Limiting Bourgain-Brezis estimates for systems: theme and variations
aa r X i v : . [ m a t h . A P ] F e b LIMITING BOURGAIN-BREZIS ESTIMATES FORSYSTEMS: THEME AND VARIATIONS
JEAN VAN SCHAFTINGEN
Abstract.
J. Bourgain and H. Brezis have obtained in 2002 some newand surprising estimates for systems of linear differential equations, deal-ing with the endpoint case L of singular integral estimates and thecritical Sobolev space W ,n ( R n ). This paper presents an overview ofthe results, further developments over the last ten years and challengingopen problems. Theme
Limiting Hodge theory for Sobolev forms.
The study of limitingestimates for systems starts from the following problem: given a function g ∈ L n ( R n ; R n ) find the best regularity that a vector field u : R n → R n such that(1.1) div u = g in R n can have.If n ≥
2, the equation (1.1) is strongly underdetermined . The standardway of finding a solution u consists in lifting the undeterminacy by solvingthe system(1.2) ( div u = g in R n , curl u = 0 in R n , where curl u = Du − ( Du ) ∗ . By the classical Calderón–Zygmund theory ofsingular integrals [30] (see also [89]), there exists a function v ∈ W ,n loc ( R n )that satisfies − ∆ v = g in R n and k D v k L n ≤ C n k g k L n . In particular, the vector-field u = ∇ v solves the problem (1.2), and thusalso the original problem (1.1), and u satisfies the estimates k Du k L n ≤ C n k g k L n . Mathematics Subject Classification.
Key words and phrases.
Critical Sobolev spaces; div-curl system; Hodge decomposi-tions for Sobolev differential forms; Calderón–Zygmund estimates; circulation integrals;Gagliardo–Nirenberg–Sobolev inequality; Hardy inequality; fractional Sobolev spaces;Lorentz-Sobolev spaces; Sobolev–Slobodetski˘ı spaces; Besov spaces; Triebel–Lizorkinspaces; boundary estimates; canceling differential operator; graded stratified nilpotentLie groups; nonisotropic Sobolev space; functions of bounded mean oscillation (BMO);strong charges.
In general, vector fields in the Sobolev space W ,n ( R n ; R n ) need not bebounded functions (see for example [2, remark 4.43; 22, remark 9.16]). Thisis also not the case for our solution u : L. Nirenberg has given as a counterex-ample the data g = − ∆ v with v ( x ) = x (log | x | ) α ζ ( x ), where ζ is a suitablecut-off function and α ∈ (0 , nn − ) [14, remark 7] (see also [2, example 4.44]).As this solution of the underdetermined system (1.2) is merely one outof infinitely many, we can still hope that (1.1) has another solution thatis bounded . J. Bourgain and H. Brezis have constructed such solutions [13,proposition 1; 14, proposition 1; 15, theorem 4; 16, theorem 5]. Theorem 1.
Let ℓ ∈ { , . . . , n − } . If g ∈ L n ( R n ; V ℓ +1 R n ) and dg = 0 inthe sense of distributions, then there exists u ∈ L ∞ ( R n ; V ℓ R n ) , such that du = g in the sense of distributions. Moreover, k u k L ∞ ≤ C k g k L n . The first part of the statement can be written, in view of the classicalHodge theory [53, 84] d (cid:0) ˙W ,n ( R n ; V ℓ R n ) (cid:1) ⊂ d (cid:0) L ∞ ( R n ; V ℓ R n ) (cid:1) , where ˙W ,n ( R n ; V ℓ R n ) denotes the homogeneous Sobolev space of weaklydifferentiable differential forms such that | Du | ∈ L n ( R n ).In the theory of lifting of fractional Sobolev maps into the unit circle [17],theorem 1 has allowed to derive some local bound on the norm of the phase k ϕ k L n/ ( n − in terms of k e iϕ k H / [14, corollary 1]. Theorem 1 was also usedto reformulate a smallness assumption on a magnetic vector potential inL ∞ ( R n ) as an assumption on the magnetic field in L n ( R n ) [1, p. 159].In comparison with the standard Hodge theory [53, 84], theorem 1 doesnot give any integrability information on the derivative Du . J. Bourgainand H. Brezis have constructed a solution that satisfies both the estimatesof theorem 1 and the classical estimates [13, theorem 1; 14, theorem 1;15, theorem 4; 16, theorem 5]. Theorem 2.
Let ℓ ∈ { , . . . , n − } . If g ∈ L n ( R n ; V ℓ +1 R n ) and dg = 0 inthe sense of distributions, then there exists u ∈ L ∞ ( R n ; V ℓ R n ) , such that du = g in the sense of distributions, u is continuous and Du ∈ L n ( R n ) .Moreover, k u k L ∞ + k Du k L n ≤ C k g k L n . The first part of the statement can be written, in view of the classicalHodge theory [53, 84] as d (cid:0) ˙W ,n ( R n ; V ℓ R n ) (cid:1) = d (cid:0) ˙W ,n ( R n ; V ℓ R n ) ∩ L ∞ ( R n ; V ℓ R n ) (cid:1) , or as˙W ,n ( R n ; V ℓ R n ) = ˙W ,n ( R n ; V ℓ R n ) ∩ L ∞ ( R n ; V ℓ R n )+ d (cid:0) ˙W ,n ( R n ; V ℓ − R n ) (cid:1) , that is, every ˙W ,n –Sobolev ℓ –form is bounded up to an exact form.When ℓ = n −
1, theorem 2 states that every g ∈ W ,n ( R n ; R n ) can bewritten as(1.3) g = R v, IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 3 with v ∈ W ,n ( R n ; R n ) ∩ L ∞ ( R n ; R n ) and the vector Riesz transform isdefined by its Fourier transform d R v ( ξ ) = iξ · v ( ξ ) / | ξ | . As noted by J.Bourgain and H. Brezis [14], the decomposition (1.3) is a refined versionof the Fefferman-Stein decomposition [42, theorem 3; 102] which states that g ∈ BMO( R n ) if and only if it can be decomposed as g = w + R v, with w ∈ L ∞ ( R n ) and v ∈ L ∞ ( R n ; R n ).Theorem 2 has allowed to obtain uniform L n/ ( n − estimates on the gradi-ent of minimizers of the Ginzburg–Landau functional [15, theorem 5; 16, the-orem 21] (see also [11, proposition 5.1; 18, theorem 11]).1.2. Bourgain–Brezis linear estimates.
The existence theorem 1 can bereformulated as a linear estimate [15; 16, theorem 1 ′ ; 104, corollary 1.4]. Theorem 3.
