elliptic estimates of Section 6 yield elliptic estimates for the pair

$\mathcal{A}=\operatorname{Curl},\qquad\mathcal{K}=\Gamma_{k}.$

In fact, we will show that there are estimates

(7.1) $\|A\|_{L^{q}(\mathbb{B}^{n})}\leqslant
C\|\operatorname{Curl}A\|_{L^{p}(\mathbb{B}^{n})}\quad\text{for }A\in
C^{\infty}_{c}(\mathbb{B}^{n},\mathcal{K}),$

for some set of exponents $q$ such that $1^{*}\leqslant q\leqslant q_{\max}$,
$p>q_{*}$, whenever $\mathcal{K}\subset\textup{int}\,\Gamma_{k}\cup\\{0\\}$ is
a closed cone. Moreover, in the terminology of Conjecture 1.1, we will prove
that

(7.2) $q_{\max}(\operatorname{Curl},\Gamma_{k})=k.$

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
$q=1^{*}$.

######  Proposition 7.1.

For $2\leqslant k\leqslant n$, there is a constant $C$ such that

$\|\rho_{k}(A)\|_{L^{1^{*}}(\mathbb{B}^{n})}\leqslant
C\|\operatorname{Curl}A\|_{L^{1}(\mathbb{B}^{n})},\quad\text{for }A\in
C^{\infty}_{c}(\mathbb{B}^{n},\Gamma_{k}).$

[Proof]By Lemma 3.4 it suffices to prove the proposition when $k=2$, as
$A\in\Gamma_{k}\subseteq\Gamma_{2}$ and we have
$0\leqslant\rho_{k}(A)\leqslant\rho_{2}(A)$. We first note that, by Lemmas 3.8
and 3.4,

$\displaystyle\rho_{2}(A)^{\frac{n}{n-1}}$
$\displaystyle=c_{n}\rho_{2}^{*}\big{(}\rho_{2}(A)\nabla\rho_{2}(A)\big{)}^{\frac{n}{n-1}}$
$\displaystyle\leqslant
c_{n}\rho_{n}^{*}\big{(}\rho_{2}(A)\nabla\rho_{2}(A)\big{)}^{\frac{n}{n-1}}=c_{n}\det\big{(}\rho_{2}(A)\nabla\rho_{2}(A)\big{)}^{\frac{1}{n-1}},$

where $c_{n}=n^{\frac{n}{n-1}}$. We apply (1.3) and Proposition 3.10 to
estimate

(7.3) $\displaystyle\int_{\mathbb{B}^{n}}\rho_{2}(A)^{\frac{n}{n-1}}\lesssim$
$\displaystyle\,\int_{\mathbb{B}^{n}}\det\big{(}\rho_{2}(A)\nabla\rho_{2}(A)\big{)}^{\frac{1}{n-1}}$
$\displaystyle\lesssim$
$\displaystyle\,\bigg{(}\int_{\mathbb{B}^{n}}\Big{|}\operatorname{Div}\Big{(}\rho_{2}(A)\nabla\rho_{2}(A)\Big{)}\Big{|}\bigg{)}^{\frac{n}{n-1}}\lesssim\,\bigg{(}\int_{\mathbb{B}^{n}}|\operatorname{Curl}A|\bigg{)}^{\frac{n}{n-1}}.$

Note that $\rho_{2}(A)\nabla\rho_{2}(A)\in\Gamma_{2}^{*}\subset\Gamma_{n}^{*}$
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 $\rho_{k}(A)$
and are therefore forced to work with fields which lie uniformly inside
$\Gamma_{k}$.

######  Proposition 7.2.

Let $2\leqslant k\leqslant n$ and $p>\frac{nk}{n+k}=k_{*}$ if $k\leqslant n-1$
or $p=\frac{n}{2}=n_{*}$ if $k=n$. For a closed convex cone
$\mathcal{K}\subset\mathrm{int}\,\Gamma_{k}\cup\\{0\\}$, we have the estimate

(7.4) $\displaystyle\|A\|_{L^{k}(\mathbb{B}^{n})}$ $\displaystyle\leqslant
C\|\operatorname{Curl}A\|_{L^{p}(\mathbb{B}^{n})}\quad\text{for }A\in
C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K})$

In fact, this improves to

(7.5) $\|A\|_{L^{k}\log L(\mathbb{B}^{n})}\leqslant
C\|\operatorname{Curl}A\|_{L^{p}(\mathbb{B}^{n})}\quad\text{for }A\in
C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K}).$

The exponent $k$ is optimal, as will be shown below in Proposition 7.6. As in
Proposition 7.1, the proof of Proposition 7.2 relies on Proposition 3.10,
which allows us to exploit a non-linear duality between the operator-cone
pairs $(\operatorname{Div},\Gamma_{k}^{*})$ and
$(\operatorname{Curl},\Gamma_{k})$.

[Proof]Let $A\in C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K})$ and recall the
identity $\rho_{k}^{*}(n\nabla\rho_{k}(A))=1$ from Lemma 3.8. By the positive
1-homogeneity of $\rho_{k}^{*}$, we find

(7.6)
$\rho_{k}(A)^{k}=n^{\frac{k}{k-1}}\rho_{k}^{*}\big{(}\rho_{k}^{k-1}(A)\nabla\rho_{k}(A)\big{)}^{\frac{k}{k-1}}.$

Suppose first that $k\leqslant n-1$. We let $r$ be such that

$\frac{1}{p}+\frac{k-2}{k}=\frac{1}{r};$

then $r>\big{(}\frac{k}{k-1}\big{)}_{*}$, since by assumption $p>k_{*}$. This
ensures that Proposition 6.1 is applicable, since in addition we have
$\rho_{k}^{k-1}(A)\nabla\rho_{k}(A)\in\Gamma_{k}^{*}$ by Lemma 3.8. Thus,
applying in order (7.6), Proposition 6.1, Corollary 3.11 and Hölder’s
inequality, we get

