# A note on the Jacobian problem of Coifman, Lions, Meyer and Semmes

Sauli Lindberg  Department of Mathematics and Statistics, University of
Helsinki, P.O. Box 68, 00014 Helsingin yliopisto, Finland
<EMAIL_ADDRESS>
###### Abstract.

Coifman, Lions, Meyer and Semmes asked in 1993 whether the Jacobian operator
and other compensated compactness quantities map their natural domain of
definition onto the real-variable Hardy space
$\mathcal{H}^{1}(\mathbb{R}^{n})$. We present an axiomatic, Banach space
geometric approach to the problem in the case of quadratic operators. We also
make progress on the main open case, the Jacobian equation in the plane.

The author was supported by the ERC Advanced Grant 834728.

##  1\. Introduction

The real-variable Hardy space $\mathscr{H}^{1}(\mathbb{R}^{n})$ often acts as
a good substitute for $L^{1}(\mathbb{R}^{n})$, mirroring many of the ways in
which $\textup{BMO}(\mathbb{R}^{n})$ substitutes $L^{\infty}(\mathbb{R}^{n})$
[15, 47]. In order to define $\mathscr{H}^{1}(\mathbb{R}^{n})$ we fix $\chi\in
C_{c}^{\infty}(\mathbb{R}^{n})$ with $\int_{\mathbb{R}^{n}}\chi(x)\,dx\neq 0$,
denote $\chi_{t}(x)\mathrel{:\mkern-0.25mu=}t^{-n}\chi(x/t)$ for every
$x\in\mathbb{R}^{n}$ and $t>0$ and set

$\mathcal{H}^{1}(\mathbb{R}^{n})\mathrel{:\mkern-0.25mu=}\left\\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})\colon\sup_{t>0}\left\lvert
f*\chi_{t}(\cdot)\right\rvert\in L^{1}(\mathbb{R}^{n})\right\\}.$

We endow $\mathscr{H}^{1}(\mathbb{R}^{n})$ with the norm $\left\lVert
f\right\rVert_{\mathscr{H}^{1}}\mathrel{:\mkern-0.25mu=}\left\lVert\sup_{t>0}\left\lvert
f*\chi_{t}(\cdot)\right\rvert\right\rVert_{L^{1}}$.

Connections between $\mathscr{H}^{1}$ integrability, commutators and weak
sequential continuity were explored by Coifman & al. in the highly influential
work [3]. Coifman & al. showed that when $n\geq 2$, the Jacobian determinants
of mappings in $\dot{W}^{1,n}(\mathbb{R}^{n},\mathbb{R}^{n})$ and several
other compensated compactness quantities belong to
$\mathscr{H}^{1}(\mathbb{R}^{n})$. The result was motivated by Müller’s higher
integrability result on Jacobians: if $u\in W^{1,n}(\Omega,\mathbb{R}^{n})$
and $J_{u}\geq 0$ a.e. in an open set $\Omega\subset\mathbb{R}^{n}$, then
$Ju\log(2+Ju)\in L^{1}_{loc}(\Omega)$ [43], in direct analogy to Stein’s
classical $L\textup{log}L$ result on $\mathscr{H}^{1}$ functions. For later
developments of the $\mathcal{H}^{1}$ theory of compensated compactness
quantities see e.g. [4, 14, 16, 19, 20, 29, 35, 36, 39, 41, 45, 48, 51].

Coifman & al. proceeded to ask whether these nonlinear quantities are
surjections onto $\mathcal{H}^{1}(\mathbb{R}^{n})$ [3, p. 258]. The most
famous open case is the following:

(1.1) $\textup{Is
}J\colon\dot{W}^{1,n}(\mathbb{R}^{n},\mathbb{R}^{n})\to\mathscr{H}^{1}(\mathbb{R}^{n})\textup{
surjective?}$

As a partial result, Coifman & al. showed that
$\mathscr{H}^{1}(\mathbb{R}^{n})$ is the smallest Banach space that contains
the range of the Jacobian. More precisely,

(1.2)
$\mathscr{H}^{1}(\mathbb{R}^{n})=\left\\{\sum_{j=1}^{\infty}Ju_{j}\colon\sum_{j=1}^{\infty}\left\lVert
Du_{j}\right\rVert_{L^{n}}^{n}<\infty\right\\}$

[3, Theorem III.2]. Further partial results were presented in [16, 18, 38].
When the domain of definition of $J$ is the _inhomogeneous_ Sobolev space
$W^{1,n}(\mathbb{R}^{n},\mathbb{R}^{n})$, the author proved non-surjectivity
in [39].

In bounded domains, the Dirichlet problem $Ju=f$ in $\Omega$, $u=\textup{id}$
on $\partial\Omega$ has a classical theory starting from the seminal works of
Moser and Dacorogna in [8, 42] and reviewed in [7]. In a setting close to
ours, when $u\in W^{1,n}_{\textup{id}}(\Omega,\mathbb{R}^{n})$ has Jacobian
$Ju\geq 0$ a.e., Müller’s higher integrability result implies that $Ju\in
L\log L(\Omega)$. As an analogue of (1.1), Hogan & al. asked whether every
non-negative $f\in L\log L(\Omega)$ with mean 1 has a solution $u\in
W^{1,n}_{\textup{id}}(\Omega,\mathbb{R}^{n})$ [23, p. 206]. Counter-examples
were recently given by Guerra & al. in [17]. For all $L^{q/n}$ data, $1<q<n$,
”pointwise” (as opposed to distributional) solutions in
$W^{1,q}(\Omega,\mathbb{R}^{n})$ with arbitrary boundary values in
$W^{1-1/q,q}(\Omega,\mathbb{R}^{n})$ were constructed in [33]. At any rate,
compared to the Dirichlet problem, different ideas are needed in the case of
$\mathbb{R}^{n}$ due to the unboundedness of the domain and the absence of
boundary conditions.

In [28], Iwaniec conjectured that for every $n\geq 2$ and $p\in[1,\infty)$ the
Jacobian operator
$J\colon\dot{W}^{1,np}(\mathbb{R}^{n},\mathbb{R}^{n})\to\mathcal{H}^{p}(\mathbb{R}^{n})$
not only is surjective but also has a continuous right inverse
$G\colon\mathscr{H}^{p}(\mathbb{R}^{n})\to\dot{W}^{1,np}(\mathbb{R}^{n},\mathbb{R}^{n})$;
recall that $\mathscr{H}^{p}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$ whenever
$1<p<\infty$. Hytönen proved in [27] the natural analogue of (1.2), so that
$L^{p}(\mathbb{R}^{n})$ is, again, the minimal Banach space containing
$J(\dot{W}^{1,np}(\mathbb{R}^{n},\mathbb{R}^{n}))$. We discuss Hytönen’s
contribution in §7.

We next briefly summarise some of the ideas presented in [28]. Whenever
$f\in\mathcal{H}^{p}(\mathbb{R}^{n})$ and the equation $Jv=f$ has a solution,
it has a _minimum norm solution_
$u\in\dot{W}^{1,np}(\mathbb{R}^{n},\mathbb{R}^{n})$, that is, $Ju=f$ and
$\int_{\mathbb{R}^{n}}\left\lvert
Du\right\rvert^{np}=\min_{Jv=f}\int_{\mathbb{R}^{n}}\left\lvert
Dv\right\rvert^{np}$. Furthermore, the range of $J$ is dense in
$\mathcal{H}^{p}(\mathbb{R}^{n})$. Iwaniec suggested a possible way of finding
a continuous right inverse $G$:

######  Strategy 1.1.

When $n\geq 2$ and $1\leq p<\infty$, the following claims would yield a
continuous right inverse of
$J\colon\dot{W}^{1,np}(\mathbb{R}^{n},\mathbb{R}^{n})\to\mathscr{H}^{p}(\mathbb{R}^{n})$:

  1. (1)

Every energy-minimal solution $u$ satisfies $\left\lVert
Du\right\rVert_{L^{np}}^{n}\lesssim\left\lVert
Ju\right\rVert_{\mathscr{H}^{p}}$.

  2. (2)

For every $f\in\mathscr{H}^{p}(\mathbb{R}^{n})$ there is a unique energy-
minimal solution $u_{f}$, modulo rotations.

  3. (3)

There exist rotations $R_{f}\in SO(n)$ such that $f\mapsto R_{f}u_{f}$ is a
continuous right inverse of $J$.

Claim (1) would easily imply the surjectivity of
$J\colon\dot{W}^{1,np}(\mathbb{R}^{n},\mathbb{R}^{n})\to\mathscr{H}^{p}(\mathbb{R}^{n})$
(see [38, p. 36]). In [16], (1) was shown to be, in fact, _equivalent_ to the
surjectivity of
$J\colon\dot{W}^{1,np}(\mathbb{R}^{n},\mathbb{R}^{n})\to\mathscr{H}^{p}(\mathbb{R}^{n})$;
this amounts to an open mapping theorem for the Jacobian. In [18], in turn,
(2) was shown to be false whenever $n=2$ and $1\leq p<\infty$. Nevertheless,
in [18], Guerra & al. found an explicit class of data whose energy-minimal
solution is, indeed, unique up to rotations. Another large class of such data
is constructed in Theorem 1.10 below. Despite the falsity of (2), claim (1)
and Iwaniec’s conjecture itself remain open. In the case $n=2$, $p=1$, a novel
Banach space geometric approach was presented in [38] to attack claim (1).

In this work, we present a natural abstract framework for the ideas of [38]
and study the surjectivity question for rather general quadratic compensated
compactness quantities, streamlining the exposition of [38] considerably. We
also make further progress on the main special case
$J\colon\dot{W}^{1,2}(\mathbb{R}^{2},\mathbb{R}^{2})\to\mathcal{H}^{1}(\mathbb{R}^{2})$.
In §7, we discuss how to adapt the methods to the case $n=2$, $1<p<\infty$.

###  1.1. Connection to commutators

Question (1.1) can be seen as a variant of the classical _factorisation
problem_ from complex analysis [46, §4.2]. Indeed, an equivalent formulation
of (1.1) in terms of differential forms is whether
$\mathscr{H}^{1}(\mathbb{R}^{n})=*\wedge_{j=1}^{n}\textup{d}\dot{W}^{1,n}(\mathbb{R}^{n})$
(since $Ju=*\textup{d}u_{1}\wedge\cdots\wedge\textup{d}u_{n}$). The
decomposition (1.2) is called a _weak factorisation_ of
$\mathscr{H}^{1}(\mathbb{R}^{n})$. In the plane, one can also deduce (1.2)
from the two-sided commutator estimate

(1.3) $c\left\lVert
b\right\rVert_{\operatorname{BMO}}\leq\left\lVert[b,T]\right\rVert_{L^{2}\to
L^{2}}\leq C\left\lVert b\right\rVert_{\operatorname{BMO}}$

for the Beurling transform $T={\mathcal{S}}\colon
L^{2}(\mathbb{C},\mathbb{C})\to L^{2}(\mathbb{C},\mathbb{C})$ and
$b\in\textup{BMO}(\mathbb{C})$ [38].