Let ℓ ∈ { , . . . , n − } . There exists C > such that for every f ∈ C ∞ c ( R n ; V ℓ R n ) and every ϕ ∈ C ∞ c ( R n ; V n − ℓ R n ) , if df = 0 , then (cid:12)(cid:12)(cid:12)Z R n f ∧ ϕ (cid:12)(cid:12)(cid:12) ≤ C k f k L k dϕ k L n . Theorem 3 would be a consequence of a critical Sobolev embedding of˙W ,n ( R n ) in L ∞ ( R n ) which is well-known to fail (see for example [22, remark9.16]). The estimate is on the integral of the form f ∧ ϕ and not of the density | f ∧ ϕ | ; it results from a compensation phenomenon which is reminiscent ofdiv-curl estimates [37, 73, 95–97].When k = 1, theorem 3 is equivalent with the classical Gagliardo–Nirenberg–Sobolev estimate [46; 74, p. 125] (see also [2, theorem 4.31; 22, theorem 9.9;69, (1.4.14); 89, V.2.5])(1.4) k u k L n/ ( n − ≤ C k Du k L . We explain how theorems 1 and 3 are equivalent [15; 16, remark 8] (seealso [104]). First by theorem 1, for every ϕ ∈ C ∞ c ( R n ; V n − ℓ R n ), thereexists u ∈ L ∞ ( R n ; V n − ℓ R n ) such that du = dϕ and k u k L ∞ ≤ C k dϕ k L n .Moreover, by the classical Calderón–Zygmund elliptic regularity estimates,there exists ζ ∈ ˙W ,n ( R n ; V n − ℓ − R n ) such that ( dζ = ϕ − u in R n ,d ∗ ζ = 0 in R n , ( d ∗ ζ denotes the exterior codifferential of the differential form ζ ). Hence, (cid:12)(cid:12)(cid:12)Z R n f ∧ ϕ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z R n f ∧ ( u + dζ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z R n f ∧ u (cid:12)(cid:12)(cid:12) ≤ C k f k L k dϕ k L n . Conversely, if g ∈ L n ( R n ; V ℓ +1 R n ), and dg = 0, by the classical Hodgetheory in Sobolev spaces, there exists v ∈ ˙W ,n ( R n ; V ℓ R n ) such that ( dv = g in R n ,d ∗ v = 0 in R n . For every ψ ∈ C ∞ c ( R n ; V n − ℓ − R n ), by theorem 1 (cid:12)(cid:12)(cid:12)Z R n dψ ∧ v (cid:12)(cid:12)(cid:12) ≤ C k dψ k L k Dv k L n ≤ C ′ k dψ k L k g k L n . JEAN VAN SCHAFTINGEN
By the classical Hahn-Banach theorem (see for example [22, corollary 1.2])and the representation of linear functionals on L (see for example [22, the-orem 4.14]), there exists u ∈ L ∞ ( R n ; V ℓ R n ) such that for every ψ ∈ C ∞ c ( R n ; V n − ℓ − R n ), Z R n dψ ∧ v = Z R n dψ ∧ u. By construction of u , we conclude that for every Z R n dψ ∧ u = Z R n dψ ∧ v = ( − n − ℓ Z R n ψ ∧ g, that is, du = g in the sense of distributions.Theorem 3 was used to show that if g ∈ W ,p (Ω , R n ) ∩ L q (Ω; R n ), q + n − p = 1 and if det( Dg ) = div µ , then the measure | µ | does not charge setsof null W ,n –capacity [23, proposition 2]. Theorem 3 yields a representationof divergence-free measures in the study of limiting div-curl lemmas [29,theorem 3.1]. Theorem 3 also allows to obtain endpoint Strichartz estimatefor the linear wave and Schrödinger equations with space divergence-freedata [35].As a consequence of theorem 1 and the classical elliptic regularity theory,we have a Gagliardo–Nirenberg–Sobolev inequality for forms [16, corollary17] (see also [63]): if ℓ ∈ { , . . . , n − } , then(1.5) k u k L n/ ( n − ≤ C (cid:0) k du k L + k d ∗ u k L (cid:1) ;the inequality still holds for ℓ ∈ { , n − } provided d ∗ u = 0 if ℓ = 1 and du if ℓ = n −
1. In particular, there is no such estimate for n = 2. Thevanishing of du or d ∗ u can be replaced by an estimate in the real Hardy space H ( R n ) [63]. The inequality (1.5) was used in the Chern–Weil theory forSobolev connections on Sobolev bundles [52, proposition 4.1]. This familyof inequalities can be extended to higher-order analogues of the exteriorderivative [61].The inequality (1.5) would be a consequence of the classical Gagliardo–Nirenberg–Sobolev inequality (1.4) and of the Gaffney inequality k Du k L ≤ C (cid:0) k du k L + k d ∗ u k L (cid:1) ;the latter inequality does not hold [16, remark 1; 75] (see also [38, 56, 57]).Theorem 1 also allows to obtain estimates when the classical Calderón–Zygmund theory fails: for every u ∈ C ∞ c ( R n ; R n ), if div u = 0, then [15,corollary 1 and remark 5; 16, theorem 2](1.6) k Du k L n/ ( n − ≤ C k ∆ u k L . Without the divergence-free condition, this estimate fails when n >
1, ascan be seen by taking u to be an approximation of the Green function ofLaplacian on R n . Even under divergence-free condition, the inequality k D u k L ≤ C k ∆ u k L does not hold [16, remark 1; 75] (see also [38, 56, 57]). The estimate (1.6)would be a consequence of the latter inequality combined with the Gagliardo–Nirenberg–Sobolev inequality 1.4. IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 5
If ˙W − k,p ( R n ) is the set of distributions which are k -th derivatives of L p functions for k ∈ N and p ∈ (1 , ∞ ), that is, the set of distributions f suchthat k f k ˙W − k,p = sup {h f, ϕ i : ϕ ∈ C ∞ c ( R n ) and k Dϕ k L p/ ( p − ≤ } < ∞ . the estimate of theorem 3 can be rewritten as k f k ˙W − ,n/ ( n − ≤ C k f k L . It is known that when n ≥
2, L ( R n ) ˙W − ,n/ ( n − ( R n ). J. Bourgainand H. Brezis have characterized by their divergence the vector fields f ∈ L ( R n ; R n ) that are in W − ,n/ ( n − ( R n ; R n ) [15, theorem 4 ′ ]. Theorem 4.
Let ℓ ∈ { , . . . , n − } . If f ∈ L ( R n ; V ℓ R n ) , then f ∈ ˙W − ,n/ ( n − ( R n ; V ℓ R n ) if and only if df ∈ ˙W − ,n/ ( n − ( R n ; V ℓ +1 R n ) . Moreover k f k ˙W − ,n/ ( n − ≤ C (cid:0) k f k L + k df k ˙W − ,n/ ( n − (cid:1) . Theorem 4 is equivalent to theorem 2 in the same way that theorem 3 isequivalent to theorem 1.Theorem 4 was used to obtain a generalized Korn type inequality in thederivation of a strain gradient theory for plasticity by homogenization ofdislocations [47].1.3.
Estimates for circulation integrals.
Theorems 1 and 3 are equiva-lent to the following geometrical inequality of J. Bourgain, H. Brezis and P.Mironescu [18, proposition 4].
Theorem 5. If Γ ⊂ R n is a closed rectifiable curve of length | Γ | and ϕ ∈ C ∞ c ( R n ; V R n ) , then (cid:12)(cid:12)(cid:12)Z Γ ϕ (cid:12)(cid:12)(cid:12) ≤ C | Γ |k Dϕ k L n . Here R Γ ϕ denotes the circulation integral of the form ϕ along the curveΓ. Again this estimate would be a consequence of the failing critical Sobolevembedding of W ,n ( R n ) into L ∞ ( R n ); it is a consequence of some compen-sation phenomenon that appears since the curve Γ is closed . When n = 2,theorem 5 is a direct consequence of the Green–Stokes integration formulaand of the classical isoperimetric inequality.Theorem 5 can be deduced from theorem 3 by applying the estimateto regularizations by convolution the divergence-free vector measure t H | Γ ,where H | Γ is the one-dimensional Hausdorff measure restricted to Γ and t is the unit tangent vector to Γ. Conversely, S. Smirnov has showed thatany divergence-free measure is the limit of convex combinations of measuresof the form t H | Γ , with a suitable control on the norms [85]; this allows todeduce theorem 3 for ℓ = n − ℓ < n − ℓ = n − JEAN VAN SCHAFTINGEN
Theorem 5 has been used to obtain L n/ ( n − bounds on minimizers of theGinzburg–Landau equation [11, proposition 5.1].The geometrical nature of the estimate of theorem 5 has raised the prob-lem of the value of optimal constants and whether they are achieved [26].Theorem 5 generalizes to surfaces [103]: if Σ is an ℓ –dimensional orientedsurface and ϕ ∈ C ∞ c ( R n ; V ℓ R n ), then (cid:12)(cid:12)(cid:12)Z Σ ϕ (cid:12)(cid:12)(cid:12) ≤ C H ℓ (Σ) k dϕ k L n . About the proofs
In this section we explain the proofs of the results presented above. First,J. Bourgain and H. Brezis have observed that the construction of the solution u in theorem 1 — and a fortiori in the stronger theorem 2 — cannot be linear [14, proposition 2]. Theorem 6.
Let ℓ ∈ { , . . . , n − } . There does not exist a linear op-erator K : L n ( R n ; V ℓ +1 R n ) → L ∞ ( R n ; V ℓ R n ) such that for every f ∈ L n ( R n ; V ℓ +1 R n ) , d ( K ( f )) = f . As the deduction of theorem 1 from theorem 3 above is based on thenonconstructive Hahn-Banach theorem on L , the corresponding map doesnot need be linear.Theorem 6 has a harmonic analysis proof and a geometric functionalanalysis proof [14]. The harmonic analysis proof begins by asssuming, by anaveraging argument, that K is a convolution operator and derives then a con-tradiction. The geometric functional analysis proof consists in noting that K ∗ ◦ d ∗ would be a factorization of the identity map from W , to L n/ ( n − through L and that such factorization is impossible by Grothendieck’s the-orem on absolutely summing operators [48; 111, theorem III.F.7].The main analytical tool in the proof of theorem 2 is an approximationlemma for functions in W ,n ( R n ) [15, theorem 6; 16, theorem 11 and (5.25)](see also [14, (5.2) and (5.3), (6.22)]). Theorem 7.