(7.7) $\displaystyle\int_{\mathbb{B}^{n}}\rho_{k}(A)^{k}\lesssim$
$\displaystyle\,\int_{\mathbb{B}^{n}}\rho_{k}^{*}\big{(}\rho_{k}^{k-1}(A)\nabla\rho_{k}(A)\big{)}^{\frac{k}{k-1}}$
$\displaystyle\lesssim$
$\displaystyle\,\bigg{(}\int_{\mathbb{B}^{n}}\Big{|}\operatorname{Div}\Big{(}\rho_{k}^{k-1}(A)\nabla\rho_{k}(A)\Big{)}\Big{|}^{r}\bigg{)}^{\frac{k}{r(k-1)}}$
$\displaystyle\lesssim$
$\displaystyle\,\Big{(}\int_{\mathbb{B}^{n}}|\operatorname{Curl}A|^{r}|A|^{r(k-2)}\Big{)}^{\frac{k}{r(k-1)}}$
$\displaystyle\lesssim$
$\displaystyle\,\|\operatorname{Curl}A\|_{L^{p}}^{\frac{k}{k-1}}\|A\|_{L^{k}}^{\frac{k(k-2)}{k-1}}.$

The conclusion follows by taking the $\frac{k-1}{k}$-th power on both sides
and recalling that, in $\mathcal{K}$, we have $|A|\lesssim\rho_{k}(A)$ 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 $p=\frac{n}{2}$ on the right.

We now establish the improvement in (7.5). Recall that $F_{k}=\rho_{k}^{k}$
and note that $F_{k}$ is a linear combination of $k\times k$ minors. It
follows that $F_{k}$ is $\operatorname{Curl}$-quasiaffine (or a null
Lagrangian), in the terminology of Appendix B. Thus we may apply Corollary
B.2, together with the estimate $|A|^{k}\lesssim F_{k}(A)$, valid for
$A\in\mathcal{K}$, to obtain

$\displaystyle\|A\|_{L^{k}\log L(\mathbb{B}^{n})}\leqslant
C\left(\|A\|_{L^{k}(\mathbb{B}^{n})}+\|\operatorname{Curl}A\|_{{\dot{W}}{{}^{-1,p^{*}}}(\mathbb{B}^{n})}\right)\leqslant
C\|\operatorname{Curl}A\|_{L^{p}(\mathbb{B}^{n})},$

where the last estimate follows from (7.4) and the embedding
$L^{p}(\mathbb{B}^{n})\hookrightarrow W^{-1,p^{*}}(\mathbb{B}^{n})$.

######  Remark 7.3 (Interpolation).

Similarly to Remark 6.3, for $1<l<k$, $l\in\mathbb{N}$, we are able to deduce
that

$\|A\|_{L^{l}(B)}\leqslant\|\operatorname{Curl}A\|_{L^{p}(B)}\quad\text{for
}A\in C^{\infty}_{c}(B,\Gamma_{k}),$

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:

$\|A\|_{L^{q}(B)}\leqslant
C\|\operatorname{Curl}A\|_{L^{p}(B)},\qquad\text{whenever }1^{*}\leqslant
q\leqslant k,\,q_{*}<p.$

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
$\rho_{2}(A)$ directly in $L^{2}$ in terms of a weaker operator than
$\operatorname{Curl}$. To do so it is convenient to first fix some notation.

######  Definition 7.4.

We introduce the notation
$\mathcal{Q}\equiv\operatorname{Div}-\textup{D}\operatorname{Tr}$, for which
we have the relation $\mathcal{Q}A=L\operatorname{Curl}(A^{\top})$ for a
linear map $L\colon\mathbb{V}\to\mathbb{R}^{n}$, where $\mathbb{V}$ is an
appropriate finite-dimensional vector space. We also have that

$\mathcal{Q}A=\operatorname{Div}(L_{\mathcal{Q}}A),\quad\text{where
}L_{\mathcal{Q}}A=A-(\operatorname{Tr}A)I_{n}\text{ for
}A\in\mathbb{R}^{n\times n}.$

The linear map $L$ is easily computed by working in coordinates, see (8.1). We
also have the bilinear inequality

(7.8) $\displaystyle\rho_{2}(A)\rho_{2}(B)\leqslant C\langle
A,L_{\mathcal{Q}}B\rangle\quad\text{for }A,\,B\in\Gamma_{2},$

see for instance [75]. We next obtain the following refinement of Proposition
7.2.

######  Proposition 7.5.

Suppose that $p>2_{*}=\frac{2n}{n+2}$. Then

(7.9) $\displaystyle\|\rho_{2}(A)\|_{L^{2}(\mathbb{B}^{n})}\leqslant
C\|{\mathcal{Q}A}\|_{L^{p}(\mathbb{B}^{n})}\leqslant
C\|\operatorname{Curl}A\|_{L^{p}(\mathbb{B}^{n})}\quad\text{for }A\in
C_{c}^{\infty}(\mathbb{B}^{n},\Gamma_{2}).$

[Proof]As in the proof of Proposition 6.1, extend $A\in
C_{c}^{\infty}(\mathbb{B}^{n},\Gamma_{2})$ by zero to $2\mathbb{B}^{n}$ and
define the function

$f=\rho_{2}(A).$

Applying Theorem 3.14, let $\psi\in C^{1,1}(\overline{2\mathbb{B}^{n}})$ be
the $k$-admissible solution of

$\begin{cases}\rho_{2}(\textup{D}^{2}\psi)=f&\text{ in }2\mathbb{B}^{n},\\\
\psi=0&\text{ on }\partial(2\mathbb{B}^{n}).\end{cases}$

By Proposition 3.16, as $p^{\prime}<\frac{2n}{n+2}$,

$\|\textup{D}\psi\|_{L^{p^{\prime}}(\mathbb{B}^{n})}\lesssim\int_{2\mathbb{B}^{n}}|\psi|\lesssim\|f\|_{L^{2}(2\mathbb{B}^{n})}=\|f\|_{L^{2}(\mathbb{B}^{n})},$

where the second inequality follows from Lemma 3.15 and we have used compact
support of $f$ in $\mathbb{B}^{n}$ in the last equality. As $0\leqslant
f=\rho_{2}(A)$, we have obtained

$\|\textup{D}\psi\|_{L^{p^{\prime}}(\mathbb{B}^{n})}\lesssim\|\rho_{2}(A)\|_{L^{2}(\mathbb{B}^{n})}.$

We then estimate using the inequality of (7.8) and integration by parts:

$\displaystyle\int_{\mathbb{B}^{n}}\rho_{2}(A)^{2}$
$\displaystyle=\int_{\mathbb{B}^{n}}\rho_{2}(A)\rho_{2}(\textup{D}^{2}\psi)\lesssim\int_{\mathbb{B}^{n}}\langle
L_{\mathcal{Q}}A,\textup{D}^{2}\psi\rangle=-\int_{\mathbb{B}^{n}}\operatorname{Div}(L_{\mathcal{Q}}A)\cdot\textup{D}\psi$
$\displaystyle\lesssim\|\textup{D}\psi\|_{L^{p^{\prime}}(\mathbb{B}^{n})}\|{\mathcal{Q}}A\|_{L^{p}(\mathbb{B}^{n})}\lesssim\left(\int_{\mathbb{B}^{n}}\rho_{2}(A)^{2}\right)^{1/2}\|{\mathcal{Q}}A\|_{L^{p}(\mathbb{B}^{n})}.$

Rearranging, the conclusion follows.

