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Compensation phenomena for concentration effects via nonlinear elliptic estimates

We study compensation phenomena for fields satisfying both a pointwise and a linear differential constraint. This effect takes the form of nonlinear elliptic estimates, where constraining the values of the field to lie in a cone compensates for the lack of ellipticity of the differential operator. We give a series of new examples of this phenomenon for a geometric class of cones and operators such as the divergence or the curl. One of our main findings is that the maximal gain of integrability is tied to both the differential operator and the cone, contradicting in particular a recent conjecture from arXiv:2106.03077. This extends the recent theory of compensated integrability due to D. Serre. In particular, we find a new family of integrands that are Div-quasiconcave under convex constraints.

1. Introduction 1.1. Nonlinear elliptic estimates. The interactions between nonlinearities, uniform estimates and weak convergence lie at the heart of many problems in the modern theory of nonlinear PDE. Famously, the Sobolev embedding theorem allows one to deduce uniform higher integrability for a sequence of functions whose gradients are bounded in an appropriate space, often leading to uniform bounds and weak (or strong) convergence for solutions of nonlinear PDE. In many cases, however, the full gradient may not be controlled, but instead one only has bounds on a weaker, non-elliptic operator, such as the divergence or the curl, for which general Sobolev inequalities fail. In this paper, we study higher integrability under the assumptions of boundedness of such a weaker operator together with a compensating pointwise constraint.
To be more precise, the main theme of this paper are nonlinear estimates of the form v L q (B n ) C Av L p (B n ) for v ∈ C ∞ c (B n , K), (1.1) the constraint set K. Here and throughout B n denotes the open unit ball. If (1.1) holds, we expect that ker A(ξ) and A(ξ) denotes the principal symbol of A. The set Λ A is known as the wave cone of A, and it describes the set of directions along which the operator is not elliptic. Since (1.1) requires (1.2), which is a type of ellipticity condition of A with respect to K, we refer to (1.1) as a nonlinear elliptic estimate for A. The simplest instance of (1.1) is when K is a linear space and, in this situation, (1.2) yields genuine ellipticity [52,Lemma 2.7]. It goes back to the work of Calderón and Zygmund that, if 1 < p < ∞ and q np n−ℓp , where ℓ is the order of A, then (1.2) implies (1.1). The case p = 1, where the Calderón-Zygmund theory breaks down, is surprisingly subtle, having been settled only much more recently by Van Schaftingen [74], after the pioneering work of Bourgain and Brezis [11]. Besides ellipticity, the estimate holds if in addition A satisfies an algebraic condition known as cancellation. In addition, also when p = 1, similar estimates were recently obtained under the assumption that K is a perturbation of a linear space [4].
The methods developed to study (1.1) in the case where K is linear (or close to linear) make fundamental use of the Fourier transform, which interacts poorly with pointwise constraints. Thus, in order to study the genuinely nonlinear case we are interested in here, completely different tools are needed.
To the best of our knowledge the only known instance of (1.1) when K is genuinely nonlinear is due to Serre [57,58]. He considered A = Div and K the set Sym + n of symmetric semi-positive definite matrices and showed that C Div A L 1 (R n ) for A ∈ C ∞ c (R n , Sym + n ). (1. 3) In particular, for closed cones contained in the interior of Sym + n we recover (1.1) with p = 1 and q = n n−1 . Subsequently, De Rosa-Tione showed in [21] that the exponent n n−1 corresponds to the maximal gain of integrability, in the sense that we cannot take q > n n−1 , even when p = ∞, see already (1.5) below. We also refer the reader to [57,58,59,60] for applications of (1.3) to hyperbolic problems and to [10] for a related result when n = 2.
In this paper we succeed in finding new examples of nonlinear elliptic estimates (1.1) for non-elliptic operators, such as Div or Curl. The first of our main results is the following: Theorem A. Let K ≡ {A ∈ R 2×2 : a 11 , a 22 0} and let A 1 , A 2 be the rows of A ∈ R 2×2 . ThenˆR In particular, writing Q + 2 (K) = {A ∈ R 2×2 : A 2 K det A}, we have A 2 c (R 2 , K ∩ Q + 2 (K)). Noting that div A 1 L 1 div A 2 L 1 Div A 2 L 1 , Theorem A should be compared with Serre's inequality (1.3) above. In particular, Theorem A shows that, in the two dimensional case, (1.3) holds for a larger cone which is not necessarily contained in the space of symmetric matrices. The cone in Theorem A is in some sense optimal, as the above estimates do not hold in the absence of one-sided constraints. However, for p > 1, such one-sided constraints are not necessary, and we have (1.4) A L p * (R 2 ) C Div A L p (R 2 ) for all A ∈ C ∞ c (R 2 , Q + 2 (K)) if and only if see Proposition 4.5 below. Moreover, not even the estimate holds. Note that if K > 1 then 2K 2K−1 < 2 so (1.4) does not hold in the range 1 < p < n = 2. The above shows that, while pointwise constraints may allow elliptic-type estimates for non-elliptic operators, the gain of integrability fails in general to be as good as in the linear elliptic theory: there is a number q max = q max (A, K) such that q > q max =⇒ there is no C such that v L q (B n ) C Av L ∞ (B n ) for v ∈ C ∞ c (B n , K). We emphasize that q max depends on both A and K and we refer to it as the maximal gain of integrability induced by A on K. Thus (1.5) q max (Div, Sym + n ) = n n − 1 , q max (Div, Q + 2 (K)) = 2K 2K − 1 .
After this paper was completed, we learned that De Philippis-Pigati had independently proved Theorem A in order to establish a Michael-Simon inequality for anisotropic varifolds [19]. Moreover, in the very interesting paper [16], Colombo-Tione have shown that Theorem A also yields regularity of certain very weak solutions of the p-Laplace equation.
Our next result shows that, in higher dimensions, there are stronger estimates for the divergence in smaller convex cones. To be precise, we consider the Gårding cones The Gårding cones, introduced in the seminal work [27], play an important role in the theory of fully nonlinear elliptic equations [13,75], of quasiconvex functions [24] and of curvature flows in differential geometry, for example in Huisken-Sinestrari [36] (see also [28,73]). Besides the Gårding cones, we will consider their dual convex cones Γ * k = {A ∈ Sym n : A, B 0 for all B ∈ Γ k }.
For instance, we have Γ n = Γ * n = Sym + n and Γ 1 = {A ∈ Sym n : Tr(A) 0}, Γ * 1 = {tI n : t 0}. The dual cones have been exploited by Krylov [45] (cf. [39]) in the study of Bellman equations and by  to derive Alexandrov-Bakelman-Pucci type estimates. We refer the reader to Section 3.1 below for further properties of these cones.
In the next statement and throughout we write p * = np n−p for the Sobolev exponent of p.
Theorem B. Let n 2, k ∈ {2, . . . , n}, p * > k k−1 (or p * n n−1 if k = n), and let K be a closed, convex cone contained in int Γ * k ∪ {0}. Then A L k k−1 (B n ) C Div A L p (B n ) for all A ∈ C ∞ c (B n , K).
Moreover, the exponent on the left hand side is the best possible, since for each ε > 0 there is a cone K ε ⊂ int Γ * k ∪ {0} such that the following estimate fails: In addition, for k < n, the restriction to the subcritical range p * > k k−1 is necessary, as the analogue of (1.3) with the appropriate nonlinear functional replacing det does not hold.
Theorem B provides a discrete interpolation between Serre's inequality, when k = n, and the classical Sobolev embedding W 1,p 0 (B n ) ֒→ L ∞ (B n ) for p > n, when k = 1. Indeed, as Γ * 1 is a half-line through the identity, fields constrained to it can be identified with scalar fields, and the divergence can be identified with the gradient. As k decreases, the cones Γ * k also decrease, and we approach the scalar case, getting higher and higher integrability. Through a nonlinear duality argument, Theorem B yields elliptic estimates for the curl: Theorem C. Let n 2, k ∈ {2, . . . , n}, p * > k (or p * = n if k = n) and let K be a closed, convex cone contained in int Γ k ∪ {0}. Then , and in fact we have the additional improvement K). Moreover, the exponent on the left hand side is the best possible, since for each ε > 0 there is a cone K ε ⊂ int Γ k ∪ {0} such that the following estimate fails: . We emphasize the nonlinear duality between Theorems B and C: on the cone Γ k (respectively Γ * k ) with operator Curl (respectively Div), the maximal gain of integrability is k (respectively k ′ = k k−1 ). As in (1.4), an interesting feature of Theorems B and C is that the exponents k k−1 and k on the left-hand side, which are optimal, depend on the cone of interest and are strictly smaller than the Sobolev exponent p * = np n−p for all p in the ranges given, just as in (1.5).
The next theorem is a higher order variant of Theorem B, and in this case we are able to obtain inequalities in a full range of exponents, including the critical case. We let p * * = np n−2p denote the second Sobolev exponent and, by convention, when p n 2 , we let p * * ≡ ∞.
Theorem D. Let n 2, k ∈ {2, . . . , n}, and let K be a closed convex cone contained in int Γ * k ∪ {0}. Then, whenever p > 1 and 1 q min{p * * , k k−1 }, Moreover, the exponent k k−1 on the left hand side is the best possible, since for each ε > 0 there is a cone K ε ⊂ int Γ k ∪ {0} such that the following estimate fails: In particular, when n = k = 2 and p = 1, Theorem D disproves [4, Conjecture 1.9], which predicts that (1.7) holds whenever q < 1 * * = ∞. A result similar to the one in Theorem D, featuring the less common operator A = Div 2 −∆ Tr, is presented in Subsection 8.2.
A rather simple but nonetheless instructive example is obtained by considering the operator A = (Div, Curl •· ⊤ ). Although its restriction to symmetric matrices is elliptic, it is not difficult to verify that A is not elliptic when acting on the full space R n×n . We have: Theorem E. Let K be a closed convex cone contained in {A ∈ R n×n : Tr(A) > 0} ∪ {0}. For 1 p < n, Thus, in Theorem E, contrary to Theorems B, C and D, we obtain an arbitrarily high gain of integrability, just as in the usual elliptic case.
Altogether, the previous results contradict the naive expectation that the inequality (1.1) behaves like a Sobolev inequality, i.e. one cannot simply take q = p * ℓ = np n−ℓp , where ℓ is the order of A. Our examples, together with those of [4,21,57], suggest that the phenomenon that we are witnessing can be described as follows: Conjecture 1.1. Let A be an ℓ-th order operator with constant coefficients and K o ⊂ V be an open convex cone such that K o ∩ Λ A = ∅. There exists a critical exponent such that, for all closed convex subcones K ⊂ K o ∪ {0} and 1 p < n, we have . Moreover, the inequality is optimal in the sense that, for any ε > 0, there is a closed convex cone K ε ⊂ K o ∪ {0} such that the following estimate fails: . For the pairs (A, K) discussed so far, the results above verify partially or completely Conjecture 1.1 in several cases, see already Table 1 below.
We expect that a theory developed around our conjecture will prove useful as a weak convergence method for nonlinear PDE [22]. We are partly motivated by Tartar's framework in continuum mechanics [68,69], whereby one couples nonlinear pointwise constraints, represented by K and corresponding to constitutive relations, with the linear PDE constraint given by A and corresponding to balance relations. The classical theorem of Vitali asserts that strong L q -convergence is equivalent to lack of oscillations, that is convergence in measure, and lack of L q -concentrations, that is q-uniform integrability. Our estimates concern the latter, a direction which has perhaps been less studied in modern PDE.
The maximal gain of integrability is particularly relevant for ruling out nontrivial concentration effects: while for q < q max we have that q-uniform integrability follows from L qmax -boundedness, an improvement to q max -uniform integrability requires additional work. For instance, the endpoint uniform integrability is achieved in Theorem C by obtaining an extra logarithm of integrability. Alternatively, one can impose in addition that the fields v j are A-free and derive L qmax+δ bounds from reverse Hölder inequalities, see Corollaries 6.6, 7.8 and 8.4.
1.2. The case of exact constraints: quasiconcavity and optimality. In the examples discussed so far, the convex cone under consideration has the structural property and then we have . Moreover, ρ k and ρ * k are respectively concave on Γ k and Γ * k , see Section 3 for further details. Given a pair (A, K) for which (1.2) and (1.8) hold, a natural question is whether there is a number α(A, K) > 1 such that G α is Λ A -concave in K, i.e. G α is concave on the segment [A, B] ⊂ K, whenever B − A ∈ Λ A . , Ă, AND M. SCHRECKER Surprisingly, if we let α max be the largest exponent with the above property, for the examples above we have that (1.9) α max (A, K) = q max (A, K).
It has been known since the work of Tartar [68] that Λ A -concavity is necessary for Aquasiconcavity, which in general is a strictly stronger property [9,29,64,68]. Nonetheless, for the examples above we manage to prove that G αmax is A-quasiconcave when restricted to fields in K, that is, identifying the torus T n with [0, 1] n , we have By definition of α max , this inequality does not hold for any exponent α > α max . Inequality (1.10) provides a higher integrability estimate, with a sharp constant, for A-free fields constrained to K. Quasiconvexity and quasiconcavity are essential notions in the vectorial Calculus of Variations, intimately related to the weak semicontinuity of multiple integrals [1,26,48,50]. To date, the interaction of quasiconvexity with pointwise constraints is not well understood, but see the contributions in [8,17,20,43,61].
To be more specific, let us start with the case (A, K) = (Div, Γ * k ), where (1.8) holds for G = ρ * k . When k = n, ρ * n = det 1/n and Serre [57] obtained the quasiconcavity inequalitŷ We succeed in extending this inequality to all natural numbers k n, obtaining new examples of restricted Div-quasiconcave integrands: The exponent k k−1 is sharp, as (ρ * k ) α is not Λ Div -concave if α > k k−1 . Equality may be achieved in Theorem F, see Section 5 for a characterisation of this case. Moreover, Theorem F trivially implies a similar result for (A, K) = (Div 2 , Γ * k ). Let us now turn to the case (A, K) = (Curl, Γ k ), where (1.8) holds for G = ρ k . The functions F k = ρ k k are null Lagrangians [9], that is, Equivalently, both F k and −F k are Curl-quasiaffine. This gives a simple proof of the fact that α max (Curl, Γ k ) = k and of the corresponding quasiconcavity of F k . Thus (1.9) also holds in this case. A related result was obtained previously by Faraco-Zhong [25], who proved that the integrands 1 Γ k F k are Curl-quasiconvex, where 1 Γ k denotes the characteristic function of Γ k . Their result generalizes a previous result of Šverák [63] for the case k = n, and in fact it served as an early inspiration for considering the cones Γ k . The case (A, K) = ((Div, Curl •· ⊤ ), {Tr 0}) is even simpler. In this case, the wave cone is not spanning: span Λ A = {Tr = 0}. In general, A-free fields are constrained to span Λ A and thus, in this case, there are no A-free fields taking values in K. Thus Tr α is trivially A-concave on K for any α 1, and so α max = +∞, confirming (1.9).
We conclude this collection of examples by returning to the example discussed in (1.4), in which case (A, K) = (Div, Q + 2 (K)). The Burkholder integrand B K : R 2×2 → R introduced in [12], see also [40], and defined by is 1-homogeneous and satisfies Q + 2 (K) = {B K 0}, but this cone is not convex. Nonetheless, it is well-known that B α K is Λ Curl -concave for α α max = q max = 2K K−1 , and it is a long-standing conjecture [40] that B αmax K is Curl-quasiconcave. Parallel to the other examples in this paper, it was recently shown in [8] that the restriction of B αmax K to Q + 2 (K) is Curl-quasiconcave, see also [5,7,31].
We collect the key facts discussed in Sections 1.1 and 1.2 for each of our examples: {Tr 0} Tr ∞  4), all of the results stated above concern smooth fields constrained to convex cones. Using the convexity of the cones, the smoothness assumption may be relaxed by a simple mollification argument, see Remark 4.3 below. It thus appears natural to ask whether or not such estimates hold for "double", non-convex cones, such as Sym + n ∪ Sym − n . While we do not give an answer to such a question here, we provide some evidence that the situation in this case is more complicated than the convex setting.
To fix ideas, consider Serre's inequality (1.3). We look for an estimate where K n ⊂ int(Sym + n ∪ Sym − n )∪{0} and BV Div = {A ∈ L 1 (R n , R n×n ) : Div A ∈ M(R n )}. However, we show in Proposition 4.14 below that in the function space BV Div with pointwise constraint Sym + n ∪ Sym − n , the density of smooth functions satisfying the pointwise constraint fails. Thus, in order to prove any estimate for functions in this space, one cannot hope to prove such an estimate for smooth functions and then pass to a limit (as we do in the case of convex constraint), and instead one must work directly with measures.
In addition, the results of Šverák [63] and more generally of Faraco-Székelyhidi [24] show that if A ∈ L 2 (B 2 , K 2 ) satisfies Div A = 0 (or Curl A = 0) and suitable boundary conditions, then A must be valued solely in either Sym + 2 or Sym − 2 , see Proposition 4.13. In other words, there is a separation between the sets Sym + 2 and Sym − 2 for such fields. The combined effect of separation for exact constraints and lack of density of smooth fields significantly complicates any attempt to study the compensated integrability phenomenon even in the simplest non-convex case of double cones.
1.4. Outline of the paper. Section 2 collects some notation that will be used throughout.
In Section 3 we develop a duality theory for the key nonlinear objects that we will consider, which are the cones Γ k and their duals Γ * k . Although the cones Γ k are standard objects in the literature of degenerate nonlinear elliptic equations, see e.g. [13,75], their duals have been much less studied. In this section we also prove a new quantitative characterization of null Lagrangians, which is the second essential element in the nonlinear duality between Theorems B and C. We also collect some relevant facts concerning weak solutions to k-Hessian equations which will be used throughout Sections 5 to 8.
In Section 4 we state and prove Theorem A: the argument is based on the classical slicing method of Gagliardo and Nirenberg. Since the setting is particularly simple, we also take the opportunity to discuss a number of issues intrinsic to our setup, such as Ornstein's non-inequalities, the necessity of entry-wise constraints, and the non-density of smooth fields in the relevant function spaces when the cones are non-convex.
In Section 5 we prove Theorem F. The proof employs the duality theory developed in Section 3 together with an auxiliary k-Hessian equation with periodic boundary condition. Besides being a result of independent interest, the proof of Theorem F gives a simple instance of the strategy that we will employ in the following sections.
In Section 6 we enrich the strategy of the previous section by employing estimates for auxiliary k-Hessian equations, and prove Theorems B and D. We also deduce higher integrability results for solenoidal fields under pointwise constraints.
In Section 7 we prove Theorem C. We further obtain higher integrability statements for irrotational fields under pointwise constraints.
Finally, Section 8 addresses three more examples of operator-cone pairs for which we obtain elliptic estimates. The first example we consider yields Theorem E. The other examples are given by a second order operator related to the composition of Div and Curl, and a weak sub-curl type operator, which only uses some components of Curl.