Estimates of the form (1.3) go back to the seminal work [5] of Coifman & al.,
with Nehari’s theorem on Hankel operators as a precursor. When
$j\in\\{1,\ldots,n\\}$ and $T=R_{j}$ is a Riesz transform, the _upper bound
estimate_ $\left\lVert[b,R_{j}]\right\rVert_{L^{2}\to L^{2}}\leq C\left\lVert
b\right\rVert_{\text{BMO}}$ [5], the formula

(1.4) $\int_{\mathbb{R}^{n}}b(\omega R_{j}\gamma+\gamma
R_{j}\omega)=\int_{\mathbb{R}^{n}}\omega[b,R_{j}]\gamma$

and $\mathcal{H}^{1}-\operatorname{BMO}$ duality imply that $\omega
R_{j}\gamma+\gamma R_{j}\omega\in\mathscr{H}^{1}(\mathbb{R}^{n})$ for all
$\omega,\gamma\in L^{2}(\mathbb{R}^{n})$. The _lower bound estimate_
$\left\lVert[b,R_{j}]\right\rVert_{L^{2}\to L^{2}}\geq c\left\lVert
b\right\rVert_{\text{BMO}}$ [5, 30, 49] gives the weak factorisation
$\mathscr{H}^{1}(\mathbb{R}^{n})=\\{\sum_{i=1}^{\infty}(\omega_{i}R_{j}\gamma_{i}+\gamma_{i}R_{j}\omega_{i})\colon\sum_{i=1}^{\infty}(\left\lVert\omega_{i}\right\rVert_{L^{2}}^{2}+\left\lVert\gamma_{i}\right\rVert_{L^{2}}^{2})<\infty\\}$
by simple functional analysis. By a result of Uchiyama, the commutator
$[b,R_{j}]$ is compact if and only if $b\in\textup{CMO}(\mathbb{R}^{n})$. Note
that (1.4) and the compactness of $[b,R_{j}]$ immediately lead to the weak-to-
weak∗ sequential continuity of the quadratic operator
$(\omega,\gamma)\mapsto\omega R_{j}\gamma+\gamma R_{j}\omega\colon
L^{2}(\mathbb{R}^{n})^{2}\to\mathscr{H}^{1}(\mathbb{R}^{n})$. These results
have been extended in numerous ways, as reviewed in [50]. In §3.1 we briefly
discuss the case of commutators with Calderón-Zygmund operators.

In Assumptions 1.2–1.3 below, we axiomatise the above-mentioned properties of
the Riesz transform $R_{j}$ and the quantity $\omega R_{j}\gamma+\gamma
R_{j}\omega$. We study the factorisation problem for quadratic weakly
continuous quantities in real-variable function spaces such as
$\mathscr{H}^{1}(\mathbb{R}^{n})$, using isometric Banach space geometry as
the main tool. Since our choice of methodology is rather unconventional in the
study of nonlinear PDE’s, we motivate it at length in §1.3. First, however, we
specify the mathematical setting of this paper.

###  1.2. The mathematical setting

We fix a real Banach space $X$ with a separable dual $X^{*}$ and a real or
complex Hilbert space $H$, and we denote the coefficient field of $H$ by
$\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$. The canonical examples are
$X=\operatorname{CMO}(\mathbb{R}^{n})$,
$X^{*}=\mathcal{H}^{1}(\mathbb{R}^{n})$,
$X^{**}=\operatorname{BMO}(\mathbb{R}^{n})$ and
$H=L^{2}(\mathbb{R}^{n},\mathbb{R}^{m})$ or
$H=L^{2}(\mathbb{R}^{n},\mathbb{C})$, where $n,m\in\mathbb{N}$.

######  Assumption 1.2.

A bilinear mapping

$(b,\omega)\mapsto T_{b}\omega\colon X^{**}\times H\to H$

satisfies the following conditions:

  1. (i)

$c\left\lVert b\right\rVert_{X^{**}}\leq\left\lVert T_{b}\right\rVert_{H\to
H}\leq C\|b\|_{X^{**}}$ for every $b\in X^{**}$,

  2. (ii)

$T_{b}$ is compact for every $b\in X$.

######  Assumption 1.3.

The bilinear mapping $(b,\omega)\mapsto T_{b}\omega$ satisfies the following
conditions for every $b\in X^{**}$:

  1. (i)

$T_{b}$ is self-adjoint.

  2. (ii)

$\left\lVert T_{b}\right\rVert_{H\to H}=\sup_{\left\lVert
f\right\rVert_{H}=1}\langle T_{b}f,f\rangle$.

######  Definition 1.4.

Given $X$, $H$ and $(b,\omega)\mapsto T_{b}f$ we define a norm-to-norm and
weak-to-weak∗ sequentially continuous mapping $Q\colon H\to X^{*}$ by

$\langle b,Q\omega\rangle_{X-X^{*}}\mathrel{:\mkern-0.25mu=}\langle
T_{b}\omega,\omega\rangle_{H}.$

Henceforth, Assumptions 1.2–1.3 will remain in place for the rest of the
introduction. Assumption 1.3 is made mainly to make the quadratic operator $Q$
real-valued and ensure that
$X^{*}=\\{\sum_{j=1}^{\infty}Q\omega_{j}\colon\sum_{j=1}^{\infty}\left\lVert\omega_{j}\right\rVert_{H}^{2}<\infty\\}$.
Whenever the map $(b,\omega)\mapsto T_{b}\omega$ satisfies Assumption 1.2, the
modified operator
$(b,(\omega,\gamma))\mapsto\tilde{T}_{b}(\omega,\gamma)\mathrel{:\mkern-0.25mu=}(T_{b}^{*}\gamma,T_{b}\omega)\colon
X^{**}\times(H\times H)\to H\times H$ satisfies Assumptions 1.2–1.3 (see
Proposition 2.10). Examples 3.5–3.6 illustrate the role of Assumption 1.3
further.

The planar Jacobian arises as follows (see Example 3.5): defining $T_{b}\colon
L^{2}(\mathbb{C},\mathbb{C})\to L^{2}(\mathbb{C},\mathbb{C})$ by
$T_{b}\omega\mathrel{:\mkern-0.25mu=}\overline{({\mathcal{S}}b-b{\mathcal{S}})\overline{{\mathcal{S}}\omega}}$,
where ${\mathcal{S}}$ is the Beurling transform, we can write
$Q\omega=\left\lvert{\mathcal{S}}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}=\left\lvert
u_{z}\right\rvert^{2}-\left\lvert u_{\bar{z}}\right\rvert^{2}=Ju$, where
$u\in\dot{W}^{1,2}(\mathbb{C},\mathbb{C})$ is the Cauchy transform of
$\omega\in L^{2}(\mathbb{C},\mathbb{C})$.

Assumptions 1.2–1.3 are not enough to determine whether $Q(H)=X^{*}$. Indeed,
$Q\colon L^{2}(\mathbb{R})\to\mathscr{H}^{1}(\mathbb{R})$,
$Q(\omega)=\omega^{2}-(H\omega)^{2}$ is non-surjective but (its Gâteaux
derivative) $\tilde{Q}\colon L^{2}(\mathbb{R})\times
L^{2}(\mathbb{R})\to\mathscr{H}^{1}(\mathbb{R})$,
$\tilde{Q}(\omega,\gamma)=\omega\gamma-H\omega H\gamma$ is surjective (see
Example 3.6). We address the following question:

######  Question 1.5.

Under which extra assumptions is $Q(H)=X^{*}$?

Below we mostly study all the operators given by Definition 1.4 in a unified
manner, as precise information on $Q(H)$ is valuable whether one intends to
prove surjectivity or non-surjectivity. However, in Assumption 1.8 we specify
a useful criterion which _holds_ for the operators
$\omega\mapsto\omega^{2}-(H\omega)^{2}\colon
L^{2}(\mathbb{R})\to\mathscr{H}^{1}(\mathbb{R})$ and
$\omega\mapsto\left\lvert{\mathcal{S}}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}\colon
L^{2}(\mathbb{C},\mathbb{C})\to\mathscr{H}^{1}(\mathbb{C})$ but _fails_ for
their Gâteaux derivatives.

###  1.3. Overall strategy and aim

We next describe the motivation behind the approach adopted in [38] and this
paper. Elementary proofs of various statements are given in §4.

$\bullet$ Whenever $f\in X^{*}$ and the equation $Q\gamma=f$ has a solution,
it has a _minimum norm solution_ $\omega\in H$, that is, $Q\omega=f$ and
$\left\lVert\omega\right\rVert_{H}=\min_{Q\gamma=f}\left\lVert\gamma\right\rVert_{H}$.

$\bullet$ We define an _energy functional_ $\mathcal{E}\colon
X^{*}\to\mathbb{R}\cup\\{\infty\\}$ by

$\mathcal{E}(f)\mathrel{:\mkern-0.25mu=}\begin{cases}\min\\{\left\lVert\omega\right\rVert_{H}^{2}\colon
Q\omega=f\\},&f\in Q(H),\\\ \infty,&f\notin Q(H).\end{cases}$

Almost by definition, $\mathcal{E}\geq\left\lVert\cdot\right\rVert_{X^{*}}$.
Claim (1) in Strategy 1.1 can be stated equivalently as the estimate
$\mathcal{E}\lesssim\left\lVert\cdot\right\rVert_{X^{*}}$.

$\bullet$ By a nonlinear version of the Banach-Schauder open mapping theorem,
recently proved by Guerra & al. in [16], for translation-invariant operators
(such as the Jacobian) we have

$\textup{either
}\mathcal{E}\lesssim\left\lVert\cdot\right\rVert_{X^{*}}\textup{ in
}X^{*}\textup{ or }\mathcal{E}=\infty\textup{ outside a meagre set.}$

In particular, the surjectivity of $Q\colon H\to X^{*}$ is _equivalent_ to the
statement that $Q(H)$ is dense in $X^{*}$ and every minimum norm solution
satisfies $\left\lVert\omega\right\rVert_{H}^{2}\lesssim\left\lVert
Q\omega\right\rVert_{X^{*}}$. This gives a metamatematical justification for
taking claim (1) as a goal.

$\bullet$ Given a minimum norm solution $\omega\in H$ of $Q\omega=f$, we may
use calculus of variations to study $\omega$ via perturbed solutions
$\omega_{\epsilon}\in H$ of $Q\omega_{\epsilon}=f$,
$\left\lVert\omega_{\epsilon}-\omega\right\rVert_{H}\to 0$. However, it tends
to be very hard to construct variations that satisfy the nonlinear constraint
$Q\omega_{\epsilon}=f$. In particular, the first variation
$\omega_{\epsilon}=\omega+\epsilon\varphi$, $\varphi\in H$, is in general
unavailable.

$\bullet$ The constraint $Q\omega_{\epsilon}=f$ could be relaxed if we found a
_Lagrange multiplier_ , that is, $b\in X^{**}$ satisfying

(1.5) $\left.\frac{d}{d\epsilon}(\langle
b,Q(f+\epsilon\varphi)\rangle_{X^{**}-X^{*}}-\left\lVert
f+\epsilon\varphi\right\rVert_{H}^{2})\right|_{\epsilon=0}=0\qquad\text{for
every }\varphi\in H.$

If, furthermore, $\|b\|_{X^{**}}\leq C$ uniformly in $f$, then setting
$\varphi=f$ in (1.5) would give the sought inequality
$\mathcal{E}\lesssim\left\lVert\cdot\right\rVert_{X^{*}}$. However, given an
arbitrary minimum norm solution, the construction of a Lagrange multiplier
$b\in X^{**}$ is a formidable task–in particular, the standard Liusternik-
Shnirelman method is not applicable.