###  7.2. The case of exact constraints: optimality and higher integrability

As in Section 6.2, by considering $\mathcal{A}$-free fields we will show that
the exponent $k$ in Proposition 7.2 is optimal.

######  Proposition 7.6.

Fix $2\leqslant k\leqslant n$ and $\varepsilon>0$. There is a closed convex
cone $\mathcal{K}^{\varepsilon}\subset\textup{int\,}\Gamma_{k}$ and a function
$\phi_{\varepsilon}\in C^{\infty}(\mathbb{B}^{n}\backslash\\{0\\})$ such that

$\textup{D}^{2}\phi_{\varepsilon}\in\mathcal{K}^{\varepsilon}\text{ a.e.\ in
}\mathbb{B}^{n},\qquad\textup{D}^{2}\phi_{\varepsilon}\in
L^{k}(\mathbb{B}^{n})\setminus L^{k+\varepsilon}(\mathbb{B}^{n}).$

The proof relies on the following:

######  Lemma 7.7.

For each $\varepsilon>0$ there exists $K=K(\varepsilon)>1$ and a function
$u_{\varepsilon}\in C^{\infty}(\mathbb{B}^{n}\backslash\\{0\\})$ such that

$\textup{D}^{2}u_{\varepsilon}\in\textup{Sym}^{+}_{n}\cap Q_{n}^{+}(K)\text{
a.e.\ in }\mathbb{B}^{n},\qquad\textup{D}^{2}u_{\varepsilon}\in
L^{n}(\mathbb{B}^{n})\setminus L^{n+\varepsilon}(\mathbb{B}^{n}).$

[Proof]Let $u_{\varepsilon}=u_{\varepsilon}(r)$ be radial. Then
$\textup{D}u_{\varepsilon}(x)=\dot{u}_{\varepsilon}(r)\nu$ (where
$\nu=\frac{x}{r}$) and

$\textup{D}^{2}u_{\varepsilon}(x)=\frac{\dot{u}_{\varepsilon}(r)}{r}\textup{Id}+\Big{(}\ddot{u}_{\varepsilon}(r)-\frac{\dot{u}_{\varepsilon}(r)}{r}\Big{)}\nu\otimes\nu.$

It follows that
$\det\textup{D}^{2}u_{\varepsilon}(x)=\ddot{u}_{\varepsilon}(r)(\frac{\dot{u}_{\varepsilon}(r)}{r})^{n-1}$
and
$\|\textup{D}^{2}u_{\varepsilon}(x)\|=\max\\{|\ddot{u}_{\varepsilon}|,|\frac{\dot{u}_{\varepsilon}}{r}|\\}$.
Fix $\varepsilon>0$ and take
$u_{\varepsilon}(r)=r^{\frac{\varepsilon}{n+\varepsilon}+1}$; note that
$u_{\varepsilon}$ is smooth away from the origin, as claimed. Substituting the
definition of $u_{\varepsilon}$ in the above formulas we obtain

$\frac{\|\textup{D}^{2}u_{\varepsilon}(x)\|^{n}}{\det\textup{D}^{2}u_{\varepsilon}(x)}=1+\frac{n}{\varepsilon}$

where we used the fact that
$\|\textup{D}^{2}u_{\varepsilon}(x)\|=\frac{\dot{u}_{\varepsilon}(r)}{r}.$ In
particular, $\textup{D}^{2}u_{\varepsilon}\in
Q_{n}^{+}(1+\frac{n}{\varepsilon}).$ Moreover, we compute

$\int_{\mathbb{B}^{n}}\|\textup{D}^{2}u_{\varepsilon}\|^{p}\operatorname{d}\\!x\approx_{n}\int_{0}^{1}r^{n-1}\left|\frac{\dot{u}_{\varepsilon}(r)}{r}\right|^{p}\operatorname{d}\\!r=\Big{(}1+\frac{\varepsilon}{\varepsilon+n}\Big{)}\int_{0}^{1}r^{\frac{n(\varepsilon+n-p)}{\varepsilon+n}-1}\operatorname{d}\\!r,$

and this integral is finite if and only if $p<n+\varepsilon.$

[Proof of Proposition 7.6] Let
$\phi_{\varepsilon}(x)=u_{\varepsilon}(x_{1},\ldots,x_{k})$, where
$u_{\varepsilon}$ is given by Lemma 7.7 for $n=k$. Then clearly
$\textup{D}^{2}\phi_{\varepsilon}\in L^{k}\setminus
L^{k+\varepsilon}(\mathbb{B}^{n},\mathcal{K}^{\varepsilon})$, where

$\mathcal{K}^{\varepsilon}\equiv\left\\{\left(\begin{matrix}A_{0}&0\\\
0&0\end{matrix}\right)\colon A_{0}\in\textup{Sym}^{+}_{k}\cap
Q_{k}(K)\right\\}$

is a closed convex cone contained in $\mathrm{int\,}\Gamma_{k}$. 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 $\Gamma_{k}$
have higher integrability:

######  Corollary 7.8.

Let $2\leqslant k\leqslant n$ and
$\mathcal{K}\subset\mathrm{int}\,\Gamma_{k}\cup\\{0\\}$ be a closed convex
cone. Consider a sequence $A_{j}\in C^{\infty}(\mathbb{R}^{n},\mathcal{K})$
which is bounded in $L^{1}_{\operatorname{loc}}(\mathbb{R}^{n})$ and such that
$\operatorname{Curl}A_{j}=0$, so $A_{j}=\textup{D}^{2}\phi_{j}$ for some
$\phi_{j}\in C^{\infty}(\mathbb{R}^{n})$. For a ball $B=B_{R}(x_{0})$ and
$\theta\in(0,1)$, we have

$\left(\frac{1}{|\theta B|}\int_{\theta
B}|A_{j}|^{k}\operatorname{d}\\!x\right)^{\frac{1}{k}}\leqslant
C(\mathcal{K},R,\theta)\frac{1}{|B|}\int_{B\setminus\theta
B}|A_{j}|\operatorname{d}\\!x.$

In particular, there is $\delta=\delta(\mathcal{K})$ such that $(A_{j})$ is
bounded in $L^{k+\delta}_{\operatorname{loc}}(\mathbb{R}^{n})$ and so
$(A_{j})$ is locally $k$-uniformly integrable.

