L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle L^{\infty }$$\end{document}-truncation of closed differential forms

In this paper, we prove that for each closed differential form u∈L1(RN;(RN)∗∧...∧(RN)∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle u \in L^1(\mathbb {R}^N;(\mathbb {R}^N)^{*} \wedge ... \wedge (\mathbb {R}^N)^{*})$$\end{document}, which is almost in L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle L^{\infty }$$\end{document} in the sense that ∫{y∈RN:|u(y)|≥L}|u(y)|dy0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle L>0$$\end{document} and a small ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \varepsilon >0$$\end{document}, we may find a closed differential form v, such that ‖u-v‖L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \Vert u - v \Vert _{L^1}$$\end{document} is again small, and v is, in addition, in L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle L^{\infty }$$\end{document} with a bound on its L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle L^{\infty }$$\end{document} norm depending only on N and L. In particular, the set {v≠u}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \{ v \ne u\}$$\end{document} has measure at most CL-1ε.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle CL^{-1} \varepsilon .$$\end{document} As an application of this theorem, we are able to prove that the A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \mathcal {A}$$\end{document}-p-quasiconvex hull of a set does not depend on p. Furthermore, we can prove a classification theorem for A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \mathcal {A}$$\end{document}-∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \infty $$\end{document}-Young measures.

(N × N )-matrix with entries ∂ i u j − ∂ j u i . Zhang's proof, which builds on the works of Liu [22] and Acerbi-Fusco [1], proceeds as follows. Denote by M f the Hardy-Littlewood maximal function of f ∈ L 1 loc (R N , R d ) and let u n = ∇w n . The estimate (1.1) implies that the sets X n = {M(∇w n ) ≥ L } have small measure for large n. One then uses (cf. [1]) that |w n (x) − w n (y)| ≤ C L |x − y| , x, y ∈ R N \X n , (1.2) i.e. w n is Lipschitz continuous on R N \X n . The fact that Lipschitz continuous functions on closed subsets of R N can be extended to Lipschitz continuous functions on R N with the same Lipschitz constant [15] yields the result.
In this paper, we show that the answer to the previously formulated question is also positive for sequences of differential forms and A = d, the operator of exterior differentiation.
Let us denote by r the r-fold wedge product of the dual space (R N ) * of R N and by d : C ∞ (R N , r ) → C ∞ (R N , r +1 ) the exterior derivative w.r.t. the standard Euclidean geometry on R N . Theorem 1.1 (L ∞ -truncation of differential forms) Suppose that we have a sequence u n ∈ L 1 (R N , r ) with du n = 0 (in the sense of distributions), and that there exists an L > 0 such thatˆ{ y∈R N : |u n (y)|>L} |u n (y)| dy −→ 0 as n → ∞. (
An analogous version of Theorem 1.1 holds if R N is replaced by the N -torus T N (cf. Theorem 5.1) or by an open Lipschitz set and functions u with zero boundary data (cf. Propostion 5.4). Moreover, the result immediately extends to R m -valued forms by taking truncations coordinatewise (cf. Proposition 5.5).
In particular, the result of Theorem 1.1 includes a positive answer to the question previously raised for the differential operator A = div after suitable identifications of N −1 and N with R N and R, respectively.
One key ingredient in the proofs is a version of the Acerbi-Fusco estimate (1.2) for simplices rather than pairs of points in Lemma 3.1. For the estimate, let us consider ω ∈ C 2 c (R N , r ) with dω = 0 and let D be a simplex with vertices x 1 , ..., x r +1 and a normal vector ν r ∈ R N ∧ ... ∧ R N (cf. Sect. 2.3 for the precise definition). Assume that Mω(x i ) ≤ L for i = 1, ..., r + 1.
The second ingredient is a geometric version of the Whitney extension theorem, which may be of independent interest, cf. Sect. 4.
Combining (1.4) and the extension theorem, one easily obtains the assertion for smooth closed forms. The general case follows by a standard approximation argument.
Before turning to an application of the truncation result, let us also mention that in Theorem 1.1 the hard part is to get the convergence in 1.1) just from the rather weak assumption (1.3). A version of Theorem 1.1 has been seen for a stronger assumption on the smallness of the sequence in [14]. Regarding solenoidal Lipschitz truncations [4,5], meaning W 1,1 -W 1,∞truncations instead of L 1 -L ∞ , the smallness corresponding to (1.3) is also assumed to be slightly different from the present setting.
Moreover, in the setting A = curl, the statement of Theorem 1.1 can be further improved as follows. If K is a compact, convex set and u n → K in L 1 , we can even get a sequence v n , such that the L ∞ -norm of dist(v n , K ) converges to 0, cf. [25]. In contrast, Theorem 1.1 only implies an L ∞ -bound on v n and convergence in measure to K . Müller's technique does not rely directly to a curl-free truncation, but on a Lipschitz truncation. It then uses suitable cut-offs and mollifications. The author does not see any obvious obstruction, why this technique should not work, if we replace the Lipschitz truncation by a general truncation statement on any potential instead of ∇. To keep the paper at a reasonable length, we however focus on A-free truncations.