Let T ∈ L ( R n ; R p ) . If ker T = { } , then for every ε > ,there exists C ε such that for every v ∈ W ,n ( R n ) there exists u ∈ C ∞ ( R n ) that satisfies k T ( ∇ u − ∇ v ) k L n ≤ ε k Dv k L n , k∇ u k L n + k u k L ∞ ≤ C ε k Dv k L n . The proof of theorem 7 is constructive and based on a Littlewood–Paleydecomposition [16]. This approximation result has been extended to somesubscale of Triebel–Lizorkin spaces when rank T = 1 [20, proposition 3.1]and to classical Sobolev spaces in the noncommutative setting of homoge-neous groups [110, lemma 1.7].It would be interesting to find a simpler proof of theorem 7.We will now state a theorem that provides bounded solutions to overde-termined systems [16, theorem 10 and 10 ′ ] reformulated in the spirit ofmore recent works [109]. We introduce therefore the notion of adcancelingopetarors. IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 7
Definition 2.1.
A homogeneous differential operator T ( D ) from E to V is adcanceling if \ ξ ∈ R n \{ } T ( ξ ) ∗ [ V ] = { } . By a classical linear algebra argument, T ( D ) is adcanceling if and only ifspan (cid:16) [ ξ =0 ker T ( ξ ) (cid:17) = E that is, there exists a basis e , . . . , e k of E and vectors ξ , . . . , ξ k in R n suchthat for every i ∈ { , . . . , k } , L ( ξ i )[ e i ] = 0 [109, §6.2].We now state the theorem of J. Bourgain and H. Brezis that providesbounded solutions to overdetermined systems [16, theorem 10 and 10 ′ ]. Theorem 8.
Let Y be a Banach space and let S : W ,n ( R n ; E ) → Y bea bounded linear operator. If S has closed range and if there exists an ad-canceling first-order homogeneous differential operator T ( D ) such that forevery ϕ ∈ C ∞ c ( R n ; E ) k S ( ϕ ) k Y ≤ k T ( D ) ϕ k L n , then for every v ∈ ˙W ,n ( R n ; E ) there exists u ∈ ˙W ,n ( R n ; E ) ∩ L ∞ ( R n ; E ) such that Su = Sv and k u k L ∞ + k Du k L n ≤ C k Dv k L n . Let us first see how theorem 2 follows from theorem 8.
Proof of theorem 2 [16, proof of theorem 5] . If df = 0 and f ∈ V ℓ +1 R n ,there exists v ∈ ˙W ,n ( R n ; V ℓ R n ) such that dv = f . We are now goingto apply theorem 8. We take E = V ℓ R n and V = V ℓ +1 R and we define S = T ( D ) = d . We observe that S (cid:0) W ,n ( R n ; V ℓ R n ) (cid:1) = (cid:8) f ∈ L n ( R n ; V ℓ R n ) : df = 0 (cid:9) is closed in Y = L n ( R n ; V ℓ R n ) and thatspan (cid:16) [ ξ =0 ker T ( ξ ) (cid:17) = span (cid:8) α ∈ V ℓ R n : there exist ξ ∈ R n such that ξ ∧ α = 0 (cid:9) = E ;the latter equality holds since ℓ ≥ (cid:3) Theorem 2 can be used to prove theorem 4 in the spirit of our proof oftheorem 3 from theorem 1 in the previous section.We now explain the proof of theorem 8 from theorem 7.
Proof of theorem 8 [16, proof of theorem 11] . Since S has closed range, bythe open mapping theorem (see for example [22, theorem 2.6], there exists w ∈ ˙W ,n ( R n ; E ) such that Sv = Sw and k Dw k L n ≤ C k Sv k Y . JEAN VAN SCHAFTINGEN
By theorem 7 and by definition 2.1, for every ε > C ε > w ∈ ˙W ,n ( R n ; E ), there exists u ∈ C ∞ ( R n ; E ) satisfying k T ( D )( w − u ) k L n ≤ ε k Dw k L n , (2.1) k Dw k L n + k w k L ∞ ≤ C ε k Dw k L n (2.2)In particular, by our assumption on T ( D ), k Sv − Su k Y = k S ( w − u ) k Y ≤ k T ( D )( w − u ) k L n ≤ ε k Dw k L n ≤ Cε k Sv k Y . If we now choose ε = C , we have k Sv − Su k Y ≤ k Sv k Y , k Du k L n + k u k L ∞ ≤ C ε C k Sv k Y . By an iterative argument as in the classical proof of the open mappingtheorem, see for example [22, proof of theorem 2.6], we can thus construct u i ∈ ˙W ,n ( R n ; E ) ∩ L ∞ ( R n ; E ) such that k Sv − Su i +1 k Y ≤ k Sv − Su i k Y , k Du i +1 − Du i k L n + k u i +1 − u i k L ∞ ≤ C ε C k Sv − Su i k Y ;this sequences converges to the desired solution. (cid:3) This solutions constructed by this iterative argument in the spirit of theclassical proof of the closed graph theorem have been studied as hierarchicalsolutions [94].The strategy of proof outlined here above relies essentially on theorem 7,which does not have yet an elementary proof. However, the weaker theorem 3has a short proof [104] (see also [63, proof of lemma 1; 72, proof of proposition2]).
Direct proof of theorem 3.
Without loss of generality, we assume that ℓ = n − ϕ ( x ) = ϕ n ( x ) dx n . For every t ∈ R , if we define ϕ tn ( y, z ) = ϕ n ( y, t ), we have for every t ∈ R the immediate bound (cid:12)(cid:12)(cid:12)Z R n − ×{ t } f ∧ ϕ n (cid:12)(cid:12)(cid:12) ≤ (cid:16)Z R n − | f | (cid:17) k ϕ tn k L ∞ . On the other hand, by the Stokes–Cartan formula, since df = 0, (cid:12)(cid:12)(cid:12)Z R n − ×{ t } f ∧ ϕ n (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z R n − × ( −∞ ,t ) f ∧ dϕ tn (cid:12)(cid:12)(cid:12) ≤ (cid:16)Z R n − × ( −∞ ,t ) | f | (cid:17) k Dϕ tn k L ∞ ≤ (cid:16)Z R n | f | (cid:17) k Dϕ tn k L ∞ By a straightforward interpolation argument, this implies that for every α ∈ (0 , (cid:12)(cid:12)(cid:12)Z R n − ×{ t } f ∧ ϕ n (cid:12)(cid:12)(cid:12) ≤ C (cid:16)Z R n − × ( −∞ ,t ) | f | (cid:17) α (cid:16)Z R n − | f | (cid:17) − α | ϕ tn | C ,α , IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 9 where the Hölder seminorm is defined by | ψ | C ,α = sup x,y ∈ R n − | ψ ( x ) − ψ ( y ) || x − y | α . In particular, if α = n , we have by the Morrey–Sobolev embedding on R n − (see for example [2, lemma 4.28; 22, theorem 9.12; 69, theorem 1.4.5 (f)]) (cid:12)(cid:12)(cid:12)Z R n − ×{ t } f ∧ ϕ n (cid:12)(cid:12)(cid:12) ≤ C ′ (cid:16)Z R n | f | (cid:17) n (cid:16)Z R n − ×{ t } | f | (cid:17) − n (cid:16)Z R n − ×{ t } | Dϕ n | n (cid:17) n . The conclusion follows by Hölder’s inequality. (cid:3)
This strategy of proof goes back to the elementary proof of the estimateon circulation integrals of theorem 5 [103]. The idea of working with hyper-planes and concluding with Hölder’s inequality is reminiscent of the originalproof of the Gagliardo–Nirenberg–Sobolev inequality [46; 74, p. 125] (seealso [22, theorem 9.9]).The two main properties of the Sobolev space ˙ W ,n that are used in thisargument are a Morrey-type embedding in a space of Hölder continuousfunctions and a Fubini-type property. The latter property is satisfied byfractional Sobolev spaces W s,p [92, 93] and by Triebel–Lizorkin spaces F s,pq [55; 82, theorem 2.3.4/2; 100, theorem 2.5.13] allowing to adapt the proofin that setting [15, remark 1; 16, remark 11; 104, remark 5; 108, proof ofproposition 2.1], but is not satisfied by the Sobolev–Lorentz spaces W , ( p,q ) if q > p [59] or by the Besov space B s,pq if q = p [101, theorem 4.4].The proof has also been adapted by constructing ϕ tn more carefully thanby a mere extension to estimates under higher-order conditions [27, lemma 2.4;105; 107], and to homogeneous groups [34].3. Variations
Boundary estimates.
The results presented above were all concernedabout the entire Euclidean space R n . They also have counterparts on thetorus T n .On a domain with a boundary, it is not clear a priori which boundaryconditions are admissible in this theory. The problem was first settled fortheorem 2 on the cube [16, theorems 5 ′ and 5 ′′ ] and then on a domain witha smooth boundary [25, lemma 4.4]. Theorem 9.