[Proof]We begin by noting that, as $\Gamma_{k}\subset\Gamma_{l}$ for each
$2\leqslant l\leqslant k$, we may define a field $B_{l}=\nabla F_{l}(A)$,
which by Proposition 3.9 takes values in a closed cone
$\mathcal{K}^{*}\subset\textup{int}\,\Gamma_{l}^{*}\cup\\{0\\}$. In addition,
Proposition 3.9 and a homogeneity argument show that

$c(\mathcal{K},l)|A|^{l-1}\leqslant|\nabla F_{l}(A)|\leqslant
C(n,l)|A|^{l-1}.$

By Proposition 3.10, $\operatorname{Div}B_{l}=0$. Thus, applying Corollary 6.6
with $\theta=\frac{1}{2}$, we have

$\|A\|_{L^{l}(B)}\lesssim\|B_{l}\|_{L^{\frac{l}{l-1}}(B)}\lesssim\|B_{l}\|_{L^{1}(2B\setminus
B)}\lesssim\|A\|_{L^{l-1}(2B\setminus B)},$

for any ball $B\subset\mathbb{R}^{n}$. Iterating these estimates from $l=2$ to
$l=k$, we obtain the desired inequality with $\theta=2^{-k}$, after relabeling
the balls; the case of general $\theta$ follows similarly. The last claim
follows from Gehring’s Lemma.  Corollary 7.8 is a higher integrability result
for sequences of Hessians of uniformly $k$-convex functions. In the case
$k=n$, the fields in Corollary 7.8 are in fact the gradients of so-called
$\delta$-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 $\theta\in(0,1)$ and
$\mathcal{K}\subset\mathrm{int}\,\Gamma_{k}\cup\\{0\\}$ be a closed convex
cone. Then

$\|\textup{D}^{2}u\|_{L^{{k}}(\theta\mathbb{B}^{n})}\leqslant
C\|\textup{D}^{2}u\|_{L^{1}(\mathbb{B}^{n}\setminus(\theta\mathbb{B}^{n}))}\quad\text{for
}u\in C^{\infty}(\bar{\mathbb{B}}^{n})\text{ with
}\textup{D}^{2}u\in\mathcal{K}.$

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 $\textup{Sym}_{n}$, to $\mathbb{R}^{n\times n}$ through
the formula

$F_{k}(A)=\frac{1}{{n\choose k}}\sum_{M}M(A),$

where the sum is over all $k\times k$ principal minors $M$. Applying Corollary
B.2, we obtain:

######  Corollary 7.10.

Let $2\leqslant k\leqslant n$, $p>k_{*}$ and
$\mathcal{K}\subset\\{A\in\mathbb{R}^{n\times n}:F_{k}(A)>0\\}$ be a closed
cone. Then

$\|A\|_{L^{k}\log L(\mathbb{B}^{n})}\leqslant
C\left(\|A\|_{L^{k}(\mathbb{B}^{n})}+\|\operatorname{Curl}A\|_{L^{p}(\mathbb{B}^{n})}\right)\quad\text{for
}A\in C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K}).$

In particular, if a sequence $\textup{D}u_{j}\in
C^{\infty}(\mathbb{R}^{n},\mathcal{K})$ is bounded in
$L^{k}_{\operatorname{loc}}(\mathbb{R}^{n})$ then it is locally $k$-uniformly
integrable.

Clearly $\Gamma_{k}\subset\\{A\in\mathbb{R}^{n\times n}:F_{k}(A)\geqslant
0\\}$ 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.

##  8\. 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 $q_{\max}=\infty$. This operator, while related to
the pair $(\operatorname{Div},\operatorname{Curl})$, is not elliptic, but has
a wave cone that is not spanning.

The second example is a second order variant of the $\operatorname{Curl}$,
similar to the $\operatorname{Div}^{2}$ operator considered above. This is the
composition of $\operatorname{div}$ with $\mathcal{Q}$, 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
$\operatorname{Curl}$, acting on functions constrained to $\textup{Sym}^{+}$.

###  8.1. A degenerate example

In this subsection we give a first order example. The cone we consider is very
large, $\mathcal{K}=\\{\operatorname{Tr}\geqslant 0\\}$, 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 $q_{\max}=\infty$.

######  Proposition 8.1.

Let $\mathcal{K}\equiv\\{A\in\mathbb{R}^{n\times
n}:\operatorname{Tr}(A)\geqslant 0\\}$.

  1. (a)

Let $1\leqslant p<n$. We have that

$\|\operatorname{Tr}A\|_{\operatorname{L}^{p^{*}}(\mathbb{R}^{n})}\leqslant
C\left(\|\operatorname{Div}A\|_{\operatorname{L}^{p}(\mathbb{R}^{n})}+\|\operatorname{Curl}(A^{\top})\|_{\operatorname{L}^{p}(\mathbb{R}^{n})}\right)\quad\text{for
}A\in C^{\infty}_{c}(\mathbb{R}^{n},\mathcal{K}).$

  2. (b)

Let $p>n$. We have that

$\|\operatorname{Tr}A\|_{\operatorname{L}^{\infty}(\mathbb{B}^{n})}\leqslant
C\left(\|\operatorname{Div}A\|_{\operatorname{L}^{p}(\mathbb{B}^{n})}+\|\operatorname{Curl}(A^{\top})\|_{\operatorname{L}^{p}(\mathbb{B}^{n})}\right)\quad\text{for
}A\in C^{\infty}_{c}(\mathbb{B}^{n},\mathcal{K}).$