A-∞ Young measures
Truncation results like the result by Zhang or Theorem 1.1 have immediate applications in the calculus of variations. In particular, they provide characterisations of the A-quasiconvex hulls of sets (cf. Sect. 6.1) and the set of Young-measures generated by sequences satisfying Au n = 0. For a precise definition of A-Young measures we refer to Sect. 6 and [11].
The classical result for Young measures generated by sequences of gradients (i.e. sequences of functions u n satisfying curl u n = 0) goes back to Kinderlehrer and Pedregal [17,18]. Here, we show the natural counterpart of their characterisation result, whenever the operator A admits the following L ∞ -truncation result: We say that A satisfies the property (ZL) if for all sequences u n ∈ L 1 (T N , R d ) ∩ ker A, such that there exists an L > 0 witĥ {y∈T N : |u n (y)|>L} |u n (y)| dy −→ 0 as n → ∞, there exists a C = C(A) and a sequence v n ∈ L 1 (T N , By Zhang [38], the property (ZL) holds for A = curl and a version of Theorem 1.1 shows this for A = d (Corollary 5.2). Further examples are shortly discussed in Example 6.2.
For the characterisation of Young measures, recall that spt ν denotes the support of a (signed) Radon measure ν ∈ M(R d ), and for f ∈ C c (R d ) If the property (ZL) holds for some differential operator A, then one is able to prove the following statement.

e. x ∈ T N and all continuous and
For further reference to classification of Ap-Young measures for p < ∞, let us shortly refer to [2,11,12,19,20].

Outline
We close the introduction with a brief outline of the paper. In Sect. 2, we introduce some notation, recall some basic facts from multilinear algebra, the theory of differential forms and Young measures. We prove the key estimate (1.4) in Sect. 3. Section 4 is devoted to the proof of the geometric Whitney extension theorem. In Sect. 5, the proof of the truncation result (and its local and periodic variant) is given. Section 6 discusses the applications to A-quasiconvex hulls and A-Young measures. The proofs of the theorems closely follow the arguments in [17] and are discussed in the last Sect. 6.3.

Notation
We consider an open and bounded Lipschitz set ⊂ R N and denote by T N the N -dimensional torus, which arises from identifying faces of [0, 1] N . We may identify functions f : T N → R d with Z N -periodic functionsf : R N → R d , and vice versa. We write B ρ (x) to denote the ball with radius ρ and centre x. Denote by L N the Lebesgue measure and, for a set X ⊂ R N , For a measure μ on R N and a μ-measurable set A ⊂ R N with 0 < μ(A) < ∞ define the average integral of a μ-measurable function f via Define the space r as the r -fold wedge product of (R N ) * , i.e.
and similarly the space r as the r -fold wedge product of R N . Then r and r are finitedimensional vector spaces. For R N denote by {e i } i∈ [N ] the standard basis and by · the standard scalar product. For (R N ) * denote by θ 1 , ..., θ N the corresponding dual basis of (R N ) * , i.e. θ i is the map y → y · e i . For k ∈ I r := {l ∈ [N ] r : l 1 < l 2 < ... < l r } the vectors e k,r = e k 1 ∧ e k 2 ∧ ... ∧ e k r (2.1) form a basis of r . Denote by · r the scalar product with respect to this basis, i.e. for k, l ∈ I r e k,r · r e l,r = 1 k = l, 0 k = l.
This also provides us with a suitable norm on r , which we denote by · r . Similarly, using the standard basis of (R n ) * , we define a basis θ k,r and a norm · r . Also note that for 0 ≤ s ≤ r there exists (up to sign) a natural map r × s → r −s (the interior product), as s is the dual space of s and r = s ∧ r −s . In particular, in the special case s = 1 for h 1 , ..., h r ∈ R N * and y ∈ R N In the case s = r and for h 1 , ..., h r ∈ (R N ) * and y 1 , ..., where S r denotes the group of permutations of {1, ..., r }. (2.3) also gives us a representation of the map r × s → r −s as for h ∈ r , x ∈ s we may consider the element of Let us shortly remark that this notation is slightly different to the usual notation for interior products.
Moreover, note that the space N is isomorphic to R via the map I N defined by

Differential forms
In the following, we will define all objects for an open set ⊂ R N , but these definitions are also valid for R N and T N respectively.
We call a map f ∈ L 1 loc ( , r ) an r -differential form on . We define the space It is well-known (c.f [7,9]) that there exists a linear map d : → , called the exterior derivative with the following properties We have the Leibniz rule: If α ∈ C ∞ ( , r ) and β ∈ C ∞ ( , s ), then We sometimes write d x to indicate that this derivative is taken in terms of a space variable x ∈ R N . This map d has the following representation in terms of the standard coordinates (cf. [9]). Let ω ∈ C ∞ ( , r ), which, for some a k ∈ C ∞ ( , R), can be written as Then ∂ l a k (y)θ l ∧ θ k,r . (2.5) with some well-known differential operator A. By definition, for r = 0, d can be identified with the gradient. For r = 1, after a suitable identification of 2 with R N ×N skew , d = curl, which is the differential operator mapping u ∈ C ∞ ( , If r = N − 1, after identifying N −1 with R N and N with R, the differential operator d becomes the divergence of a vector field which is defined for

Lemma 2.2
We have the following product rules for d: where we define ∇ y ω(y) · (y − z) ∈ C( , 1 ) as follows: Proof (i) simply follows from a calculation, i.e., if as mentioned which is what we claimed. (ii) then follows from (i) and using (2.2). Likewise, (iii) then follows from ii).