Let Ω ⊂ R n be a smooth domain and let ≤ k ≤ n − . Forevery v ∈ W ,n (Ω; V k R n ) , there exist u ∈ W ,n (Ω; V k − R n ) ∩ C (Ω; V k R n ) and w ∈ W ,n (Ω; V k − R n ) such that u = v + dw. satisfying k u k W ,n + k u k L ∞ + k w k W ,n ≤ C k v k W ,n . Moreover, if t v = 0 on ∂ Ω , then u = 0 and w = 0 on ∂ Ω and if u = 0 on ∂ Ω , then Dw = 0 on ∂ Ω . Here t v denotes the tangential component on ∂ Ω of the form v [84,(1.2.25)]. The inequalities on the boundary are interpreted in the senseof traces. The presentation of theorem 9 differs from that of theorem 2. In the caseof a domain with nontrivial topology writing u = v + dw is stronger than du = dv ; the latter statement was however more convenient to state theweaker theorem 1.If Ω is a cube, the proof of theorem 9 relies on the counterpart of theorem 8on a cube which is based on the counterpart of theorem 7 on a cube [16,corollary 15]. The surjective of the trace and of normal derivative operator[65] allows to obtain u = 0 on ∂ Ω (see also [2, theorem 5.19; 14, proof oftheorem 2]). A partition of the unity allows to pass to a general smoothdomain [25, lemma 3.3].Similarly, the counterpart of theorem 3 is
Theorem 10.
Let Ω ⊂ R n be a smooth domain and let ℓ ∈ { , . . . , n − } .There exists C > such that for every f ∈ C ∞ ( ¯Ω; V ℓ R n ) and every ϕ ∈ C ∞ ( ¯Ω; V n − ℓ R n ) , if df = 0 and either t f = 0 or t ϕ = 0 , then (cid:12)(cid:12)(cid:12)Z R n f ∧ ϕ (cid:12)(cid:12)(cid:12) ≤ C k f k L (cid:0) k dϕ k L n + k ϕ k L n (cid:1) . Theorem 10 is a consequence of the counterparts of theorem 4 where thequantity k df k W − ,n/ ( n − is replaced by a suitable dual quantity [25, lemmas3.11 and 3.16]: Theorem 11.
Let Ω ⊂ R n be a smooth domain and let ℓ ∈ { , . . . , n − } .There exists C > such that for every f ∈ C ∞ ( ¯Ω; V ℓ R n ) and every ϕ ∈ C ∞ ( ¯Ω; V n − ℓ R n ) , then (cid:12)(cid:12)(cid:12)Z Ω f ∧ ϕ (cid:12)(cid:12)(cid:12) ≤ C (cid:16) k f k L + sup n(cid:12)(cid:12)(cid:12)Z Ω f ∧ dζ (cid:12)(cid:12)(cid:12) : ζ ∈ C ∞ ( ¯Ω) and k D ζ k L n ≤ o(cid:17) k Dϕ k L n . If moreover t ϕ = 0 on ∂ Ω , then (cid:12)(cid:12)(cid:12)Z Ω f ∧ ϕ (cid:12)(cid:12)(cid:12) ≤ C (cid:16) k f k L + sup n(cid:12)(cid:12)(cid:12)Z Ω f ∧ dζ (cid:12)(cid:12)(cid:12) : ζ ∈ C ∞ ( ¯Ω) , t ζ = 0 on ∂ Ω and k D ζ k L n ≤ o(cid:17) k Dϕ k L n . Theorem 10 can also be deduced from theorem 3: first theorem 10 isproved on a half-space by a reflection argument (see also [7–9]), then it isextended to a general domain by local charts and partition of the unity[25, remark 3.3] (see also [112]).3.2.
Other Sobolev spaces.
Besides the Sobolev space ˙W ,n ( R n ) , thereare other spaces that just miss the embedding in L ∞ ( R n ) : the Sobolevspaces ˙W k,n/k ( R n ) for k < n , the Sobolev–Lorentz spaces ˙W k,n/k,q ( R n ) ,the fractional Sobolev–Slobodetski˘ı spaces ˙W s,n/s ( R n ) , the Besov spaces ˙B s,n/sq ( R n ) and the Triebel–Lizorkin spaces ˙F s,n/sq ( R n ) for s > and q ≥ .Let us first observe that in theorems 3 and 5, the ˙W ,n norm can bereplaced by any stronger norm; the same is true for theorem 1 with any spacethat is embedded in L n . The other cases are not covered straightforwardly.An inspection of the proof of theorem 3 shows that the main ingredients arean embedding on hyperplanes into Hölder-continuous functions and a Fubini IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 11 type theorem. The latter property for Triebel–Lizorkin spaces [12, théorème2; 82, theorem 2.3.4/2; 100, Theorem 2.5.13] allows to extend theorem 3[15, remark 1; 16, Remark 11; 104, remark 5; 108, proposition 2.1].
Theorem 12.
Let ℓ ∈ { , . . . , n − } , s > and q > . There exists C > such that for every f ∈ C ∞ c ( R n ; V ℓ R n ) and every ϕ ∈ C ∞ c ( R n ; V n − ℓ R n ) ,if df = 0 , then (cid:12)(cid:12)(cid:12)Z R n f ∧ ϕ (cid:12)(cid:12)(cid:12) ≤ C k f k L k ϕ k ˙F s,n/sq . The result can then be extended by classical embedding theorems toSobolev–Lorentz spaces W ,n,q ( R n ) and Besov spaces ˙B s,pq ( R n ) with q < ∞ [72; 106, remark 4.2; 108]. It is not known whether the results can be ex-tended to the case q = ∞ . Open Problem 3.1 (Critical estimate in Besov spaces [108, open problem1; 109, open problem 8.2]) . Does there exist a constant
C > such that forevery f ∈ C ∞ c ( R n ; V ℓ R n ) and every ϕ ∈ C ∞ c ( R n ; V n − ℓ R n ) , if df = 0 , then (cid:12)(cid:12)(cid:12)Z R n f ∧ ϕ (cid:12)(cid:12)(cid:12) ≤ C k f k L k ϕ k ˙B s,n/s ∞ ? Open Problem 3.2 (Critical estimate in Sobolev–Lorentz spaces [16, openproblem 1; 108, open problem 2; 109, open problem 8.3]) . Is there a constant
C > such that for every f ∈ C ∞ c ( R n ; V ℓ R n ) and every differential form ϕ ∈ C ∞ c ( R n ; V n − ℓ R n ) , if df = 0 , then (cid:12)(cid:12)(cid:12)Z R n f ∧ ϕ (cid:12)(cid:12)(cid:12) ≤ C k f k L k Dϕ k L n, ∞ ? When ℓ = 1 , by the embeddings of the Sobolev space ˙W , ( R n ) into theBesov space B s,n/ ( n +1 − s )1 ( R n ) [58, corollary 1] and into the Lorentz space L nn − , ( R n ) [5, 98], the inequalities holds [108].A positive answer to open problem 1 or open problem 2 would imply somelimiting Sobolev type inequalities into L ∞ which have been proved since [21;71, proposition 3; 109, p. 911].The extension of theorems 2 and 4 to the fractional case is more delicate.Theorem 2 has been extended when ℓ = n − to some scale of Triebel–Lizorkin spaces [20]. Theorem 13.
Let s ∈ ( , n ] and q ∈ [2 , n/s ] . If g ∈ F s − ,n/sq ( R n ; V n R n ) and dg = 0 in the sense of distributions, then there exists u ∈ L ∞ ( R n ; V n − R n ) ∩ F s,n/sq ( R n ; V n − R n ) , such that u is continuous and du = g in the sense ofdistributions. Moreover, k u k L ∞ + k u k F s,n/sq ≤ C k g k F s − ,n/sq . When s = n and q = 2 , the result goes back to V. Maz ′ ya [67] (see also[70, 71]).3.3. Hardy inequalities.
Another question is whether the Sobolev inequal-ity (1.5) has a corresponding Hardy inequality. A positive answer has been given by V. Maz ′ ya [68] (see also [19, 21]):(3.1) Z R n | u ( x ) || x | d x ≤ C Z R n | du | + | d ∗ u | . H. Castro, J. Dávila and Wang Hui have obtained another family of Hardyinequalities with cancellation phenomena [31–33]: Z R n − × R + (cid:12)(cid:12)(cid:12) D (cid:16) u ( y ) y n (cid:17)(cid:12)(cid:12)(cid:12) dy ≤ C Z R n − × R + | D u | ; their work is concerned in boundary singularities in the potential whereaswe are concerned with point singularities.3.4. Larger classes of operators.