We remark that in the case that
$\widetilde{\mathcal{K}}\subset\mathrm{int}\,\mathcal{K}\cup\\{0\\}$ is a
closed convex cone, we have the higher integrability estimates

$\|A\|_{\operatorname{L}^{p^{*}}(\mathbb{R}^{n})}\leqslant
C\left(\|\operatorname{Div}A\|_{\operatorname{L}^{p}(\mathbb{R}^{n})}+\|\operatorname{Curl}(A^{\top})\|_{\operatorname{L}^{p}(\mathbb{R}^{n})}\right)\quad\text{for
}A\in C^{\infty}_{c}(\mathbb{R}^{n},\widetilde{\mathcal{K}}).$

We note that the operator
$\mathcal{P}A\equiv(\operatorname{Div}A,\operatorname{Curl}(A^{\top}))$ is
both elliptic and canceling whenever $A\in\textup{Sym}_{n}$, 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 $\mathcal{P}(\xi)(\xi\otimes\xi^{\bot})=0$, where
$\xi^{\bot}$ is any vector orthogonal to $\xi$. Therefore Proposition 8.1 is
genuinely different from the previously mentioned results.

[Proof]Consider first the case $1\leqslant p<n$. Let us take $A\in
C^{\infty}_{c}(\mathbb{R}^{n},\mathcal{K})$. Writing $A=(a_{ij})$ and
$A^{\top}=(a^{\top}_{ij})$, and $A_{i},A_{i}^{\top}$ for their rows, we have

(8.1)
$\displaystyle({\operatorname{Div}A-\textup{D}\operatorname{Tr}(A)})_{i}=\sum_{j}\partial_{j}a_{ij}-\partial_{i}a_{jj}=\sum_{j}\partial_{j}a^{\top}_{ji}-\partial_{i}a^{\top}_{jj}=\sum_{j}(\operatorname{curl}A^{\top}_{j})_{ij}.$

It then follows from the usual Gagliardo–Nirenberg inequality that

$\|\operatorname{Tr}(A)\|_{L^{p^{*}}}\lesssim\|\textup{D}\operatorname{Tr}(A)\|_{L^{p}}\leqslant\|\operatorname{Div}A\|_{L^{p}}+\|\operatorname{Curl}A^{\top}\|_{L^{p}},$

as 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
$\operatorname{L}^{p^{*}}$-norm can be replaced by the Lorentz
$\operatorname{L}^{p^{*},p}$-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 $p,\,q\geqslant 1$. We have that

$\|\rho_{2}(A)\|_{L^{q}(\mathbb{B}^{n})}\leqslant
C\|\operatorname{Div}^{2}A-\Delta\operatorname{Tr}A\|_{L^{p}(\mathbb{B}^{n})}\quad\text{for
}A\in C_{c}^{\infty}(\mathbb{B}^{n},\Gamma_{2}),$

whenever

  1. (a)

$p>1$, $n\geqslant 4$, and $q=\min\\{p^{**},2\\}$,

  2. (b)

$p=1$, $q<1^{**}$, $q\leqslant 2$.

As before, if $\mathcal{K}\subset\mathrm{int}\,\Gamma_{2}\cup\\{0\\}$ is a
closed convex cone then, for the same ranges of $p$ and $q$, we have

$\|A\|_{L^{q}(\mathbb{B}^{n})}\leqslant
C\|\operatorname{Div}^{2}A-\Delta\operatorname{Tr}A\|_{L^{p}(\mathbb{B}^{n})}\quad\text{for
}A\in C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K}).$

We recall that here $p^{**}=(p^{*})^{*}=\frac{np}{n-2p}$ is the Sobolev
exponent. Note that in the special cases $n=2,\,3$, we can take $p=1$, $q=2$.
We also have

(8.2)
$\operatorname{Div}^{2}A-\Delta\operatorname{Tr}A=\operatorname{Div}^{2}(L_{\mathcal{Q}}A)=\operatorname{div}\mathcal{Q}A,$

where we recall Definition 7.4.

[Proof]Let $A\in C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K})$ and let $\psi\in
C^{1,1}(\overline{\mathbb{B}^{n}})$ be the $k$-admissible solution of

$\begin{cases}\rho_{2}(\textup{D}^{2}\psi)=f\equiv\rho_{2}(A)^{q-1}&\text{ in
}\mathbb{B}^{n},\\\ \psi=0&\text{ on }\partial\mathbb{B}^{n}.\end{cases}$

In particular, $\textup{D}^{2}\psi\in\Gamma_{2}$ a.e. in $\mathbb{B}^{n}$ and
we have the estimate

$\|\psi\|_{L^{p^{\prime}}(\mathbb{B}^{n})}\lesssim\|f\|_{L^{q^{\prime}}(\mathbb{B}^{n})}=\left(\int_{\mathbb{B}^{n}}\rho_{2}(A)^{q}\right)^{1/q^{\prime}}$

by 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:

$\displaystyle\int_{\mathbb{B}^{n}}\rho_{2}(A)^{q}=$
$\displaystyle\int_{\mathbb{B}^{n}}\rho_{2}(A)\rho_{2}(\textup{D}^{2}\psi)\lesssim\int_{\mathbb{B}^{n}}\langle
L_{\mathcal{Q}}A,\textup{D}^{2}\psi\rangle=\int_{\mathbb{B}^{n}}\psi\operatorname{Div}^{2}(L_{\mathcal{Q}}A)$
$\displaystyle\lesssim\|\psi\|_{L^{p^{\prime}}(\mathbb{B}^{n})}\|\operatorname{Div}^{2}(L_{\mathcal{Q}}A)\|_{L^{p}(\mathbb{B}^{n})}\lesssim\left(\int_{\mathbb{B}^{n}}\rho_{2}(A)^{q}\right)^{1/q^{\prime}}\|\operatorname{Div}^{2}(L_{\mathcal{Q}}A)\|_{L^{p}(\mathbb{B}^{n})}.$

The first inequality follows from (8.2) and the second is also proved after
noting that for $A\in\mathcal{K}$ we have $|A|\lesssim\rho_{2}(A)$.

The next result shows that the exponent $q=2$ in Proposition 8.2 is optimal.

