling. IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 25
Proof. We fi rst prove that A ( D ) is elliptic. Indeed, if A ( ξ )[ v ] = 0 , then there exists j ∈ { , . . . , m + 1 } such that for i = j , a i ( ξ ) = 0 . We have thus for i = j , w i · v = 0 , which implies v = 0 .Now we show that A ( D ) is canceling. For every i ∈ { , . . . , m + 1 } , one can fi nd ξ ∈ R n \ { } such that a i ( ξ ) = 0 . This proves that if e ∈ T ξ ∈ R n \{ } A ( ξ )[ V ] ,then e i = 0 . Since this is true for every i ∈ { , . . . , m + 1 } , A ( D ) is canceling. (cid:3) The construction of proposition 6.13 is always possible given any n ≥ and V .Indeed take ξ , . . . , ξ m +1 to be unit vectors of R n such that | ξ i · ξ j | = 0 if i = j and set for ξ ∈ R n , a i ( ξ ) = | ξ | − ( ξ i · ξ ) . Since for an elliptic canceling lineardi ff erential operator A ( D ) on R n from V to E one needs to have dim E > dim V ,this construction is the most economic in terms of the number of components of thesecond order derivative that are taken.In view of theorem 1.3, for every u ∈ C ∞ c ( R n ; V ) , k Du k L nn − ≤ C (cid:16) m +1 X i =1 k a i ( D ) w i · u ) k L (cid:17) . In particular for every u ∈ C ∞ c ( R ) , k∇ u k L ≤ C (cid:0) k ∂ u k L + k ∂ u k L (cid:1) . (6.2)This inequality is originally due to V. A. Solonnikov [35, theorem 3]. This estimateis quite striking because there is no estimate of the form k∇ u k L ≤ C k ∂ u + ∂ u k L (6.3)as one can see by inspection of the fundamental solution of − ∆ on R nor of theform k D u k L ≤ C (cid:0) k ∂ u k L + k ∂ u k L (cid:1) . (6.4)(this was the original motivation of D. Ornstein’s work [30]). The inequality (6.2)also explains why the construction of D. Ornstein to disprove (6.4) had to go beyondthe study of the fundamental solutions, as one does to disprove (6.3).7. Partially canceling operators7.1. Partially canceling operators.
If an operator A ( D ) is not canceling, there isstill a weaker inequality. Theorem 7.1.
Let n ≥ , let A ( D ) be an elliptic linear homogeneous differentialoperator on R n from V to E and let T ∈ L ( E ; F ) . The estimate k D k − u k L nn − ≤ C k A ( D ) u k L holds for every u ∈ C ∞ c ( R n ; V ) such that T ◦ A ( D ) u = 0 if and only if \ ξ ∈ R n \{ } A ( ξ )[ V ] ∩ ker T = { } . Remark . The estimate does not imply ellipticity. Indeed, take A ( D ) on R from R to R de fi ned by A ( D )[ u ] = ( ∂ u , ∂ u , ∂ u ) and T ∈ L ( R ; R ) de fi nedby T ( v ) = ( v , v ) . If u ∈ C ∞ c ( R ; R ) and T ◦ A ( D ) u = 0 , then u = 0 .Therefore the estimate holds trivially. On the other hand A ( D ) is not elliptic as A (1 , , . Estimates for partially cocanceling operators.
In order to prove theorem 7.1we shall need an extension of theorem 1.4 to partially cocanceling operators.
Proposition 7.2.
Let L ( D ) be a homogeneous linear differential operator of order k on R n from E to F and let Q ∈ L ( E ; E ) be a projector. If ker Q = \ ξ ∈ R n \{ } ker L ( ξ ) , then for every f ∈ L ( R n ; E ) such that L ( D ) f = 0 and ϕ ∈ C ∞ c ( R n ; E ) , Z R n ( Q ◦ f ) · ϕ ≤ C k Q ◦ f k L k Dϕ k L n . Proof. De fi ne ˜ L ( D ) to be the linear homogeneous di ff erential operator on R n from Q ( E ) to F de fi ned by restriction of L ( D ) . Since T ξ ∈ R n \{ } ker L ( ξ ) ⊆ ker Q , ˜ L ( D ) is cocanceling. Moreover, since ker Q ⊂ T ξ ∈ R n \{ } ker L ( ξ ) and Q is aprojector, for every ξ ∈ R n , (id − Q )( E ) = ker Q ⊆ ker L ( ξ ) . Hence, one has L ( ξ ) = L ( ξ ) ◦ Q = ˜ L ( ξ ) ◦ Q . Assume now that L ( D ) f = 0 . One has then ˜ L ( D )( Q ◦ f ) = 0 . Since ˜ L is cocanceling, theorem 1.4 applies to Q ◦ f and givesthe estimate. (cid:3) There is a converse statement to proposition 7.2
Proposition 7.3.
Let L ( D ) be a homogeneous linear differential operator from E to F and let Q ∈ L ( E ; F ) . If for every f ∈ L ( R n ; E ) such that L ( D ) f = 0 , onehas Q ◦ f ∈ ˙ W − , nn − ( R n ; E ) , then \ ξ ∈ R n \{ } ker L ( ξ ) ⊆ ker Q. Proof.