Definition 2.3
For ω ∈ L 1 loc ( , r ) and u ∈ L 1 loc ( , r +1 ) we say that dω = u in the sense Note that this definition is equivalent to the following formula:

Stokes' theorem on simplices
We want to establish a suitable notion of Stokes' theorem for differential forms on simplices. Let 1 ≤ r ≤ N and x 1 , ..., x r +1 ∈ R N . Define the simplex Sim(x 1 , ..., x r +1 ) as the convex hull of x 1 , ..., x r +1 . We call this simplex degenerate, if its dimension is strictly less than r .

The maximal function
The Hardy-Littlewood maximal function for u ∈ L 1 loc (R N , R d ) is defined by Again, we can also define the maximal function for functions on the torus using the identification with periodic functions.

A geometric estimate for closed differential forms
In this section we prove a key lemma for our main theorem.

Lemma 3.1 There exists a constant C
This lemma can be seen as a natural analogue of Lipschitz continuity on the set where the maximal function is small. In particular, it has been proven (for example in [1]) that for Hence, one should view Lemma 3.1 as a generalisation of this result.
Proof For simplicity write |ω| := ω r . Recall that It suffices to show that there exists z ∈ R N such that The equation (3.2) can be verified by Stokes' theorem (2.11), using that boundary terms with a simplex with vertex z cancel out on the left-hand side of (3.2) (Fig. 1).
We now prove to be the r -dimensional hyperplane going through x 1 , ..., x r and z. This is well-defined if z is not in the (r − 1) dimensional hyperplane F going through x 1 , ..., x r . Note that for z,z / ∈ F z ∈ E(z) ⇔z ∈ E(z). x 1 it also follows by Fubini and (3.3) Using that Sim(x 1 , ..., Choose now μ * = 2(r +1)b r b N R r λC −1 1 . Plugging this into (3.4), we see that the measure of this set is smaller than R N (2(r +1)) −1 . Repeating this procedure for all (r −1)-dimensional faces of Sim(x 1 , ..., x r +1 ), we get that for i > 1 , .
such that all the summands of (3.1) are smaller than μ * = ((2(r This is what we wanted to prove.

A Whitney-type extension theorem
First, let us recall the following Lipschitz extension theorem.

the Lipschitz constant does not depend on X ).
Of course, there are several ways to prove such a theorem, even with C(N ) = 1 [15]. However, Whitney's proof [36] plays with the geometry of R N quite nicely. Similar geometric ideas lies behind our proof for closed differential forms. First, let us define an analogue of (4.1).
Suppose that X is a closed subset of R N , such that

Remark 4.3
The constant C does not depend on the choice of u or X , it is only important that the pair (u, X ) satisfies (4.2). The assumption that X C is bounded makes the proof easier, but may be dropped. It is not clear, whether the assumption that |∂ X | = 0 is necessary for the statement to hold or not.

Remark 4.4
As one can see in the proof, the assumption u ∈ C ∞ c (R N , r ) can be weakened to u ∈ C 2 c (R N , r ), as we only need the first two derivatives of u. However, it is important to remember that we cannot prove Lemma 4.2 for the even weaker assumption u ∈ L 1 loc , as (4.2) is not well-defined.
For the proof we follow the classical approach by Whitney with a few little twists. First, we will define the extension in (4.4). Then we prove that v satisfies properties (i)-(iii). (ii) and (iii) are quite easy to see from the definition of v, however it is hard to verify that (i) holds. On the one hand, we show that the strong derivative of v exists almost everywhere, namely in R N \∂ X and that dv = 0 almost everywhere, where we use the assumption that the boundary of X is a null-set. On the other hand, we then prove that the distributional derivative dv is in fact also an L 1 function, yielding that dv = 0 in the sense of distributions.
We now start with the definition of the extension. Let us recall (cf. [31]) that for X ⊂ R N closed we can find a collection of pairwise disjoint open cubes Choose 0 < ε < 1/4 and define another collection of cubes by • For all i ∈ N, the number of cubes Q j such that Q i ∩ Q j = ∅ is bounded by a dimensional constant C(N ); • In particular, all x ∈ R N are only contained in at most C(N ) cubes Q i ; • The distance to the boundary is again comparable to the sidelength, i.e.
Note that if X is Z N -periodic, then also Q i can be chosen to be Z N periodic (initially, we have a collection of dyadic cubes) (Fig. 2).
Define the partition of unity on X C by Note that 0 ≤ ϕ j ≤ 1 and that there exists a constant C > 0 such that for all j ∈ N For each cube Q i , we may find an x ∈ X such that dist We now define the differential form α ∈ L 1 (R N , r ) by Note that in this setting is the function satisfying all the properties of Lemma 4.2.
Lemma 4.5 The differential form α defined in (4.3) satisfies α ∈ L 1 (X C , r ) and the sum in (4.3) converges pointwise and in L 1 .
Proof Pointwise convergence is clear, as for fixed y ∈ X C only finitely many summands are nonzero in a neighbourhood of y (ϕ i is only nonzero in Q i and any point is only covered by at most C(N ) cubes). For L 1 convergence fix some i 1 ∈ N. Note that there are at most Moreover, we can bound ν r by Hence, we can bound the L ∞ -norm of a nonzero summand of (4.3) by C u L ∞ , as |G(I )| ≤ u L ∞ . As the support of the summand is contained in Q i 1 , we have that its L 1 norm is bounded by Remember that any point in X C is covered by only C(N ) cubes, such that the sum of |Q i | is bounded by C(N )|X C |. Hence, the sum in (4.3) converges absolutely in L 1 and its L 1 norm is bounded by C(N ) r +1 C u L ∞ |X C |.