The Korn–Sobolev inequality of M. J.Strauss [91](3.2) k u k L n/ ( n − ≤ C k Du + ( Du ) ∗ k is a variant of the Gagliardo–Nirenberg–Sobolev inequality that plays a rolein the study of maps of bounded deformation [6, 99]. The components ofthe deformation tensor Eu = ( Du + ( Du ) ∗ ) / do not satisfy any first-orderdifferential condition, so that theorem 3 cannot be applied. However, thetensor field Eu satisfies the Saint-Venant compatibility conditions: ∂ k ∂ l ( Eu ) ij + ∂ i ∂ j ( Eu ) kl = ∂ k ∂ j ( Eu ) il + ∂ i ∂ l ( Eu ) s kj . In view of this H. Brezis has suggested that theorem 3 should hold when thederivative is replaced by higher-order conditions. This was proved for a classof second-order conditions [105] (yielding an alternative proof to the Korn–Sobolev inequality (3.2) [16, corollary 26; 105]) and for a class of higher-orderconditions [16, corollary 14]. The differential operators for which theorems 3and 4 hold have been characterized [109, theorem 1.4, proposition 2.1 andtheorem 9.2].
Theorem 14.
Let 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 ) , (cid:12)(cid:12)(cid:12)Z R n f · ϕ (cid:12)(cid:12)(cid:12) ≤ C k f k L k Dϕ k L n , (ii) 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, k f k ˙W − ,n/ ( n − ≤ C (cid:0) k f k L + k L ( D ) f k ˙W − − k,n/ ( n − (cid:1) , (iii) for every f ∈ L ( R n ; E ) such that L ( D ) f = 0 Z R n f = 0 , (iv) for every e ∈ E , if L ( D ) ( δ e ) = 0 , then e = 0 ,(v) L ( D ) is cocanceling. The cocancellation condition is a new condition that was introduced inorder to solve this problem [109, definition 1.3].
IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 13
Definition 3.1.
Let L ( D ) be a homogeneous linear differential operator on R n from E to F . The operator L ( D ) is cocanceling if \ ξ ∈ R n \{ } ker L ( ξ ) = { } . Theorem 14 implies that the elements of the kernel of a differential opera-tor acting vector measures from on R n define linear functionals on ˙W ,n ( R n ) if and only if the kernel does not contain any Dirac measure. In particular,if such a measure does not charge points, it does not charge sets of null W ,n –capacity.The main analytical difficulty in theorem 14 is proving (ii). When E = R it can be proved by theorem 8; as the latter deals naturally only with first-order differential operators, this requires decomposing in a suitable fashionhigher-order differential operators [107, proof of theorem 8]. The weaker(i) can be obtained directly when E = R following the lines of the proofof theorem 3 [107, proof of theorem 5]; this argument has been adapted tothe fractional case [107, (4)]. The passage to the vector case is done by analgebraic construction [109, lemma 2.5].The inequality (i) appears with L ( D ) = div in the homogenization ofstiff heterogeneous plates [28, lemma 15].An interesting consequence is the characterization of the operators suchthat a Gagliardo–Nirenberg–Sobolev inequality or a Hardy inequality hold[21; 109, theorem 1.3 and proposition 6.1]. Theorem 15.
Let A ( D ) be an elliptic homogeneous linear differential oper-ator of order k on R n from V to E . The following conditions are equivalent(i) for every u ∈ C ∞ c ( R n ; V ) , k D k − u k L nn − ≤ C k A ( D ) u k L , (ii) for every u ∈ C ∞ c ( R n ; V ) , Z R n | D k − u ( x ) || x | d x ≤ C k A ( D ) u k L , (iii) for every u ∈ C ∞ ( R n ; V ) , if supp A ( D ) u is compact, then Z R n A ( D ) u = 0 , (iv) A ( D ) is canceling. In (iii) it is essential to consider vector fields u that do not have compactsupport and do not tend to too fast at infinity. The ellipticity conditionis the classical notion of ellipticity for overdetermined differential operators[49, theorem 1; 88, definition 1.7.1] (when dim V = 1 , see also S. Agmon[3, §7; 4, definition 6.3]). Definition 3.2.
A homogeneous linear differential 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 cancellation is a new condition dual to cocancellation which wasintroduced to characterize such operators [109, definition 1.2]. Definition 3.3.
A homogeneous linear differential operator A ( D ) on R n from V to E is canceling if \ ξ ∈ R n \{ } A ( ξ )[ V ] = { } . Theorem 15 gives in particular the Hodge–Sobolev inequality (3.2) and theKorn–Sobolev inequality (1.5) as the corresponding differential operators arecanceling [109, propositions 6.4 and 6.6]. Further examples of higher-ordercanceling operators which are a higher-order analogue of the Hodge complexwere recently given [62].The proof of (i) and (ii) are based on the construction of a differential op-erator such that for every ξ ∈ R n \ { } , ker L ( ξ ) = A ( ξ )[ V ] [109, proposition4.2]. The latter is a combination of a classical commutative algebra result ofL. Ehrenpreis [41; 60, theorem 2; 88, theorem 1.5.5] which states that everysubmodule of the module of differential operators is finitely generated, andthe application of ellipticity.These estimate generalize to fractional Sobolev spaces and Sobolev Lorentzspace as the Hodge estimates above. In particular, since the derivative on R n is canceling if and only n ≥ , we recover that the inequality [18, lemma D.1;36, theorem 1.4; 58, theorem 4; 83, proposition 4; 87, theorem 2; 109, corol-lary 8.2] k u k ˙W s,n/ ( n − (1 − s )) ≤ C k Du k L holds for every u ∈ C ∞ c ( R n ) if and only if n ≥ .If A ( D ) is elliptic and canceling, then [21] k D k − n u k L ∞ ≤ C k A ( D ) u k L . The cancellation condition is not necessary [21; 109, remark 5.1]. Indeed,the derivative D on R is not canceling but the inequality k u k L ∞ ≤ k u ′ k holds nevertheless.The ellipticity assumption in theorem 15 is necessary for the first-orderSobolev inequality of (i) [109, theorem 1.3 and corollary 5.2] and for Hardy–Sobolev inequalities [21]; it is not necessary for Hardy inequalities of theform (ii) [21] nor for higher-order Sobolev inequalities [109, proposition 5.4].For the Hodge operator, A ( D ) = ( δ, d ) , theorem 11 implies that if eitherthe tangential or the normal component vanishes on the boundary (either t u = 0 or n u = 0 on ∂ R n + ), then k u k L n/ ( n − ≤ C (cid:0) k du k + k d ∗ u k (cid:1) . It would be interesting to define a class of canceling boundary conditionsthat ensure the validity of Sobolev estimates on half-spaces.
Open Problem 3.3. If A ( D ) is an elliptic canceling operator, under whatboundary conditions on ∂ R n + = R n − × { } does the inequality k D k − u k L nn − ( R n + ) ≤ C k A ( D ) u k L ( R n + ) hold?It would be natural to investigate boundary conditions that satisfy theLopatinski˘ı–Shapiro ellipticity condition (also known as coerciveness or cov-ering conditions) [50, definition 20.1.1; 66; 86]. IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 15
The reader will have noted that we did not state in this section anycounterpart of theorem 1 and 2.
Open Problem 3.4.
State necessary and sufficient conditions on K ( D ) such that for every v ∈ ˙W ,n ( R n ; E ) , there exists u ∈ ˙W ,n ( R n ; E ) suchthat K ( D ) u = K ( D ) v and k Du k L n + k u k L ∞ ≤ C k Dv k L n . Noncommutative situations.