$\bullet$ Nevertheless, many minimum norm solutions do possess a Lagrange
multiplier. Indeed, (1.5) says that $\omega$ is a critical point of the
functional $I_{b}\colon H\to\mathbb{R}$,

$I_{b}(\theta)\mathrel{:\mkern-0.25mu=}\langle
b,Q\theta\rangle_{X^{**}-X^{*}}-\left\lVert\theta\right\rVert_{H}^{2}=\langle
T_{b}\theta,\theta\rangle_{H}-\left\lVert\theta\right\rVert_{H}^{2}.$

Now, if $b\in X$ and $\left\lVert T_{b}\right\rVert_{H\to H}=1$, then
$\sup_{\left\lVert\theta\right\rVert_{H}=1}I_{b}(\theta)=0$ is attained at
some $\omega\in\mathbb{S}_{H}$, so that $b$ is a Lagrange multiplier of
$\omega$ and $\left\lVert\omega\right\rVert_{H}^{2}\lesssim\left\lVert
Q\omega\right\rVert_{X^{*}}$.

$\bullet$ As every $b\in X$ with $\left\lVert T_{b}\right\rVert_{H\to H}=1$ is
a Lagrange multiplier, we use the norm of $X$ given by

$\left\lVert
b\right\rVert_{X_{Q}}\mathrel{:\mkern-0.25mu=}\sup_{\left\lVert\omega\right\rVert_{H}=1}\langle
b,Q\omega\rangle_{X-X^{*}}=\left\lVert T_{b}\right\rVert_{H\to H},$

endow $X^{*}$ with the dual norm

$\left\lVert
f\right\rVert_{X^{*}_{Q}}\mathrel{:\mkern-0.25mu=}\sup_{\left\lVert
b\right\rVert_{X_{Q}}=1}\langle f,b\rangle_{X^{*}-X}$

and denote the resulting Banach spaces by $X_{Q}$ and $X^{*}_{Q}$. Thus every
$b\in\mathbb{S}_{X_{Q}}$ is a Lagrange multiplier of some
$\omega\in\mathbb{S}_{H}$ with $Q\omega\in\mathbb{S}_{X_{Q}^{*}}$.

$\bullet$ As a consequence, the set

$\mathscr{A}\mathrel{:\mkern-0.25mu=}\\{\omega\in\mathbb{S}_{H}\colon
Q\omega\in\mathbb{S}_{X^{*}_{Q}}\\}\subset\mathbb{S}_{H}$

and its image $Q(\mathscr{A})\subset\mathbb{S}_{X^{*}_{Q}}$ are rather large.
The motivation above leads to the following more precise variant of Question
1.5, presented for the planar Jacobian in [38]:

######  Question 1.6.

Is $Q(\mathscr{A})=\mathbb{S}_{X^{*}_{Q}}$? Equivalently, is
$\mathcal{E}=\left\lVert\cdot\right\rVert_{X^{*}}$?

As already noted, there exist some natural cases where the answer to Question
1.6 is negative. A positive answer holds for the simple operator
$(b,(\omega,\gamma))\mapsto(b\omega,-b\gamma)\colon\mathbb{R}\times\mathbb{R}^{2}\to\mathbb{R}^{2}$.
It is hoped that suitable further properties of $T$, beyond Assumptions
1.2–1.3, will yield an answer to Question 1.6 for natural classes of
operators. In this paper we present several partial results.

###  1.4. Main results

As a first largeness criterion on $Q(\mathscr{A})$ we mention that

(1.6) $Q(\mathscr{A})\supset\operatorname{ext}(\mathbb{B}_{X^{*}_{Q}}),$

where $\operatorname{ext}(\mathbb{B}_{X^{*}_{Q}})$ is the set of extreme
points of $\mathbb{B}_{X_{Q}^{*}}$. Since $X^{*}_{Q}$ is a separable dual
space, the Bessaga-Pelczynski Theorem (Theorem 2.7) implies that
$\overline{\operatorname{co}}(\operatorname{ext}(\mathbb{B}_{X^{*}_{Q}}))=\mathbb{B}_{X^{*}_{Q}}$.
The inclusion (1.6) is contained in Theorem 5.1. We will find more refined
information on $Q(\mathscr{A})$ by using the set-valued duality mapping.

######  Definition 1.7.

The _duality mapping_ $D\colon\mathbb{S}_{X_{Q}}\to
2^{\mathbb{S}_{X^{*}_{Q}}}$ is defined by

$D(b)\mathrel{:\mkern-0.25mu=}\\{h\in\mathbb{S}_{X^{*}_{Q}}\colon\langle
b,h\rangle_{X_{Q}-X_{Q}^{*}}=1\\}.$

We denote the set of _norm-attaining points_ by
$\textit{NA}_{\left\lVert\cdot\right\rVert_{X_{Q}}}\mathrel{:\mkern-0.25mu=}\cup_{b\in\mathbb{S}_{X_{Q}}}D(b)$.

By the Bishop-Phelps Theorem [12, Theorem 3.54],
$NA_{\left\lVert\cdot\right\rVert_{X_{Q}}}$ is dense in
$\mathbb{S}_{X^{*}_{Q}}$. Since $Q(\mathscr{A})$ is closed, Question 1.6
reduces to the question whether $D(b)\cap Q\mathscr{A}=D(b)$ for every
$b\in\mathbb{S}_{X_{Q}}$. Before presenting partial results we formulate a
useful extra assumption which is satisfied by the planar Jacobian. Its aim is
to quantify the symmetries of the class $\mathscr{A}$ (see Remark 5.6).

######  Assumption 1.8.

If $\omega,\gamma\in\mathscr{A}$ satisfy $Q\omega=Q\gamma$, then
$Q^{\prime}_{\omega}(c\gamma)\neq 0$ for some $c\in\mathbb{S}_{\mathbb{K}}$.

The following result collects partial results on Question 1.6; for relevant
definitions see §2.1. The parts (iv)–(vi) are new also in the case of the
Jacobian.

######  Theorem 1.9.

Under Assumptions 1.2–1.3, the following statements hold:

  1. (i)

For every $b\in\mathbb{S}_{X_{Q}}$, the convex set $D(b)$ has finite affine
dimension.

  2. (ii)

For every $b\in\mathbb{S}_{X_{Q}}$, $D(b)\cap Q(\mathscr{A})$ contains
$\operatorname{ext}(D(b))$ and is path-connected.

  3. (iii)

The norm and relative weak∗ topologies coincide in $Q(\mathscr{A})$.

Under the further Assumption 1.8, the following statements hold:

  1. (iv)

For every $b$ in a dense, relatively open subset of $\mathbb{S}_{X_{Q}}$,
$D(b)=\\{Q\omega\\}$ for some $\omega\in\mathscr{A}$ and
$\left\lVert\cdot\right\rVert_{X_{Q}}$ is Fréchet differentiable at $b$.

  2. (v)

$D\colon\mathbb{S}_{X_{Q}}\to 2^{\mathbb{S}_{X_{Q}^{*}}}$ is a cusco map.

  3. (vi)

The norm and relative weak∗ topologies also coincide in
$\textit{NA}_{\left\lVert\cdot\right\rVert_{X_{Q}}}$.

The proof is presented in §6. The second main result concerns uniqueness of
minimum norm solutions and is proved in §5.

######  Theorem 1.10.

Suppose Assumptions 1.2–1.3 and 1.8 hold, and let
$f\in\operatorname{ext}(\mathbb{S}_{X_{Q}^{*}})$. Then the minimum norm
solution $\omega\in\mathscr{A}$ of $Q\omega=f$ is unique up to multiplication
by $c\in\mathbb{S}_{\mathbb{K}}$.

In view of (1.6) it is natural to ask whether $X_{Q}^{*}$ is strictly convex,
that is, $\textup{ext}(\mathbb{B}_{X_{Q}^{*}})=\mathbb{S}_{X_{Q}^{*}}$. In the
case of the planar Jacobian, the answer is negative, as an immediate
consequence of Theorems 1.9–1.10 and the non-uniqueness of general minimum
norm solutions:

######  Corollary 1.11.

In the case of $\omega\mapsto
Q\omega=\left\lvert{\mathcal{S}}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}\colon
L^{2}(\mathbb{C},\mathbb{C})\to\mathcal{H}^{1}(\mathbb{C})$ we have
$\textup{ext}(\mathbb{B}_{X_{Q}^{*}})\subsetneq\mathbb{S}_{X_{Q}^{*}}$.

It is unclear to the author whether an analogue of Corollary 1.11 holds for
the Gâteaux derivative $(\omega,\gamma)\mapsto Q^{\prime}_{\omega}\gamma\colon
L^{2}(\mathbb{C},\mathbb{C})^{2}\to\mathscr{H}^{1}(\mathbb{C})$.

##  2\. Preliminaries

The main tools of this work come from isometric Banach space geometry. We
collect some notions and results and refer to [11] and [12] for most of the
proofs.

###  2.1. Smoothness properties of norms and duality mappings

In this subsection, $X$ is a real Banach space. Suppose $U\subset Z$ is open
and $g\colon U\to Z$ is convex, where $Z$ is a real Banach space. We say that
$g$ is _Gâteaux differentiable_ at $x\in U$ if there exists
$L\in\mathscr{B}(X,Z)$ such that

(2.1) $\lim_{t\to
0}\left\lVert\frac{g(x+th)-g(x)}{t}-Lh\right\rVert=0\quad\textup{for every
}h\in X.$

The operator $L$ is then denoted by $g^{\prime}_{x}$ and called the _Gâteaux
derivative_ of $g$ at $x$. If the limit in (2.1) is uniform in
$h\in\mathbb{S}_{X}$, then $g$ is said to be _Fréchet differentiable_ at $x$.

Note that the convex function $g(x)=\left\lVert x\right\rVert_{X}$ is Gâteaux
(Fréchet) differentiable at $x\in\mathbb{S}_{X}$ if and only if it is Gâteaux
(Fréchet) differentiable at $\lambda x$ for every
$\lambda\in\mathbb{R}\setminus\\{0\\}$. We will use the following theorem of
Asplund and Lindenstrauss (see [12, Theorem 8.21]) on the norm of $X$.

######  Theorem 2.1.

Suppose $X^{*}$ is separable and $g\colon X\to\mathbb{R}$ is a continuous
convex function. Then $g$ is Fréchet differentiable in a dense $G_{\delta}$
subset of $X$.

Recall that the duality mapping $D\colon\mathbb{S}_{X}\to
2^{\mathbb{S}_{X^{*}}}$ is defined by

$D(x)\mathrel{:\mkern-0.25mu=}\\{x^{*}\in\mathbb{S}_{X^{*}}\colon\langle
x^{*},x\rangle_{X^{*}-X}=1\\}.$

If $g(x)=\left\lVert x\right\rVert_{X}$ is Gâteaux differentiable at
$x\in\mathbb{S}_{X}$, then $D(x)=\\{g^{\prime}_{x}\\}$.

######  Definition 2.2.

Suppose $Y$ and $Z$ are Hausdorff topological spaces. A mapping $F\colon Y\to
2^{Z}\setminus\\{\emptyset\\}$ is said to be _upper semicontinuous_ at $y\in
Y$ if for any open set $V\supset F(y)$, there exists a neighbourhood $U$ of
$y$ in $Y$ such that $F(U)\subset V$.

We recall characterisations of the norm-to-norm upper semicontinuity of the
duality mapping [13, 24].

######  Theorem 2.3.

Let $x\in\mathbb{S}_{X}$. The following statements are equivalent.

  1. (i)

$D$ is norm-to-norm upper semicontinuous at $x$ and $D(x)$ is norm compact.

  2. (ii)

For every sequence of points $x_{j}^{*}$ in $\mathbb{S}_{X^{*}}$ such that
$\langle x_{j}^{*},x\rangle\to 1$, there exists a subsequence convergent to
some $x^{*}\in D(x)$.

  3. (iii)

The weak∗ and norm topologies agree on $\mathbb{S}_{X^{*}}$ at points of
$D(x)$.