Lemma 4.6
The function v is strongly differentiable almost everywhere and satisfies dv(y) = 0 for all y ∈ R N \∂ X.
Proof Note that u ∈ C ∞ c (R N , r ) and hence v is strongly differentiable in X \∂ X . Furthermore, the sum in (4.3) is a finite sum in a neighbourhood of y for all y ∈ X C . As the summands are also C ∞ , the sum is C ∞ in the interior of X C .
By assumption du = 0, hence it remains to prove that dα(y) = 0 for all y ∈ X C . Note that in a neighbourhood of y ∈ X C again only finitely many summands are nonzero. Using that d 2 = 0 and the Leibniz rule, we get ν r (x i 1 , ..., x i r+1 ))). (4.5) Observe that this term does not converge in L 1 and hence this identity is only valid pointwise.
Pick some j ∈ N such that y ∈ Q j . As all ϕ i sum up to 1 in X C , we have Replace dϕ j in the sum in (4.5) by − We apply Stokes' theorem (2.11) to the r -form u and the simplex with vertices x j , x i 1 , ..., x i r+1 , use that du = 0 and conclude that this term is 0, i.e.
Hence, the pointwise derivative equals 0 almost everywhere.
It is important to note that the sum (4.3) in the definition of α converges in L 1 , but in general does not converge in W 1,1 , and thus we have no information on the behaviour at the boundary of X C . However, it suffices to show that the distribution dv for v given by (4.4) is actually an L 1 function. If dv ∈ L 1 , we can conclude with Lemma 4.6 that dv = 0 in the sense of distributions.

Lemma 4.7
The distributional exterior derivative of v defined in (4.4) In view of the definition of α, this expression is given by: We use the splitting G(I ) = (G(I ) − u(·)) + u(·) and write ( * ) as (4.6) Note that (I) defines a distribution given by an L 1 function. Indeed, the sum To see this, one can repeat the proof of Lemma 4.5 and use that there are additional factors in the estimate of the norms. For this, note that if z ∈ Q i 1 One gets improved regularity and may integrate by parts to eliminate the derivative of ψ. Term (II) is not so easy to handle. We prove the following claims: Claim 1 Let 1 ≤ s ≤ r and I = (i s , ..., i r +1 ) ∈ N r −s+2 . There exists h s ∈ L 1 (R N , r +1 ) such that (4.7) Here we use the notation that ν 0 ( (4.8) Note that Claim 2 follows from Claim 1 by an inductive argument. The domain of integration in (4.8) can be replaced by X C as well, as all ϕ i j are supported in X C .
First, let us conclude the proof under the assumption that Claim 1 holds true. Using (4.6) and Claim 2 we see that there is an Recall that du = 0 in the sense of distributions and therefore −ˆX We conclude that there exists an L 1 function h ∈ L 1 (R N , r +1 ) such that Thus, dv is an L 1 function.
It remains to prove Claim 1. Note that This can be verified using that the wedge product is alternating and explicitly writing the right-hand side of (4.9).
Using this identity, we may split the right-hand side of (4.7) (denoted by (III)), i.e.
Arguing as in Lemma 4.5, we see that the sum is in fact convergent in L 1 . Moreover, the index i j only appears once in this sum. Recall that for y ∈ X C i s ∈N dϕ i s (y) = 0. Thus, For (IIIb) note that i 1 ∈N ϕ i s = 1 X C and, by the same argument as for (IIIa), we can write We can now integrate by parts to eliminate the exterior derivative in front of ϕ i s+1 . Applying Lemma 2.2, using d 2 = 0, the Leibniz rule and the fact that Arguing similarly to Lemma 4.5 and as for term (I), we can show that and that this sum is convergent in W 1,1 . Hence, we have shown that there exists h s ∈ L 1 (R N , r +1 ) such that every summand can be bounded by C L due to (4.2) and the estimate |dϕ j | ≤ C dist(Q j , X ) −1 . Again, we get the L ∞ bound, as only finitely many summands are nonzero for every y ∈ X C . With slight modifications one is able to prove the following variants.