A nilpotent homogeneous group G isa connected and simply connected Lie group whose Lie algebra g of left-invariant vector fields is graded, nilpotent and stratified:(a) g = g ⊕ g ⊕ · · · ⊕ g p ,(b) [ g i , g j ] ⊂ g i + j for i + j ≤ p and [ V i , V j ] = { } if i + j > p ,(c) g generates g by Lie brackets.These spaces are a good framework for defining homogeneous Sobolev spaces[44], Hardy spaces [45] and studying singular integrals [90, theorem XII.4].The homogeneous dimension Q = P pj =1 jm j plays an important role inproperties of G . In particular, the following Sobolev embedding holds for p < Q for the nonisotropic Sobolev space N ˙W ,p ( G ) [43, 44, 77]: N ˙W ,p ( G ) = { u ∈ L p ( G ) : D b u ∈ L p ( G ) } ⊂ L QpQ − p ( G ) , where the horizontal derivative D b u is the pointwise restriction of Du tothe horizontal bundle T b G = g . As these embeddings are counterparts ofthe classical Sobolev embedding in the Euclidean space, theorem 3 has acounterpart on homogeneous groups [34, theorem 1]. Theorem 16. If f ∈ C ∞ c ( G ; T b G ) is a section and for every ψ ∈ C ∞ c ( G ) , Z G h D b ϕ, f i = 0 , and if ϕ ∈ C ∞ c ( G ; T ∗ b G ) is a section, then (cid:12)(cid:12)(cid:12)Z G h ϕ, f i (cid:12)(cid:12)(cid:12) ≤ C k f k L ( G ) k∇ b ϕ k L Q ( G ) . Theorem 16 is proved following the strategy of the proof of theorem 3;the main point is to split with Jerison’s machinery for analysis on nilpotenthomogeneous groups [54] a function on a normal subgroup into a functionwhich is controlled in L ∞ and another which is the restriction of functionwith a horizontal derivative controlled in L ∞ [34, lemma 2.1]. The proofalso gives fractional estimates [34, theorem 4]. Theorem 16 gives Gagliardo–Nirenberg–Sobolev inequalities for (0 , q ) forms in the ¯ ∂ b complex of classesof CR manifolds [113, theorems 2 and 3] and for involutive structures [51].Following the ideas of [107], theorem 16 has a higher–order analogue inwhich f is a symmetric tensor-field and the condition is replaced by a higherorder condition [34, theorem 5 and lemma 5.3]. Theorem 17. If f ∈ C ∞ c ( G ; ⊗ k T b G ) is a section and for every ψ ∈ C ∞ c ( G ) , Z G h D kb ϕ, f i = 0 , and if ϕ ∈ C ∞ c ( G ; Sym k ( T b G )) is a section, then (cid:12)(cid:12)(cid:12)Z G h ϕ, f i (cid:12)(cid:12)(cid:12) ≤ C k f k L ( G ) k∇ b ϕ k L Q ( G ) . Here
Sym k ( T b G ) denotes the bundle of symmetric k -linear maps on thehorizontal bundle. It is not known whether the symmetry assumption isnecessary for this inequality to hold [34, open problem 1]. This result wasused to obtain the Gagliardo–Nirenberg inequality for forms in the Rumincomplex [78–81] for forms on the Heisenberg groups H and H [10]. Sincethe Rumin complex contains higher-order differential operators, the higher-order estimates play a crucial role in the proof. The method would requireheavy explicit computations of complexes to be generalized to higher-orderHeisenberg groups.The counterparts of the stronger theorems 2 and 4 have been obtainedby Yi Wang and Po-Lam Yung [110] following the strategy of Bourgain andBrezis [16].The study of canceling and cocanceling operators on noncommutativehomogeneous groups remains widely open.4. Coda: characterizing functions satisfying the estimates
The relationship with bounded mean oscillation.
The resultsabove can all be thought as substitutes for the failing embedding of ˙W ,n ( R n ) into L ∞ ( R n ) when n ≥ . A classical substitute for this embedding isthe embedding of ˙W ,n ( R n ) into the space of functions of bounded meanoscillations BMO( R n ) [24, example 1].H. Brezis has suggested investigating the relationship between this embed-ding and theorem 3 by studying for ℓ ∈ { , . . . , n − } the space D ℓ ( R n ) ofmeasurable functions ϕ : R n → R for which the semi-norm | ϕ | D ℓ = sup n(cid:12)(cid:12)(cid:12)Z R n ϕf ∧ e (cid:12)(cid:12)(cid:12) : f ∈ C ∞ c (cid:0) R n ; V ℓ R n (cid:1) , e ∈ V n − ℓ R n and Z R n | e || f | ≤ o is finite [106, definition 2.4]. Theorem 3 is equivalent with the embedding ˙W ,n ( R n ) ⊂ D n − ( R n ) . By an observation of F. Bethuel, G. Orlandi and D. Smets, this embeddingis strict [11, remark 5.4].These new spaces form a monotone family between the classical spaces L ∞ ( R n ) and BMO( R n ) [106, proposition 2.9, theorem 3.1, proposition 4.6,proposition 5.1, theorem 5.3], L ∞ ( R n ) ( D n − ( R n ) ( · · · ( D ( R n ) ( BMO( R n ) . The embeddings are strict since log (cid:16) ℓ X i =1 | x i | (cid:17) ∈ D k ( R n ) if and only if ℓ ≤ n − k [106, proposition 4.6]. It can be also noticed that VMO( R n ) is not a subset of D ( R n ) [106, proposition 5.1]. IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 17
The conclusion of this study is that the estimates of theorem 3 are strongerthan the embedding of ˙W ,n ( R n ) into BMO( R n ) .4.2. Strong charges.
It is possible to characterize the distributions F suchthat there exists f ∈ C ( R n ; V n − R n ) such that df = F [40, theorem 4.8;76, chapter 11]. Theorem 18.
Let F : R n → V n R n be a distribution. There exists f ∈ C ( R n ; V n − R n ) such that F = df if and only if F is a strong charge. Strong charges were introduced to solve this problem [40; 76, chapter 10].
Definition 4.1.
The distribution F : R n → V n R n is a strong charge if forevery ε > and every R > , there exists C > such that if ϕ ∈ C ∞ c ( B R ) , (cid:12)(cid:12)(cid:12)Z R n F ∧ ϕ (cid:12)(cid:12)(cid:12) ≤ C k ϕ k L + ε k dϕ k L . In particular, functions in L n ( R n ) define strong charges [40, proposition2.9]. Theorem 18 provides thus an alternate proof of theorem 1. The caseof ℓ forms with ℓ ∈ { , . . . , n − } has also been studied [39]. References [1] L. Abatangelo and S. Terracini,
Solutions to nonlinear Schrödinger equations withsingular electromagnetic potential and critical exponent , J. Fixed Point Theory Appl. (2011), no. 1, 147–180.[2] R. A. Adams and J. J. F. Fournier, Sobolev spaces , 2nd ed., Pure and AppliedMathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003.[3] S. Agmon,
The L p approach to the Dirichlet problem. I. Regularity theorems , Ann.Scuola Norm. Sup. Pisa (3) (1959), 405–448.[4] , Lectures on elliptic boundary value problems , Van Nostrand MathematicalStudies, No. 2, Van Nostrand, Princeton, N.J. – Toronto – London, 1965.[5] A. Alvino,
Sulla diseguaglianza di Sobolev in spazi di Lorentz , Boll. Un. Mat. Ital. A(5) (1977), no. 1, 148–156.[6] L. Ambrosio, A. Coscia, and G. Dal Maso, Fine properties of functions with boundeddeformation , Arch. Rational Mech. Anal. (1997), no. 3, 201–238.[7] C. Amrouche and H. H. Nguyen,
New estimates for the div, curl, grad operators andelliptic problems with L -data in the half-space , Appl. Math. Lett. (2011), no. 5,697–702.[8] , New estimates for the div-curl-grad operators and elliptic problems with L -data in the whole space and in the half-space , J. Differential Equations (2011),no. 7, 3150–3195.[9] , Elliptic problems with L -data in the half-space , Discrete Contin. Dyn. Syst.Ser. S (2012), no. 3, 369–397.[10] A. Baldi and B. Franchi, Sharp a priori estimates for div-curl systems in Heisenberggroups , J. Funct. Anal. (2013), no. 10, 2388–2419.[11] F. Bethuel, G. Orlandi, and D. Smets,
Approximations with vorticity bounds for theGinzburg-Landau functional , Commun. Contemp. Math. (2004), no. 5, 803–832.[12] G. Bourdaud, Calcul fonctionnel dans certains espaces de Lizorkin–Triebel , Arch.Math. (Basel) (1995), no. 1, 42–47.[13] J. Bourgain and H. Brezis, Sur l’équation div u = f , C. R. Math. Acad. Sci. Paris (2002), no. 11, 973–976.[14] , On the equation div Y = f and application to control of phases , J. Amer.Math. Soc. (2003), no. 2, 393–426.[15] , New estimates for the Laplacian, the div-curl, and related Hodge systems ,C. R. Math. Acad. Sci. Paris (2004), no. 7, 539–543. [16] ,
