Potentials for $\mathcal{A}$-quasiconvexity

We show that each constant rank operator $\mathcal{A}$ admits an exact potential $\mathbb{B}$ in frequency space. We use this fact to show that the notion of $\mathcal{A}$-quasiconvexity can be tested against compactly supported fields. We also show that $\mathcal{A}$-free Young measures are generated by sequences $\mathbb{B}u_j$, modulo shifts by the barycentre.


Introduction
A challenging question in the study of non-linear partial differential differential equations is to find which non-linear functionals are well-behaved with respect to weak convergence, which represents the typical topology consistent with physical measurements and has satisfactory compactness properties. In the context of the Calculus of Variations, answering this question amounts, roughly speaking, to describing semi-continuity properties of functionals (1) with respect to weak convergence in certain weakly closed, convex subsets C, say, of L p -spaces, 1 < p < ∞), under growth conditions 0 f c(| · | p + 1) on the integrands f . Such subsets C can account for differential constraints and boundary conditions. Modulo terms removed for simplicity of exposition, such functionals could model, for instance, the energy arising from the deformation of a solid body Ω, viewed as a sufficiently regular open subset of R n , where f is a continuous energy density map characterized by the constitutive properties of the material. In accordance with the Direct Method in the Calculus of Variations, imposing a suitable bound from below on f ensures existence and weak compactness of minimizing sequences w j . The appropriate continuity property of E in this case is that of lower semi-continuity with respect to weak convergence in L p which, if satisfied, implies existence of a minimizer w ∈ C.
It is well-known that if C consists of the whole of L p , then E is weakly sequentially lower semi-continuous if and only if f satisfying (2) is convex. Of course, convexity of f is sufficient for lower semi-continuity (always understood as weakly sequential throughout this note) in any reasonable class C, but it is hardly necessary in general. For instance, if C is the space of weak gradients in L 2 and f is a quadratic form, then one can easily show that f being positive on rank-one matrices implies lower semi-continuity. This example, that we will later come back to in more generality, is of particular relevance, as it provides the insight for a second convexity condition, which is necessary for lower semi-continuity with the constraint w = ∇u: if E is lower semi-continuous, then f is convex along rank-one lines. In particular, for integrands f of class C 2 , this is equivalent to the so-called Legendre-Hadamard ellipticity condition where summation over repeated indices is adopted. From this point of view, lower semi-continuity of E acting on gradients reflects a semi-convexity condition on f . Indeed, it was shown by Morrey in [23] that lower semi-continuity of E is equivalent with quasiconvexity of f , i.e., the Jensen-type inequality holds for all η and all smooth maps u with compact support in the open cube Q.
On one hand, the quasiconvexity assumption is a plausible constitutive relation for energy functionals arising in solid mechanics [5]; on the other hand, it is but a minor improvement of the lower semi-continuity concept, which makes it particularly difficult to check in applications. The counterexample of Šverák [33] rules out the possibility of quasiconvexity being a type of directional convexity (see also [7,Ex. 3.5] for the case of higher order gradients). A tractable sufficient condition for quasiconvexity is polyconvexity, i.e., f is a convex functions of the minors, also introduced by Morrey in [23] in connection with lower semi-continuity and used by Ball to obtain existence theorems under very mild growth conditions, giving very satisfactory existence results in non-linear elasticity [4]. The fact that quasiconvexity does not imply polyconvexity is much easier to see, at least in higher dimensions, and follows from an old observation of Terpstra concerning quadratic forms [38] (see also [6,2] and the references therein). The above considerations show that a considerable amount of work was devoted to the treatment of lower semi-continuity in the case when C consists of gradients (see [1,20] and the monographs [14,28]). However, for instance in continuum mechanics, it is often the case that C consists of those L p -fields w that satisfy a linear, typically under-determined, partial differential constraint, say Aw = 0, assumption that we