Let e ∈ T ξ ∈ R n \{ } ker L ( ξ ) . By assumption if f ∈ L ( R n ; R ) , one has f Q ( e ) ∈ ˙ W − , nn − , and then necessarily R R n f Q ( e ) = 0 . By choosing f such that R R n f = 1 , we conclude that Q ( e ) = 0 . (cid:3) An example of partially cocanceling operator operator.
An example of par-tially cocanceling operator is given by the
Curl Div operator:
Proposition 7.4.
Let L ( D ) be the homogeneous linear differential operator of order on R n from L ( R n ; R n ) to V R n defined for ξ ∈ R n ≃ V R n and e ∈L ( R n ; R n ) by L ( ξ )[ e ] = ξ ∧ e ( ξ ) , One has \ ξ ∈ R n \{ } ker L ( ξ ) = R id . Proof.
If for every ξ ∈ R n , L ( ξ )[ e ] = 0 , then for every ξ ∈ R n , there exists λ ∈ R \ { } such that e ( ξ ) = λξ . Since e is linear, there exists λ ∈ R such that e = λ id . (cid:3) By the application of proposition 7.2, we deduce
IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 27
Corollary 7.5 (M. Briane and J. Casado-Diaz, 2010 [12]) . If f ∈ L ( R n ; L ( R n ; R n )) and L ( D ) f = 0 , then f − (tr f ) id ∈ ˙ W − , nn − (cid:0) R n ; L ( R n ; R n ) (cid:1) and k f − (tr f ) id k ˙ W − ,n/ ( n − ≤ C k f − (tr f ) id k L . This result is used in the study of some Navier–Stokes equation [12].7.4.
Proof of the Sobolev estimate.
We now have the proof of the su ffi ciency partof theorem 7.1. We shall prove a quantitative version Proposition 7.6.
Let n ≥ and let A ( D ) be an elliptic linear homogeneous differ-ential operator on R n from V to E and let P ∈ L ( E ; E ) be a projector on \ ξ ∈ R n \{ } A ( ξ )[ V ] . For every u ∈ C ∞ c ( R n ; E ) , one has k D k − u k L nn − ≤ C (cid:0) k (id − P ) ◦ A ( D ) u k L + k P ◦ A ( D ) u k ˙ W − ,n/ ( n − (cid:1) . The interpretation is that the image of A ( D ) has some bad directions T ξ ∈ R n \{ } A ( ξ )[ V ] .If one has some better control in these directions, one can have a control on k D k − u k L n/ ( n − . Proof of proposition 7.6. If L ( D ) is given by proposition 4.2, one has \ ξ ∈ R n \{ } ker L ( ξ ) = \ ξ ∈ R n \{ } A ( ξ )[ V ] = P ( E ) = ker(id − P ) . In view of proposition 7.2, one has k (id − P ) ◦ A ( D ) u k ˙ W − ,n/ ( n − ≤ C k (id − P ) ◦ A ( D ) u k L . Hence, k A ( D ) u k ˙ W − ,n/ ( n − ≤ C (cid:0) k (id − P ) ◦ A ( D ) u k L + k P ◦ A ( D ) u k ˙ W − ,n/ ( n − (cid:1) . One concludes by using the ellipticity of A ( D ) as in the proof of proposition 4.6that k D k − u k L n/ ( n − ≤ C ′ (cid:0) k (id − P ) ◦ A ( D ) u k L + k P ◦ A ( D ) u k ˙ W − ,n/ ( n − (cid:1) . (cid:3) The necessity condition for the estimate. We fi nally sketch the proof of thenecessity part of theorem 7.1 Proposition 7.7.
Let n ≥ and let A ( D ) be an elliptic linear homogeneous differen-tial operator on R n from V to E and let T ∈ L ( E ; E ) . If for every u ∈ C ∞ c ( R n ; E ) such that T ◦ A ( D ) u = 0 k D k − u k L nn − ≤ C k A ( D ) u k L then \ ξ ∈ R n \{ } A ( ξ )[ V ] ∩ ker T = { } . Proof.
The proof follows the proof of proposition 5.5. One chooses e ∈ T ξ ∈ R n \{ } A ( ξ )[ V ] ∩ ker T and one checks that by construction of u λ , T ◦ A ( D ) u λ = 0 . (cid:3) An example of partially canceling operator.
We consider the Hodge–Sobolevinequality in the case that was not treated corresponding to (1.4)
Proposition 7.8.
Let n ≥ and A ( D ) = ( d, d ∗ ) be the homogeneous linear dif-ferential operator of order on R n from V R n to V R n × V R n such that forevery ξ ∈ R n and v ∈ V R n A ( ξ )[ v ] = (cid:0) ξ ∧ v, ∗ ( ξ ∧ ∗ v ) (cid:1) . The operator A ( D ) is elliptic.One has \ ξ ∈ R n \{ } A ( ξ ) (cid:2)V R n (cid:3) = { } × V R n . By theorem 7.1, we have the inequality obtained by J. Bourgain and H. Brezis[8, theorem 2; 9, corollary 12; 26, main theorem (b); 41, theorem 1.1]: for every u ∈ C ∞ c ( R n ) with d ∗ u = 0 , k u k L n/ ( n − ≤ C k du k L . If we use the quantitative version of of proposition 7.6, this gives
Corollary 7.9.