######  Proposition 8.3.

Let $n\geqslant 2$. For each $\varepsilon>0$ there is a closed convex cone
$\mathcal{K}^{\varepsilon}\subset\mathrm{int}\,\Gamma_{2}\cup\\{0\\}$ such
that the inequality

$\|v\|_{L^{2+\varepsilon}(\mathbb{B}^{n})}\leqslant
C\|\mathcal{A}v\|_{L^{\infty}(\mathbb{B}^{n})}\quad\text{for }v\in
C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K}^{\varepsilon})$

fails, where
$\mathcal{A}=\operatorname{div}\mathcal{Q}=\operatorname{Div}^{2}-\Delta\operatorname{Tr}$.

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 $L^{q}$,
$q<\infty$, on the right hand side, and so the proposition falsifies that
conjecture.

[Proof]Let $u(x)=u_{\varepsilon}(x_{1},x_{2})$, where $u_{\varepsilon}$ is
given by Lemma 7.7 for $n=2$. Then clearly $\textup{D}^{2}u\in L^{2}\setminus
L^{2+\varepsilon}_{\operatorname{loc}}(\mathbb{B}^{n},\mathcal{K}^{\varepsilon})$,
where

$\mathcal{K}^{\varepsilon}=\left\\{\left(\begin{matrix}A_{0}&0\\\
0&0\end{matrix}\right)\colon A_{0}\in\textup{Sym}_{2}\cap
Q_{2}^{+}(K)\right\\}$

is contained in $\mathrm{int\,}\Gamma_{2}\cup\\{0\\}$ and $K=K(\varepsilon)$.
By (8.2) and Definition 7.4, we have that $\mathcal{A}\textup{D}^{2}u=0$. The
same cut-off argument as in the proof of Theorem B, noting that

$\operatorname{Div}^{2}L_{\mathcal{Q}}(\rho\textup{D}^{2}u)=\langle
L_{\mathcal{Q}}\textup{D}^{2}\rho,\textup{D}^{2}u\rangle,$

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
$\mathcal{Q}A_{j}=\operatorname{Div}L_{\mathcal{Q}}A_{j}=0$ for each $j$.

######  Corollary 8.4.

Let $\mathcal{K}\subset\mathrm{int}\,\Gamma_{2}\cup\\{0\\}$ be a closed convex
cone. Consider a sequence $A_{j}\in C^{\infty}(\mathbb{R}^{n},\mathcal{K})$
which is bounded in $L^{1}_{\operatorname{loc}}(\mathbb{R}^{n})$ and such that
$\mathcal{Q}A_{j}=0$ for all $j$. There is $\delta=\delta(\mathcal{K})$ such
that $A_{j}$ is bounded in
$L^{2+\delta}_{\operatorname{loc}}(\mathbb{R}^{n}).$ In particular, $(A_{j})$
is locally $2$-uniformly integrable.

[Proof]For a scalar field $\rho\in C_{c}^{\infty}(2B)$ we have

$\displaystyle\operatorname{Div}^{2}(L_{\mathcal{Q}}(\rho A))$
$\displaystyle=\operatorname{div}(\rho\operatorname{Div}(L_{\mathcal{Q}}A)+(L_{\mathcal{Q}}A)\textup{D}\rho)$
$\displaystyle=\operatorname{Div}(L_{\mathcal{Q}}A)\cdot\textup{D}\rho+\langle\textup{D}^{2}\rho,L_{\mathcal{Q}}A\rangle=\langle\textup{D}^{2}\rho,L_{\mathcal{Q}}A\rangle.$

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 $n\geqslant 3$. In particular, we prove the end-point
estimate for $p=1$ in the setting of (1.1), for fields restricted to
$\textup{Sym}^{+}_{n}$ and an operator which is related to but strictly weaker
than $\operatorname{Curl}$. To fix the notation for this operator, we
distinguish between the cases $n=2k$ is even and $n=2k+1$ is odd.

In even dimensions $n=2k$, we define the operator $\mathcal{P}_{n}$ which
takes a matrix field $A=(a_{ij})$ to the $2\times k$ matrix formed of curls
acting in $\partial_{j}$, $\partial_{j+1}$ on principal sub-matrices
$A_{j,j+1}$ with $j$ odd:

$\mathcal{P}_{n}A=\begin{pmatrix}\partial_{2}a_{11}-\partial_{1}a_{12}&\cdots&\partial_{n}a_{n-1,n-1}-\partial_{n-1}a_{n-1,n}\\\
\partial_{2}a_{21}-\partial_{1}a_{22}&\cdots&\partial_{n}a_{n,n-1}-\partial_{n-1}a_{nn}\end{pmatrix}.$

Note that this has exactly $2k=n$ entries, i.e. it has the same number of
entries as $\operatorname{Div}$. One easily checks that
$\Lambda_{\mathcal{P}_{n}}$ consists of matrices of the form

$\begin{pmatrix}a_{1}\otimes(\xi_{1},\xi_{2})&*&*\\\ *&\ddots&*\\\
*&*&a_{k}\otimes(\xi_{n-1},\xi_{n})\end{pmatrix}$

where $a_{1},\dots,a_{k}\in\mathbb{R}^{2}$ and $*$ denotes arbitrary entries.

When $n=2k+1$, we define $\mathcal{P}_{n}$ as the operator that takes a matrix
field $A=(a_{ij})$ to the $2\times n$ matrix formed of curls acting in
$\partial_{j}$, $\partial_{j+1}$ on principal sub-matrices $A_{j,j+1}$ with
$j=1,\dots,n$, the last pair being taken modulo $n$:

$\mathcal{P}_{n}A=\begin{pmatrix}\partial_{2}a_{11}-\partial_{1}a_{12}&\partial_{3}a_{22}-\partial_{2}a_{23}\cdots&\partial_{1}a_{n,n}-\partial_{n}a_{n,1}\\\
\partial_{2}a_{21}-\partial_{1}a_{22}&\partial_{3}a_{32}-\partial_{2}a_{33}\cdots&\partial_{1}a_{1,n}-\partial_{n}a_{11}\end{pmatrix}.$

######  Proposition 8.5.

Take $K\geqslant 1$ and let $A\in
C_{c}^{\infty}(\mathbb{R}^{n},\textup{Sym}^{+}_{n}\cap Q_{n}^{+}(K))$. Then

$\|A\|_{L^{1^{*}}(\mathbb{R}^{n})}\leqslant
C_{K}\|\mathcal{P}_{n}A\|_{L^{1}(\mathbb{R}^{n})}.$

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 $2\times 2$
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
$A\in\textup{Sym}^{+}_{n}$ has positive diagonal entries; secondly, under our
assumptions, given any $2\times 2$ principal minor $M$, there is a constant
$c=c(K)$ such that $M(A)\geqslant c|A|^{2}$. The rest of the proof is split
into two cases.