make henceforth. Examples arise in elasticity, plasticity, elasto-plasticity, electromagnetism, and others. The A-free framework originates in the pioneering work of Murat and Tartar in compensated compactness [24,34,35] and can be correlated with the question of finding energy functionals that are continuous with respect to weak convergence in C [25]. The latter question was also studied in generality by Ball, Currie, and Olver in [7], leading to the generalization of polyconvexity to the case where energy functionals depend on higher order derivatives. In this case, the definition of quasiconvexity extends mutatis mutandis [21]. As to the question of lower semi-continuity, the analysis of the case when f is a quadratic form (see, e.g., [36,Ch. 17] for all η and all bounded w such that´Q w = 0 and Aw = 0. However, it is not clear whether this condition is necessary. More recently, Fonseca and Müller showed in [16] that if one assumes in addition that the fields w are periodic, in which case f is called A-quasiconvex, then one indeed obtains a necessary and sufficient condition 1 (under suitable growth assumptions on f ). Their result holds under the assumption that the symbol map A(·) of A is a constant rank matrixvalued field away from 0. This condition, introduced in [31, Def. 1.5] to prove coerciveness inequalities for non-elliptic systems, was first used in the context of compensated compactness by Murat and ensures, as noted on [24, p.502], the continuity of the map making tools from pseudo-differential calculus available. In the absence of the constant rank assumption, little is known about the lower semi-continuity problem.
One of the few results in this direction was proved by Müller in [26], answering a long standing question of Tartar (see also [19] for a generalization).
In the proof of the main result of [16], considerable difficulty is encountered when proving sufficiency of A-quasiconvexity. One reason for this is the absence of potential functions for A, which, if available, should allow one to test with compactly supported functions in the definition of A-quasiconvexity and, perhaps, use more standard methods.
The main result of the present work is to show that the existence of such a potential in Fourier space is equivalent with the constant rank condition. Theorem 1. Let A be a linear, homogeneous differential operator with constant coefficients on R n . Then A has constant rank if and only if there exists a linear, homogeneous differential operator B with constant coefficients on R n such that ker A(ξ) = im B(ξ) (4) for all ξ ∈ R n \ {0}.
Here A(·), B(·) denote the (tensor-valued) symbol maps of, respectively, A, B. We say that A has constant rank if the map 0 = ξ → rank A(ξ) is constant (see Section 2 for detailed notation). We will regard B as the potential and A as the annihilator, although this terminology is not standard.
It is important to mention that the algebraic relation (4) does not, in general, imply for vector fields w that To see this, simply take A = ∇ k . In turn, if we impose restrictions on w that allow for usage of the Fourier transform, (5) can be shown to hold (Lemma 5). As a consequence, standard arguments in the Calculus of Variations lead to the fact that a map f is A-quasiconvex if and only if For comparison, see also Seregin's work [32] in incompressible linearized elasticity, where the methods used to project on solenoidal fields do not require Fourier analysis.
for all η and all smooth vector fields u supported in an open cube Q (Corollary 6). It is also the case that under the constant rank condition, the notions of A-quasiconvexity [16, Def. 3.1] and Dacorogna's A-B-quasconvexity [12, Eq. (A.12)] coincide. In particular, one can define A-quasiconvexity via integration over arbitrary domains (Lemma 8). As a consequence, the lower semi-continuity properties of functionals (1) in the topologies considered in [16,3], which are natural from the point of view of compensated compactness theory, rely only on the structure of B.
In fact, we will show that the A-quasiconvex relaxation of a continuous integrand can be described in terms of B only. From this point of view, it is natural to investigate the Young measures generated by sequences satisfying differential constraints [16,Sec. 4], as they efficiently describe the minimization of energies that are not lower semi-continuous. We recall that the role of parametrized measures for non-convex problems in the Calculus of Variations was first recognized by Young in the pioneering works [40,41,42]. See the monographs [27,28] for a modern, detailed exposition.