For every u ∈ C ∞ c ( R n ; V R n ) , one has k u k L n/ ( n − ≤ C (cid:0) k du k L + k d ∗ u k ˙ W − ,n/ ( n − (cid:1) . By the embedding of the real Hardy space H ( R n ) in ˙ W − ,n/ ( n − ( R n ) , corol-lary 7.9 also implies the estimate of L. Lanzani and E. Stein [26, main theorem (b)] k u k L n/ ( n − ≤ C (cid:0) k du k L + k d ∗ u k H (cid:1) .
8. F ra ctional and Lorentz estimates8.1. Sobolev estimates in fractional and Lorentz spaces. If A ( D ) is a homoge-neous linear di ff erential operator of order k on R n from V to E , one has the in-equality k D k − u k L nn − ≤ C k A ( D ) u k L . This estimate can be improved in various fractional cases.8.1.1.
Sobolev–Slobodecki˘ı spaces.
In the case of fractional Sobolev–Slobodecki ˘ ı spaces,we have Theorem 8.1.
Let n ≥ and let A ( D ) be a homogeneous linear differential operatorof order k on R n from V to E and let s ∈ (0 , and p ∈ (1 , ∞ ) be such that p − sn = 1 − n . The estimate k D k − u k ˙ W s,p ≤ C k A ( D ) u k L , holds for every u ∈ C ∞ c ( R n ; V ) if and only if A ( D ) is elliptic and canceling. Here, k v k ˙ W s,p is the homogeneous fractional Sobolev–Slobodecki ˘ ı semi-norm,that is k v k p ˙ W s,p = Z R n Z R n | v ( x ) − v ( y ) | p | x − y | n + sp dy dx. The su ffi ciency part of theorem 8.1 is not a consequence of theorem 1.3.Recall that the derivative operator is canceling if and only if n ≥ (proposi-tion 6.3). This allows us to recover the classical result [10, appendix D; 32, proposi-tion 4] IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 29
Corollary 8.2.
Let n ≥ , s ∈ (0 , and p ∈ (1 , ∞ ) be such that p − sn = 1 − n . The estimate k u k ˙ W s,p ≤ C k Du k L holds for every u ∈ C ∞ c ( R n ) if and only if n ≥ . The su ffi ciency part of corollary 8.2 also follows from the inequality k u k B s,p ≤ C k∇ u k L for every u ∈ C ∞ c ( R n ) obtained in the more general context of anisotropic Sobolevspaces [35, theorem 2] by V. I. Kolyada [24, theorem 4] or the estimate k u k B s,p ≤ C k∇ u k sL k u k − sL dd − obtained by A. Cohen, W. Dahmen, I. Daubechies and R. DeVore [15, theorem 1.4](see also J. Bourgain, H. Brezis and P. Mironescu [10, lemma D.2]) together withstandard embeddings between Besov spaces and the identi fi cation of Besov spaceswith fractional Sobolev–Slobodecki˘ ı spaces [39, 2.3.2(5), 2.3.5(3) and 2.5.7(9)]. Acounterexample when n = 1 can be obtained by taking regularizations of a charac-teristic function [32].8.1.2. Triebel–Lizorkin spaces.
Theorem 1.3 also extends to Triebel–Lizorkin spaces,as it was already the case for the Hodge–Sobolev inequality (1.5) [45, theorem 1].In the scale of Triebel–Lizorkin spaces, we have
Theorem 8.3.
Let n ≥ and let A ( D ) be a homogeneous linear differential operatorof order k on R n from V to E , let s ∈ ( k − nn − , k ) and p ∈ (1 , ∞ ) be such that p − sn = 1 − kn , and let q ∈ (0 , ∞ ] . The estimate k u k ˙ F sp,q ≤ C k A ( D ) u k L , holds for every u ∈ C ∞ c ( R n ; V ) if and only if A ( D ) is elliptic and canceling. We need the restriction s > k − nn − to prove the ellipticity. As discussed at theend of section 5.1, the theorem fails for s ≤ k − . This raises the problem Open Problem 8.1.
Let n ≥ . Does theorem 8.3 fail for s ∈ ( k − , k − nn − ] ?8.1.3. Besov spaces.
The extension of the Hodge–Sobolev inequality in Besov spaces[28, proposition 1; 45, theorem 1] to homogeneous linear differential operators is
Theorem 8.4.
Let n ≥ and let A ( D ) be a homogeneous linear differential operatorof order k on R n from V to E , let s ∈ ( k − nn − , k ) and p ∈ (1 , ∞ ) be such that p − sn = 1 − kn , and let q ∈ (1 , ∞ ) . The estimate k u k ˙ B sp,q ≤ C k A ( D ) u k L , holds for every u ∈ C ∞ c ( R n ; V ) if and only if A ( D ) is elliptic and canceling. In the case q = ∞ , the ellipticity alone is necessary and sufficient (see proposi-tion 8.22). When q = 1 , the ellipticity and the cancellation are necessary, but as forthe Hodge–Sobolev estimate [45, open problem 1] we do not know whether theyare sufficient: Open Problem 8.2.