The norm-to-norm upper semicontinuity of $D$ can also be used to characterise
Fréchet differentiability of the norm (see [11, Corollary 7.16]).

######  Theorem 2.4.

The norm $\left\lVert\cdot\right\rVert_{X}$ is Fréchet differentiable at
$x\in\mathbb{S}_{X}$ if and only if $D(x)$ is a singleton and $D$ is norm-to-
norm upper semicontinuous at $x$.

######  Definition 2.5.

Let $Y$ and $Z$ be Hausdorff topological spaces. A set-valued map $F\colon
Y\to 2^{Z}\setminus\\{\emptyset\\}$ is said to be _cusco_ (convex upper
semicontinuous nonempty compact-valued) if it is norm-to-norm upper
semicontinuous and $F(y)$ is compact for every $y\in Y$.

###  2.2. Extreme points of the unit ball of a separable dual

When $C$ is a convex subset of a real Banach space $Y$, a point $y\in C$ is an
_extreme point_ of $C$ if there exists no proper line segment that contains
$y$ and lies in $C$.

######  Definition 2.6.

A Banach space is said to have the _Krein-Milman property_ if every closed,
bounded, convex set is the closed convex hull of its extreme points.

We recall a result of C. Bessaga and A. Pełczynski (see e.g. [9, p. 198]).

######  Theorem 2.7.

Every separable dual space has the Krein-Milman property.

Recall that in Assumptions 1.2–1.3, the dual space $X^{*}$ is separable. The
real-variable Hardy space
$\mathcal{H}^{1}(\mathbb{R}^{n})=(\operatorname{CMO}(\mathbb{R}^{n}))^{*}$
satisfies this criterion (see [38, Proposition 2.15]). We also recall Milman’s
theorem on extreme points which is formulated as follows (see [11, Theorem
3.66]).

######  Theorem 2.8.

Let $K$ be a non-empty subset of a Hausdorff locally convex space such that
$\overline{\operatorname{co}}(K)$ is compact. Then every extreme point of
$\overline{\operatorname{co}}(K)$ lies in $\overline{K}$.

###  2.3. Self-adjoint variants of non-self-adjoint operators

When Assumption 1.2 holds but Assumption 1.3 does not, we use a natural self-
adjoint modification of $T_{b}$. Its existence and uniqueness are guaranteed
by the following simple lemma.

######  Lemma 2.9.

Suppose $A\colon H\to H$ is $\mathbb{K}$-linear. Then there exists a unique
self-adjoint $\mathbb{K}$-linear operator $B\colon H\times H\to H\times H$
such that

$\langle B(x,y),(x,y)\rangle_{H\times H}=2\mathbf{Re}\langle
Ax,y\rangle_{H}\quad\text{for all }x,y\in H.$

The operator $B$ is of the form $B(x,y)=(A^{*}y,Ax)$ and satisfies
$\left\lVert B\right\rVert_{H\times H\to H\times H}=\left\lVert
A\right\rVert_{H\to H}$.

######  Proposition 2.10.

If a bilinear mapping $T\colon X^{**}\times H\to H$ satisfies Assumption 1.2,
then the modified operator

$(b,f,g)\mapsto\widetilde{T_{b}}(f,g)\colon X^{**}\times(H\times H)\to H\times
H,\qquad\widetilde{T_{b}}(f,g)\mathrel{:\mkern-0.25mu=}(T_{b}^{*}g,T_{b}f)$

satisfies Assumptions 1.2–1.3.

If $T$ satisfies Assumptions 1.2–1.3, then $\tilde{T}$ is related to the
Gâteaux derivative of $Q$ by

(2.2) $\langle\tilde{T}_{b}(f,g),(f,g)\rangle_{H\times H}=\langle
b,Q^{\prime}_{f}g\rangle_{X^{**}-X^{*}}\qquad\text{for all }f,g\in H,\,b\in
X^{**}.$

We illustrate Proposition 2.10 in Examples 3.5–3.6.

##  3\. Operators satisfying Assumptions 1.2–1.3

In this section we discuss two classes of operators which satisfy assumption
1.2: commutators of Calderón-Zygmund operators with $\operatorname{BMO}$
functions and paracommutators. Specific examples are then given in §3.3.

###  3.1. Commutators of Calderón–Zygmund operators and $\operatorname{BMO}$
functions

Recall that

$\textup{BMO}(\mathbb{R}^{n})\mathrel{:\mkern-0.25mu=}\left\\{b\in
L^{1}_{loc}(\mathbb{R}^{n})\colon\left\lVert
b\right\rVert_{\operatorname{BMO}}\mathrel{:\mkern-0.25mu=}\sup_{Q}-\nobreak\hskip-10.8pt\nobreak\int_{Q}\left\lvert
b(x)-b_{Q}\right\rvert\,dx<\infty\right\\},$

where the supremum is taken over cubes $Q\subset\mathbb{R}^{n}$ and
$b_{Q}\mathrel{:\mkern-0.25mu=}-\nobreak\hskip-8.9pt\nobreak\int_{Q}b(y)\,dy$.
The closure of $C_{c}^{\infty}(\mathbb{R}^{n})$ in
$\textup{BMO}(\mathbb{R}^{n})$ is called $\operatorname{CMO}(\mathbb{R}^{n})$.
By classical results by Fefferman and by Coifman and Weiss, respectively, we
have the dualities
$[\mathcal{H}^{1}(\mathbb{R}^{n})]^{*}=\textup{BMO}(\mathbb{R}^{n})$ and
$[\operatorname{CMO}(\mathbb{R}^{n})]^{*}=\mathscr{H}^{1}(\mathbb{R}^{n})$.

Coifman & al. showed in [5] that when $T$ is a Calderón-Zygmund operator with
a suitable smooth kernel $\Omega$, we have
$\left\lVert[b,T]\right\rVert_{L^{2}\to L^{2}}\lesssim\left\lVert
b\right\rVert_{\operatorname{BMO}}$ for all
$b\in\textup{BMO}(\mathbb{R}^{n})$. Specifically, $\Omega$ was assumed to be
homogeneous of degree zero with vanishing mean over $S^{n-1}$ and to satisfy
$\left\lvert\Omega(x)-\Omega(y)\right\rvert<\left\lvert x-y\right\rvert$ for
all $x,y\in S^{n-1}$. They also showed that if $[b,R_{j}]\colon
L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$ is bounded for every
$j\in\\{1,\ldots,n\\}$, then $b\in\operatorname{BMO}(\mathbb{R}^{n})$ and
$\left\lVert b\right\rVert_{\operatorname{BMO}}\lesssim\max_{1\leq j\leq
n}\left\lVert[b,R_{j}]\right\rVert_{L^{2}\to L^{2}}$. Uchiyama [49] and Janson
[30] showed independently that in order to obtain
$b\in\textup{BMO}(\mathbb{R}^{n})$, it suffices to show boundedness of $[b,T]$
in $L^{2}(\mathbb{R}^{n})$ for only one of the kernels $\Omega\not\equiv 0$ in
the result of [5]. Uchiyama also showed that $[b,T]$ is compact if and only is
$b\in\operatorname{CMO}(\mathbb{R}^{n})$. Thus
$T_{b}f\mathrel{:\mkern-0.25mu=}[b,T]f$ satisfies Assumption 1.2.

The two-sided estimate (1.3) has been extended in numerous ways (to multi-
parameter and weighted spaces etc.); see e.g. [22, 27, 50] and the references
contained therein. In the case of commutators with Calderón-Zygmund operators,
Hytönen recently proved the estimate $\left\lVert[b,T]\right\rVert_{L^{2}\to
L^{2}}\simeq\left\lVert b\right\rVert_{\operatorname{BMO}}$ under the very
weak assumption that the kernel is ”non-degenerate”. We recall relevant
definitions from [27].

######  Definition 3.1.

A measurable function $K\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$
is called an _$\omega$ -Calderón-Zygmund kernel_ if $K(x,y)\leq
c_{K}\left\lvert x-y\right\rvert^{-n}$ whenever $x\neq y$ and

$\left\lvert K(x,y)-K(x^{\prime},y)\right\rvert+\left\lvert
K(y,x)-K(y,x^{\prime})\right\rvert\leq\frac{1}{\left\lvert
x-y\right\rvert^{n}}\omega\left(\frac{\left\lvert
x-x^{\prime}\right\rvert}{\left\lvert x-y\right\rvert}\right)$

whenever $\left\lvert x-x^{\prime}\right\rvert<\left\lvert x-y\right\rvert/2$,
where $\omega\colon[0,1)\to[0,\infty)$ is increasing.

######  Definition 3.2.

A measurable function $K\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$
is called a _non-degenerate Calderón-Zygmund kernel_ if $K$ satisfies at least
one of the following two conditions:

  1. (1)

$K$ is an $\omega$-Calderón-Zygmund kernel with $\omega(t)\to 0$ as $t\to 0$
and for every $y\in\mathbb{R}^{n}$ and $r>0$, there exists $x\in\complement
B(y,r)$ such that $\left\lvert K(x,y)\right\rvert\leq cr^{-d}$.

  2. (2)

$K$ is an homogeneous Calderón-Zygmund kernel with $\Omega\in
L^{1}(S^{n-1})\setminus\\{0\\}$.

For compactness of commutators of CMO functions and Calderón-Zygmund operators
on (possibly weighted) $L^{2}(\mathbb{R}^{n})$ we refer to [1, 2, 21, 26, 34,
49]. In particular, when an homogeneous kernel $\Omega\in
L^{1}(S^{n-1})\setminus\\{0\\}$ has vanishing mean and satisfies

$\sup_{\zeta\in
S^{n-1}}\int_{S^{n-1}}\left\lvert\Omega(\eta)\right\rvert\left(\textup{ln}\frac{1}{\left\lvert\eta\cdot\zeta\right\rvert}\right)^{\theta}\textup{d}\eta<\infty$

for some $\theta>2$, the corresponding commutator satisfies Assumption 1.2 [1,
27]. Assumption 1.2 is verified for commutators with the Cauchy integral
operator in [37].

In many natural instances, an operator $T_{b}$ satisfying Assumption 1.2 is
not precisely a commutator but, for instance, the composition of a commutator
with a Calderón-Zygmund operator. A natural general framework for such
operators is provided by _paracommutators_.

###  3.2. Paracommutators

We briefly recall basic definitions and results from the theory of
paracommutators and refer to [31]. Paracommutators are operators $T_{b}(A)$ of
the form

(3.1)
$\widehat{T_{b}(A)\omega}(\xi)\mathrel{:\mkern-0.25mu=}(2\pi)^{-n}\int_{\mathbb{R}^{n}}\hat{b}(\xi-\eta)A(\xi,\eta)\hat{\omega}(\eta)\,d\eta,\quad\xi\in\mathbb{R}^{n},$

where $A\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ and
$\omega\colon\mathbb{R}^{n}\to\mathbb{R}$ are functions and
$b\colon\mathbb{R}^{n}\to\mathbb{R}$ is called the _symbol of $T_{b}(A)$_.

As the most basic example, $A(\xi,\eta)\equiv 1$ yields (under suitable
integrability assumptions) the multiplication operator $T_{b}\omega=b\omega$.
The commutators $T_{b}(A)=[b,R_{j}]$ arise via
$A(\xi,\eta)=\xi_{j}/\left\lvert\xi\right\rvert-\eta_{j}/\left\lvert\eta\right\rvert$.
More generally, if $T$ is a Calderón-Zygmund singular integral operator with
Fourier symbol $m$ and $A(\xi,\eta)=m(\xi)-m(\eta)$, then $T_{b}(A)=[b,T]$.
Several further examples are presented in [31, pp. 469-473] and [45, pp.
513-519].