Roughly speaking, for 1 < p < ∞, we consider a sequence w j converging weakly in L p which is asymptotically A-free and generates a Young measure ν. Technically speaking, it suffices to take Aw j to be strongly compact in W −k,p loc , where k is the order of A. This is (slightly more general than) the topology considered in [16, Rk. 4.2(i)] and is consistent with the topology considered in compensated compactness (see, e.g., [36,Thm. 17.3], which essentially deals with the case of linear Euler-Lagrange equations). In this setting, we will show that the Young measure ν is generated by a sequence of smooth maps Bu j , modulo a shift by the barycentre.
To sum up, under the constant rank condition on the annihilator A, the objects characterizing the lower semi-continuous relaxation of functionals defined on Afree vector fields (i.e., A-quasiconvex envelopes and A-free Young measures) can be described only in terms of the potential B constructed in Theorem 1. From this point of view, it is the author's opinion that the study of functionals is essentially dual (strictly under the constant rank condition). See also [13] and the Appendix of [12].
Since testing with the appropriate quantity is fundamental in the study of partial differential equations, we hope that the observations made in this work will increase the flexibility of analyzing functionals in either class described above. On the other hand, the functional F seems better suited for incorporating boundary conditions. This will be pursued elsewhere.
This paper is organized as follows: In Section 2 we prove the main Theorem 1, in Section 3 we prove that A-quasiconvexity can be tested with compactly supported fields w = Bu (Corollary 6), and in Section 4 we prove that A-free Young measures are shifts of Young measures generated by sequences Bu j .
Acknowledgement. The author is grateful to Jan Kristensen for introducing him to the problem and for offering insightful comments and helpful suggestions. This work was supported by Engineering and Physical Sciences Research Council Award EP/L015811/1. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 757254 (SINGULARITY).

Proof of Theorem 1
We take a moment to clarify notation. By a k-homogeneous, linear differential operator A on R n from W to X we mean where A α ∈ Lin(W, X) for all multi-indices α such that |α| = k, for finite dimensional inner product spaces W, X. We also define the (Fourier) symbol map We also recall the condition mentioned above that A is of constant rank if there exists a natural number r such that As to the resolution of Theorem 1, we recall the notion of (Moore-Penrose) generalized inverse, introduced independently in [22,8,29], to which we refer plainly as the pseudo-inverse, although the terminology is not standard. For a matrix M ∈ R N ×m , its pseudo-inverse M † is the unique m × N matrix defined by the relations where M * denotes the adjoint (transpose) of M . Equivalently, the pseudo-inverse is determined by the geometric property that M M † and M † M are orthogonal projections onto im M and (ker M ) ⊥ respectively. We refer the reader to the monograph [10] for more detail on generalized inverses.
With these considerations in mind, it is easy to see that the projection map P ∈ C ∞ (R n \ {0}, Lin(W, W )) defined in (3) can be represented as The smoothness of P is well-known [16,Prop. 2.7]; for a proof using pseudoinverses see [30,Sec. 4]. By the basic properties of pseudo-inverses, it is easy to see that, with the choice B = P, we have that (4) holds; however, the tensorvalued map P is 0-homogeneous, hence not polynomial in general. In particular, P cannot define a differential operator.
On the other hand, motivated by a similar construction in [39, Rk. 4.1], one can speculate that P and, in fact, A † (·) are rational functions. This is indeed the case, as a consequence of the main result of Decell in [15], building on the fundamental result of Penrose [29, Thm. 2] and the Cayley-Hamilton Theorem.