Let k ≥ n and A ( D ) be a homogeneous linear differentialoperator of order k on R n from V to E . Assume that A ( D ) is elliptic and cancelingand that s ∈ ( k − n, n ) and p ∈ (1 , ∞ ) satisfy p − sn = 1 − kn . Does one have forevery u ∈ C ∞ c ( R n ; V ) , k u k ˙ B sp, ≤ C k A ( D ) u k L ? The answer is positive in the scalar case V = R [24, corollary 1]. The questionis already open for the Hodge–Sobolev inequality [45, open problem 1].8.1.4. Lorentz spaces.
Finally, in the framework of Lorentz spaces, we have, as forthe Hodge–Sobolev estimate [45, theorem 3]
Theorem 8.5.
Let n ≥ and let A ( D ) be a homogeneous linear differential operatorof degree k on R n from V to E and q ∈ (1 , ∞ ) . The estimate k D k − u k L n/ ( n − ,q ≤ C k A ( D ) u k L , holds for every u ∈ C ∞ c ( R n ; V ) if and only if A ( D ) is elliptic and canceling. Again, when q = ∞ , the ellipticity alone is necessary and sufficient (see propo-sition 8.24). If q = 1 , the ellipticity and the cancellation are necessary, but as forthe Hodge–Sobolev estimate [45, open problem 2] we could not determine whetherthey are sufficient Open Problem 8.3.
Let k ≥ n and A ( D ) be a homogeneous linear differentialoperator of order k on R n from V to E . Assume that A ( D ) is elliptic and canceling.Does one have for every u ∈ C ∞ c ( R n ; V ) , k D k − u k L nn − , ≤ C k A ( D ) u k L ? This property is true when one considers the gradient in Sobolev spaces forLorentz spaces [4].Since k D k − n u k L ∞ ≤ C k I n − ( k − s ) k ˙ B s nk − s , ∞ k u k ˙ B s nn − ( k − s ) , and k D k − n u k L ∞ ≤ C k I n − ( k − ℓ ) k L nk − ℓ , ∞ k D ℓ u k L nn − ( k − ℓ ) , , where I α is the Riesz potential of order α ∈ (0 , n ) defined for x ∈ R n \ { } by I α ( x ) = π n/ α Γ( α/ n − α ) / | x | n − α , a positive answer to either open problem 8.2 or openproblem 8.3 would imply the estimate k D k − n u k L ∞ ≤ C k A ( D ) u k L . This motivates the problem
Open Problem 8.4.
Let k ≥ n and A ( D ) be a homogeneous differential operatorof order k on R n from V to E . Assume that A ( D ) is elliptic and canceling. Doesone have for every u ∈ C ∞ c ( R n ; V ) , k D k − n u k L ∞ ≤ C k A ( D ) u k L ? (8.1) IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 31
The answer is positive in the scalar case: for every u ∈ C ∞ c ( R n ) , k u k L ∞ ≤ k D n u k L . A nontrivial vector example is given by the estimate k∇ u k L ∞ ≤ C k ∆ ∇ u k L (8.2)for every u ∈ C ∞ c ( R ) . This estimate was obtained by J. Bourgain and H. Brezis[8, remark 5; 9, theorem 3] (see also [11, theorem 2.1; 43, corollary 4.9]). For analternative proof, note that ∇ u = ( ∇ div)∆ − (∆ ∇ u ) , = (cid:16) ∂ − ∂ ∂ ∂ ∂ ∂ ∂ − ∂ (cid:17) ∆ − (∆ ∇ u ) . (8.3)If G denotes the fundamental solution of ∆ in R , P. Mironescu has shown that ∂ G − ∂ G and ∂ ∂ G are bounded [27, proposition 1]. The estimate (8.2) thenfollows.More generally, if n is even, one has ∇ u = ( ∇ div)∆ − n − (∆ n ∇ u ) , = 1 n − n ∇ div − ∆)∆ − n − (∆ n ∇ u ) . If G denotes the Green function of ∆ n +1 on R n , nD G − ∆ G id ∈ L ∞ [27,proposition 3], and therefore k u k L ∞ ≤ C k ∆ n ∇ u k L . Also note that as noticed in remark 5.1, canceling is not necessary for (8.1).8.2. L estimates and cocanceling operators. In order to prove the fractional andLorentz space estimates, we first extend the results of section 2 concerning cocancel-ing operators
Proposition 8.6.
Let L ( D ) be a homogeneous differential operator from E to F ,let s ∈ (0 , and p ∈ (1 , ∞ ) be such that sp = n . If L ( D ) is cocanceling, f ∈ L ( R n ; E ) and L ( D ) f = 0 in the sense of distributions, then for every ϕ ∈ C ∞ c ( R n ; E ) , Z R n f · ϕ ≤ C k f k L k ϕ k ˙ W s,p . Proof.
The proof is similar to the proof of proposition 8.6, it relies on the counterpartof proposition 2.4 for fractional Sobolev–Slobodecki ˘ ı spaces [44, (4)]. (cid:3) The cocancellation condition is here necessary (see the proof of proposition 2.2).One can also use the same kind of arguments in order to obtain a counterpartof proposition 2.4 for Triebel–Lizorkin spaces [45, proof of proposition 2.1]. Thisshows that one can replace in the statement of proposition 8.6 ˙ W s,p ( R n ; E ) by ˙ F sp,q ( R n ; E ) for every q ≥ . This can also be deduced from proposition 8.6 bystandard embeddings between fractional spaces [39, theorem 2.7.1 and §5.2.5]: Proposition 8.7.