Corollary 4.8
Let u ∈ C ∞ (R N , r ) with du = 0, let L > 0, and let X ⊂ R N be a nonempty closed set such that u L ∞ (X ) ≤ L and for all x 1 , ..., x r +1 ∈ X we have Suppose further that |∂ X | = 0. There exists a constant C = C (N , r ) such that for all u ∈ C ∞ (R N , r ) and X meeting these requirements there exists v ∈ L 1 loc (R N , r ) with This statement is proven in the same way as Lemma 4.2, but all the statements are only true locally (e.g. the L 1 bounds on α are replaced by bounds in L 1 loc (X C , r )). If we choose u and X to be Z N periodic we get a suitable statement for the torus.

Corollary 4.9
Let u ∈ C ∞ (T N , r ) with du = 0, let L > 0, and let X ⊂ R N be a nonempty, closed, Z N -periodic set (which can be viewed as a subset of T N ) such that u L ∞ (X ) ≤ L and for all x 1 , ..., x r +1 ∈ X we have There exists a constant C = C(N , r ) such that for all u ∈ C ∞ (T N , r ) and X meeting these requirements there exists v ∈ L 1 (T N , r ) with As mentioned before, we can choose the cubes Q j to be rescaled dyadic cubes. As the set X is periodic, the set of cubes (and hence also the partition of unity) and their projection points may also be chosen to be Z N -periodic. By definition then also the extension will be Z N -periodic.

L ∞ -truncation
Now we prove the main result of this paper on the L ∞ -truncation of closed forms. Theorem 5.1 (L ∞ -truncation of differential forms) There exist constats C 1 , C 2 > 0 such that for all u ∈ L 1 (T N , r ) with du = 0 and all L > 0 there exists v ∈ L ∞ (T N , r ) with dv = 0 and y∈T N : |u(y)|>L} |u(y)| dy.
Given the Whitney-type extension obtained in Lemma 4.9 and Lemma 4.2 combined with Lemma 3.1, the proof now roughly follows Zhang's proof for Lipschitz truncation in [38]. First, we prove the statement in the case that v is smooth directly using our extension theorem for the set X = {Mu ≤ L}. After calculations similar to [38] we are able to show that this extension satisfies the properties of Theorem 5.1. Afterwards, we prove the statement for u ∈ L 1 (T N , r ) by a standard density argument.
Using the weak-L 1 estimate for the maximal function (Proposition 2.4), we get This is what we wanted to show. Note that the proof only uses u ∈ C ∞ (T N , r ) to define v and nowhere else, hence estimate (5.3) is valid for all u ∈ L 1 (T N , r ).
For general u ∈ L 1 (T N , r ), one may consider a sequence u n ∈ C ∞ (T N , r ) with du n = 0 and u n → u in L 1 and pointwise almost everywhere. This sequence can be easily constructed by convolving with standard mollifiers.
Letting n → ∞, by a) this sequence converges, up to extraction of a subsequence, weakly * to some v ∈ L ∞ (T N , r ). The weak * -convergence implies dv = 0. Moreover, by construction, the set {y ∈ T N : v n = u n } is contained in the set {y ∈ T N : Mu n (y) ≥ 2λ}. As u n → u pointwise a.e. and in L 1 , we get using (5.5) that v = u on the set {y ∈ T N : Mu(y) ≤ λ}. (If v n converges to u in measure on a set A and v n weakly to some v, then v = u on A.) Hence, v defined as the weak * limit of v n satisifies There exists a C 1 = C 1 (N , r ) and a sequence v n ∈ L 1 (T N , r ) with dv n = 0 and This directly follows by applying Theorem 5.1. The proof of Theorem 5.1 also works if L 1 is replaced by L p for 1 < p < ∞. Furthermore, we do not need to restrict us to periodic functions on R N , the statement is also valid for nonperiodic functions.
As described, the proof is pretty much the same as for Theorem 5.1. We may also want to truncate closed forms supported on an open bounded subset ⊂ R N (cf. [4,5]). This is possible, but we may lose the property, that they are supported in this subset. Let us, for simplicity, consider balls = B ρ (0) and, after rescaling, ρ = 1.