Theorem 2 (Decell [15,Thm. 3]). Let M ∈ R N ×m and denote by Proof of Theorem 1 (sufficiency). Suppose that A has constant rank. We put M := A(ξ) in the above Theorem for ξ ∈ R n \ {0}, and abbreviate H(ξ) := A(ξ)A * (ξ). The first, perhaps most crucial, observation is that r(ξ), as defined by (8), equals the number of non-zero eigen-values of M M * , which equals the number of singular values of M . This is, in turn, equal to rank M , which is independent of ξ by the constant rank assumption on A. Therefore, if r(ξ) = r = 0, we have that A(ξ) = 0 N ×m , A † (ξ) = 0 m×N , so we can simply choose B(ξ) = Id W , which satisfies (4) and gives rise to a linear, 0-homogeneous differential operator. Otherwise, if r(ξ) = r > 0, we obtain It is easy to see that H(·) is a tensor-valued polynomial in ξ. The scalar fields a j , j = 1 . . . r, are such that a j (ξ) is a coefficient of the characteristic polynomial of H(ξ), hence a linear combination of minors. In particular, a j are scalar-valued polynomials in ξ.
It then follows that, with P as in (3), defines a tensor-valued polynomial that satisfies (4). In particular, (9) gives rise to a linear differential operator. To check that it is homogeneous, it suffices to see that a r (·) is a linear combination of minors of the same order of H(·), which is homogeneous since A(·) is.
The necessity of the constant rank condition in Theorem 1 follows from the following Lemma and the Rank-Nullity Theorem.
Lemma 3. Let S ⊂ R n be a set of positive Lebesgue measure and P, Q be two matrix-valued polynomials on R n . Suppose that there exists s such that rank P (ξ) + rank Q(ξ) = s for ξ ∈ S.
Then both P and Q have constant rank in S.
Proof. We abbreviate R P := rank P , R Q := rank Q and assume for contradiction that R P is not constant in S. Say R P (S) = {r 1 , r 1 + 1 . . . , r 2 } for natural numbers r 1 < r 2 . We also write M d for the map that has input a matrix and returns (a vector of) all its minors of order d. In particular, M d P , M d Q are vector-valued polynomials on R n . We then have that On the other hand, which is Lebesgue-null by the same argument. Since It is natural to ask the reversed question, whether a constant rank operator B admits an exact annihilator A. This is indeed the case, as can be shown by a simple modification of the argument above: Remark 4. Let B be a linear, homogeneous, differential operator of constant rank on R n from V to W . Then, we can choose M := B(ξ) for ξ ∈ R n \ {0} in Theorem 2, so that for ξ ∈ R n satisfies (4) and gives rise to a differential operator. In particular, the formula is consistent with [39,Eq. (4.3)]. This fact can be used to extend the L 1 -estimates in [39,9] to constant rank operators.
We conclude the discussion of algebraic properties with two remarks: Firstly, it is quite convenient that the two constructions presented are explicitly computable. On the other hand, performing the computations on simple examples, e.g., involving only div, grad, curl, one easily notices that the operators constructed via our formulas are often overcomplicated. Perhaps more computationally efficient methods, e.g., in the spirit of [39, Sec. 4.2] can be developed.

A-quasiconvexity
The relevance of Theorem 1 for analysis can be seen, for instance, from the fact that periodic A-free fields have differential structure: Lemma 5. Let A, B be linear, homogeneous, differential operators of constant rank with constant coefficients on R n from W to X, and from V to W , respectively. Assume that (4) holds. Then for all w ∈ C ∞ (T n , W ) such that Aw = 0 and Tn w(x) dx = 0, there exists u ∈ C ∞ (T n , V ) such that w = Bu. Similarly, for all w ∈ S (R n , W ) such that Aw = 0, there exists u ∈ S (R n , V ) such that w = Bu.
Here T n denotes the n-dimensional torus, identified in an obvious way with (a quotient of) [0, 1] n . The Fourier transform is defined aŝ for ξ ∈ Z n and u ∈ C ∞ (T n ). Also, S (R n ) denotes the Schwartz class of rapidly decreasing functions on R n , where the Fourier transform is defined also by (10), with the amendment that the integral is taken over R n .