Let L ( D ) be a homogeneous differential operator from E to F , let s ∈ (0 , and p ∈ (1 , ∞ ) be such that sp = n and let q ∈ [1 , ∞ ] . If L ( D ) iscocanceling, f ∈ L ( R n ; E ) and L ( D ) f = 0 in the sense of distributions, then forevery ϕ ∈ C ∞ c ( R n ; E ) , Z R n f · ϕ ≤ C k f k L k ϕ k ˙ F sp,q . The cocancellation condition is still necessary for Triebel–Lizorkin spaces.For Besov spaces, one has
Proposition 8.8.
Let L ( D ) be a homogeneous differential operator on R n from E to F , let s ∈ (0 , and p ∈ (1 , ∞ ) be such that sp = n and let q ∈ (1 , ∞ ] . If L ( D ) is cocanceling, f ∈ L ( R n ; E ) and L ( D ) f = 0 in the sense of distributions, thenfor every ϕ ∈ C ∞ c ( R n ; E ) , Z R n f · ϕ ≤ C k f k L k ϕ k ˙ B sp,q . This proposition is deduced from proposition 8.6 or from proposition 8.7. Thecase q = 1 is a consequence of the estimate k ϕ k L ∞ ≤ C k ϕ k ˙ B sp, , the cocancellation condition is not necessary in this case (see proposition 8.17). Inthe other cases, it is necessary.The case q = ∞ is open. The current arguments fail in this case because propo-sition 2.4 relies on a Fubini-type property that is only present in Triebel–Lizorkinspaces. Proposition 2.4 can thus only be proved in those spaces; the Nikol’ski ˘ ı spaces B sp, ∞ do not embed in this scale of spaces.We remark that a counterexample cannot be constructed by taking for ϕ a regu-larization of x ∈ R n log | x | Proposition 8.9.
Let L ( D ) be a homogeneous differential operator of order k on R n from E to F . If L ( D ) is cocanceling, f ∈ L ( R n ; E ) and L ( D ) f = 0 in thesense of distributions, then for every ϕ ∈ C ∞ c ( R n ; E ) , Z R n f · ϕ ≤ C k f k L k X ℓ =1 sup x ∈ R n | x | ℓ | D ℓ ϕ ( x ) | . Proof.
We extend the argument proposed in the case where L ( D ) is the divergenceoperator [43, proposition 4.3]. Let K α be given by lemma 2.5 and define P : R n ( E ; F ) for x ∈ R n by P ( x ) = X α ∈ N n | α | = k x α α ! K ∗ α . One has in view of (2.1), for every x ∈ R n , (cid:0) L ( D ) ∗ P (cid:1) ( x ) = X α ∈ N n | α | = k ∂ α x α α ! L ∗ α ◦ K ∗ α = id . Therefore, since L ( D ) f = 0 , Z R n f · ϕ = Z R n f · (cid:0) L ( D ) ∗ P (cid:1) [ ϕ ] = Z R n f · (cid:0) ( L ( D ) ∗ P )[ ϕ ] − L ( D ) ∗ ( P [ ϕ ]) (cid:1) . IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 33
One concludes by noting that for every x ∈ R n , (cid:12)(cid:12)(cid:0) L ( D ) ∗ P (cid:1) ( x )[ ϕ ( x )] − (cid:0) L ( D ) ∗ ( P [ ϕ ]) (cid:1) ( x ) (cid:12)(cid:12) ≤ C k X ℓ =1 | x | ℓ | D ℓ ϕ ( x ) | . (cid:3) The estimate of proposition 2.3 becomes in the framework of Lorentz spaces
Proposition 8.10.
Let L ( D ) be a homogeneous differential operator from E to F and q ∈ [1 , ∞ ) . If L ( D ) is cocanceling, f ∈ L ( R n ; E ) and L ( D ) f = 0 in thesense of distributions, then for every ϕ ∈ C ∞ c ( R n ; E ) , Z R n f · ϕ ≤ C k f k L k Dϕ k L n,q . Again the cocancellation condition is necessary if q > and the case q = ∞ isopen.8.3. Proofs of the Sobolev estimates.
The proof of the Sobolev estimates in thefractional and Lorentz spaces can be done as in section 4. First one note thatthe results in section 4.1 extend to fractional Sobolev–Slobodecki ˘ ı spaces, Triebel–Lizorkin, Besov and Lorentz–Sobolev spaces by standard multiplier theorems adaptedto these spaces [39, theorem 2.3.7].Our previous approach extends to fractional Sobolev–Slobodecki ˘ ı spaces: by us-ing proposition 8.6 instead of (1.4) and the counterpart of proposition 4.1 in frac-tional Sobolev–Slobodecki ˘ ı spaces, we obtain the sufficiency part of theorem 8.1 Proposition 8.11.