Acknowledgments.
A. Guerra was supported by the Infosys Membership at the Institute for Advanced Study. M. Schrecker's research is supported by the EPSRC Early Career Fellowship EP/S02218X/1. We thank Luc Nguyen for helpful discussions concerning k-Hessian equations and Denis Serre for his interesting comments.

Notation
Throughout, we will work with the L p spaces, 1 p ∞ and will denote by p ′ the Hölder conjugate of an exponent p. Given p ∈ [1, n ℓ ), we write p * ℓ = np n−ℓp for its ℓ-th order Sobolev exponent, which is denoted by p * or p * * when ℓ = 1 or 2, respectively. We also define ( n ℓ ) * ℓ = ∞. For q n n−1 , we write q * = nq n + q for the exponent such that (q * ) * = q.
Given a matrix A ∈ R n×n , we write A = (a ij ) n i,j=1 and we denote by A i = (a ij ) n j=1 its rows. In particular, if A : R n → R n×n is a matrix field, we have Div A = (div A i ) n i=1 . We write Diag n for the space of diagonal matrices in R n×n , diag(λ 1 , . . . , λ n ) for the diagonal matrix with entries λ 1 , . . . , λ n , Sym n ≡ {A ∈ R n×n : A = A ⊤ } for the space of symmetric matrices, and for the cones of symmetric positive semi-definite and negative semi-definite matrices. We also use the notation for the Euclidean and the operator norms of A ∈ R n×n . The Euclidean norm is induced from the inner product A, B ≡ Tr(AB ⊤ ). The cones of orientation-preserving and orientation-reversing K-quasiconformal matrices will also play an important role in this paper: given K 1, we write If A ∈ Q + n (1) then we say that A is conformal, and if A ∈ Q − n (1) it is said to be anticonformal. When n = 2, the cones Q ± 2 (1) are linear spaces and we may also write, see e.g. [6, (16.33)], Finally, for any function F : R n×n → R, we will write ∇F (A) for the gradient of F with respect to the matrix variables A, reserving the notation D for the gradient operator in the spatial variables on R n .