Proof. Let w ∈ C ∞ (T n , W ) have zero average and satisfy Aw = 0, so that for x ∈ T n , where the coefficientsŵ(ξ) ∈ ker A(ξ) decay faster than any polynomial as |ξ| → ∞. We define u(x) := ξ∈Z n \{0} B † (ξ)ŵ(ξ) e 2π i x·ξ , for x ∈ T n , which is smooth by homogeneity of B † (·): say B has order l, then B † (·) is (−l)-homogeneous. We can thus differentiate the sum term by term to obtain where the exactness relation (4) is used in the second equality, along with the geometric properties of the pseudo-inverse. The proof of the first case is complete.
We give an analogous argument for the case when w ∈ S (R n , W ) is A-free. We have the pointwise relation A(ξ)ŵ(ξ) = 0, so that (4) implies that w ∈ im B(ξ) and we can defineû which satisfies the required properties.
We conclude this Section by showing that one can test with compactly supported smooth maps in the definition of A-quasiconvexity. Corollary 6. Let A, B be as in Lemma 5 and f : W → R be Borel measurable and locally bounded. Then are equal for all η ∈ W . Moreover, if B has order l and α ∈ [0, 1), we have for any η ∈ W and ε > 0.
The proof follows standard arguments; in particular we follow [14,Prop. 5.13] and [18,Thm. 4.2] and include the proof for completeness of the present work.
Proof. It is obvious that Q A f Q B f . To prove the opposite inequality, let ε > 0, η ∈ W , and w be a periodic field as in the definition of Q A f (η). We will construct v ∈ C ∞ c ((0, 1) n , V ) such that By Lemma 5, we have that w = Bu for a periodic field u ∈ C ∞ (T n , V ). Say, as before, that B has order l and define u N (x) := N −l u(N x) for N sufficiently large. This does not change the value of the integral over the cube. Next, let δ > 0 be sufficiently small and truncate to obtain u δ N : . . l and some constant C > 0. We impose δN ≥ 1 and leave δ to be determined. It follows, for c 1 ≥ 1 depending on B only, that which implies (12) with v := u δ N . To prove the equality of the two envelopes, we distinguish two cases: so that we can conclude by (12) that To prove (11), we need only show that the infimum is smaller than the envelope. Firstly, note as above that by replacing u with u N (x) = N −l u(N x), where u is extended by periodicity to R n , the value of the integral does not change. It suffices to choose N large enough so that u N has small C l−1,α -norm. Note that for j = 0 . . . l − 1 we have which can clearly be made arbitrarily small.
Finally, to check the Hölder bound, say that {z i + [0, N −1 ] n } N n i=1 is a covering of [0, 1] n by cubes of side-length N −1 that can only touch at their boundaries and let x, y ∈ [0, 1] n . If x, y lie in the same cube z i + [0, N −1 ] n , we have that which can be made small since 1 − α > 0. If x, y lie in different cubes, which we label Q x , Q y . Letx ∈ ∂Q x ∩(x, y),ȳ ∈ ∂Q y ∩(x, y), so that |x−y| ≥ |x−x|+|y−ȳ|, |x −x|, |y −ȳ| √ nN −1 , and all derivatives of u N vanish nearx,ȳ. Using these facts and the previous step we get where the last inequality follows by concavity and monotonicity of 0 t → t α . The proof is complete. Remark 7. Using the argument in Corollary 6, one can show for constant rank operators A that A-quasiconvexity, as defined by Fonseca and Müller in [16,Def. 3.1], coincides with A-B-quasiconvexity, as introduced by Dacorogna in [12,13] (to be precise, in the original definition of A-B-quasiconvexity, the operator B is assumed to be of first order, but this is only a minor technical restriction). In this case, it is not difficult to prove that [13,Thm. 4] is essentially unconditional. A proof of this fact will be given elsewhere.
We also have that A-quasiconvexity can be defined by integrals over arbitrary domains, instead of cubes. Proof. Fix η ∈ W , v ∈ C ∞ c (Ω, V ), extended by zero to R n . By the argument in the proof of Corollary 6, we write C := (0, 1) n and have that . For sufficiently small ε > 0, we can find x 0 ∈ R n such that Rearranging the terms we obtain the conclusion.