Let A ( D ) be a homogeneous linear differential operator of order k on R n from V to E and let s ∈ (0 , and p ∈ (1 , ∞ ) be such that p − sn = 1 − n .If A ( D ) is elliptic and canceling, then for every u ∈ C ∞ c ( R n ; V ) , k D k − u k ˙ W s,p ≤ C k A ( D ) u k L . Similarly, in Triebel–Lizorkin spaces, one has the sufficiency part of theorem 8.3:
Proposition 8.12.
Let A ( D ) be a homogeneous linear differential operator of order k on R n from V to E , let s ∈ ( k − n, k ) and p ∈ (1 , ∞ ) be such that p − sn = 1 − kn , and let q ∈ [1 , ∞ ] . If A ( D ) is elliptic and canceling, then for every u ∈ C ∞ c ( R n ; V ) , k u k ˙ F sp,q ≤ C k A ( D ) u k L . Proof.
For q > , the proof goes as the proof of proposition 4.6, using proposition 8.7instead of theorem 1.4 and the counterpart of proposition 4.1 in Triebel–Lizorkinspaces. One can then treat the case q ∈ (0 , by embeddings between Triebel–Lizorkin spaces [31, proposition 2.2.3; 39, theorem 2.7.1]: if t ∈ ( s, n ) and r ∈ (1 , ∞ ) are such that r − un = 1 − kn and u ∈ (0 , ∞ ] , then k u k ˙ F sp,q ≤ C k u k ˙ F tr,u . (cid:3) In the case of the Besov spaces, one has the sufficiency part of theorem 8.4
Proposition 8.13.
Let A ( D ) be a homogeneous linear differential operator of order k on R n from V to E , let s ∈ ( k − n, k ) and p ∈ (1 , ∞ ) be such that p − sn = 1 − kn , and let q ∈ (1 , ∞ ] . If A ( D ) is elliptic and canceling, then for every u ∈ C ∞ c ( R n ; V ) , k u k ˙ B sp,q ≤ k A ( D ) u k L . When q = ∞ , the ellipticity alone is sufficient (proposition 8.22). The proof issimilar to the proof of proposition 8.12, except that the counterpart of (8.3) onlyholds if u ≤ q .Finally, we have in Lorentz spaces Proposition 8.14.
Let A ( D ) be a homogeneous linear differential operator of degree k on R n from V to E and q ∈ (1 , ∞ ) . If A ( D ) is elliptic and canceling, then forevery u ∈ C ∞ c ( R n ; V ) , k D k − u k L n/ ( n − ,q ≤ k A ( D ) u k L , Necessity of the ellipticity.
The proof of proposition 5.1 applies to fractionalspaces and yields
Proposition 8.15.
Let A ( D ) be a homogeneous differential operator on R n of order k from V to E . Let s ∈ ( k − nn − , k ) , p ≥ and q > be such that q − sn = p − kn .If for every u ∈ C ∞ c ( R n ; V ) , k u k ˙ W s,q ≤ C k A ( D ) u k L p , then A ( D ) is elliptic. For s ∈ [ k − , k ) , this is a consequence of corollary 5.2 by classical embeddingstheorems for fractional Sobolev–Slobodecki ˘ ı spaces [1, theorem 7.57]. Proof.
One begins as the in proof of proposition 5.1. One notes then that k u λ k ˙ W s,q = Cλ n − | v | + o ( λ n − ) . (cid:3) In Triebel–Lizorkin spaces, one has
Proposition 8.16.
Let A ( D ) be a homogeneous differential operator on R n of order k from V to E . Let s ∈ ( k − nn − , k ) , p ≥ , r > and q > be such that q − sn = p − kn . If for every u ∈ C ∞ c ( R n ; V ) , k u k ˙ F sq,r ≤ C k A ( D ) u k L p , then A ( D ) is elliptic. Proposition 8.16 can be obtained either by a direct proof or by deduction fromproposition 8.15 by standard embedding theorems and the characterization of frac-tional Sobolev–Slobodecki ˘ ı spaces as Triebel–Lizorkin spaces [39, theorems 2.5.7and 2.7.1].For Besov spaces we have Proposition 8.17.
Let A ( D ) be a homogeneous differential operator on R n of order k from V to E . Let s ∈ ( k − nn − , k ) , p ≥ , r > and q > be such that q − sn = p − kn . If for every u ∈ C ∞ c ( R n ; V ) , k u k ˙ B sq,r ≤ C k A ( D ) u k L p , then A ( D ) is elliptic. Proposition 8.17 cannot be deduced from proposition 5.1. Such an argumentwould in fact impose the additional restriction that r ≤ q that does not appear withthe direct argument. IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 35
Finally, for Lorentz spaces, one has
Proposition 8.18.
Let A ( D ) be a homogeneous differential operator on R n of order k from V to E . Let q > be such that q − sn = p − kn . If for every u ∈ C ∞ c ( R n ; V ) , k D k − u k L q,r ≤ C k A ( D ) u k L p , then A ( D ) is elliptic. When r ≤ q , this is an immediate consequence of proposition 5.1. When r > q ,the proof of proposition 5.1 applies and gives the conclusion.8.5. Necessity of the cancellation.
Concerning fractional spaces, the proof of propo-sition 5.5 allows to prove
Proposition 8.19.