In [31], Janson and Peetre gave conditions on the function $A$ under which the
boundedness of $T_{b}(A)\colon L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$
is equivalent to the condition $b\in\operatorname{BMO}(\mathbb{R}^{n})$. In
order to state the result we need some definitions. We do not motivate them
here but refer to [31] for more information.

######  Definition 3.3.

Let $U,V\subset\mathbb{R}^{n}$. The space $M(U\times V)$ consists of functions
$\varphi\in L^{\infty}(U\times V)$ that admit a representation

(3.2) $\varphi(\xi,\eta)=\int_{X}\alpha(\xi,x)\beta(\eta,x)\,d\mu(x)$

for some $\sigma$-finite measure space $(X,\mu)$ and measurable functions
$\alpha\colon U\times X\to\mathbb{R}$ and $\beta\colon V\times X\to\mathbb{R}$
which satisfy

(3.3)
$\int_{X}\left\lVert\alpha(\cdot,x)\right\rVert_{L^{\infty}(U)}\left\lVert\beta(\cdot,x)\right\rVert_{L^{\infty}(V)}\,d\mu(x)<\infty.$

Furthermore, $M(U\times V)$ is a Banach algebra in the norm given by
minimising the left hand side of (3.3) over all representations (3.2).

For every $j\in\mathbb{Z}$ we denote
$\Delta_{j}\mathrel{:\mkern-0.25mu=}\\{\xi\in\mathbb{R}^{n}\colon
2^{j}\leq\left\lvert\xi\right\rvert<2^{j+1}\\}$. We list some of the
assumptions of [31] and retain their numbering.

(A0) There exists $r>1$ such that $A(r\xi,r\eta)=A(\xi,\eta)$ for all
$\xi,\eta\in\mathbb{R}^{n}$.

(A1) $\left\lVert A\right\rVert_{M(\Delta_{j}\times\Delta_{k})}\leq C$ for all
$j,k\in\mathbb{Z}$.

(A2) There exist $A_{1},A_{2}\in M(\mathbb{R}^{n}\times\mathbb{R}^{n})$ and
$\delta>0$ such that

$\displaystyle A(\xi,\eta)=A_{1}(\xi,\eta)\quad\text{when
}\left\lvert\eta\right\rvert<\delta\left\lvert\xi\right\rvert,$ $\displaystyle
A(\xi,\eta)=A_{2}(\xi,\eta)\quad\text{when
}\left\lvert\xi\right\rvert<\delta\left\lvert\eta\right\rvert.$

(A3($\alpha$)): There exist $\alpha,\delta>0$ such that whenever
$r<\delta\left\lvert\xi_{0}\right\rvert$, we have

$\left\lVert A\right\rVert_{M(B(\xi_{0},r),B(\xi_{0},r))}\leq
C\left(\frac{r}{\left\lvert\xi_{0}\right\rvert}\right)^{\alpha}.$

(A5) For every $\xi_{0}\neq 0$ there exist $\delta>0$ and
$\eta_{0}\in\mathbb{R}^{n}$ such that $1/A(\xi,\eta)\in M(U\times V)$, where
$U=\\{\xi\colon\left\lvert\xi/\left\lvert\xi\right\rvert-\xi_{0}/\left\lvert\xi_{0}\right\rvert\right\rvert<\delta\text{
and }\left\lvert\xi\right\rvert>\left\lvert\xi_{0}\right\rvert\\}$ and
$V=B(\eta_{0},\delta\left\lvert\xi_{0}\right\rvert))$.

We collect the results of [31] that are most relevant to this paper.

######  Theorem 3.4.

Suppose $A\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ satisfies
(A0), (A1), (A2), (A3($\alpha$)) and A(5), where $\alpha>0$. Then $T_{b}(A)$
satisfies Assumption 1.2.

By a result of Peng, under the assumptions (A0), (A1), (A3($\alpha$)) and
(A5), compactness of $T_{b}\colon L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})$ conversely implies $b\in\textup{CMO}(\mathbb{R}^{n})$
[44]. We also mention that under (A0)–(A5), Assumption 1.3 is not satisfied in
general. Indeed, denoting $\tilde{b}(x)\mathrel{:\mkern-0.25mu=}b(-x)$ and
abusing notation, $\langle
T_{b}(A(\xi,\eta))\omega,\gamma\rangle=\langle\omega,T_{\tilde{b}}(A(\eta,\xi))\gamma\rangle$
for all $\omega,\gamma\in L^{2}(\mathbb{R}^{n})$.

If the assumptions of Theorem 3.4 and Assumption 1.3 hold, we may use
Definition 1.4 to define a quadratic operator $Q\colon
L^{2}(\mathbb{R}^{n})\to\mathscr{H}^{1}(\mathbb{R}^{n})$. Its form is

(3.4)
$Q_{\tilde{A}}(\omega,\gamma)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\hat{\omega}(\xi)\hat{\gamma}(\eta)\tilde{A}(\xi,\eta)e^{-i(\xi+\eta)\cdot
x}\,d\xi\,d\eta,$

where $\tilde{A}(\eta,-\xi)=A(\xi,\eta)$. Conversely, $Q_{\tilde{A}}$, defined
via (3.4), gives rise to a paracommutator $T_{b}(A)$ defined by (3.1). These
and many other relations between $T_{b}(A)$ and $Q_{\tilde{A}}$ are explored
in [45].

###  3.3. Specific operators

We list some concrete quadratic operators which arise via Assumption 1.2;
numerous further examples are presented in [31, 45].

######  Example 3.5.

Let $u\in\dot{W}^{1,2}(\mathbb{C},\mathbb{C})$. When
$u_{z}\mathrel{:\mkern-0.25mu=}2^{-1}(\partial_{x}-i\partial_{y})(u_{1}+iu_{2})$
and
$u_{\bar{z}}\mathrel{:\mkern-0.25mu=}2^{-1}(\partial_{x}+i\partial_{y})(u_{1}+iu_{2})$
are the Wirtinger derivatives of $u$ and $\mathcal{S}\colon
L^{2}(\mathbb{C},\mathbb{C})\to L^{2}(\mathbb{C},\mathbb{C})$ is the Beurling
transform, we have $\mathcal{S}u_{\bar{z}}=u_{z}$. Therefore, the Jacobian
$Ju=|u_{z}|^{2}-|u_{\bar{z}}|^{2}$ corresponds to
$Q\omega\mathrel{:\mkern-0.25mu=}|{\mathcal{S}}\omega|^{2}-|\omega|^{2}$ via
the isometric isomorphism $u\mapsto
u_{\bar{z}}\mathrel{=\mkern-0.25mu:}\omega\colon\dot{W}^{1,2}(\mathbb{C},\mathbb{C})\to
L^{2}(\mathbb{C},\mathbb{C})$ whose inverse is the Cauchy transform
$\mathcal{C}$. (We identify elements of $\dot{W}^{1,2}(\mathbb{C},\mathbb{C})$
that differ by a constant and set $\left\lVert
u\right\rVert_{\dot{W}^{1,2}}\mathrel{:\mkern-0.25mu=}\left\lVert
u_{\bar{z}}\right\rVert_{L^{2}}=\left\lVert Du\right\rVert_{L^{2}}/2$.) The
quadratic operator $Q$ arises via the formula $\langle
b,Q\omega\rangle_{\operatorname{BMO}-\mathcal{H}^{1}}=\langle
T_{b}\omega,\omega\rangle_{L^{2}}$, where
$T_{b}\omega\mathrel{:\mkern-0.25mu=}\overline{(Sb-bS)\overline{S\omega}}$
satisfies Assumption 1.2 for $H=L^{2}(\mathbb{C},\mathbb{C})$ and
$X=\textup{CMO}(\mathbb{C})$ (see [38]).

It is instructive to also consider the planar Jacobian in real variable
notation. Define

(3.5) $T_{b}\mathrel{:\mkern-0.25mu=}R_{1}[b,R_{2}]-R_{2}[b,R_{1}]\colon
L^{2}(\mathbb{R}^{2})\to L^{2}(\mathbb{R}^{2}).$

Now $T_{b}^{*}=-T_{b}\neq 0$ and thus $T_{b}$ is not self-adjoint. However,
the operator $\widetilde{T_{b}}\colon L^{2}(\mathbb{R}^{2},\mathbb{R}^{2})\to
L^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$, defined in Proposition 2.10, generates
the Jacobian:

$\displaystyle\int_{\mathbb{R}^{2}}\widetilde{T_{b}}(\omega_{1},\omega_{2})\cdot(\omega_{1},\omega_{2})$
$\displaystyle=2\langle
b,R_{1}\omega_{1}R_{2}\omega_{2}-R_{2}\omega_{1}R_{1}\omega_{2}\rangle_{\operatorname{BMO}-\mathcal{H}^{1}}$
$\displaystyle=2\langle
b,\partial_{1}v_{1}\partial_{2}v_{2}-\partial_{2}v_{1}\partial_{1}v_{2}\rangle_{\text{BMO}-\mathcal{H}^{1}},$

where
$v=\Lambda\omega\mathrel{:\mkern-0.25mu=}(-\Delta)^{-1/2}\omega\in\dot{W}^{1,2}(\mathbb{R}^{2})$.
This example serves to further motivate Assumption 1.3.

######  Example 3.6.

On the real line, two quadratic $\mathcal{H}^{1}$-integrable quantities arise
as follows. Isomorphically,
$\mathcal{H}^{p}(\mathbb{R})\cong\mathcal{H}^{p}(\mathbb{R},\mathbb{C})\mathrel{:\mkern-0.25mu=}\\{g+iHg\colon
g,Hg\in L^{p}(\mathbb{R})\\}$ for all $p\in(0,\infty)$, where $H$ is the
Hilbert transform. Given $\omega\in L^{2}(\mathbb{R})$ we set

$(\omega+iH\omega)^{2}=\omega^{2}-(H\omega)^{2}+2i\omega
H\omega\mathrel{=\mkern-0.25mu:}Q_{1}\omega+iQ_{2}\omega.$

As is well-known, $Q_{1},Q_{2}\colon
L^{2}(\mathbb{R})\to\mathcal{H}^{1}(\mathbb{R})$ are not surjective but their
Gâteaux derivatives $(Q_{1})_{\omega}^{\prime}\gamma=\omega\gamma-H\omega
H\gamma$ and $(Q_{2})^{\prime}_{\omega}\gamma=\omega H\gamma+\gamma H\omega$
are–in other words,

(3.6)
$\\{\alpha^{2}\colon\alpha\in\mathcal{H}^{2}(\mathbb{R},\mathbb{C})\\}\subsetneq\\{\alpha\beta\colon\alpha,\beta\in\mathcal{H}^{2}(\mathbb{R},\mathbb{C})\\}=\mathcal{H}^{1}(\mathbb{R},\mathbb{C}).$

We give a proof of (3.6) in Appendix A for the reader’s convenience.

The operator $Q_{1}$ arises via the bounded linear operator $T_{b}\colon
L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$,
$T_{b}\mathrel{:\mkern-0.25mu=}[H,b]H$ by the formula $\langle
b,Q_{1}\omega\rangle_{\operatorname{BMO}-\mathcal{H}^{1}}\mathrel{:\mkern-0.25mu=}\int_{\mathbb{R}}\omega
T_{b}\omega$. Now $T_{b}$ satisfies Assumptions 1.2–1.3 but $Q_{1}\colon
L^{2}(\mathbb{R})\to\mathcal{H}^{1}(\mathbb{R})$ is not surjective. This
illustrates the fact that the modified operator $\tilde{T}_{b}$ (see
Proposition 2.10) might be needed to ensure surjectivity even if $T_{b}$
satisfies Assumptions 1.2–1.3.