Proposition 5.4
Let 1 ≤ p < ∞. There exist constants C 1 , C 2 > 0 such that, for all u ∈ L p (R N , r ) with du = 0 and spt(u) ⊂ B 1 (0) and all L > 0, there exists v ∈ L p (R N , r ) with dv = 0 and y∈R N : |u(y)|>L} |u(y)| p dy; (iv) spt(v) ⊂ B R (0), where R only depends on the L p -norm of u and on L.
Again, this proof is very similar to the proof of Theorem 5.1. Property 5.4) comes from the fact that if a function u is supported in B 1 (0), then its maximal function Mu(y) decays fast as y → ∞. Regarding the construction made in Sect. 4 and Lemma 3.1, it is not clear, how to avoid the rather weak statement 5.4), i.e. we cannot directly deal with arbitrary boundary values and need to modify the truncation.
Let us mention that this result also holds for vector-valued differential forms, i.e. u ∈ L p (R N , r × R m ), where the exterior derivative is taken componentwise.
This statement follows directly from the proof of Theorem 5.1 by simply truncating every component of u. Likewise, similar statements as in Propositions 5.2, 5.3 and 5.3 follow for vector-valued differential forms.

Applications to A-quasiconvexity and Young measures
In the following, we consider a linear and homogeneous differential operator of first order, i.e. we are given A : where A k : R d → R l are linear maps. We call a continuous function f : holds true. Fonseca and Müller showed that [11], if the constant rank condition seen below holds, then A-quasiconvexity is a necessary and sufficient condition for weak * lowersemicontinuity of the functional I : Define the symbol A : The operator A is said to satisfy the constant rank property (cf. [27]) if for some fixed r ∈ {0, ..., d} and all ξ ∈ S N −1 = {ξ ∈ R N : |ξ | = 1} dim(ker A(ξ )) = r .
We call a homogeneous differential operator B : which is not necessarily of order one, the potential of A if i.e. if ψ =û(λ)e −2πi x·λ for λ ∈ R N \ {0}, then Aψ = 0 if and only if there isŵ(λ), such that ψ = B(ŵ(λ)e −2πi x·λ ). Recently, Raiţa showed that A has such a potential if and only if A satisfies the constant rank property ( [28]). In the following, we always assume that A satisfies the constant rank property and that B is the potential of A. Definition 6. 1 We say that A satisfies the property (ZL) if for all sequences u n ∈ L 1 (T N , R d ) ∩ ker A such that there exists an L > 0 witĥ there exists a C = C(A) and a sequence v n ∈ L 1 (T N , Our goal now is to show that (ZL) implies further properties for the operator A. We first look at a few examples. Example 6.2 (a) As shown by Zhang [38], the operator A = curl has the property (ZL). This is shown by using that its potential is the operator B = ∇. In fact, most of the applications here have been shown for B = ∇ relying on (ZL), but can be reformulated for A satisfying (ZL).
and define the operator as taking the curl on the last component ofũ, i.e. for I ∈ [N ] k−1 Note that this operator has the potential ∇ k : To the best of the author's knowledge the proof of the property (ZL) is in this setting not written down anywhere explicitly, but basically combining the works [1,13,31,38] yields the result. (c) In this work, it has been shown that the exterior derivative d satisfies the property (ZL).
The most prominent example is A = div. (d) The result is also true, if we consider matrix-valued functions instead (cf. Proposition 5.4).
For example, (ZL) also holds if we consider div : be two differential operators satisfying (ZL). Then also the operator defined componentwise for u = (u 1 , u 2 ) by satisfies the property (ZL). The truncation is again done separately in the two components. The most prominent example, which is also covered by the result of this paper, is A 1 = curl and A 2 = div, which is highly significant in elasticity and in the framework of compensated compactness.
An overview of the results one is able to prove using property (ZL) can be found in the lecture notes [26,Sec. 4] and in the book [29,Sec. 4,7], where they are formulated for the case of (curl)-quasiconvexity.
In view of the separation theorem for convex sets in Banach spaces we define (cf. [8,33,34]) the A-quasiconvex hull of a set K ⊂ R d by The Ap-quasiconvex hull for 1 ≤ p < ∞ can be alternatively defined via if q ≤ q . In [8] it is shown that equality holds for A being the symmetric divergence of a matrix, K compact and 1 < q, q < ∞. The proof can be adapted for different A, but uses the Fourier transform and is not suitable for the cases p = 1 and p = ∞. Here, the property (ZL) comes into play.
For a compact set K we define the set K Aapp (cf. [26]) as the set of all x ∈ R d such that there exists a bounded sequence u n ∈ L ∞ (T N , R d ) ∩ ker A with dist(x + u n , K ) −→ 0 in measure, as n → ∞.