A-free Young measures
We recall the definition of oscillation Young measures, while also giving a simplified variant of the Fundamental Theorem of Young measures.
Theorem 9 (FTYM, [27,28]). Let Ω ⊂ R n be a bounded, open set and z j ∈ L 1 (Ω, R d ) be a bounded sequence in L 1 . Then there exists a subsequence (not relabeled) and a weakly-* measurable map ν : Ω → P(R d ) (or parametrized measure if and only if the sequence f ( ·, z j ) is uniformly integrable.
Above, P(R d ) denotes the space of probability measures on R d . In the notation of Theorem 9, we say that z j generates the Young measure ν (in symbols, z j Y → ν). We also recall that a sequence z j is said to be uniformly integrable if and only if for all ε > 0, there exists δ > 0 such that for all borel sets E ⊂ Ω, we have that If |z j | p is uniformly integrable, we say that z j is p-uniformly integrable. Lemma 10 ([16,Prop. 2.4]). Let z j generate a Young measure ν andz j →z in measure. Then z j +z j generates the Young measure µ given by µ x = ν x ⋆ δz (x) for L n a.e. x, i.e., The following is an extension of [16,Lem. 2.15]. The first two steps of the present proof are almost a repetition of their arguments, which we include since the original proof only covers first order annihilators A.
Proposition 11. Let A, B be as in Lemma 5 and have orders k, l, respectively, Ω ⊂ R n be a bounded Lipschitz domain, and 1 < p < ∞. Let w j , w ∈ L p (Ω, W ) be such that Then there exists a sequence u j ∈ C ∞ c (Ω, V ) such that Bu j ⇀ 0 in L p (Ω, W ), Moreover, u j can be chosen such that (Bu j ) j is p-uniformly integrable.
A Young measure ν satisfying the assumptions of Proposition 11 is said to be an A-free Young measure.
Proof. By Lemma 10 and linearity we can assume that w = 0. We will identify maps defined on Ω with their extensions by zero to full-space without mention. Uniform integrability considerations strictly refer to sequences defined on Ω.
Recall the truncation operators, defined for α > 0 by which are clearly Carathéodory integrands. By Theorem 9, we have that so that we can choose a diagonal subsequence α j ↑ ∞ such that´Ω |τ α j w j | p dx equals the p-th moment of ν. It also follows from Theorem 9 that (τ α j w j ) j is p-uniformly integrable.
We now show that τ α j w j generates ν. Since w j converges weakly in L p (Ω, W ), it converges weakly in L 1 , hence is uniformly integrable, so that τ α j w j − w j → 0 in measure. It also follows by elementary manipulations that τ α j w j − w j ⇀ 0 in L p , so that, indeed, τ α j w j generates ν by Lemma 10.
Let 1 < q < p. We have that so that Aτ α j w j → 0 in W −k,q loc (Ω, X). We also record that τ α j w j is precompact in We can therefore choose a sequence of cut-off functions ρ j ∈ C ∞ c (Ω, [0, 1]) such that ρ j ↑ 1 in Ω and ρ j Aτ α j w j W −k,q (R n ,X) → 0 and where B m are fixed bi-linear pairings given by the Leibniz rule. To see that this is possible, consider Ω j := {x ∈ Ω : dist(x, ∂Ω) < j}, where s j ↓ 0 will be determined. We require that ρ j = 1 in Ω\Ω s j , ρ j = 0 in Ω 2s j and |D m ρ j | cs −m j , m = 1, . . . , k. It is easy to see that the sum above is controlled in W −k,q by so that it suffices to choose any s j ≥ max m=1,...,k D k−m τ α j w j 1/(2m) W −k,q ↓ 0 as j → ∞. Alternatively, one can consider a different cut-off sequence ρ i ↑ 1 and employ a diagonalization argument.