######  Example 3.7.

In [51], Wu considered higher-dimensional variants of the operators $Q_{j}$ of
Example 3.6 in Clifford algebras. When
$\Omega=\omega+\sum_{j=1}^{n}R_{j}\omega\mathbf{e}_{j}\in
L^{2}(\mathbb{R}^{n},\mathbb{C}_{(n)})$ and
$\Gamma=\gamma+\sum_{j=1}^{n}R_{j}\gamma\mathbf{e}_{j}\in
L^{2}(\mathbb{R}^{n},\mathbb{C}_{(n)})$, their product can be written as

(3.7) $\Omega\Gamma=\omega\gamma-\sum_{j=1}^{n}R_{j}\omega
R_{j}\gamma+\sum_{j=1}^{n}(\omega R_{j}\gamma+\gamma
R_{j}\omega)\mathbf{e}_{j}+\sum_{j<k}(R_{j}\omega R_{k}\gamma-R_{k}\omega
R_{j}\gamma)\mathbf{e}_{jk}.$

Each component of $\Omega\Gamma$ then belongs to
$\mathcal{H}^{1}(\mathbb{R}^{n})$ [51, Theorem 12.3.1]. Note that when $n=2$,
the last term of (3.7) is a Jacobian in disguise: $R_{1}\omega R_{2}\gamma-
R_{2}\omega R_{1}\gamma=Jv$ for
$v=(\Lambda^{-1}\omega,\Lambda^{-1}\gamma)\in\dot{W}^{1,2}(\mathbb{R}^{2},\mathbb{R}^{2})$.
In analogy to (3.6), Wu conjectured that
$\mathcal{H}^{1}(\mathbb{R}^{n})=\\{\omega\gamma-\sum_{j=1}^{n}R_{j}\omega
R_{j}\gamma\colon\omega,\gamma\in L^{2}(\mathbb{R}^{n})\\}$ [51, p. 237].

Wu also studied more general combinations of Riesz transforms that arise via
$A(\xi,\eta)=1-(\xi\cdot\eta)^{m}/(\left\lvert\xi\right\rvert\left\lvert\eta\right\rvert)^{m}$,
$m\in\mathbb{N}$, and showed that each such $A$ satisfies the assumptions of
Theorem 3.4. As an example, $T_{b}\mathrel{:\mkern-0.25mu=}[R,b]\cdot R\colon
L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$ generates

$Qf=\left\lvert\omega\right\rvert^{2}-\left\lvert
R\omega\right\rvert^{2}=\left\lvert\Lambda u\right\rvert^{2}-\left\lvert\nabla
u\right\rvert^{2},$

where $\Lambda u=f$. In this case,
$A(\xi,\eta)=1-\eta\cdot\xi/(\left\lvert\eta\right\rvert\left\lvert\xi\right\rvert)$.
Thus Wu’s conjecture says that the Gâteaux derivative
$Q^{\prime}_{\omega}\gamma=2(\omega\gamma-R\omega\cdot R\gamma)=2(\Lambda
u\Lambda v-\nabla u\cdot\nabla v)$ gives a surjective bilinear operator
$L^{2}(\mathbb{R}^{n})\times
L^{2}(\mathbb{R}^{n})\to\mathcal{H}^{1}(\mathbb{R}^{n})$.

The planar Monge-Ampère equation arises as follows. Define $T_{b}\colon
L^{2}(\mathbb{R}^{2})\to L^{2}(\mathbb{R}^{2})$ by

$T_{b}\omega\mathrel{:\mkern-0.25mu=}\frac{[R_{11},b]R_{22}\omega+[R_{22},b]R_{11}\omega-2[R_{12},b]R_{12}\omega}{2}.$

Then $Q\colon L^{2}(\mathbb{R}^{2})\to\mathcal{H}^{1}(\mathbb{R}^{2})$ is of
the form $Q\omega=R_{11}\omega R_{22}\omega-R_{12}\omega R_{21}\omega$. Using
the isomorphism $-\Delta\colon\dot{W}^{2,2}(\mathbb{R}^{2})\to
L^{2}(\mathbb{R}^{2})$ we may write the Hessian as
$\mathcal{H}=Q\circ(-\Delta)^{-1}\colon\dot{W}^{2,2}(\mathbb{R}^{2})\to\mathcal{H}^{1}(\mathbb{R}^{2})$.
In this case,
$A(\xi,\eta)=1-(\xi\cdot\eta)^{2}/(\left\lvert\xi\right\rvert^{2}\left\lvert\eta\right\rvert^{2})$.

##  4\. Banach space geometric considerations

This subsection is devoted to proving various claims presented in §1.3. We
assume that $T$ satisfies Assumptions 1.2–1.3.

###  4.1. Minimum norm solutions

We begin by noting that if the equation $Q\omega=f$ has a solution, then the
direct method gives a minimum norm solution.

######  Proposition 4.1.

Suppose $f\in X^{*}_{Q}$, $\gamma\in H$ and $Q\gamma=f$. Then there exists
$\omega\in H$ with $Q\omega=f$ and
$\left\lVert\omega\right\rVert_{H}=\min_{Q\gamma=f}\left\lVert\gamma\right\rVert_{H}$.

###### Proof.

Choose a minimising sequence so that $Q\omega_{j}=f$ for every
$j\in\mathbb{N}$ and
$\lim_{j\to\infty}\left\lVert\omega_{j}\right\rVert_{H}=\inf_{Q\gamma=f}\left\lVert\gamma\right\rVert_{H}$.
Since $\mathbb{B}_{H}$ is sequentially weakly compact and $Q$ is weak-to-weak∗
continuous, we have $\omega_{j}\rightharpoonup\omega$ and
$Q\omega_{j}\overset{*}{\rightharpoonup}Q\omega$ for a subsequence, and
$\omega$ is the sought minimum norm solution. ∎

We then show that surjectivity of $Q\colon H\to X^{*}_{Q}$ would follow from
the weak∗ density of the range $Q(H)$ in $X^{*}_{Q}$ combined with a suitable
a priori estimate.

######  Proposition 4.2.

Suppose that $\overline{Q(H)}^{\operatorname{w}-*}=X^{*}_{Q}$ and that
$\left\lVert\omega\right\rVert_{H}^{2}\lesssim\left\lVert
Q\omega\right\rVert_{X^{*}_{Q}}$ for every minimum norm solution. Then
$Q(H)=X^{*}_{Q}$.

###### Proof.

Let $f\in X^{*}$ and choose minimum norm solutions $\omega_{j}\in H$ with
$Q\omega_{j}\overset{*}{\rightharpoonup}\omega$. For large enough
$j\in\mathbb{N}$ we have
$\left\lVert\omega_{j}\right\rVert_{H}^{2}\lesssim\left\lVert
f\right\rVert_{X^{*}}+1$. After passing to a subsequence, a weak limit
$\omega$ of $(\omega_{j})_{j=1}^{\infty}$ satisfies $Q\omega=f$. ∎

We also mention a version of the Banach-Schauder open mapping theorem for
multilinear (and more general) operators from [16]. We say that $Q$ enjoys
_generalised translation invariance_ if there exist isometric isomorphisms
$\sigma_{j}\in H^{H}$ and $\sigma_{j}\in(X_{Q}^{*})^{X_{Q}^{*}}$,
$j\in\mathbb{N}$, such that $Q\circ\sigma_{j}=\sigma_{j}\circ Q$ for all
$j\in\mathbb{N}$ and $\sigma_{j}f\overset{*}{\rightharpoonup}0$ as
$j\to\infty$ for all $f\in X_{Q}^{*}$. The example one should have in mind is
$\sigma_{j}\omega(x)=\omega(x-je)$ and
$\sigma_{j}f(x)\mathrel{:\mkern-0.25mu=}f(x-je)$ for a non-zero vector $e$.

######  Theorem 4.3.

Suppose $Q$ enjoys generalised translation invariance. Then one of the
following claims holds:

  1. (1)

$Q(H)=X_{Q}^{*}$ and every minimum norm solution satisfies
$\left\lVert\omega\right\rVert_{H}^{2}\lesssim\left\lVert
Q\omega\right\rVert_{X_{Q}^{*}}$.

  2. (2)

$Q(H)$ is meagre in $X_{Q}^{*}$.

Theorem 4.3 is primarily a practical tool for showing _non_ -surjectivity of
$Q\colon H\to X^{*}_{Q}$; disproving the _a priori_ estimate
$\left\lVert\omega\right\rVert_{H}^{2}\lesssim\left\lVert
Q\omega\right\rVert_{X_{Q}^{*}}$ is a markedly less daunting task than
disproving surjectivity [16].

###  4.2. Definition and existence of Lagrange multipliers

When $\omega\in H$ is a minimum norm solution, it is natural to look for a
Lagrange multiplier of $\omega$, as discussed in §1.3. The Lagrange multiplier
condition (1.5) can be written more concisely as follows: $b\in X^{**}_{Q}$ is
a Lagrange multiplier of $\omega\in H$ if

(4.1) $2\mathbf{Re}\langle\omega,\varphi\rangle_{H}=\langle
b,Q^{\prime}_{\omega}\varphi\rangle_{X_{Q}^{**}-X_{Q}^{*}}\quad\text{ for
every }\varphi\in H.$

Yet more consicely, $(Q_{\omega}^{\prime})^{*}b=2\omega$. The existence of $b$
then means that
$\omega\in\operatorname{ran}((Q_{f}^{\prime})^{*})=\ker(Q_{f}^{\prime})^{\perp}$

The standard tool for showing the existence of a Lagrange multiplier in a
Banach space is the Liusternik-Schnirelman theorem which, in this setting,
says that if $Q_{\omega}^{\prime}$ maps $H$ onto $X_{Q}^{*}$, then $\omega$
possesses a Lagrange multiplier. However, in the cases of interest to us, the
Liusternik-Schnirelman theorem is not available, as shown belov.

######  Proposition 4.4.

Suppose $X^{*}_{Q}$ is not isomorphic to a Hilbert space. Then we have
$\\{Q^{\prime}_{\omega}\gamma\colon\gamma\in H\\}\subsetneq X^{*}_{Q}$ for
every $\omega\in H$.

###### Proof.

Seeking a contradiction, suppose $Q^{\prime}_{\omega}H=X^{*}$. Thus
$Q^{\prime}_{\omega}\colon\operatorname{ker}(Q^{\prime})^{\perp}\to X^{*}$ is
an isomorphism from a Hilbert space onto $X^{*}$. ∎

######  Proposition 4.5.

Every $b\in\mathbb{S}_{X_{Q}}$ is a Lagrange multiplier of some
$\omega\in\mathcal{A}$.

###### Proof.

Choose a maximizing sequence: $\langle
T_{b}\omega_{j},\omega_{j}\rangle_{H}\to 1$. For a subsequence,
$\omega_{j}\rightharpoonup\omega$ in $H$, and so $T_{b}\omega_{j}\to
T_{b}\omega$, giving $1=\langle T_{b}\omega,\omega\rangle_{H}=\langle
b,Q\omega\rangle_{X_{Q}-X_{Q}^{*}}\leq\left\lVert
Qf\right\rVert_{X_{Q}^{*}}\leq\left\lVert f\right\rVert_{H}^{2}\leq 1$, so
that $\omega\in\mathscr{A}$. ∎

######  Definition 4.6.

When $Y$ is a real Banach space, a set $A\subset\mathbb{S}_{Y^{*}}$ is call a
_James boundary of $Y$_ if for every $y\in\mathbb{S}_{Y}$ there exists
$y^{*}\in A$ such that $\langle y^{*},y\rangle_{Y^{*}-Y}=1$.