Symmetric cones and k-Hessian equations
In Section 3.1 we develop a comprehensive duality theory for the symmetric cones. In particular, we establish some new duality results which go beyond the standard theory as described in [75]. Section 3.2 contains a new quantitative characterisation of null Lagrangians that will be essential for the nonlinear duality between Div and Curl. In Section 3.3 we collect known results about existence and regularity of solutions to k-Hessian equations. The results of this section will be extensively used in Sections 5, 6 and 7.
3.1. Symmetric polynomials and the cones Γ k and Γ * k . We are interested in the cones Γ k ≡ {λ ∈ R n : σ j (λ) 0 for j = 1, . . . , k}, where For each k = 1, . . . , n, the cone Γ k is clearly closed, convex and symmetric. Note that Γ 1 = {λ ∈ R n : i λ i 0} is a half-space, while Γ n = {λ ∈ R n : λ i 0 for all i} is the positive cone. In general, we have the following alternative characterizations of Γ k , see for instance [37,70]: Moreover, Γ k is the closure of the component of {σ k > 0} which contains Γ n .
Given a cone Γ ⊂ R n , its dual cone is the closed, convex cone Γ * ≡ {µ ∈ R n : λ · µ 0 for all λ ∈ Γ}. For l > k we have Γ l ⊂ Γ k and thus Γ * l ⊃ Γ * k . It is easy to see that Γ * 1 = {t(1, . . . , 1) : t 0} is a ray, while Γ n is self-dual, i.e. Γ n = Γ * n . For our purposes, it is convenient to consider the functions ρ k , defined for λ ∈ Γ k by We have defined the cones Γ k , Γ * k as subsets of R n . However, it is also natural to view them as subsets of Sym n : we say that a symmetric matrix A is in Γ k if λ(A) ∈ Γ k , where λ(A) is the vector of eigenvalues of A, arranged in descending order, and similarly for Γ * k . We also extend ρ k to Sym n by (3.2) ρ k (A) ≡ ρ k (λ(A)). Recalling (1.6), note that, for A ∈ Γ k , where F k is, up to a multiplicative constant, the sum of the principal k × k-minors of A.
In particular, F k is a null Lagrangian (see Appendix B). At this point it is reassuring to observe that the cones Γ * k ⊂ Sym n are exactly what one would expect: Proof. By definition we have Γ * k = {B ∈ Sym n : λ(A) · λ(B) 0 for all A ∈ Γ k }. Let Σ be the last set in the statement of the lemma. For any A ∈ Sym n we have A ∈ Γ k if and only if QAQ ⊤ ∈ Γ k for all Q ∈ O(n). It follows that the sets Γ * k and Σ have the same property. Since Γ * k ∩ Diag n = Σ ∩ Diag n , the conclusion follows. Associated to the dual cones Γ * k ⊂ R n , we have the dual functions which are defined for µ ∈ Γ * k . The definition of ρ * k can of course be extended to µ ∈ R n \Γ * k but for such µ one always has ρ * k (µ) = −∞. As above, we also define ρ * k (A) ≡ ρ * k (λ(A)) for A ∈ Sym n . The following classical result is a fundamental property of these functions and we refer the reader to [13] for a proof: The case k = 2 also admits an explicit description as ∇F 2 is linear: elementary calculations reveal that The next lemma shows that the families {ρ k } n k=1 and {ρ * k } n k=1 are ordered: Lemma 3.4. For all 1 l < k n, we have Proof. The first inequality is exactly Maclaurin's inequality, see e.g. [35], and the second inequality follows from the first. Indeed, if B ∈ Γ k ⊂ Γ l is such that ρ k (B) 1, then also ρ l (B) 1, so that Taking the infimum over B ∈ Γ k such that ρ k (B) 1, we have We have the following alternative characterizations of the dual functions: In order to prove the proposition, we first establish the following lemma: Lemma 3.6. For k ∈ {1, . . . , n} and B ∈ Γ * k , we have Proof. The case k = 1 can be verified directly, so fix k 2. We use the fact that for a convex subset K ⊆ R n we have conv(∂K) = K if and only if K is not an affine subspace or the intersection of an affine subspace with a closed halfspace, see for instance [56,Lemma 1.4.1]. This identity can be applied to Proof of Proposition 3.5. We claim that the infimum of A, B over the set of A ∈ Γ k such that ρ k (A) = 1 is attained. Indeed, according to von Neumann's trace inequality, see e.g. [49], we have the coercivity inequality Note that λ min (B) > 0 since B, being in the interior of Γ * k ⊂ Γ n , is a positive definite matrix. The claim now follows from (3.6) and a compactness argument.
Let A 0 be the point where the infimum of A, B over the set {A ∈ Γ k : ρ k (A) = 1} is attained; we necessarily have B = c∇ρ k (A 0 ) for some Lagrange multiplier c ∈ R. It follows that A 0 and B are simultaneously diagonalizable, so we may assume that both of them are diagonal matrices. Therefore To prove the reverse inequality we note that, as above, the right-hand is in fact attained at a matrix A 1 ∈ Γ k with ρ k (A 1 ), which we may assume to be diagonal.
A, B , since both Γ k and ρ k are invariant under the operation A → Q ⊤ AQ. The conclusion now follows from Lemma 3.6.
As an immediate consequence of Proposition 3.5, we obtain the main result of this subsection, which will play a key role in sections 5 and 6. Proposition 3.7 (Duality pairing). For k = 1, . . . , n and any A ∈ Γ k , B ∈ Γ * k , we have Moreover, for each B ∈ Γ * k there is some A ∈ Γ k such that equality is attained. Proof. We may assume that ρ k (A), ρ * k (B) > 0 as otherwise, according to Lemma 3.2, there is nothing to prove. In this case B ∈ Γ * k \∂Γ * k and, by homogeneity, we may also assume that ρ k (A) = 1. The proposition now follows from Proposition 3.5.
The first part of Proposition 3.7 was stated in [46, Proposition 2.1], although their proof omits a few details.
The following lemma establishes a useful relationship between ρ k and its dual function: Proof. Let A,Ã ∈ Γ k be arbitrary. Using the concavity of ρ k provided by Proposition 3.3 we have where the last equality follows from Euler's identity for homogeneous functions; thus We first note that (3.7) shows that 0 ∇ρ k (A),Ã for allÃ ∈ Γ k : thus ∇ρ k (A) ∈ Γ * k . Let us now assume that ρ k (Ã) = 1. Taking the infimum over suchÃ in (3.7) gives 1 ρ * k (n∇ρ k (A)), where used the characterization of ρ * k given by Proposition 3.5. To prove the opposite inequality, assume that ρ k (A) = 1; this is justified by homogeneity, By combining Proposition 3.7 with Lemma 3.8, we obtain: Clearly F k is smooth and so, to prove that it is a local diffeomorphism, it is enough to show that ∇F k (A) is non-singular for all A ∈ Γ k \∂Γ k . By Lemmas 3.4 and 3.8, recalling > 0, and so by the Inverse Function Theorem ∇F k is a local diffeomorphism. It remains to prove that this map is bijective.
Let us begin with surjectivity. Applying Proposition 3.7, for each By Lemma 3.8 and the Lagrange multiplier argument in the proof of Proposition 3.5, we have .
To prove injectivity, we note that it is sufficient to show that the set {ρ k 1} is strictly convex, as then ∇ρ k is the normal to {ρ k = 1} and hence it is injective. By Lemma 3.8, if ∇F k (A) = ∇F k (Ã) for A,Ã ∈ int Γ k , then ρ k (A) = ρ k (Ã) and so ∇ρ k (A) = ∇ρ k (Ã) as well. Thus the injectivity claim follows.
The strict convexity of {ρ k 1} follows from [27,Theorem 5] (see also the comment at end of page 964 there): a positive level set of a hyperbolic polynomial P bounds a strictly convex region, modulo the lineality space L P . In our case, P = F k and we have L F k = {0}: in Gårding's terminology, F k is complete, as it depends on all variables.