New estimates for elliptic equations and Hodge type systems , J. Eur. Math.Soc. (JEMS) (2007), no. 2, 277–315.[17] J. Bourgain, H. Brezis, and P. Mironescu, Lifting in Sobolev spaces , J. Anal. Math. (2000), 37–86.[18] , H / maps with values into the circle: minimal connections, lifting, and theGinzburg-Landau equation , Publ. Math. Inst. Hautes Études Sci. (2004), 1–115.[19] P. Bousquet and P. Mironescu, An elementary proof of an inequality of Maz ′ ya involv-ing L vector fields , Nonlinear Elliptic Partial Differential Equations (D. Bonheure,M. Cuesta, E. J. Lami Dozo, P. Takáč, J. Van Schaftingen, and M. Willem, eds.),Contemporary Mathematics, vol. 540, American Mathematical Society, Providence,R. I., 2011, pp. 59-63.[20] P. Bousquet, P. Mironescu, and E. Russ, A limiting case for the divergence equation ,Math. Z. (2013), no. 1-2, 427–460.[21] P. Bousquet and J. Van Schaftingen,
Hardy-Sobolev inequalities for vector fields andcanceling linear differential operators , available at .[22] H. Brezis,
Functional analysis, Sobolev spaces and partial differential equations , Uni-versitext, Springer, New York, 2011.[23] H. Brezis and H.-M. Nguyen,
The Jacobian determinant revisited , Invent. Math. (2011), no. 1, 17–54.[24] H. Brezis and L. Nirenberg,
Degree theory and BMO . I:
Compact manifolds withoutboundaries , Selecta Math. (N.S.) (1995), no. 2, 197–263.[25] H. Brezis and J. Van Schaftingen, Boundary estimates for elliptic systems with L -data , Calc. Var. Partial Differential Equations (2007), no. 3, 369–388.[26] , Circulation integrals and critical Sobolev spaces: problems of optimal con-stants , Perspectives in Partial Differential Equations, Harmonic Analysis and Appli-cations (D. Mitrea and M. Mitrea, eds.), Proc. Sympos. Pure Math., vol. 79, Amer.Math. Soc., Providence, RI, 2008, pp. 33–47.[27] M. Briane and J. Casado-Díaz,
Estimate of the pressure when its gradient is thedivergence of a measure. Applications , ESAIM Control Optim. Calc. Var. (2011),no. 4, 1066–1087.[28] , Homogenization of stiff plates and two-dimensional high-viscosity Stokesequations , Arch. Ration. Mech. Anal. (2012), no. 3, 753–794.[29] M. Briane, J. Casado-Díaz, and F. Murat,
The div-curl lemma “trente ans après”: anextension and an application to the G -convergence of unbounded monotone operators ,J. Math. Pures Appl. (9) (2009), no. 5, 476–494.[30] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals , ActaMath. (1952), 85–139.[31] H. Castro and H. Wang, A Hardy type inequality for W m, (0 , functions , Calc. Var.Partial Differential Equations (2010), no. 3-4, 525–531.[32] H. Castro, J. Dávila, and H. Wang, A Hardy type inequality for W , (Ω) functions ,C. R. Math. Acad. Sci. Paris (2011), no. 13-14, 765–767.[33] , A Hardy type inequality for W m, (Ω) functions , J. Eur. Math. Soc. (JEMS) (2013), no. 1, 145–155.[34] S. Chanillo and J. Van Schaftingen, Subelliptic Bourgain-Brezis estimates on groups ,Math. Res. Lett. (2009), no. 3, 487–501.[35] S. Chanillo and P.-L. Yung, An improved Strichartz estimate for systems with diver-gence free data , Comm. Partial Differential Equations (2012), no. 2, 225–233.[36] A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore, Harmonic analysis of thespace BV , Rev. Mat. Iberoamericana (2003), no. 1, 235–263.[37] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness andHardy spaces , J. Math. Pures Appl. (9) (1993), no. 3, 247–286.[38] S. Conti, D. Faraco, and F. Maggi, A new approach to counterexamples to L esti-mates: Korn’s inequality, geometric rigidity, and regularity for gradients of separatelyconvex functions , Arch. Ration. Mech. Anal. (2005), no. 2, 287–300.[39] T. De Pauw, L. Moonens, and W. F. Pfeffer, Charges in middle dimensions , J. Math.Pures Appl. (9) (2009), no. 1, 86–112. IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 19 [40] T. De Pauw and W. F. Pfeffer,
Distributions for which div v = F has a continuoussolution , Comm. Pure Appl. Math. (2008), no. 2, 230–260.[41] L. Ehrenpreis, A fundamental principle for systems of linear differential equationswith constant coefficients, and some of its applications , Proc. Internat. Sympos. Lin-ear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem, 1961, pp. 161–174.[42] C. Fefferman and E. M. Stein, H p spaces of several variables , Acta Math. (1972),no. 3-4, 137–193.[43] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups , Ark.Mat. (1975), no. 2, 161–207.[44] G. B. Folland and E. M. Stein, Estimates for the ¯ ∂ b complex and analysis on theHeisenberg group , Comm. Pure Appl. Math. (1974), 429–522.[45] , Hardy spaces on homogeneous groups , Mathematical Notes, vol. 28, Prince-ton University Press, Princeton, N.J., 1982.[46] E. Gagliardo,
Proprietà di alcune classi di funzioni in più variabili , Ricerche Mat. (1958), 102–137.[47] A. Garroni, G. Leoni, and M. Ponsiglione, Gradient theory for plasticity via ho-mogenization of discrete dislocations , J. Eur. Math. Soc. (JEMS) (2010), no. 5,1231–1266.[48] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques ,Bol. Soc. Mat. São Paulo (1953), 1–79.[49] L. Hörmander, Differentiability properties of solutions of systems of differential equa-tions , Ark. Mat. (1958), 527–535.[50] , The analysis of linear partial differential operators . III:
Pseudodifferen-tial operators , Grundlehren der Mathematischen Wissenschaften, vol. 274, Springer,Berlin, 1985.[51] J. Hounie and T. Picon,
Local Gagliardo-Nirenberg estimates for elliptic systems ofvector fields , Math. Res. Lett. (2011), no. 4, 791–804.[52] T. Isobe, Topological and analytical properties of Sobolev bundles . II:
Higher dimen-sional cases , Rev. Mat. Iberoam. (2010), no. 3, 729–798.[53] T. Iwaniec, Integrability Theory, and the Jacobians , Lecture Notes, Universität Bonn,1995.[54] D. Jerison,
The Poincaré inequality for vector fields satisfying Hörmander’s condi-tion , Duke Math. J. (1986), no. 2, 503–523.[55] G. A. Kaljabin, Descriptions of functions from classes of Besov-Lizorkin-Triebeltype , Studies in the theory of differentiable functions of several variables and itsapplications, VIII, Trudy Mat. Inst. Steklov., vol. 156, 1980, pp. 82–109 (Russian).[56] B. Kirchheim and J. Kristensen,
Automatic convexity of rank–1 convex functions , C.R. Math. Acad. Sci. Paris (2011), no. 7–8, 407–409.[57] ,
On rank one convex functions that are homogeneous of degree one . in prepa-ration.[58] V. I. Kolyada,
On the embedding of Sobolev spaces , Mat. Zametki (1993), no. 3, 48–71, 158 (Russian); English transl., Math. Notes (1993), no. 3-4, 908–922 (1994).[59] , On Fubini type property in Lorentz spaces , Recent Advances in HarmonicAnalysis and Applications (D. Bilyk, L. De Carli, A. Petukhov, A. M. Stokolos,and B. D. Wick, eds.), Springer proceedings in mathematics & statistics, vol. 25,Springer, 2013, pp. 171-179.[60] H. Komatsu,
Resolutions by hyperfunctions of sheaves of solutions of differentialequations with constant coefficients , Math. Ann. (1968), 77–86.[61] L. Lanzani,
Higher order analogues of exterior derivative , Bull. Inst. Math. Acad.Sin. (N.S.) (2013), no. 3, 389–398.[62] L. Lanzani and A. S. Raich, On div-curl for higher order , Advances in Analysis:The Legacy of Elias M. Stein (C. Fefferman, A. D. Ionescu, D. H. Phong, and S.Wainger, eds.), Princeton Mathematical Series, Princeton University Press, Prince-ton, 2014, pp. 245-272.[63] L. Lanzani and E. M. Stein,
A note on div curl inequalities , Math. Res. Lett. (2005), no. 1, 57–61. [64] E. H. Lieb and M. Loss, Analysis , 2nd ed., Graduate Studies in Mathematics, vol. 14,American Mathematical Society, Providence, RI, 2001.[65] J.-L. Lions and E. Magenes,