Proposition 4.5 then says that $Q(\mathscr{A})$ is a James boundary of
$X_{Q}$. By Godefroy’s Theorem (see [12, Theorem 3.46]), if
$A\subset\mathbb{S}_{Y^{*}}$ is a separable James boundary of $Y$, then
$\overline{\operatorname{co}}(A)=\mathbb{B}_{Y^{*}}$.

######  Corollary 4.7.

$\overline{\operatorname{co}}(Q(\mathscr{A}))=\mathbb{B}_{X^{*}_{Q}}$.

######  Corollary 4.8.

$\left\lVert
f\right\rVert_{X_{Q}^{*}}=\inf\\{\sum_{j=1}^{\infty}\left\lVert\omega_{j}\right\rVert_{H}^{2}\colon
f=\sum_{j=1}^{\infty}Q\omega_{j}\\}$ for all $f\in X_{Q}^{*}$.

###  4.3. Further properties of Lagrange multipliers

The Lagrange multiplier condition (4.1) has several useful equivalent
characterisations, and we collect two of them in the following proposition.

######  Proposition 4.9.

Whenever $b\in\mathbb{S}_{X_{Q}^{**}}$ and $\omega\in\mathbb{S}_{H}$, the
following conditions are equivalent:

  1. (i)

$b$ is a Lagrange multiplier of $\omega$, i.e., (4.1) holds.

  2. (ii)

$\langle b,Q\omega\rangle_{X_{Q}^{**}-X_{Q}^{*}}=1$.

  3. (iii)

$\omega\in\ker(I-T_{b})$.

If (i)–(iii) hold, then $\omega$ is a minimum norm solution and belongs to
$\mathscr{A}$.

###### Proof.

We first prove (i) $\Leftrightarrow$ (ii). Suppose (ii) holds and fix
$\varphi\in H$ and small $\delta>0$. The function
$I\colon[-\delta,\delta]\to\mathbb{R}$,
$I(\epsilon)\mathrel{:\mkern-0.25mu=}\langle
b,Q(\omega+\epsilon\varphi)\rangle_{X_{Q}^{**}-X_{Q}^{*}}/\left\lVert\omega+\epsilon\varphi\right\rVert_{H}^{2}$
is maximised at $\epsilon=0$, and therefore $I^{\prime}(0)=\langle
b,Q^{\prime}_{\omega}\varphi\rangle_{X_{Q}^{**}-X_{k}^{*}}-2\mathbf{Re}\langle\omega,\varphi\rangle_{H}=0$,
giving (i). The direction (i) $\Rightarrow$ (ii) is proved by setting
$\varphi=\omega$. We then prove (ii) $\Leftrightarrow$ (iii). First, (ii)
gives $\langle\omega,T_{b}\omega\rangle_{H}=1$. Since $\left\lVert
T_{b}\right\rVert_{H\to H}=1$ and $H$ is strictly convex, we conclude that
(iii) holds. On the other hand, if (iii) holds, then $\langle
b,Q\omega\rangle_{X_{Q}^{**}-X_{Q}^{*}}=\langle\omega,T_{b}\omega\rangle_{H}=\langle\omega,\omega\rangle_{H}=1$.
∎

In many cases, conditions (i)–(iii) can be supplemented by Euler-Lagrange
equations for a suitable potential. In the case of the planar Jacobian,
denoting $u_{\bar{z}}=\omega$ as before, (i)–(iii) are equivalent to
$u_{z\bar{z}}=(bu_{\bar{z}})_{z}-(bu_{z})_{\bar{z}}$ [38, Proposition 4.8].

######  Lemma 4.10.

$(X^{**},\left\lVert T_{\cdot}\right\rVert_{H\to H})=(X_{Q})^{**}$
isometrically.

###### Proof.

Let $b\in X^{**}$. Since
$Q(\mathbb{B}_{H})\subset\mathbb{B}_{X_{Q}^{*}}=\overline{\operatorname{co}}(Q\mathscr{A})$,
we get

$\left\lVert T_{b}\right\rVert_{H\to H}=\sup_{\left\lVert
f\right\rVert_{H}=1}\langle
b,Qf\rangle_{X^{**}-X^{*}}=\sup_{h\in\operatorname{co}(Q\mathscr{A})}\langle
b,h\rangle_{X^{**}-X^{*}}=\left\lVert
b\right\rVert_{(X^{*},\left\lVert\cdot\right\rVert_{X_{Q}^{*}})^{*}}.$

∎

######  Proposition 4.11.

Suppose $b\in\mathbb{S}_{X_{Q}^{**}}$. The following conditions are
equivalent.

  1. (i)

$b$ is a Lagrange multiplier of some $\omega\in\mathscr{A}$.

  2. (ii)

$b$ is norm-attaining.

###### Proof.

If $b$ is a Lagrange multiplier of $\omega\in\mathscr{A}$, then
$T_{b}\omega=\omega$ and therefore $\langle
b,Q\omega\rangle_{X_{Q}^{**}-X_{Q}^{*}}=\langle
T_{b}\omega,\omega\rangle_{H}=1$.

Conversely, if $b$ is norm-attaining, denote
$D^{-1}(b)\mathrel{:\mkern-0.25mu=}\\{f\in\mathbb{S}_{X_{Q}^{*}}\colon\langle
b,f\rangle_{X_{Q}^{**}-X_{Q}^{*}}=1\\}$. Since $D^{-1}(b)$ is closed, bounded
and convex, Theorem 2.7 implies that $D^{-1}(b)$ contains an extreme point
$f$. The definition of $D^{-1}(b)$ implies that $f$ is also an extreme point
of $\mathbb{S}_{X_{Q}^{*}}$, and Lemma 5.1 then gives $f\in Q(\mathscr{A})$. ∎

We also characterise elements of $\mathbb{S}_{H}$ that possess a Lagrange
multiplier.

######  Proposition 4.12.

Let $\omega\in\mathbb{S}_{H}$. The following conditions are equivalent.

  1. (i)

$\omega$ has a Lagrange multiplier $b\in\mathbb{S}_{X_{Q}^{**}}$.

  2. (ii)

$\omega\in\mathscr{A}$.

###### Proof.

If $\omega\in\mathbb{S}_{H}$ has a Lagrange multiplier
$b\in\mathbb{S}_{X_{Q}^{**}}$, then $\langle
b,Q\omega\rangle_{X_{Q}^{**}-X_{Q}^{*}}=\langle
T_{b}\omega,\omega\rangle_{H}=\left\lVert\omega\right\rVert_{H}^{2}=1$ so that
$\omega\in\mathscr{A}$.

Conversely, let $\omega\in\mathscr{A}$. Since
$Q\omega\in\mathbb{S}_{X_{Q}^{*}}$, there exists $b\in\mathbb{S}_{X_{Q}^{**}}$
such that $\langle b,Q\omega\rangle_{X_{Q}^{**}-X_{Q}^{*}}=1$. Now $b$ is a
Lagrange multiplier of $\omega$. ∎

Using the results above we collect many equivalent formulations of Question
1.6.

######  Corollary 4.13.

If $Q(H)$ is dense in $X_{Q}^{*}$, the following conditions are equivalent:

  1. (i)

$Q(\mathscr{A})=\mathbb{S}_{X_{Q}^{*}}$.

  2. (ii)

$\mathcal{E}=\left\lVert\cdot\right\rVert_{X_{Q}^{*}}$.

  3. (iii)

Every minimum norm solution satisfies
$\left\lVert\omega\right\rVert_{H}^{2}=\left\lVert
Q\omega\right\rVert_{X_{Q}^{*}}$.

  4. (iv)

Every minimum norm solution has a Lagrange multiplier in
$\mathbb{S}_{X_{Q}^{**}}$.

##  5\. The proof of Theorem 1.10

We get the proofs of Theorems 1.9 and 1.10 underway by proving (1.6).

######  Lemma 5.1.

If Assumptions 1.2–1.3 holds, then
$Q(\mathscr{A})\supset\operatorname{ext}(\mathbb{B}_{X^{*}_{Q}})$.
Furthermore, $Q(\mathscr{A})$ is closed in the relative weak∗ topology of
$\mathbb{S}_{X^{*}_{Q}}$. In particular, it is norm closed.

###### Proof.

We first show that $Q(\mathscr{A})$ is relatively weak∗ (sequentially) closed
in $\mathbb{S}_{X_{Q}^{*}}$. Since $X_{Q}$ is separable, it suffices to
consider sequences instead of nets. Suppose $\omega_{j}\in\mathscr{A}$ for
every $j\in\mathbb{N}$ and
$Q\omega_{j}\overset{*}{\rightharpoonup}f\in\mathbb{S}_{X_{Q}^{*}}$. We claim
that $f=Q\omega$ for some $\omega\in\mathscr{A}$.

Passing to a subsequence, $\omega_{j}\rightharpoonup\omega\in\mathbb{B}_{H}$,
since $\mathbb{B}_{H}$ is weakly compact. Thus
$Q\omega_{j}\overset{*}{\rightharpoonup}Q\omega$, and so it suffices to show
that $\omega\in\mathscr{A}$. The chain of inequalities
$\left\lVert\omega\right\rVert_{H}^{2}\leq\liminf_{j\to\infty}\left\lVert\omega_{j}\right\rVert_{H}^{2}=1=\left\lVert
Q\omega\right\rVert_{X^{*}_{Q}}\leq\left\lVert\omega\right\rVert_{H}^{2}$
implies that $\omega\in\mathscr{A}$.

We then show that every extreme point of $\mathbb{B}_{X_{Q}^{*}}$ lies in
$Q(\mathscr{A})$. In Theorem 2.8, choose $K$ to be the weak∗ closure of
$Q(\mathscr{A})$ in $\mathbb{B}_{X_{Q}^{*}}$. By Corollary 4.7,
$\overline{\operatorname{co}}^{\operatorname{w}^{*}}(K)=\mathbb{B}_{X_{Q}^{*}}$.
Theorem 2.8 implies that $\operatorname{ext}(\mathbb{B}_{X_{Q}^{*}})\subset
K$. Let now $f\in\operatorname{ext}(B_{X_{Q}^{*}})$ and choose
$\omega_{j}\in\mathscr{A}$ such that
$Q\omega_{j}\overset{*}{\rightharpoonup}f$. Since $Q(\mathscr{A})$ is
relatively weak∗ closed in $\mathbb{S}_{X_{Q}^{*}}$, we obtain $f\in
Q(\mathscr{A})$. ∎

We recall the statement of Theorem 1.10.

######  Claim 5.2.

Suppose Assumptions 1.2–1.3 and 1.8 hold, and let
$f\in\operatorname{ext}(\mathbb{B}_{X_{Q}^{*}})$. Then the minimum norm
solution $\omega\in\mathscr{A}$ of $Q\omega=f$ is unique up to multiplication
by $c\in\mathbb{S}_{\mathbb{K}}$.