Case 1: even dimension. Assume $n=2k$. For each sub-matrix $A_{j,j+1}$ with
$j$ odd, $j=1,3,...,n-1$, we may apply Corollary 4.6 to get

$\displaystyle\int_{\mathbb{R}^{2}}M_{j,j+1}(A)\operatorname{d}\\!x_{j}\operatorname{d}\\!x_{j+1}\leqslant\Big{(}\int_{\mathbb{R}^{2}}|\mathcal{P}_{n}A|(x)\operatorname{d}\\!x_{j}\operatorname{d}\\!x_{j+1}\Big{)}^{2}.$

Using the lower bound $M_{j,j+1}\geqslant c|A|^{2}$, we take square root of
this inequality and integrate in the other variables to obtain

$\displaystyle\int_{\mathbb{R}^{n-2}}\Big{(}\int_{\mathbb{R}^{2}}|A|^{2}(x)\operatorname{d}\\!x_{j}\operatorname{d}\\!x_{j+1}\Big{)}^{\frac{1}{2}}\operatorname{d}\\!x_{1}\cdots\operatorname{d}\\!x_{j-1}\operatorname{d}\\!x_{j+2}\cdots\operatorname{d}\\!x_{n}\leqslant
C\|\mathcal{P}_{n}A\|_{L^{1}(\mathbb{R}^{n})}.$

Applying Minkowski’s integral inequality to re-order the integrals, this
implies

$\displaystyle\|A\|_{L^{\mathbf{p}_{j}}(\mathbb{R}^{n})}$
$\displaystyle\equiv\,\int_{\mathbb{R}^{n-(j+1)}}\Big{(}\int_{\mathbb{R}^{2}}\Big{(}\int_{\mathbb{R}^{j-1}}|A|\operatorname{d}\\!x_{1}...\operatorname{d}\\!x_{j-1}\Big{)}^{2}\operatorname{d}\\!x_{j}\operatorname{d}\\!x_{j+1}\Big{)}^{\frac{1}{2}}\operatorname{d}\\!x_{j+2}...\operatorname{d}\\!x_{n}$
$\displaystyle\leqslant C\|\mathcal{P}_{n}A\|_{L^{1}(\mathbb{R}^{n})};$

here $\mathbf{p}_{j}\in\mathbb{R}^{n}$ 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

$\||A|^{k}\|_{L^{\mathbf{r}}}\leqslant\prod_{\begin{subarray}{c}j=1\\\ j\text{
odd}\end{subarray}}^{n-1}\|A\|_{L^{\mathbf{p}_{j}}}\leqslant
C^{k}\|\mathcal{P}_{n}A\|_{L^{1}}^{k},$

where $\mathbf{r}$ is computed component-wise by

$\frac{1}{\mathbf{r}}=\sum_{\begin{subarray}{c}j=1\\\ j\text{
odd}\end{subarray}}^{n-1}\frac{1}{\mathbf{p}_{j}}=(k-1+\frac{1}{2},\ldots,k-1+\frac{1}{2}).$

Thus, the norm on the left-hand side is simply the $L^{r}$-norm with
$r=\frac{2}{2k-1}=\frac{2}{n-1}$, and so we may rewrite the previous
inequality as

$\Big{(}\int_{\mathbb{R}^{n}}|A|^{\frac{n}{n-1}}\operatorname{d}\\!x\Big{)}^{\frac{n-1}{2}}\leqslant
C\|\mathcal{P}_{n}A\|_{L^{1}(\mathbb{R}^{n})}^{k}.$

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 $j=1,2,3,\ldots,n-1$, the estimate

$\|A\|_{L^{\mathbf{p}_{j}}}\leqslant C\|\mathcal{P}_{n}A\|_{L^{1}},$

where $\mathbf{p}_{j}$ is as before. We now work with the sub-matrix $A_{1,n}$
on the $x_{1}$-$x_{n}$ plane to obtain

$\displaystyle\int_{\mathbb{R}^{n-2}}\Big{(}\int_{\mathbb{R}^{2}}|A|^{2}(x)\operatorname{d}\\!x_{1}\operatorname{d}\\!x_{n}\Big{)}^{\frac{1}{2}}\operatorname{d}\\!x_{2}\cdots\operatorname{d}\\!x_{n-1}\leqslant
C\|\mathcal{P}_{n}A\|_{L^{1}},$

leading to

$\|A\|_{L^{\mathbf{p}_{n}}}\leqslant C\|\mathcal{P}_{n}A\|_{L^{1}},$

where $\mathbf{p}_{n}\in\mathbb{R}^{n}$ 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 $\frac{1}{\mathbf{p}_{j}}$ consists of $n-2+2\times\frac{1}{2}$,
so that

$\sum_{j=1}^{n}\frac{1}{\mathbf{p}_{j}}=(n-1,\ldots,n-1).$

Thus, applying the generalised Hölder’s inequality,

$\displaystyle\||A|^{n}\|_{L^{\frac{1}{n-1}}(\mathbb{R}^{n})}\leqslant\prod_{j=1}^{n}\|A\|_{L^{\mathbf{p}_{j}}(\mathbb{R}^{n})}\leqslant
C^{n}\|\mathcal{P}_{n}A\|_{L^{1}(\mathbb{R}^{n})}^{n}.$

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

$\rho_{k}(\textup{D}^{2}u)-\rho_{k}(\textup{D}^{2}v)=0,$

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 $\psi$, where we write $\psi\equiv\frac{1}{2}Sx\cdot
x+\varphi$.