Problemi ai limiti non omogenei. III , Ann. Scuola Norm.Sup. Pisa (3) (1961), 41–103.[66] Ya. B. Lopatinski˘ı, On a method of reducing boundary problems for a system ofdifferential equations of elliptic type to regular integral equations , Ukrain. Mat. Ž. (1953), 123–151 (Russian); English transl., Thirteen Papers on Differential Equa-tions, Amer. Math. Soc. Transl. (2) (1970), 149-183.[67] V. Maz ′ ya, Bourgain–Brezis type inequality with explicit constants , Interpolationtheory and applications (L. De Carli and M. Milman, eds.), Contemp. Math., vol. 445,Amer. Math. Soc., Providence, RI, 2007, pp. 247–252.[68] ,
Estimates for differential operators of vector analysis involving L -norm , J.Eur. Math. Soc. (JEMS) (2010), no. 1, 221–240.[69] , Sobolev spaces with applications to elliptic partial differential equations , 2nded., Grundlehren der Mathematischen Wissenschaften, vol. 342, Springer, Heidel-berg, 2011.[70] V. Maz ′ ya and T. Shaposhnikova, A collection of sharp dilation invariant integralinequalities for differentiable functions , Sobolev spaces in mathematics. I (V. Maz ′ yaand V. Isakov, eds.), Int. Math. Ser. (N. Y.), vol. 8, Springer, New York, 2009,pp. 223–247.[71] P. Mironescu, On some inequalities of Bourgain, Brezis, Maz ′ ya, and Shaposhnikovarelated to L vector fields , C. R. Math. Acad. Sci. Paris (2010), no. 9–10, 513–515.[72] I. Mitrea and M. Mitrea, A remark on the regularity of the div-curl system , Proc.Amer. Math. Soc. (2009), no. 5, 1729–1733.[73] F. Murat,
Compacité par compensation , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1978), no. 3, 489–507.[74] L. Nirenberg, On elliptic partial differential equations , Ann. Scuola Norm. Sup. Pisa(3) (1959), 115–162.[75] D. Ornstein, A non-equality for differential operators in the L norm , Arch. RationalMech. Anal. (1962), 40–49.[76] W. F. Pfeffer, The divergence theorem and sets of finite perimeter , Pure and AppliedMathematics, CRC Press, Boca Raton, Flor., 2012.[77] L. P. Rothschild and E. M. Stein,
Hypoelliptic differential operators and nilpotentgroups , Acta Math. (1976), no. 3-4, 247–320.[78] M. Rumin,
Formes différentielles sur les variétés de contact , J. Differential Geom. (1994), no. 2, 281–330.[79] , Differential geometry on C-C spaces and application to the Novikov-Shubinnumbers of nilpotent Lie groups , C. R. Acad. Sci. Paris Sér. I Math. (1999),no. 11, 985–990.[80] ,
Sub-Riemannian limit of the differential form spectrum of contact manifolds ,Geom. Funct. Anal. (2000), no. 2, 407–452.[81] , Around heat decay on forms and relations of nilpotent Lie groups , Sémin.Théor. Spectr. Géom., vol. 19, Univ. Grenoble I, Saint, 2001, pp. 123–164.[82] T. Runst and W. Sickel,
Sobolev spaces of fractional order, Nemytskij operators, andnonlinear partial differential equations , de Gruyter Series in Nonlinear Analysis andApplications, vol. 3, de Gruyter, Berlin, 1996.[83] B. J. Schmitt and M. Winkler,
On embeddings between BV and ˙W s,p , Preprint no. 6,Lehrstuhl I für Mathematik, RWTH Aachen, Mar. 15, 2000.[84] G. Schwarz, Hodge decomposition—a method for solving boundary value problems ,Lecture Notes in Mathematics, vol. 1607, Springer, Berlin, 1995.[85] S. K. Smirnov,
Decomposition of solenoidal vector charges into elementary solenoids,and the structure of normal one-dimensional flows , Algebra i Analiz (1993), no. 4,206–238; English transl., St. Petersburg Math. J. (1994), no. 4, 841–867.[86] V. A. Solonnikov, Overdetermined elliptic boundary value problems , Zap. Naučn.Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) (1971), 112–158 (Russian);English transl., J. Sov. Math. (1973), no. 4, 477-512. IMITING BOURGAIN-BREZIS ESTIMATES FOR SYSTEMS 21 [87] ,
Inequalities for functions of the classes ˙W ~mp ( R n ), Zapiski Nauchnykh Sem-inarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. SteklovaAkademii Nauk SSSR (1972), 194-210 (Russian); English transl., J. Sov. Math. (1975), 549-564.[88] D. C. Spencer, Overdetermined systems of linear partial differential equations , Bull.Amer. Math. Soc. (1969), 179–239.[89] E. M. Stein, Singular integrals and differentiability properties of functions , PrincetonMathematical Series, vol. 30, Princeton University Press, Princeton, N.J., 1970.[90] ,
Harmonic analysis: real-variable methods, orthogonality, and oscillatory in-tegrals , Princeton Mathematical Series, vol. 43, Princeton University Press, Prince-ton, NJ, 1993.[91] M. J. Strauss,
Variations of Korn’s and Sobolev’s equalities , Partial differential equa-tions (Univ. California, Berkeley, Calif., 1971) (D. C. Spencer, ed.), Proc. Sympos.Pure Math., vol. 23, Amer. Math. Soc., Providence, R.I., 1973, pp. 207–214.[92] R. S. Strichartz,
Multipliers on fractional Sobolev spaces , J. Math. Mech. (1967),1031–1060.[93] , Fubini-type theorems , Ann. Scuola Norm. Sup. Pisa (3) (1968), 399–408.[94] E. Tadmor, Hierarchical solutions for linear equations: a constructive proof of theclosed range theorem , available at arXiv:1003.1525 .[95] L. Tartar,
Une nouvelle méthode de résolution d’équations aux dérivées partiellesnon linéaires , Journées d’Analyse Non Linéaire (Besançon, 1977), Lecture Notes inMath., vol. 665, Springer, Berlin, 1978, pp. 228–241.[96] ,
Compensated compactness and applications to partial differential equations ,Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes inMath., vol. 39, Pitman, Boston, Mass., 1979, pp. 136–212.[97] ,
Homogénéisation et compacité par compensation , Séminaire Goulaouic-Schwartz (1978/1979), École Polytech., Palaiseau, 1979, pp. Exp. No. 9, 9.[98] ,
Imbedding theorems of Sobolev spaces into Lorentz spaces , Boll. Unione Mat.Ital. Sez. B Artic. Ric. Mat. (8) (1998), no. 3, 479–500.[99] R. Temam and G. Strang, Functions of bounded deformation , Arch. Rational Mech.Anal. (1980/81), no. 1, 7–21.[100] H. Triebel, Theory of function spaces , Monographs in Mathematics, vol. 78,Birkhäuser, Basel, 1983.[101] ,
The structure of functions , Monographs in Mathematics, vol. 97, Birkhäuser,Basel, 2001.[102] A. Uchiyama,
A constructive proof of the Fefferman-Stein decomposition of BMO ( R n ), Acta Math. (1982), 215–241.[103] J. Van Schaftingen, A simple proof of an inequality of Bourgain, Brezis andMironescu , C. R. Math. Acad. Sci. Paris (2004), no. 1, 23–26.[104] ,
Estimates for L -vector fields , C. R. Math. Acad. Sci. Paris (2004),no. 3, 181–186.[105] , Estimates for L vector fields with a second order condition , Acad. Roy.Belg. Bull. Cl. Sci. (6) (2004), no. 1-6, 103–112.[106] , Function spaces between BMO and critical Sobolev spaces , J. Funct. Anal. (2006), no. 2, 490–516.[107] ,
Estimates for L vector fields under higher-order differential conditions , J.Eur. Math. Soc. (JEMS) (2008), no. 4, 867–882.[108] , Limiting fractional and Lorentz space estimates of differential forms , Proc.Amer. Math. Soc. (2010), no. 1, 235–240.[109] ,
Limiting Sobolev inequalities for vector fields and canceling linear differen-tial operators , J. Eur. Math. Soc. (JEMS) (2013), no. 3, 877–921.[110] Y. Wang and P.-L. Yung, A subelliptic Bourgain-Brezis inequality . to appear in J.Eur. Math. Soc. (JEMS).[111] P. Wojtaszczyk,
Banach spaces for analysts , Cambridge Studies in Advanced Math-ematics, vol. 25, Cambridge University Press, Cambridge, 1991.[112] X. Xiang, L / -estimates of vector fields with L curl in a bounded domain , Calc.Var. Partial Differential Equations (2013), no. 1-2, 55–74. [113] P.-L. Yung, Sobolev inequalities for (0 , q ) forms on CR manifolds of finite type , Math.Res. Lett. (2010), no. 1, 177–196. Université catholique de Louvain, Institut de Recherche en Mathématiqueet Physique (IRMP), Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
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