3.2.
A quantitative characterisation of null Lagrangians. For an arbitrary minor M , the next proposition establishes a form of nonlinear duality between the divergence of ∇M and the curl of the matrix field on which it acts: Recall that we use the notation ∇ for derivatives with respect to the A variables and reserve D for spatial derivatives.
Proof. The proposition is most easily proved through differential forms. Let A i ∈ R n be the rows of the matrix field A ∈ R n×n ; we think of A i as a 1-form given by A i = n j=1 a ij dx j and so we may identify Curl A with (dA i ) n i=1 , since We first deal with the case where k = n and M = det. We consider the linear map Λ n−1 A : Λ n−1 (R n ) → Λ n−1 (R n ), defined as usual on decomposable elements through (3.8) Λ n−1 (A)(e 1 ∧ · · · ∧ e i ∧ · · · ∧ e n ) = A 1 ∧ · · · ∧ A i ∧ · · · ∧ A n , where· means that the corresponding term is omitted. We may identify the above (n − 1)form with the i-th row of cof A, which as expected we denote by (cof A) i : indeed, writing (e i ) for the canonical basis of R n , where A ij is the (n − 1) × (n − 1) matrix obtained from A by deleting the i-th row and the j-th column. Taking exterior derivatives in (3.8), we compute Thus, applying the triangle inequality, we obtain where we used the fact that, for any indices i 1 < · · · < i n−2 , This last inequality can be deduced as follows: similarly to (3.9), the left-hand side is the norm of the vector M i 1 ...i n−2 (A) of those (n − 2) × (n − 2) minors of A which only depend on the i 1 , . . . , i n−2 rows of A; the right-hand side is the norm of the vector of all (n − 2) × (n − 2) minors of A. Hence (3.10) | Div(cof(A))| C(n)| Curl A||∇ 2 det A| and since ∇ det = cof, the proof of this case is complete. We now prove the general case. Without loss of generality we assume that M is the k × k minor which depends on the first k rows and columns. The main observation is that , where the subscripts denote the variables with respect to which one takes the divergence and ∇ (k) is the gradient operator with respect only to the matrix variables (a ij ) k i,j=1 , thus ∇ (k) M : R n×n → R k×k . To prove (3.11), we write π k : R n×n → R k×k for the linear operator such that π k (A) ij = a ij if 1 i, j k, and so We see that ∇M (A) ∈ R n×n is such that (∇M (A)) ij = 0 unless 1 i, j k, and thus (3.11) is proved.
It follows from (3.11) that, to estimate Div ∇M (A) at a pointx = (x 1 , . . . ,x n ), we may consider the restriction of A to the copy of R k given by {(x k+1 , . . . , x n ) = (x k+1 , . . . ,x n )}, and apply (3.11) and (3.10) to obtain and also, as in the proof of (3.11) above, we conclude the proof of the proposition.
Proof. Up to a multiplicative constant, ρ k k (A) is the sum of the k × k principal minors of A. Since ρ k−1 k ∇ρ k = 1 k ∇ρ k k , the conclusion follows from Proposition 3.10 by noting that, for a k × k minor M , we have |∇ 2 M (A)| C|A| k−2 by Hadamard's inequality.
To end this subsection, we note that Proposition 3.10 yields a quantitative version of the well-known fact (see e.g. [9]) that a C 1 integrand F : if and only if F can be written as a linear combination of minors. Indeed, we have: be an integrand and k ∈ [2, ∞). The following, are equivalent: (a) k n is an integer and F is a linear combination of k × k minors.
In particular, the estimate above can be viewed as a perturbed Euler-Lagrange equation for homogeneous null Lagrangians.
Proof. If we assume that F is a homogeneous null Lagrangian, the estimate follows at once from Corollary 3.11. Conversely, if the estimate holds, we can first plug in A = Du for u ∈ C 2 (R n , R n ) to see that F is a null Lagrangian. By the fact mentioned above, F is therefore an affine combination of minors, of possibly different orders. We then replace A with tA for t > 0 in the inequality of (b) and notice that the right hand side is (k − 1)homogeneous, so the left hand side must be too, as it is a sum of homogeneous terms. It follows that F must be k-homogeneous, which concludes the proof.
3.3. k-Hessian equations. We will frequently make use of the existence and regularity theory for the so-called k-Hessian equation, and so we collect here a few results in this direction.
For k = 1, . . . , n, the k-Hessian equation is the equation where B ⊂ R n is a ball and u is k-admissible. Note that this is a slight variant of the usual definition in the literature, where typically one uses F k instead of ρ k . The equations are, of course, equivalent up to taking k-th powers and adjusting the integrability assumptions on f accordingly. We say that a function When restricted to k-admissible solutions, equation In order to consider weak solutions of (3.12), it is important to extend the notion of kadmissibility to solutions which are not necessarily C 2 . For our purposes it will be enough to consider continuous weak solutions.
The definition of k-admissible functions coincides with the notion of the inequality ρ k (D 2 u) 0 holding in the viscosity sense, see for instance [70] or [75, Definition 8.1] for further details.
It is natural to consider the Dirichlet problem for (3.12). The classical existence theory, originally due to Caffarelli-Nirenberg-Spruck [13], see also [38], yields in particular the following result: We also refer the reader to Theorem 3.4 in [75] and the ensuing remarks. The next result can be found in [71, Lemma 2.1, Theorem 2.2]: We will also use the following integral gradient estimate due to Trudinger-Wang [70], see also Corollary 7.1 and the remarks following Definition 8.1 in [75]: Then, for all q ∈ [1, nk n−k ) and all Ω ⋐ B, there exists C > 0 such that Besides Theorem 3.14, we require the following existence result for k-Hessian equations on the torus: The case k = n was proved by Li [47] and we note that the case k = 1 is elementary, as then (3.13) is just the Poisson equation. To the best of our knowledge, Theorem 3.17 has not been proved for general k in the literature, although it follows from well-known arguments and results; the interested reader can find a proof in Appendix A.

Elliptic estimates in two dimensions
In this section, we prove our first elliptic estimates for the divergence and the curl operators under pointwise constraints. Here we will discuss only the case where n = 2 and p = 1: thus, the estimates we prove can be thought of as a version of the Gagliardo-Nirenberg inequality for certain non-elliptic operators. We also note that the methods we employ here are different from and more elementary than those used in later sections.
In Section 4.1 we will prove the main result of this section, Theorem 4.1, which in particular contains Theorem A. In Section 4.2 we deduce from Theorem 4.1 a Gagliardo-Nirenberg inequality for the curl and show an ad-hoc Ornstein non-inequality demonstrating the nonequivalence of these estimates. We also discuss whether a general version of Ornstein's L 1 non-inequality [55] may hold under pointwise constraints. Section 4.3 collects examples which show that Theorem 4.1 is essentially optimal. In Section 4.4 we discuss in some detail the possibility of extending our results to non-convex cones.
4.1. Elliptic estimates for the divergence. Given a matrix A ∈ R n×n , recall that we denote by A i its rows and a ij its entries. The following is the main result of this section, from which Theorem A follows immediately: In fact, it will be clear from the proof that there are two continuous one-parameter families of admissible cones for which estimate (4.1) holds. The specific examples K ± i stated above are the ones having the clearest characterisation.
By restricting the cones in Theorem 4.1, we readily obtain a Gagliardo-Nirenberg-type inequality for the divergence: be as in Theorem 4.1 and consider a cone K ⊂ Q ± 2 (K)∩K ± i for some K 1 and i = 1, . . . , 4. There is a constant C = C(K) > 0 such that, for any Notice that, in the above results, we restrict ourselves to smooth matrix fields. We now remark on the possibility of extending these inequalities to larger classes of rough fields, a point which we will also discuss in further detail in Section 4.4 below.

Remark 4.3 (Extension to rough fields). Let K ±
i be the cones in Theorem 4.1. Since these cones are convex, a standard mollification argument shows that estimate (4.1) holds . In order to extend estimate (4.1) to fields which are only in L 1 (R 2 ), we need to constrain the target further. For instance, if A ∈ L 1 (R 2 , Sym + 2 ) vanishes outside a compact set, and if moreover Div A ∈ M (R 2 ), then estimate (4.1) still holds. Indeed, using the convexity of Sym + 2 , a simple mollification shows that there is a sequence A ε ∈ C ∞ c (R 2 , Sym + 2 ) such that Div A ε → Div A in M (R 2 ) in the strict topology and A ε → A in L 1 (R 2 ). In particular, A ε → A a.e. in R 2 . Since Sym + 2 ⊂ {det 0} we may apply Fatou's lemma to deduce that More generally, the same conclusion holds if we replace Sym + 2 with any convex subcone of some K ± i on which the determinant has a fixed sign. Similarly, Corollary 4.2 also extends to compactly supported matrix fields which a priori are only in L 1 (R 2 , K), as long as K ⊂ Q ± 2 (K) ∩ K ± i is assumed to be convex. Remark 4.4 (Boundary conditions). The estimate in Corollary (4.2) is equivalent to the more general estimate whenever Ω ⊂ R 2 is a bounded Lipschitz domain with unit normal ν. Clearly this last estimate recovers the one in Corollary 4.2, but it can also be deduced from it by a standard argument, which we include here for the sake of completeness. We begin by extending A to a matrix fieldÃ which is zero outside Ω. Take R large enough so that Ω ⋐ B R (0). ThenÃ = 0 on a neighborhood of ∂B R (0) and moreover Thus there is no loss of generality in assuming that Ω = B R (0) is a ball, hence ν = x R . Now consider a Lipschitz radial cutoff defined by Clearly χ ε A ∈ K a.e. and so we may apply Corollary 4.2 to find As ε → 0, we have Thus, sending ε → 0 in the previous inequality, we have the estimatê We are now ready to proceed with the proof of Theorem 4.1.
Proof of Theorem 4.1. We begin by observing that it suffices to establish (4.1) when A takes values in K + 1 . Indeed, this implies immediately an analogous inequality if A takes values in K − 1 , noting that A ∈ K + 1 if and only ifÂ ∈ K − 1 , where the rows ofÂ are defined asÂ 1 = A 1 andÂ 2 = −A 2 . Thus it is enough to show that the inequality for fields in K ± 1 implies the same inequality for fields in K ± i , i = 2, 3, 4.
where R θ is a rotation matrix by angle θ. We haveÃ j (x) = R −θ A j (R θ x) and so In particular, we see that both sides of the inequality (4.1) are kept unchanged under the transformation A →Ã. Noting that , the conclusion follows.
Thus our task is to show (4.1) for K + 1 . We are going to work in conformal/anti-conformal coordinates, that is, we pick the following orthogonal basis of R 2×2 : Note that E 1 , E 4 span the space of conformal matrices, while E 2 , E 3 span the space of anti-conformal matrices. We will write and we record the formulae Using the fundamental theorem of calculus, we get and also Note that the assumption A ∈ K + 1 ensures that b 1 |b 3 |; hence we can take the product of the last two inequalities and integrate over R 2 to obtain First, we observe that II = III = 0 as the first factor in II (respectively III) is independent of x 1 (respectively x 2 ) and A is compactly supported. Moreover, we check for I that . Finally, as b 2 and b 4 have compact support, we integrate by parts twice in IV to obtain Collecting everything, we find the estimatê as wished.
We conclude this subsection with a version of Theorem 4.1 for p > 1: Proposition 4.5. Fix K 1 and let p ∈ (1, 2) be such that p * < 2K K−1 . There is a constant such that, for all compactly supported fields A ∈ L 1 loc (C, Q + 2 (K)) with Div A ∈ L p (C),

This estimate fails for
Proof. Let us identity A ∈ R 2×2 with a pair (a + , a − ) ∈ C 2 according to the usual rule Az = a + z + a −z . The condition A ∈ Q 2 (K) is equivalent to the equation where R is large enough so that supp A ⊂ B R (0). Let us denote by ∂,∂ the usual Wirtinger derivatives acting on complex-valued functions and by S the Beurling-Ahlfors transform, which is a singular integral operator with the characteristic property S •∂ =∂ • S = ∂ in D ′ (C), see e.g. [6, §4] for further information on this operator. We also note that S −1 = S· and that both S and S −1 are bounded in L q (C) for 1 < q < ∞.
Under the obvious identifications, we compute 2 Div A =∂a + + ∂a − =∂(a + + Sa − ). Thus, as∂ is elliptic (but not canceling), if p ∈ (1, 2) we have Since both a + and a − are compactly supported, from (4.2) we easily deduce the representation formulas Here we note that the operator I − µS is invertible on L q (C) if and only if 2K K+1 < q < 2K K−1 [6, §14]. Thus, by our assumption, Since 2 = 1 * < p * < 2K K−1 , the conclusion follows. The failure of the estimate at the endpoint p = 1 will be proven in Lemma 4.10 below. For the failure at the endpoint p * = 2K K−1 , it suffices to consider the spherically symmetric map and one easily deduces that Elliptic estimates for the curl. We begin this subsection by rewriting Theorem 4.1 using the curl operator instead of the divergence.
Proof. The corollary follows from the identity curl A i = div A ⊥ i and the fact that the family of cones . Although the previous estimate follows at once from Theorem 4.1, it is nonetheless a genuinely different inequality. Indeed, we have the following Ornstein non-inequality (note that Sym + 2 ⊂ K + 1 ):

Proof. Consider the matrix fields
.