Theorem 6.3 Suppose that K is compact and A is an operator satisfying (ZL). Then
Proof We first prove K Aapp ⊂ K Aqc ∞ . Let x ∈ K Aapp and take an arbitrary A-quasiconvex function f : R d → [0, ∞) with f |K = 0. We claim that then f (x) = 0.
Take a sequence u n from the definition of K Aapp . As f is continuous and hence locally bounded, f (x +u n ) → 0 in measure and 0 ≤ f (x +u n ) ≤ C. Quasiconvexity and dominated convergence yield The proof of the inclusion {x ∈ R d : Q A (dist(·, K ))(x) = 0} ⊂ K Aapp uses (ZL). If As K is compact, there exists R > 0 such that K ⊂ B(0, R). Moreover, as x ∈ K Aqc ∞ , also x ∈ B(0, R). This implies that lim n→∞ˆT N ∩{|ϕ n |≥6R} |ϕ n | dy = 0.
We may apply (ZL) and find a sequence Hence, x ∈ K Aapp . Remark 6.4 Theorem 6.3 shows that for all 1 ≤ p < ∞ This follows directly, as all the sets K Aqc p are nested and, conversely, all the hulls of the distance functions are admissible f in the definition of K Aqc ∞ . Remark 6.5 Such a kind of theorem is not true for general unbounded closed sets K . As a counterexample one may consider A = curl (i.e. usual quasiconvexity) and look at the set of conformal matrices K = {λQ : λ ∈ R + , Q ∈ SO(n)} ⊂ R n×n . If n ≥ 2 is even, by [24], there exists a quasiconvex function F : On the other hand, let n ≥ 4 be even and F : R n×n → R be a rank-one convex function with F |K = 0 and for some p < n/2 Then F = 0 by [37].
A reason for the nice behaviour of compact sets is that for such sets all distance functions are coercive, i.e. dist(v, K ) p ≥ |v| p − C, which is obviously not true for unbounded sets. Coercivity of a function is often needed for relaxation results (c.f [6]).
We call such a map ν : E → M(R d ) the Young measure generated by the sequence u j k . One may show that every weak * measurable map E → M(R d ) satisfying (i) is generated by some sequence u j k .

Remark 6.8
If u k generates a Young measure ν and v k → 0 in measure (in particular, if v k → 0 in L 1 ), then the sequence (u k + v k ) still generates ν.
If u : T N → R d is a function, we may consider the oscillating sequence u n (x) := u(nx). This sequence generates the homogeneous (i.e. ν x = ν a.e.) Young measure ν defined by

Question 6.9
What happens to the Young measure generated by a sequence u j k if we impose further conditions on it, for instance Au j k = 0?
sup j∈N sup E⊂ : |E|<εˆE |v j (y)| p dy = 0. Definition 6.10 Let 1 ≤ p ≤ ∞. We call a map ν : → R d an Ap-Young measure if there exists a p-equi-integrable sequence {v j } ⊂ L p ( , R d ) (for p = ∞ a bounded sequence), such that v j generates ν and satisfies Av j = 0.
For 1 ≤ p < ∞ the set of Ap Young measures was classified by Fonseca and Müller in [11] and for the special case A = curl already in [18].
(iii) for a.e. x ∈ T N and all continuous g with |g(v)| ≤ C(1 + |v| p ) we have is weakly compact, that averages of (non-homogeneous) A-infty Young measures are in H A (K ) and that the set H x A (K ) = {ν ∈ H A : ν, id = x} is weak* closed and convex. The characterisation theorem then follows by using Hahn-Banachs separation theorem and showing that any μ ∈ M Aqc cannot be separated from H A (K ), i.e. for all f ∈ C(K ) and for all μ ∈ M Aqc (K ) with μ, id = 0 Proposition 6.14 then can be shown using Proposition 6.13 and a localisation argument.

On the proofs of Propositions 6.13 and 6.14
In this section, we present the proof of Proposition 6.13, basing on its counterpart for gradient Young measures in [26]. After that we shortly sketch the proof of 6.14, which is then done by a standard technique of approximation on small cubes. The property (ZL) is helpful due to the following two observations: is a homogeneous A-∞-Young measure, then by using (ZL) we can find a sequence generating ν with an L ∞ -bound only depending on |K | ∞ := sup y∈K |y| (cf.

Lemma 6.16) 2 A Young measure ν is an
Remark 6. 15 Moreover, note that, if a sequence u n ∈ L ∞ (T N , R d ) ∩ ker A generates a homogeneous Young measure ν, we can find v n ∈ C ∞ c ((0, 1) N , R d ) ∩ ker A with v n L ∞ ≤ C u n L ∞ and u n − v n L 1 → 0. In particular, v n still generates the same homogeneous Young measure.
To find such a sequence, recall that there is a potential B of order k B to the differential operator A. Let us, for simplicity, assume that all u n have zero average. Then we can write u n = BU n with U N W k B ,q ≤ C q u n L q ≤ C q u n L ∞ for all 1 < q < ∞ and a constant C q > 0. Let us define for a suitable sequence of cut-offs ϕ j → 1 in L 1 ((0, 1) N , R). Picking suitable subsequences i(n) and j(n) we obtain a sequence u n,i(n), j(n) bounded in L ∞ , still generating ν, but with compact support in (0, 1) N . Convolution with a standard mollifier gives a sequence v n that is also in C ∞ c ((0, 1) N , R d ) Proof (i) follows from the definition of H A (K ). The uniform bound on the L ∞ norm of u j can be guaranteed by (ZL) and vi) in Theorem 6.7. For the weak * compactness note that H A (K ) is contained in the weak * compact set P(K ) of probability measures on K . As the weak * topology is metrisable on P(K ) it suffices to show that H A (K ) is sequentially closed. Hence, we consider a sequence ν k ⊂ H A (K ) with ν k * ν and show that ν ∈ H A (K ).
Due to the definition of Young measures, we may find sequences u j,k ∈ L ∞ (T N , R d ) ∩ ker A such that u j,k generates ν k for j → ∞. Recall that the topology of generating Young measures is metrisable on bounded set of L ∞ (T N , R d ) (c.f. Remark 6.6). We may find a subsequence u j k ,k which generates ν. As we know that u j k ,k L ∞ ≤ C|K | ∞ , ν ∈ H A (K ) and hence H A (K ) is closed. Lemma 6.17 Let ν be an A-∞-Young measure generated by a bounded sequence u k ∈ L ∞ (T N , R d ) ∩ ker A. Then the measureν defined via duality for all f ∈ C 0 (R d ) by Note that by Theorem 6.7 ii) we also havê Due to metrisability on bounded sets (Remark 6.6), we can find a subsequence u Thus,ν ∈ H A (K ). Proof Weak * -closedness is clear by Lemma 6.16. For convexity, let ν 1 , ν 2 be A-∞-Young measures. By an argumentation following Remark 6.15, we can find sequences v n ∈ C ∞ c ((0, λ) × (0, 1) N −1 , R d ) and w n C ∞ c ((λ, 1) × (0, 1) N −1 , R d ) that generate ν 1 and ν 2 , respectively. Define u n = v n in (0, λ) × (0, 1) N −1 , w n in (λ, 1) × (0, 1) N −1 .