Let $x_{0},x_{1}$ be such that $\psi(x_{0})=\min_{\mathbb{T}^{n}}\psi$,
$\psi(x_{1})=\max_{\mathbb{T}^{n}}\psi$. Let
$\tilde{\psi}=\psi-\max_{2\mathbb{T}^{n}}\psi$, which is non-positive in
$2\mathbb{T}^{n}$. With $R=|x_{0}-x_{1}|\leqslant 1$, and according to Theorem
9.1 in [75],

$-\tilde{\psi}(x_{0})\lesssim\Lambda-\tilde{\psi}(x_{1})\implies\tilde{\psi}(x_{1})-\tilde{\psi}(x_{0})=\textup{osc}_{\mathbb{T}^{n}}\psi\lesssim\Lambda,$

where $\lambda\leqslant f\leqslant\Lambda$. Thus we deduce the estimate

$\sup|\psi|\lesssim\Lambda.$

This implies a $C^{1}$ estimate on $\psi$ by [75, Thm. 4.1], which in turn
implies a $C^{2}$ estimate on $\psi$ (see for instance [30]). From here we can
apply the Evans–Krylov Theorem, see e.g. [14], to deduce that

$\|\psi\|_{C^{2,\alpha}}\leqslant
C(\Lambda,\lambda)\implies\|\varphi\|_{C^{2,\alpha}}\leqslant
C(\Lambda,\lambda)+C\|S\|.$

We conclude the proof by applying the continuity method, as in [47]: we let

$S_{t}(\varphi)=F_{k}(S+D^{2}\varphi)-tf^{k}-(1-t)F_{k}(S)$

which maps $C^{2,\alpha}_{\sharp}$ to $C^{\alpha}_{\sharp}$; here the
subscript denotes mean-free functions. The crucial point now is that, for
$\varphi$ in $C^{2,\alpha}$, if $S+D^{2}\varphi\in\Gamma_{k}$, the linearized
operator $S^{\prime}_{t}$ is elliptic. It follows in particular from standard
Schauder theory that $S_{t}^{\prime}\colon C^{2,\alpha}_{\sharp}\to
C^{\alpha}_{\sharp}$ is bijective, i.e. the linearized equation can be solved
uniquely for a given datum. Thus the continuity method applies.

##  Appendix B $\mathcal{A}$-quasiaffine integrands and $L\log L$ estimates

Given finite dimensional inner product spaces $\mathbb{V}$ and $\mathbb{W}$
and an $\ell$-th order homogeneous differential operator

$\mathcal{A}=\sum_{|\alpha|=\ell}L_{\alpha}\partial^{\alpha}\quad\text{ where
}L_{\alpha}\in\textup{Lin}(\mathbb{V},\mathbb{W}),$

we say that a locally bounded, Borel measurable integrand
$F\colon\mathbb{V}\rightarrow\mathbb{R}$ is $\mathcal{A}$-quasiconvex if

$\int_{\mathbb{T}^{n}}F(v(x))\operatorname{d}\\!x\geqslant
F\left(\int_{\mathbb{T}^{n}}v(x)\operatorname{d}\\!x\right)\quad\text{for all
}v\in C^{\infty}(\mathbb{T}^{n},\mathbb{V})\text{ with }\mathcal{A}v=0,$

and we say that $F$ is $\mathcal{A}$-quasiaffine if both $\pm F$ are
$\mathcal{A}$-quasiconvex. The latter is the class of weakly continuous
functionals defined on $\mathcal{A}$-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 $\mathcal{A}$, with symbol

$\mathcal{A}(\xi)=\sum_{|\alpha|=\ell}\xi^{\alpha}L_{\alpha}\in\operatorname{Lin}(\mathbb{V},\mathbb{W}),$

is said to have constant rank if $\mathrm{rank\,}\mathcal{A}(\xi)$ is
independent of $\xi\in\mathbb{R}^{n}\setminus\\{0\\}$, and is said to have a
spanning wave cone if

$\mathrm{span\,}\Lambda_{\mathcal{A}}=\mathbb{V},\quad\text{where
}\Lambda_{\mathcal{A}}=\bigcup_{\xi\in\mathbb{R}^{n}\setminus\\{0\\}}\ker\mathcal{A}(\xi).$

Under the spanning wave cone condition it is classical that
$\mathcal{A}$-quasiaffine integrands are polynomials. We then have:

######  Theorem B.1 ([33, Thm. C]).

Let $\mathcal{A}$ be a constant rank operator with spanning wave cone, and let
$F\colon\mathbb{V}\rightarrow\mathbb{R}$ be an $\mathcal{A}$-quasiaffine
polynomial of degree $s\in\\{2,\ldots,\min\\{n,\dim\mathbb{V}\\}\\}$. Let
$r>s$ be a real number. Then

$\|F(v)\|_{L\log L(\mathbb{B}^{n})}\leqslant
C\left(\|v\|_{L^{s}(\mathbb{B}^{n})}+\|\mathcal{A}v\|_{{\dot{W}}{{}^{-\ell,r}}(\mathbb{B}^{n})}\right)\quad\text{for
}v\in C_{c}^{\infty}(\mathbb{B}^{n},\mathbb{V})\text{ with }F(v)\geqslant 0.$

This result can be used to give inequalities in the spirit of (1.1):

######  Corollary B.2.

In the setting of Theorem B.1, let $\mathcal{K}=\\{z\in\mathbb{V}\colon
c|z|^{s}\leqslant F(z)\\}$, for some constant $c>0$. Then

$\|v\|_{L^{s}\log L(\mathbb{B}^{n})}\leqslant
C\left(\|v\|_{L^{s}(\mathbb{B}^{n})}+\|\mathcal{A}v\|_{{\dot{W}}{{}^{-\ell,r}}(\mathbb{B}^{n})}\right)\quad\text{for
}v\in C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K}).$

We therefore obtain sharp uniform integrability statements:

######  Corollary B.3.

In the setting of Corollary B.2, if a sequence $(v_{j})\subset
C_{c}^{\infty}(\mathbb{B}^{n},\mathcal{K})$ is bounded in
$L^{s}(\mathbb{B}^{n})$ and $(\mathcal{A}v_{j})$ is bounded in
${\dot{W}}{{}^{-\ell,r}}(\mathbb{B}^{n})$, we have that $(v_{j})$ is
$s$-uniformly integrable.

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(André Guerra) School of Mathematics, Institute for Advanced Study, 1 Einstein
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(Bogdan Raiţă) Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale
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(Matthew Schrecker) Department of Mathematics, University College London,
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