One easily checks that
.
Note that A = B = Id on ∂([0, 2π] 2 ). In order to get a compactly supported example, it is enough to consider a smooth cutoff η with support in [0, 2π] 2 which is equal to 1 in [π/2, 3π/2] 2 . We may replace the fields A ε , B ε by ηA ε , ηB ε : they still take values in Sym + 2 and their divergence and curl is affected only by a constant which is independent of ε.
It would be interesting to establish an analogue of Ornstein's result [55] for constrained fields. We propose a formulation of such a result in the next remark.
Remark 4.8 (Ornstein's non-inequality with constraints). Given a finite-dimensional vector space V, let K ⊂ V be a closed, convex cone which spans V. Let P 1 , P 2 be two ℓ-th order, constant coefficient partial differential operators on R n from V to W 1 , W 2 , respectively. If there is a constant C such that , is it the case that |P 1 ϕ| C ′ |P 2 ϕ| pointwise for some constant C ′ and all ϕ ∈ C ∞ c (R n , K)? An inspection of the standard proofs in the unconstrained case in [18,23,42,55] shows that they do not extend to this setting in an obvious way.

Optimality. In this subsection we collect several examples and remarks which show that Theorem 4.1 is essentially optimal.
The first point we address is whether one can hope for a gain of higher integrability if one assumes that the divergence of the matrix field is itself more integrable. This question was posed already by Serre [57] and answered negatively by De Rosa-Tione [21]: Proposition 4.9 (Necessity of p = 1). Fix p ∈ (1, n). Let A ∈ L p (B n , Sym + n ) be such that A = 0 near S n−1 and Div A ∈ L p (B n ). In general, (det A) 1/n ∈ L p * (B n ). In fact, for 1 < p < 1 * , there is no ε > 0 such that we always have (det A) 1/n ∈ L 1 * +ε (B n ).
A different question, which was also briefly addressed by Serre [58, §4], is the extent to which the entry-wise constraints are truly necessary. One could imagine, for instance, that Proposition 4.9 may still hold if we replace K + 1 with the (non-convex) cone of matrices with non-negative determinant; however, this is not the case: Lemma 4.10 (Necessity of entry-wise constraints). There is no constant C such that, for Proof. Let G(x) = 1 2π log |x| be the Green's function for the planar Laplacian. We define, for |x| < 1, the field Note that A takes values in the two-dimensional linear space Q + 2 (1) of conformal matrices, cf. (2.2). We have A ∈ L 2,∞ \L 2 (B 2 ) and moreover A. GUERRA, B. RAIT , Ă, AND M. SCHRECKER Take A ε ≡ η(A * ρ ε ), where ρ ε is the standard mollifier and η ∈ C ∞ c (B 2 ) is a non-negative cutoff with η = 1 near 0. Clearly A ε still respects the constraint and for some constant C independent of ε, as wished.
Remark 4.11. An example related to the previous lemma can be obtained by swapping the two rows of A in (4.3): by doing so, we get a matrix fieldÃ which shows that there is also no estimate for fields constrained to take values in Sym 2 ∩ {det 0}. Note that K − 2 ∩ Sym 2 is a two-dimensional half-space, whileÃ takes values in the full linear space containing this half-space. Thus entrywise constraints are in some sense necessary for an estimate.
In fact, for fields satisfying the exact constraint Curl A = 0, we have the following rigidity result, which is a particular case of a theorem of Šverák [63], see also the remarkable generalisations due to Faraco-Székelyhidi [24] as well as [65,66,67] for related results: Proposition 4.13. Let A ∈ L 2 (B 2 , Sym 2 ) be such that Curl A = 0 and A| S 1 is constant. Then either A ∈ Sym + 2 a.e. in Ω or A ∈ Sym − 2 a.e. in Ω. As before, one can convert the constraint Curl A = 0 into the constraint DivÃ = 0 through the map which interchanges Sym + 2 and Sym − 2 . A question related to Question 4.12 is whether it is possible to approximate rough fields (with or without exact differential constraints) by smooth fields that retain the pointwise constraint. As mentioned before, when the target is a convex cone, this is easily achieved by a straightforward mollification argument. However, when convexity fails, the situation is far more complex, and Lemma 4.10 shows that in general it is not possible to do so. We now give an example which shows that the same phenomenon happens in the setting of Question 4.12.
In other words, there is no smooth sequence approximating A in the strict topology in BV Div that preserves the pointwise constraint.
Proof. Let A 1 ∈ K 0 , A 2 ∈ −K 0 be such that the line segment [A 1 , A 2 ] does not pass through the origin. Define the function A by One easily sees that Div A = (A 2 − A 1 )e 1 H 1 x 1 =0 , and hence Div A M = 2|(A 2 − A 1 )e 1 |. Moreover, by construction of A 2 and A 1 , Suppose for a contradiction that there exists a smooth sequence A ε as in the statement of the proposition. We fix small auxiliary parameters δ 0 , λ > 0 which will be chosen later, as well as a number δ ∈ ( δ 0 2 , δ 0 ).
is a smooth manifold}. By Sard's Theorem, this set has full measure in R. By the co-area formula,

) is a smooth manifold and (4.7)
H for otherwise the integral in the right-hand side of (4.6) would be infinite. Let us write R δ ≡ [−δ, δ] × [−1 + δ, 1 − δ] and consider the sets According to the General Position Lemma the set E 1 has full measure [34], and by (4.5) and Fubini's theorem the same holds for E 2 , so we take Strictly speaking, to apply the General Position Lemma, we would need the sets R δ to be smooth, while they are just Lipschitz. This technicality can be dealt with by rounding off the corners of R δ a little bit and dealing with the new terms generated by the round parts as in (4.12) below. We leave the details to the interested reader and instead we pretend that R δ is a smooth manifold. More importantly for our purposes here, note that by construction of E 1 the relatively open sets We now estimate the L 1 norm of Div A ε as follows. First, we break the region of integration into two pieces: Working on Ω + , we estimatê where we have used that ∂Ω + is Lipschitz and A ε is smooth to perform the integration by parts; here ν denotes the unit normal to ∂Ω + . We divide the boundary of ∂Ω + into four portions: Note that, equivalently, we have λ |A 2 | by Chebyshev's inequality and the definition of E 2 ; note that, as |v 1 | We estimate the integral of A ε ν on each of the components of ∂Ω + separately and we use throughout the fact that v ∈ E 2 . Starting with the first term, we have (4.10) where we used (4.9). For the second term, by (4.8), The third term is estimated using that H 1 (Σ + 3 ) 4δ as (4.12) Finally, on Σ + 4 , we have that f ε = b, and hence, as K ⊂ Sym + 2 ∪ Sym − 2 , we have the simple estimate |A ε | |f ε | = b. So using (4.7) we obtain (4.13) Thus, combining estimates (4.10)-(4.13), we obtain Arguing similarly on Ω − , we obtain By (4.4), if we choose δ and λ sufficiently small, we conclude that Div A ε L 1 → 2|(A 2 − A 1 )e 1 | = Div A M , a contradiction to the strict convergence.