Lemma 6.18 Define the set H x
and u n,i via u n,i (x) = u n (i x). Then proceeding as in Lemma 6.17, picking a suitable subsequence i(n) yields that u n,i(n) generates λν 1 + (1 − λ)ν 2 .
We proceed with the proof of the characterisation of homogeneous A-∞-Young measures.
Proof of Theorem 6. 13 We have that H A (K ) ⊂ M Aqc due to the fundamental theorem of Young measures: ν ≥ 0 and spt ν ⊂ K are clear by i) and iii) of Theorem 6.7. The corresponding inequality follows by A-quasiconvexity, i.e. if u n ∈ L ∞ (T N , R d )∩ker A generates the Young measure ν, then To prove M Aqc (K ) ⊂ H A (K ), w.l.o.g. consider a measure such that ν, id = 0. We just proved that H 0 A (K ) is weak * closed and convex. Remember that C(K ) is the dual space of the space of signed Radon measures M(K ) with the weak * topology (see e.g. [30]). Hence, by Hahn-Banach separation theorem, it suffices to show that for all f ∈ C(K ) and all μ ∈ M Aqc (K ) with μ, id = 0 To this end, fix some f ∈ C(K ), consider a continuous extension to C 0 (R d ) and let We claim that lim If we show (6.6), μ satisfies finishing the proof. For the identity μ, f = μ, f k recall that μ is supported in K and dist(x, K ) = 0 for x ∈ K . Hence, suppose that (6.6) is wrong. As f k is strictly increasing, there exists δ > 0 such that Using the definition of the A-quasiconvex hull (6.2), we get u k ∈ L ∞ (T N , R d ) ∩ ker A withˆT N u k (y) dy = 0 andˆT N f k (u k (y)) dy ≤ −δ. (6.7) We may assume that u k 0 in L 2 (T N , R d ) and also that dist 2 (u k , K ) → 0 in L 1 (T N ). By property (ZL), there exists a sequence v k ∈ ker A bounded in L ∞ (T N , R d ) with u k − v k L 1 → 0. v k generates (up to taking subsequences) a Young measure ν with spt ν x ⊂ K .
Then for fixed j ∈ N, using Lemma 6.17 and thatν ∈ H A (K ) ⊂ M Aqc (K ), But this is a contradiction to (6.7), as f k ≥ f j if k ≥ j.
Let us finally outline the strategy of the proof for Proposition 6.14. For details we refer to [17,26].
Proof of Propostion 6.14 (Sketch) Necessity of condition (i)-(iii) is established by the following argument. (i) and (ii) follow directly from the fact that the Young-measure μ is generated by an A-free sequence that, up to a subsequence, has a weak- * -limit u. iii) follows from the lower-semicontinuity statement of Fonseca and Müller [11].
To prove sufficiency of these conditions, one needs to construct a sequence generating the Young-measure ν. Let us suppose that u = 0, otherwise we define the Young-measurẽ ν = ν − u. Then we find a sequence v n generatingν and, consequently, v n + u generates ν.
To find such a sequence one divides T N into subcubes and approximates ν by maps ν n : T N → M(R d ), which are constant on the subcubes. For each subcube Q one then constructs a sequence v Q n,m ∈ L ∞ (Q, R d ) ∩ ker A, m ∈ N, that generates ν n and satisfies v Q n,m ∈ C ∞ c (Q, R d ).
These v Q n,m then give a sequence v n,m generating ν n and taking a suitable diagonal sequence one may find a sequence generating ν (cf. [26,Proof of Theorem 4.7]).