Let A ( D ) be an elliptic homogeneous linear differential operatorof order k on R n from V to E , let s ∈ (0 , , p ≥ and ℓ ∈ { , . . . , n − } suchthat ℓ ≥ n − k and p − ℓ + sn = 1 − n . If for every u ∈ C ∞ c ( R n ; V ) , k D ℓ u k ˙ W s,p ≤ k A ( D ) u k L , then A ( D ) is canceling.Proof. One proceeds as in the proof of proposition 5.5, using the fact that if (5.6)is satisfied, then u α does not have finite fractional Sobolev–Slobodecki ˘ ı norm andapplying the Fatou property in fractional Sobolev–Slobodecki ˘ ı spaces: if ∂ α u λ → u α almost everywhere as λ → ∞ , then Z R n Z R n | u α ( x ) − u α ( y ) || x − y | n + sp dx dy ≤ lim inf λ →∞ Z R n Z R n | ∂ α u λ ( x ) − ∂ α u λ ( y ) || x − y | n + sp dx dy. (cid:3) Proposition 8.20.
Let A ( D ) be an elliptic homogeneous linear differential operatorof order k on R n from V to E , let p ∈ (1 , ∞ ) and s ∈ ( k − n, k ) be such that p − sn = 1 − kn and let q ∈ (0 , ∞ ] . If for every u ∈ C ∞ c ( R n ; V ) , k u k ˙ F sp,q ≤ C k A ( D ) u k L , then A ( D ) is canceling. When s ≥ , this is a consequence of proposition 8.19, classical embeddings be-tween Triebel–Lizorkin spaces [39, theorem 2.7.1] and the equivalence between frac-tional Sobolev–Slobodecki ˘ ı spaces and Triebel–Lizorkin spaces [39, theorem 2.5.7] Proof of proposition 8.20.
Follow the proof of proposition 5.5 till (5.6) with ( − ∆) s u instead of ∂ α u . Define u s ( x ) = lim λ →∞ ( − ∆) s u λ ( x ) . One has in place of (5.6) for each x ∈ R n \ { } and t ∈ (0 , ∞ ) u s ( tx ) = u s ( x ) t n − ( k − s ) (8.4)Therefore, u s F p,q ( R n ; V ) if and only if u s [31, lemma 2.3.1/1].Since ( − ∆) s u λ ≤ C | x | n − ( k − s ) , one has ( − ∆) s u λ → u s as λ → ∞ in L ( R n ; V ) . By the Fatou propertyfor Triebel–Lizorkin spaces [19] (see also [31, proposition 2.1.3/2]), k u λ k ˙ F sp,q is notbounded as λ → ∞ . One concludes as in the proof of proposition 5.5. (cid:3) Similarly, one can prove in Besov spaces
Proposition 8.21.
Let A ( D ) be an elliptic homogeneous linear differential operatorof order k on R n from V to E , let p ∈ (1 , ∞ ) and s ∈ ( k − n, k ) be such that p − sn = 1 − kn and q ∈ (0 , ∞ ) . If for every u ∈ C ∞ c ( R n ; V ) , k u k ˙ B sp,q ≤ C k A ( D ) u k L , then A ( D ) is canceling. The restriction q < ∞ comes from the fact that (8.4) is not incompatible with u α ∈ B sp, ∞ ( R n ; V ) . This restriction is essential as shows Proposition 8.22.
Let A ( D ) be an elliptic linear homogeneous differential operatorof order k on R n from V to E and let s ∈ ( k − n, k ) and p ∈ [1 , ∞ ) be such that p − sn = 1 − kn . For every u ∈ C ∞ c ( R n ; V ) , k u k ˙ B sp, ∞ ≤ C k A ( D ) u k L . Proof.
Define G : R n \ { } → L ( E ; V ) such that for every ξ ∈ R n \ { } , b G ( ξ ) = | ξ | s (cid:0) A ( ξ ) ∗ ◦ A ( ξ ) (cid:1) − ◦ A ∗ ( ξ ) . Since b G is homogeneous of degree − ( k − s ) , G is homogeneous of degree − ( n − ( k − s )) and therefore G ∈ ˙ B p, ∞ ( R n ; L ( V ; E )) . Since k·k ˙ B sp, ∞ is a norm, byconvexity, k u k B sp, ∞ = k G ∗ ( A ( D ) u ) k B p, ∞ ≤ k G k B p, ∞ k A ( D ) u k L . (cid:3) An alternative argument would be to use the estimate [31, theorem 2.2.2; 34,theorem 3.1.1] k A ( D ) u k B , ∞ ≤ C k A ( D ) u k L . together with the theory of Fourier multipliers on Besov spaces [31, proposition2.1.6/5; 39, theorem 2.3.7] and the embeddings between Besov spaces [31, theorem2.2.3; 39, theorem 2.7.1].The argument of proposition 5.5 still applies to Lorentz space estimates Proposition 8.23.
Let A ( D ) be an elliptic homogeneous linear differential operatorof order k on R n from V to E and let q ∈ [1 , ∞ ) . If for every u ∈ C ∞ c ( R n ; V ) , k u k L nn − ,q ≤ C k A ( D ) u k L , then A ( D ) is canceling. This only follows from proposition 5.5 when q ≤ nn − . The proof is similar tothat of proposition 8.21, using the Fatou property for Lorentz spaces, and the factthat for q ∈ [1 , ∞ ) , there are no nonzero homogeneous functions.Again the restriction q < ∞ is optimal, as one has Proposition 8.24.