We definew where η ε(j) denotes a standard sequence of (radial, positive) mollifiers and ε(j) ↓ 0 is such thatw j ∈ C ∞ c (Ω, W ) and, therefore, Aw j → 0 in W −k,q (R n , X). The latter inequality follows since, for all ϕ ∈ C ∞ c (R n , W ) with ϕ W k,q 1, It is also clear that w j − τ α j w j L p → 0, so thatw j is p-uniformly integrable, converges weakly to 0 in L p , and generates ν.
Step II. We projectw j on the kernel of A in R n and show that Pw j are puniformly integrable in Ω, converge weakly to zero in L p , and generate ν. Here the L 2 -orthogonal projection operator P is given by the multiplier in (7), Pw(ξ) := P(ξ)ŵ(ξ) = [Id W −A † (ξ)A(ξ)]ŵ(ξ) for w ∈ S (R n , W ).
Since the symbol P(·) is homogeneous of degree zero, P is a singular integral operator of convolution type; in particular P maps Schwartz functions to Schwartz functions. Moreover, we have that F (w j − Pw j ) (ξ) = B † (ξ)B(ξ)Fw j (ξ) = A † ξ |ξ| Aw j (ξ) |ξ| k , so that, by boundedness of singular integrals on L q w j − Pw j L q (R n ,W ) c F −1 Aw j | · | k L q (R n ,X) = c Aw j W −k,q (R n ,X) → 0.
It immediately follows by Lemma 10 that Pw j generates ν. To see that Pw j ⇀ 0 in L p (Ω, W ), we note that, since P is (pointwisely) self-adjoint, we have, for any g ∈ L p/(p−1) (Ω, W ),ˆΩ g, Pw j dx =ˆΩ Pg,w j dx → 0, since Pg ∈ L p/(p−1) (Ω, W ) by boundedness of singular integrals. To see that Pw j is p-uniformly integrable, we use the idea in [16, Lem. 2.14.(iv)]. We first note, by boundedness of P on L p , that sup j Pw j − Pτ αwj L p (R n ,W ) c sup j w j − τ αwj L p (R n ,W ) → 0 as α → ∞ by p-uniform integrability ofw j . Note that for each fixed α, Pτ αwj is bounded in L r for any p < r < ∞, hence is p-uniformly integrable. Let ε > 0. We choose α > 0 such that sup j Pw j − Pτ αwj L p (R n ,W ) < ε and also choose δ > 0 such that for each Borel set E ⊂ Ω with L n (Ω) < δ, we have that´E |Pτ αwj | p dx < ε for all j. It follows that for all such E, E |Pw j | p dx 2 p−1 sup jˆE |Pw j − Pτ αwj | p dx + sup jˆE |Pτ αwj | p dx < (2ε) p , where the right hand side is independent of j. The second step is concluded.
Since B has order l, we first note that D l u(ξ) = B † (ξ) Bu(ξ) ⊗ ξ ⊗l , so that Bu → D l u is a singular integral operator of convolution type. It follows that D l u j is bounded in L p (R n ) (recall here that Bu j = Pw j is bounded in L p as w j ∈ C ∞ c (Ω, W ) is a weakly convergent sequence), so u j is bounded in W l,p (Ω, V ). By compactness of the embedding W l,p (Ω) ֒→ W l−1,p (Ω), we have that u j → u in W l−1,p (Ω, V ). Since Bu j ⇀ 0, we have that Bu = 0. On the other hand, u = F −1 [B † (·)] ⋆ (Bu) = 0, so that D l−m u j → 0 in L p (Ω) for m = 1, . . . , l.
We now proceed similarly to Step I. Let ρ ∈ C ∞ c (R n ) be such that ρ j = 1 in Ω \ Ω s j and |D m ρ j | cs which tends to zero by p-uniform integrability of Bu j and the choice of s j . Here B m is another collection of bi-linear pairings given by the product rule. It then follows that B(ρ j u j ) converges weakly to zero in L p (Ω, W ), is p-uniformly integrable, and generates ν. The proof is complete.