New examples of div-quasiconcave integrands
The purpose of this section is to prove that appropriate powers of the integrands ρ * k : Γ * k → [0, +∞), as defined in Section 3.1, are Div-quasiconcave. When k = n we recover inequality (1.11), due to Serre [57,58], and for k = n we obtain new examples of compensated integrability for divergence-free fields constrained to the cones Γ * k : The inequality also holds for Div 2 -free fields. Moreover, the inequality is sharp: in the class of smooth fields A ∈ C ∞ (T n , Γ * k ), equality is attained if and only if A = ∇F k (S + D 2 ϕ) for a periodic ϕ ∈ C ∞ (T n ) and S ∈ Γ k such that S + D 2 ϕ ∈ Γ k .
The authors would like to thank D. Serre for pointing out to us that such matrix fields achieve equality in Theorem 5.1.
We recall that Div 2 is an abbreviation for the operator div • Div. Note that, when k = 1, fields in Γ * 1 are scalar multiples of the identity, so the condition of being Div-free is equivalent to being constant. Hence the inequality is trivial in this case.
Proof. Since the cones Γ * k are convex, it is enough to consider the case where A ∈ C ∞ (T n , Γ * k ) is Div-free as else we may mollify. We write f ≡ ρ * k (A) k/(k−1) . We may also assume that A is uniformly inside Γ * k , i.e. that f is uniformly positive, as otherwise we replace A with A ε ≡ A + ε Id, apply the inequality to A ε , and then send ε → 0.
Since a Div-free field is also Div 2 -free, it suffices to prove the inequality in the latter case. Take an arbitrary S ∈ Γ k such that Sx · x + ϕ be the solution of the equation ρ k (D 2 ψ) = f 1/k on T n provided by Theorem 3.17; in particular,´T n ϕ = 0. Using Proposition 3.7, we then estimatê whereĀ ≡´T n A dx and we have used that A is Div 2 -free to integrate by parts twice in the ϕ term. Note that, by convexity of Γ * k , we haveĀ ∈ Γ * k . We now choose S, according to Proposition 3.7, so that 1 It follows that ˆT .
Recalling that f = ρ * k (A) k/(k−1) and the definition ofĀ, the inequality follows by taking the k k−1 power.
To show the equality case, we inspect the above proof. In order to have equality in (5.1), we must have that By the proof of Proposition 3.5, at any point x 0 ∈ T n where this equality is achieved, Moreover, by the same proof, this implies that Recalling now the definition of ψ, we have that by Lemma 3.8. Solving for c, we have obtained Moreover, by the proof of the inequality above, any A of this form achieves equality in (5.1), as wished.
Remark 5.2 (k = 2). When k = 2 one can prove a stronger version of Theorem 5.1 using standard arguments, as we now show. We consider the quadratic form which extends the representation (3.5) of (ρ * 2 ) 2 to any A ∈ Sym n : in particular, we have It turns out that F 2 is Div-quasiconcave in Sym n , regardless of pointwise constraints. To prove this, we recall a celebrated theorem due to Tartar [68] which asserts that a quadratic form is Div-quasiconcave if and only if it is non-positive on Λ Div ≡ {A ∈ Sym n : det A = 0}. To check this last condition, let A ∈ Λ Div have eigenvalues (λ 1 (A), . . . , λ n−1 (A), 0). Then a straightforward calculation shows that We now show that the exponents obtained in Theorem 5.1 are optimal for the quasiconcavity of the integrands ρ * k . Our argument is based on the classical fact [68] that if an integrand F is Div-quasiconcave then it is necessarily concave along any line parallel to a singular matrix, that is, for any A ∈ Sym n , t → F (A + tX) is concave, whenever det X = 0.
Although this fact is typically proved in the case where the matrix fields are unconstrained, the proof extends to the case where the integrand is Div-quasiconcave only with respect to fields which take values in Γ * k , see e.g. [57, §1.2] for a similar argument. We will prove the following result: Proof. It suffices to prove the 'only if' part of proposition, as the 'if' part is a consequence of Theorem 5.1. It is enough to show that if (ρ * k ) α is Λ Div -concave on diagonal matrices then α ≤ k k−1 . Let us consider, for t 0, the straight line (A t ) t 0 , where A t = diag(1+t, 1, . . . , 1) ∈ Γ n . For any A ∈ Γ k , we have It follows that the image of the line (A t ) t 0 under ρ k−1 k (·)∇ρ k (·) is a straight line parallel to Λ Div = {A ∈ Sym n : det A = 0}. Indeed, n k ∇F k (A t ) = diag (a n,k , a n,k + q n,k t, . . . , a n,k + q n,k t) where a n,k = n − 1 According to Lemma 3.8, we have where c n,k ≡ n k 1 k and b n,k , m n,k > 0 are appropriately chosen constants. By the previous paragraph, if (ρ * k ) α is Λ Div -concave then the function t → ρ * k (ρ k (λ t ) k−1 ∇ρ k (λ t )) α is concave, and so the function t → (b n,k + m n,k t) (k−1)α/k must be sublinear. Thus we must have α k k−1 , as wished.

Elliptic estimates for the divergence
In the previous section we saw that divergence-free fields constrained to take values in the cones Γ * k have improved integrability. A natural question, especially in light of Serre's inequality (1.3), see [57,58], is whether a similar result holds for fields which do not satisfy the exact constraint Div A = 0, but instead have divergence bounded in some Lebesgue space. In other words, do we have elliptic estimates for More precisely, for A as above, do we have estimates for some set of exponents 1 * q q max and p 1 sufficiently large? As we will see below, the answer is positive. Through a nonlinear duality argument with solutions of an appropriate k-Hessian equation, we will show that (6.1) holds in the full range of exponents when A = Div 2 and a partial set of exponents (including both q max and 1 * ) when A = Div, see Remark 6.3. Moreover, in the terminology of Conjecture 1.1, we show that The second equality in particular implies that Conjecture 1.9 in [4] is false. Lemma 3.4. Note that estimate (6.3) already yields (6.1) when q = 1 * .
Proof. Let A ∈ C ∞ c (B n , Γ * k ) and extend A by zero to the ball 2B n . We let and apply Theorem 3.14 to obtain a k-admissible function ψ ∈ C 1,1 (2B n ) such that In particular, D 2 ψ ∈ Γ k a.e in 2B n . By Proposition 3.16, as p ′ < k * = nk n−k by assumption, we have By Lemma 3.15, ψ satisfies the estimatê where we have used that f is compactly supported inside B n in the first equality. We then estimate using Proposition 3.7 and integration by parts: The first inequality in the statement follows. The second inequality follows by observing that for A ∈ K we have |A| ρ * k (A). In fact, the range of exponents in Proposition 6.1 is optimal, as we show next. Proposition 6.2. Let 2 k < n and p = nk nk−(n−k) = k k−1 * . Then, for any j ∈ N, there exists a matrix field Proof. Let η(x) 0 be a standard radial mollifier. Given ε > 0, consider the solution to the problem which, according to [72,Equations (3.2), (3.16)], is given by Thus w ε k is smooth, radial, and D 2 w ε k (x) ∈ int Γ k for all x ∈ B n . Moreover, writing v ε k (|x|) ≡ w ε k (x), for r ε we have the estimates where the constants depend on n and k but are independent of ε < 1. In particular, using the identity we find that which is well-defined as a map into the interior of Γ * k by Lemma 3.8. For δ > 0, consider the radial, scalar function (6.6) φ δ (x) ≡ log log(1 + 1 |x| ) |x| δ, log log(1 + 1 δ ) |x| < δ, and, without risk of confusion, let us also denote by φ δ a smooth approximation to this function. We now set A ε,δ (x) ≡ φ δ (x) A ε (x) and observe that, when |x| δ, for i = 1, . . . , n as Div A ε = 0 by Proposition 3.10, while for |x| < δ, Div A ε,δ ≡ 0. We assume in addition that 0 < ε < δ. Using the bounds from (6.5) and the k − 1 homogeneity of ∇F k , we estimate where we have used the definition of p to simplify the exponent and the fact that p > 1 to see convergence of the integral with a bound independent of ε > 0. Note that it is in this last step that we crucially use the assumption k < n: when k = n, we have p = 1 and the integral diverges.
In addition we calculate , where we have used the homogeneity of ρ * k and Lemma 3.8. Therefore, taking a nonnegative continuous test function ϕ ∈ C 0 c (B n ), and recalling that 0 < ε < δ, we test where we have used that the support of η ε is B ε , on which φ δ is constant. Then, as η ε is an approximation to the identity, by taking ε = ε(δ) sufficiently small (note φ δ is uniformly integrable), we deduce that and thus we conclude that It is interesting to inspect formally the above proof when k = 1, as it served as motivation for the general case k < n. We have that ∇F 1 is constant and thus A ε,δ can be identified with the scalar field φ δ defined in (6.6). Formally, when k = 1 the relation p * = k k−1 specifies p = n, and thus the above argument yields the failure of the embedding On the other hand, in the other limiting case, when k = n, this counterexample no longer holds, as the constructed field will not have an L 1 divergence due to the failure of integrability of r −1 log(1 + 1 r ) −1 on (0, 1). This reflects the fact that the Monge-Ampère equation (the dual equation when k = n) enjoys an unusual gain of regularity: when written in 1-homogeneous form, with suitable boundary condition, the solution u typically satisfies a Lipschitz bound that improves the usual failure of the embedding W 2,n ֒→ W 1,∞ that obstructs such an estimate in the linear case (and, indeed, for k < n). Remark 6.3 (Interpolation). Since the space {A ∈ L p (B n , Γ * k ) : Div A ∈ L p (B n )} is nonlinear, we are unable to interpolate between the end-point estimates in Proposition 6.1 and Serre's inequality (6.3), and so we do not obtain the conjectural estimates Nonetheless, the monotonicity ρ * k ρ * l for k l implies a partial result towards (6.7): for each k l n, l ∈ N, we have the estimate where p > l l−1 * . In the case l = n, we may take the critical estimate with p = 1. A key difficulty in proving (6.7) directly, i.e. by the method of proof used to treat the end-point cases, is that the needed gradient estimates for solutions of k-Hessian equations are not known. When A = Div 2 , this can be remedied by integrating by parts twice: by doing so, we only require L p estimates on the solution of the k-Hessian equation, and not on its gradient. Proposition 6.4. Let 2 k n. We have the estimate , whenever either of the following conditions hold: (a) p > 1, k n 2 , and q = min{p * * , k k−1 }; K). It may seem that the restriction k n 2 is arbitrary in (a), but in fact, when k > n 2 , we automatically have k k−1 < 1 * * , so there is no loss in restricting to the case p = 1 and we fall in case (b). We also note that the restriction q k k−1 is necessary, as can be seen from Proposition 6.5 below.
Proof. We begin by setting f = ρ 2 (A) q−1 and we apply Theorem 3.14 to find a k-admissible function ψ ∈ C 1,1 (B n ) solving In particular, D 2 ψ ∈ Γ k a.e. in B n . We now apply the L p estimates of Lemma 3.15, noting q ′ k by assumption, to estimate the assumed restrictions on the exponents p and q ensure that Lemma 3.15 is applicable. We then estimate using Proposition 3.7 and integration by parts twice: The first inequality now follows directly and the second is a consequence of the fact that if K is a closed convex cone in int Γ * k ∪ {0}, then |A| ρ * k (A). 6.2. The case of exact constraints: optimality and higher integrability. The quasiconcavity result in Theorem 5.1 suggests that, in order to see that the exponent k k−1 in both Propositions 6.1 and 6.4 is optimal, it should be enough to consider Div-free fields. This is indeed the case: Proposition 6.5. Let 2 k n and ε > 0. There is a closed convex cone K ε ⊂ int Γ * k ∪{0} and a matrix field A ε ∈ C ∞ (B n \{0}, K ε ) such that Proof. With r = |x|, let u ε = u ε (r) be a convex function and take is, up to a multiplicative constant, the sum of the k × k principal minors, Proposition 3.10 shows that A ε is divergence-free, since D 2 u ε is evidently Curl-free. Writing ν = x r , we have and so in particular the eigenvalues of D 2 u ε (x) are (ü ε (r),u ε (r)/r, . . . ,u ε (r)/r). Recalling (3.1), it follows that We take α ≡ n k + ε(k − 1) and we split the analysis into two cases.
Case 1: 2 k n 2 . In this case we have 1 α < 2 and so we may take u ε (x) = −r 2−α , which is clearly convex. Using Lemma 3.8 again we see that In particular, using spherical coordinates we calculatê by the definition of α. Since ρ k (A ε ) C(n, k)|A ε |, the conclusion follows. Case 2: n 2 < k n. In this case we have α 2 and so we may take u ε (x) = r 2−α , which is convex. Repeating the arguments of the previous case the conclusion follows.
Proof of Theorems B and D. The proof of Theorem B follows almost directly from Propositions 6.1 and 6.5, while the last claim in the statement is the content of Proposition 6.2. To verify the first non-inequality, we observe that, given an ε > 0, we may multiply the function A ε of Proposition 6.5 by a test function ρ ∈ C ∞ c (B n , [0, 1]) such that ρ ≡ 1 on 1 2 B n . Then, as A ε is smooth and bounded on the annulus B n \ 1 but ρA ε ∈ L k/(k−1)+ε (B n ). Taking a smooth sequence of approximations to A ε , we conclude. In particular, (6.2) holds.
The proof of Theorem D follows along very similar lines, using Proposition 6.4 instead of Proposition 6.1.
We conclude this section by noting that, despite Proposition 6.5, solenoidal fields uniformly inside Γ * k have higher integrability: Corollary 6.6. Let K ⊂ int Γ * k ∪ {0} be a closed convex cone. Let A j ∈ C ∞ (R n , K) be a sequence bounded in L 1 loc (R n ) and such that Div A j = 0. For a ball B = B R (x 0 ) and θ ∈ (0, 1), we have In particular, there exists δ = δ(K) such that (A j ) is bounded in L k/(k−1)+δ loc (R n ), and so (A j ) is locally k k−1 -uniformly integrable. Proof. Fix a ball B ⊂ R n . If Div A = 0, we have, for a scalar field ρ ∈ C ∞ c (B), Div 2 (ρA) = div(ρ Div A + ADρ) = Div A · Dρ + D 2 ρ, A = D 2 ρ, A .
We therefore have that Div 2 (ρA j ) ∈ C ∞ c (B) is bounded uniformly in L 1 (B). Fix a constant τ ∈ (0, 1) to be chosen later and choose ρ non-negative and such that ρ = 1 in τ B. Thus, letting 1 < q < 1 * * be such that q k k−1 , Proposition 6.4 implies with a constant that depends only on the radius of B and τ . If k k−1 < 1 * * we are done. If k k−1 1 * * , we take someτ ∈ (0, τ ) andρ ∈ C ∞ c (τ B) such thatρ ≡ 1 onτ B. Repeating the argument above we obtain, forq = min{q * * , k k−1 }, again with a constant depending only on the radius of B and τ,τ . Iterating this argument finitely many times until we arrive at the exponent k k−1 , and choosing τ,τ , . . . , appropriately, we obtain the desired inequality. The last claim in the statement of the corollary follows from Gehring's Lemma, see for instance [41].