Let A ( D ) be a linear homogeneous elliptic operator of order k on R n from V to E . For every u ∈ C ∞ c ( R n ; V ) , k D k − u k L nn − , ∞ ≤ C k A ( D ) u k L . IMITING SOBOLEV INEQUALITIES AND CANCELING OPERATORS 37
Proof.
The proof is similar to the proof of proposition 8.22; an alternate proof wouldstart from a weak L estimate for the elliptic operator together with Sobolev em-beddings in the framework of Marcinkiewicz spaces. (cid:3)
9. S trong B ourgain –B rezis estimates If A ( D ) is a linear homogeneous di ff erential operator of order k on R n , one hasthe estimates k D k − u k L nn − ≤ C k A ( D ) u k L (9.1)and k D k − u k L nn − ≤ C k A ( D ) u k ˙ W − ,n/ ( n − . (9.2)In view of these estimates, one can wonder whether one can obtain a strongerstatement using a weaker norm k A ( D ) u k L + ˙ W − ,n/ ( n − .J. Bourgain and H. Brezis [6, (8); 7, lemma 1; 8, remark 6; 9, corollary 12] haveobtained such results for the gradient and the exterior derivative. Relying on theirabstract results, we prove a similar counterpart of proposition 4.6 in which a weakernorm of A ( D ) u is taken. Theorem 9.1.
Let A ( D ) be a linear homogeneous differential operator of order k on R n from V to E . If A ( D ) is elliptic and canceling, then for every u ∈ C ∞ c ( R n ; V ) , k D k − u k L nn − ≤ C k A ( D ) u k L + ˙ W − ,n/ ( n − . These estimates are not a consequence of (9.1) and (9.2). Indeed, from the defini-tion of k A ( D ) u k L + ˙ W − ,n/ ( n − , there exists f ∈ C ∞ c ( R n ; E ) such that A ( D ) u − f ∈ ˙ W − , nn − ( R n ; E ) and k f k L + k A ( D ) u − f k ˙ W − , nn − ≤ k A ( D ) u k L + ˙ W − , nn − . but nothing says that f can be written as f = A ( D ) w with w ∈ C ∞ c ( R n ; V ) withthe useful estimates.It is not known whether theorem 9.1 holds in any other Sobolev space [9, openproblem 2], that is, whether, given s = 1 and p ∈ (1 , ∞ ) such that p − sn = 1 − kn , if A ( D ) is elliptic and canceling, one has for every u ∈ C ∞ c ( R n ; V ) , k u k ˙ W s,p ≤ C k A ( D ) u k L + ˙ W k − s,p . The main ingredient in the proof of theorem 9.1 is the following variant on theo-rem 1.4
Theorem 9.2.
Let L ( D ) be a linear homogeneous differential operator of order k on R n from E to F . If L ( D ) is cocanceling, then for every f ∈ L ( R n ; E ) , one has f ∈ ˙ W − , nn − ( R n ; E ) if and only if L ( D ) f ∈ ˙ W − − k, nn − ( R n ; F ) . Moreover, if f ∈ L ( R n ; E ) and L ( D ) f ∈ ˙ W − − k, nn − ( R n ; F ) , one has k f k ˙ W − ,n/ ( n − ≤ C (cid:0) k f k L + k L ( D ) f k ˙ W − − k,n/ ( n − (cid:1) , Proof.
The proof follows the lines of the proof of proposition 2.3, it relies on astrengthened version of proposition 2.4 [44, theorem 9]. (cid:3)
Whereas the sufficiency part of theorem 9.2 is much stronger than theorem 1.4,its proof relies on a difficult construction of J. Bourgain and H. Brezis [9] whiletheorem 1.4 relies on proposition 2.4 that is proved by elementary methods. As it was mentioned for theorem 9.2, the result of J. Bourgain and H. Brezis has not beenextended to other critical Sobolev spaces.We can now prove theorem 9.1
Proof of theorem 9.1.
The necessity part follows from theorem 1.3.For the sufficiency part, choose f ∈ C ∞ c ( R n ) such that k f k L + k A ( D ) u − f k ˙ W − , nn − ≤ k A ( D ) u k L + ˙ W − ,n/ ( n − . Let L ( D ) be the homogeneous differential operator of order ℓ given by proposi-tion 4.2. Since A ( D ) is canceling, L ( D ) is cocanceling. In view of theorem 9.2,since L ( D ) f = L ( D ) (cid:0) f − A ( D ) u (cid:1) , k f k ˙ W − , nn − ≤ C (cid:0) k f k L + k L ( D ) (cid:0) f − A ( D ) u (cid:1) k ˙ W − − ℓ, nn − (cid:1) ≤ C ′ (cid:0) k f k L + k f − A ( D ) u k ˙ W − , nn − (cid:1) . We have thus k A ( D ) u k ˙ W − , nn − ≤ C ′′ k A ( D ) u k L + ˙ W − ,n/ ( n − . We conclude by proposition 4.1 as in the proof of theorem 1.3. (cid:3)
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