Elliptic estimates for the curl
In this section we present nonlinear duality arguments which show that the elliptic estimates of Section 6 yield elliptic estimates for the pair In fact, we will show that there are estimates , for some set of exponents q such that 1 * q q max , p > q * , whenever K ⊂ int Γ k ∪ {0} is a closed cone. Moreover, in the terminology of Conjecture 1.1, we will prove that (7.2) q max (Curl, Γ k ) = k. In particular, combining the above results, we prove Theorem C. 7.1. Elliptic estimates. We begin by establishing the critical version of (7.1) at the endpoint q = 1 * . Note that ρ 2 (A)∇ρ 2 (A) ∈ Γ * 2 ⊂ Γ * n by Lemma 3.8, so (1.3) is indeed applicable. We next establish (7.1) at the other endpoint, i.e. when q = k. Unfortunately, in this case we are unable to estimate the nonlinear quantity ρ k (A) and are therefore forced to work with fields which lie uniformly inside Γ k . Proposition 7.2. Let 2 k n and p > nk n+k = k * if k n − 1 or p = n 2 = n * if k = n. For a closed convex cone K ⊂ int Γ k ∪ {0}, we have the estimate In fact, this improves to (7.5) A L k log L(B n ) C Curl A L p (B n ) for A ∈ C ∞ c (B n , K).
then r > k k−1 * , since by assumption p > k * . This ensures that Proposition 6.1 is applicable, since in addition we have ρ k−1 k (A)∇ρ k (A) ∈ Γ * k by Lemma 3.8. Thus, applying in order (7.6), Proposition 6.1, Corollary 3.11 and Hölder's inequality, we get The conclusion follows by taking the k−1 k -th power on both sides and recalling that, in K, we have |A| ρ k (A) in order to absorb the A L k term on the left-hand side.
In the case k = n, observe that inequality (1.3) holds for the critical exponents and not just in the sub-critical range. The same proof as above now allows us to conclude with exactly p = n 2 on the right. We now establish the improvement in (7.5). Recall that F k = ρ k k and note that F k is a linear combination of k × k minors. It follows that F k is Curl-quasiaffine (or a null Lagrangian), in the terminology of Appendix B. Thus we may apply Corollary B.2, together with the estimate |A| k F k (A), valid for A ∈ K, to obtain where the last estimate follows from (7.4) and the embedding L p (B n ) ֒→ W −1,p * (B n ).

Remark 7.3 (Interpolation)
. Similarly to Remark 6.3, for 1 < l < k, l ∈ N, we are able to deduce that , for p > l * . If we knew that (6.7) holds then, by repeating the arguments above, we would obtain a full range of elliptic estimates: whenever 1 * q k, q * < p.
In the special case when k = 2, we can actually obtain a slightly sharper form of Proposition 7.2 and in fact we manage to estimate the quantity ρ 2 (A) directly in L 2 in terms of a weaker operator than Curl. To do so it is convenient to first fix some notation.
Definition 7.4. We introduce the notation Q ≡ Div −D Tr, for which we have the relation QA = L Curl(A ⊤ ) for a linear map L : V → R n , where V is an appropriate finitedimensional vector space. We also have that The linear map L is easily computed by working in coordinates, see (8.1). We also have the bilinear inequality ρ 2 (A)ρ 2 (B) C A, L Q B for A, B ∈ Γ 2 , (7.8) see for instance [75]. We next obtain the following refinement of Proposition 7.2.
Proof. As in the proof of Proposition 6.1, extend A ∈ C ∞ c (B n , Γ 2 ) by zero to 2B n and define the function Applying Theorem 3.14, let ψ ∈ C 1,1 (2B n ) be the k-admissible solution of By Proposition 3.16, as p ′ < 2n n+2 , where the second inequality follows from Lemma 3.15 and we have used compact support of f in B n in the last equality. As 0 f = ρ 2 (A), we have obtained We then estimate using the inequality of (7.8) and integration by parts: Rearranging, the conclusion follows.
7.2. The case of exact constraints: optimality and higher integrability. As in Section 6.2, by considering A-free fields we will show that the exponent k in Proposition 7.2 is optimal.
Proposition 7.6. Fix 2 k n and ε > 0. There is a closed convex cone K ε ⊂ int Γ k and a function φ ε ∈ C ∞ (B n \{0}) such that The proof relies on the following: Lemma 7.7. For each ε > 0 there exists K = K(ε) > 1 and a function u ε ∈ C ∞ (B n \{0}) such that Proof. Let u ε = u ε (r) be radial. Then Du ε (x) =u ε (r)ν (where ν = x r ) and It follows that det D 2 u ε (x) =ü ε (r)(u ε(r) r ) n−1 and D 2 u ε (x) = max{|ü ε |, |u ε r |}. Fix ε > 0 and take u ε (r) = r ε n+ε +1 ; note that u ε is smooth away from the origin, as claimed. Substituting the definition of u ε in the above formulas we obtain where we used the fact that D 2 u ε (x) =u ε(r) r . In particular, D 2 u ε ∈ Q + n (1+ n ε ). Moreover, we computê Proof of Proposition 7.6. Let φ ε (x) = u ε (x 1 , . . . , x k ), where u ε is given by Lemma 7.7 for n = k. Then clearly is a closed convex cone contained in int Γ k . The proof of (7.2) is similar to the proof of Proposition 6.5 and we omit the details.
Despite Proposition 7.6, irrotational fields uniformly inside Γ k have higher integrability: Corollary 7.8. Let 2 k n and K ⊂ int Γ k ∪ {0} be a closed convex cone. Consider a sequence A j ∈ C ∞ (R n , K) which is bounded in L 1 loc (R n ) and such that Curl A j = 0, so A j = D 2 φ j for some φ j ∈ C ∞ (R n ). For a ball B = B R (x 0 ) and θ ∈ (0, 1), we have In particular, there is δ = δ(K) such that (A j ) is bounded in L k+δ loc (R n ) and so (A j ) is locally k-uniformly integrable.
Proof. We begin by noting that, as Γ k ⊂ Γ l for each 2 l k, we may define a field B l = ∇F l (A), which by Proposition 3.9 takes values in a closed cone K * ⊂ int Γ * l ∪ {0}. In addition, Proposition 3.9 and a homogeneity argument show that c(K, l)|A| l−1 |∇F l (A)| C(n, l)|A| l−1 .
By Proposition 3.10, Div B l = 0. Thus, applying Corollary 6.6 with θ = 1 2 , we have for any ball B ⊂ R n . Iterating these estimates from l = 2 to l = k, we obtain the desired inequality with θ = 2 −k , after relabeling the balls; the case of general θ follows similarly. The last claim follows from Gehring's Lemma.
Corollary 7.8 is a higher integrability result for sequences of Hessians of uniformly kconvex functions. In the case k = n, the fields in Corollary 7.8 are in fact the gradients of so-called δ-monotone mappings and in this case the improved integrability claimed above follows from the general theory of these mappings, as developed in [44] and [6, §3.11].
We also deduce the reverse Hölder inequality: Corollary 7.9. Let θ ∈ (0, 1) and K ⊂ int Γ k ∪ {0} be a closed convex cone. Then Finally, we observe that a weaker version of Corollary 7.8 holds outside the set of symmetric matrices. Indeed, extend the integrands F k , defined in (3.3) for matrices in Sym n , to R n×n through the formula where the sum is over all k × k principal minors M . Applying Corollary B.2, we obtain: Corollary 7.10. Let 2 k n, p > k * and K ⊂ {A ∈ R n×n : F k (A) > 0} be a closed cone. Then In particular, if a sequence Du j ∈ C ∞ (R n , K) is bounded in L k loc (R n ) then it is locally k-uniformly integrable.
0} and note that, when k = n, this last cone coincides with the closure of {det > 0}.
Proof of Theorem C. The proof of Theorem C now follows similar lines to the proof of Theorem B, where we now combine Propositions 7.2 and 7.6. In particular, (7.2) holds.

Other examples
In this final section, we treat three examples slightly different from those considered above. The first is an example in which we obtain the maximal gain of integrability exponent q max = ∞. This operator, while related to the pair (Div, Curl), is not elliptic, but has a wave cone that is not spanning.
The second example is a second order variant of the Curl, similar to the Div 2 operator considered above. This is the composition of div with Q, the sub-curl type operator defined above in Definition 7.4, and we obtain the same q max as in that case.
The final example is a sub-curl operator strictly weaker than the Curl, acting on functions constrained to Sym + . 8.1. A degenerate example. In this subsection we give a first order example. The cone we consider is very large, K = {Tr 0}, and the operator is almost elliptic. The example is somewhat degenerate in the sense that the wave cone of the operator is not spanning. However, in the notation of Conjecture 1.1, we obtain q max = ∞. (a) Let 1 p < n. We have that We remark that in the case that K ⊂ int K ∪ {0} is a closed convex cone, we have the higher integrability estimates for A ∈ C ∞ c (R n , K). We note that the operator PA ≡ (Div A, Curl(A ⊤ )) is both elliptic and canceling whenever A ∈ Sym n , in which case Proposition 8.1 follows from the work of Van Schaftingen [74] and Stolyarov [62]. However, for non-symmetric matrices, this operator is not elliptic: it is easy to verify that P(ξ)(ξ ⊗ ξ ⊥ ) = 0, where ξ ⊥ is any vector orthogonal to ξ. Therefore Proposition 8.1 is genuinely different from the previously mentioned results.
Proof. Consider first the case 1 p < n. Let us take A ∈ C ∞ c (R n , K). Writing A = (a ij ) and A ⊤ = (a ⊤ ij ), and A i , A ⊤ i for their rows, we have It then follows from the usual Gagliardo-Nirenberg inequality that wished. An analogous argument proves part (b).
We note that, by using the refined Gagliardo-Nirenberg-Sobolev inequality we can actually deduce a stronger form of Proposition 8.1, where the L p * -norm can be replaced by the Lorentz L p * ,p -norm [3,54].

8.2.
A second order example. In this subsection we consider an example of a second order operators and a cone for which we have the full resolution of Conjecture 1.1. The results are similar in spirit to Theorem D.
Proposition 8.2. Let p, q 1. We have that (a) p > 1, n 4, and q = min{p * * , 2}, As before, if K ⊂ int Γ 2 ∪ {0} is a closed convex cone then, for the same ranges of p and q, we have . We recall that here p * * = (p * ) * = np n−2p is the Sobolev exponent. Note that in the special cases n = 2, 3, we can take p = 1, q = 2. We also have where we recall Definition 7.4.

Proof.
Let A ∈ C ∞ c (B n , K) and let ψ ∈ C 1,1 (B n ) be the k-admissible solution of In particular, D 2 ψ ∈ Γ 2 a.e. in B n and we have the estimate Lemma 3.15, where the restrictions on p and q follow from those in the statement of the lemma. We then estimate using inequality (7.8) and integration by parts twice: The first inequality follows from (8.2) and the second is also proved after noting that for A ∈ K we have |A| ρ 2 (A).
The next result shows that the exponent q = 2 in Proposition 8.2 is optimal.
In particular, in Proposition 8.3, we can take n = 2. In this case, the estimate predicted by [4, Conjecture 1.9] should hold for any L q , q < ∞, on the right hand side, and so the proposition falsifies that conjecture.
Proof. Let u(x) = u ε (x 1 , x 2 ), where u ε is given by Lemma 7.7 for n = 2. Then clearly is contained in int Γ 2 ∪ {0} and K = K(ε). By (8.2) and Definition 7.4, we have that AD 2 u = 0. The same cut-off argument as in the proof of Theorem B, noting that allows us to conclude.
The consequence pertaining to Hessians in Corollary 7.8 for k = 2 can be improved using Proposition 8.2 instead of Proposition 7.2. The improvement concerns the operator; rather than imposing that we have a sequence of exact Hessians, we instead require only that QA j = Div L Q A j = 0 for each j.
Corollary 8.4. Let K ⊂ int Γ 2 ∪ {0} be a closed convex cone. Consider a sequence A j ∈ C ∞ (R n , K) which is bounded in L 1 loc (R n ) and such that QA j = 0 for all j. There is δ = δ(K) such that A j is bounded in L 2+δ loc (R n ). In particular, (A j ) is locally 2-uniformly integrable.
Applying Proposition 8.2 iteratively, the rest of the proof now follows the same lines as that of Corollary 6.6.

8.3.
A first order example. It is natural to extend the slicing methods of Section 4 to higher dimensions, and so we conclude this section with an inequality which generalizes Corollary 4.6 to dimensions n 3. In particular, we prove the end-point estimate for p = 1 in the setting of (1.1), for fields restricted to Sym + n and an operator which is related to but strictly weaker than Curl. To fix the notation for this operator, we distinguish between the cases n = 2k is even and n = 2k + 1 is odd.
Note that this has exactly 2k = n entries, i.e. it has the same number of entries as Div.
Proposition 8.5. Take K 1 and let A ∈ C ∞ c (R n , Sym + n ∩ Q + n (K)). Then A L 1 * (R n ) C K P n A L 1 (R n ) .
Note that Proposition 8.5 yields an example of the situation posed in Conjecture 1.1 in which we may take p = 1 and also have the critical Sobolev gain of integrability q = 1 * . This result is therefore complementary to many of the other results of this paper, in which we typically obtain the sub-critical estimate q = q max with p > (q max ) * . However, Proposition 8.5 should be compared to Proposition 7.2 in which we obtain the same estimate for a stronger operator in a larger family of cones.
The proof of Proposition 8.5 is based on a higher dimensional slicing argument.
Proof. Throughout, we will use the notation A j,k for the 2×2 matrix obtained by deleting all rows and columns from A except the j-th and k-th and M j,k for the determinant of this matrix. We begin with two simple observations. Firstly, any matrix A ∈ Sym + n has positive diagonal entries; secondly, under our assumptions, given any 2 × 2 principal minor M , there is a constant c = c(K) such that M (A) c|A| 2 . The rest of the proof is split into two cases.
Applying Minkowski's integral inequality to re-order the integrals, this implies here p j ∈ R n has the j-th and (j + 1)-th entries equal to 2 and all other entries equal to 1. We refer the reader to the book of Adams and Fournier for further details concerning these mixed-norm spaces [2].
We take the product of all these k inequalities (for j = 1, 3, 5, ..., n − 1) and apply the generalised Hölder's inequality, see e.g. [2,Theorem 2.49], to obtain |A| k L r n−1 j=1 j odd where r is computed component-wise by Thus, the norm on the left-hand side is simply the L r -norm with r = 2 2k−1 = 2 n−1 , and so we may rewrite the previous inequality as C P n A k L 1 (R n ) .
Taking the k-th root, the desired inequality follows. Case 2: odd dimension. Assume n = 2k + 1. Arguing as above, i.e. by considering the M j,j+1 minor, we easily find, for j = 1, 2, 3, . . . , n − 1, the estimate where p j is as before. We now work with the sub-matrix A 1,n on the x 1 -x n plane to obtain R n−2 ˆR 2 |A| 2 (x) dx 1 dx n 1 2 dx 2 · · · dx n−1 C P n A L 1 ,

leading to
A L pn C P n A L 1 , where p n ∈ R n has the first and last entries equal to 2 and all other entries equal to 1. We note that each component of the sum of the terms 1 p j consists of n − 2 + 2 × 1 2 , so that n j=1 1 p j = (n − 1, . . . , n − 1).
Thus, applying the generalised Hölder's inequality, A L p j (R n ) C n P n A n L 1 (R n ) .
Taking the n-th root, we obtain the desired inequality.
Appendix A. Proof of Theorem 3.17 In this Appendix we include a proof of Theorem 3.17. We begin by proving the uniqueness claim: we observe that if u and v are both solutions, then u − v satisfies the uniformly elliptic equation ρ k (D 2 u) − ρ k (D 2 v) = 0, to which the comparison principle applies. It follows that u − v is constant.
To prove the existence claim we rely on argument similar to the one used by YanYan Li in [47]. The crucial point is to obtain an a priori estimate on the oscillation of ψ, where we write ψ ≡ 1 2 Sx · x + ϕ.
Appendix B. A-quasiaffine integrands and L log L estimates Given finite dimensional inner product spaces V and W and an ℓ-th order homogeneous differential operator we say that a locally bounded, Borel measurable integrand F : for all v ∈ C ∞ (T n , V) with Av = 0, and we say that F is A-quasiaffine if both ±F are A-quasiconvex. The latter is the class of weakly continuous functionals defined on A-free sequences [26,32,53], which benefits from surprising improvements in integrability [15,32,33,51]; we will recall a general result below. To this end, recall that the differential operator A, with symbol A(ξ) = |α|=ℓ ξ α L α ∈ Lin(V, W), is said to have constant rank if rank A(ξ) is independent of ξ ∈ R n \ {0}, and is said to have a spanning wave cone if ker A(ξ).
Under the spanning wave cone condition it is classical that A-quasiaffine integrands are polynomials. We then have: for v ∈ C ∞ c (B n , K).
We therefore obtain sharp uniform integrability statements: Corollary B.3. In the setting of Corollary B.2, if a sequence (v j ) ⊂ C ∞ c (B n , K) is bounded in L s (B n ) and (Av j ) is bounded inẆ −ℓ,r (B n ), we have that (v j ) is s-uniformly integrable.