###### Proof.

Seeking a contradiction, suppose
$f\in\operatorname{ext}(\mathbb{B}_{X_{Q}^{*}})$ and $Q\omega=Q\gamma=f$,
where $\omega,\gamma\in\mathscr{A}$ are not equal up to a rotation. Thus we
may write $\gamma=s\omega+t\omega^{\perp}$, where $s,t\in\mathbb{K}$ with
$\left\lvert s\right\rvert^{2}+\left\lvert t\right\rvert^{2}=1$, $t\neq 0$ and
$\omega^{\perp}\in\mathbb{S}_{H}\cap\\{\omega\\}^{\perp}$. We intend to show
that

(5.1) $Q(\omega^{\perp})=Q\omega.$

Once (5.1) is proved, Assumption 1.8 yields $c\in\mathbb{S}_{\mathbb{K}}$ such
that $Q^{\prime}_{\omega}(c\omega^{\perp})\neq 0$. Denoting
$\psi^{\pm}\mathrel{:\mkern-0.25mu=}(\omega\pm
c\omega^{\perp})/\sqrt{2}\in\mathbb{S}_{H}$ we therefore get
$Q\omega=(Q\psi^{+}+Q\psi^{-})/2$ but $\mathbb{B}_{X_{Q}^{*}}\ni
Q\psi^{\pm}=Q\omega\pm Q^{\prime}_{\omega}(c\omega^{\perp})/2\neq Q\omega$,
thereby obtaining the sought contradiction.

Formula (5.1) clearly holds if $s=0$, so assume $s\neq 0$. We denote
$\tilde{\omega}^{\perp}\mathrel{:\mkern-0.25mu=}(\left\lvert
st\right\rvert/s\bar{t})\omega^{\perp}$ and
$\tilde{\gamma}\mathrel{:\mkern-0.25mu=}(\bar{s}/\left\lvert
s\right\rvert)\gamma=\left\lvert s\right\rvert\omega+\left\lvert
t\right\rvert\tilde{\omega}^{\perp}\in\mathscr{A}$. Now
$Q\omega=Q\tilde{\gamma}=\left\lvert s\right\rvert^{2}Q\omega+\left\lvert
t\right\rvert^{2}Q\tilde{\omega}^{\perp}+\left\lvert s\right\rvert\left\lvert
t\right\rvert Q^{\prime}_{\omega}\tilde{\omega}^{\perp}$, which yields

(5.2) $Q\tilde{\omega}^{\perp}=Q\omega-\frac{\left\lvert
s\right\rvert}{\left\lvert
t\right\rvert}Q^{\prime}_{\omega}\tilde{\omega}^{\perp}.$

Since $Q\tilde{\omega}^{\perp}=Q\omega^{\perp}$, it therefore suffices to show
that $Q^{\prime}_{\omega}\tilde{\omega}^{\perp}=0$.

Whenever $\delta,\epsilon\in\mathbb{R}$ with $\delta^{2}+\epsilon^{2}=1$, we
have

(5.3)
$Q(\delta\omega+\epsilon\tilde{\omega}^{\perp})=\delta^{2}Q_{\omega}+\delta\epsilon
Q^{\prime}_{\omega}\tilde{\omega}^{\perp}+\epsilon^{2}Q\tilde{\omega}^{\perp}=Q\omega+(\delta\epsilon-\epsilon^{2}\left\lvert
s\right\rvert\left\lvert
t\right\rvert^{-1})Q^{\prime}_{\omega}\tilde{\omega}^{\perp}.$

By choosing any $r>0$ and letting $\delta_{\pm}=\epsilon(\left\lvert
s\right\rvert/\left\lvert t\right\rvert\pm r)$ with
$\delta_{\pm}^{2}+\epsilon^{2}=1$ we get

(5.4) $\displaystyle\delta_{+}\epsilon-\epsilon^{2}\left\lvert
s\right\rvert/\left\lvert
t\right\rvert=-(\delta_{-}\epsilon-\epsilon^{2}\left\lvert
s\right\rvert/\left\lvert t\right\rvert)\neq 0,$ (5.5) $\displaystyle
Q(\delta_{+}\omega+\epsilon_{1}\tilde{\omega}^{\perp})+Q(\delta_{-}\omega+\epsilon_{2}\tilde{\omega}^{\perp})=2Q\omega.$

Since $Q\omega\in\operatorname{ext}(\mathbb{B}_{X_{Q}^{*}})$, we conclude from
(5.3)–(5.5) that $Q^{\prime}_{\omega}\tilde{\omega}^{\perp}=0$, and now (5.2)
and the definition of $\tilde{\omega}^{\perp}$ imply (5.1). ∎

######  Corollary 5.3.

Suppose Assumptions 1.2–1.3 and 1.8 hold and $b\in\mathbb{S}_{X_{Q}}$ is a
point of Fréchet differentiability. Then $\ker(I-T_{b})$ is one-dimensional.

######  Proposition 5.4.

The operator $\omega\mapsto
Q\omega=\left\lvert{\mathcal{S}}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}\colon
L^{2}(\mathbb{C},\mathbb{C})\to\mathcal{H}^{1}(\mathbb{C})$ satisfies
Assumption 1.8, whereas
$(\omega,\gamma)\mapsto\tilde{Q}(\omega,\gamma)\mathrel{:\mkern-0.25mu=}G^{\prime}_{\omega}\gamma\colon
L^{2}(\mathbb{C},\mathbb{C})^{2}\to\mathscr{H}^{1}(\mathbb{C})$ does not.

###### Proof.

We prove the first statement by contradiction. Suppose
$\omega,\gamma\in\mathscr{A}$ satisfy $Q\omega=Q\gamma$ and
$Q_{\omega}^{\prime}(c\gamma)=2\mathbf{Re}(S\omega\overline{S(c\gamma)}-\omega\overline{c\gamma})=0$
for every $c\in\mathbb{S}_{\mathbb{K}}$. Thus
$S\omega\overline{S\gamma}-\omega\overline{\gamma}=0$. Fix $\delta>0$ and
denote
$X_{\delta}\mathrel{:\mkern-0.25mu=}\\{z\in\mathbb{C}\colon\left\lvert\omega(z)\right\rvert\in[\delta,1/\delta]\\}$.
In $X_{\delta}$ we use the formula
$S\omega\overline{S\gamma}-\omega\overline{\gamma}=0$ to write

$\left\lvert\mathcal{S}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}=\left\lvert\mathcal{S}\gamma\right\rvert^{2}-\left\lvert\gamma\right\rvert^{2}=-\frac{\left\lvert\mathcal{S}\gamma\right\rvert^{2}}{\left\lvert\omega\right\rvert^{2}}(\left\lvert\mathcal{S}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}),$

so that
$\left\lvert\mathcal{S}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}=0$
in $X_{\delta}$. Thus

$\int_{\omega\neq
0}\left\lvert\left\lvert\mathcal{S}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}\right\rvert=\lim_{\delta\to
0}\int_{X_{\delta}}\left\lvert\left\lvert\mathcal{S}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}\right\rvert=0,$

and therefore
$\int_{\omega=0}\left\lvert\mathcal{S}\omega\right\rvert^{2}=1-\int_{\omega\neq
0}\left\lvert\mathcal{S}\omega\right\rvert^{2}=1-\int_{\omega\neq
0}\left\lvert\omega\right\rvert^{2}=0$. We conclude that
$\left\lvert\mathcal{S}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}=0$,
which yields the desired contradiction.

We show a strengthened version of the second statement: whenever
$(\omega,\gamma)\in\mathscr{A}$, there exists $(\varphi,\psi)\in\mathscr{A}$
such that $\tilde{Q}(\omega,\gamma)=\tilde{Q}(\varphi,\psi)$ but
$\tilde{Q}^{\prime}_{(\omega,\gamma)}[c(\varphi,\psi)]=0$ for all $c\in
S^{1}$. Given $(\omega,\gamma)$, we choose
$(\varphi,\psi)=(-\overline{{\mathcal{S}}\gamma},\overline{{\mathcal{S}}\omega})$
and use the facts that ${\mathcal{S}}\colon L^{2}(\mathbb{C},\mathbb{C})\to
L^{2}(\mathbb{C},\mathbb{C})$ is an isometry and
${\mathcal{S}}\overline{{\mathcal{S}}\eta}=\bar{\eta}$ for all $\eta\in
L^{2}(\mathbb{C},\mathbb{C})$. ∎

An almost identical proof applies to Example 3.6, but instead of
$(\varphi,\psi)=(-\overline{{\mathcal{S}}\gamma},\overline{{\mathcal{S}}\omega})$
we set $(\varphi,\psi)=(-H\omega,H\gamma)$.

######  Proposition 5.5.

The operator $\omega\mapsto\omega^{2}-(H\omega)^{2}\colon
L^{2}(\mathbb{R})\to\mathcal{H}^{1}(\mathbb{R})$ satisfies Assumption 1.8,
whereas $(\omega,\gamma)\mapsto\omega\gamma-H\omega H\gamma\colon
L^{2}(\mathbb{R})^{2}\to\mathscr{H}^{1}(\mathbb{R})$ does not.

######  Remark 5.6.

The proof of Proposition 5.4 uses a central difference of the operator
$\omega\mapsto
Q\omega=\left\lvert{\mathcal{S}}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}$
and its Gâteaux derivative $\tilde{Q}$. Given $\omega\in\mathscr{A}$, the set
$\\{\varphi\in\mathscr{A}\colon Q\varphi=Q\omega\\}$ sometimes consists only
of rotations of $\omega$, whereas all the corresponding sets
$\\{(\varphi,\psi)\in\tilde{\mathscr{A}}\colon\tilde{Q}(\varphi,\psi)=\tilde{Q}(\omega,\gamma)\\}$
are invariant under the linear transformation
$L(\omega,\gamma)\mathrel{:\mkern-0.25mu=}(-\overline{S\gamma},\overline{{\mathcal{S}}\omega})$.
Note that $L\circ L=-\textup{id}$. Similar remarks apply to the sets of
minimum norm solutions for a given datum $f\in\mathscr{H}^{1}(\mathbb{C})$.

##  6\. The proof of Theorem 1.9

###  6.1. Claims (i)–(iii)

In this subsection, Assumptions 1.2–1.3 are in effect. Before recalling Claims
(i)–(iii) we note the following consequence of Corollary 4.13.

######  Lemma 6.1.

Let $b\in\mathbb{S}_{X_{Q}}$. Then $\\{f\in\mathscr{A}\colon Qf\in
D(b)\\}=\ker(I-T_{b})\cap\mathbb{S}_{H}$.

######  Claim 6.2.

For every $b\in\mathbb{S}_{X_{Q}}$, $D(b)\cap Q(\mathscr{A})$ contains
$\textup{ext}(D(b))$ and is path-connected.

###### Proof.

Lemma 5.1 directly implies that $D(b)\cap Q(\mathscr{A})$ contains
$\operatorname{ext}(D(b))$. The path-connectedness of $D(b)\cap
Q(\mathscr{A})$ follows from Lemma 6.1, the path-connectedness of
$\ker(I-T_{b})\cap\mathbb{S}_{H}$ and the continuity of $Q$. ∎

######  Claim 6.3.

For every $b\in\mathbb{S}_{X_{Q}}$, the convex set $D(b)$ has finite affine
dimension.

###### Proof.

Since $D(b)\cap Q(\mathscr{A})$ contains $\operatorname{ext}(D(b))$, it
suffices to show that $D(b)\cap Q(\mathscr{A})$ has finite affine dimension.
By Lemma 6.1, $D(b)\cap Q(\mathscr{A})\subset Q(\ker(I-T_{b}))$. Since $K_{b}$
is compact and self-adjoint, $\ker(I-T_{b})$ is finite-dimensional, and so
$Q(\ker(I-T_{B}))$ has finite affine dimension. Precisely, when
$\\{\omega_{1},\ldots,\omega_{n}\\}$ is an orthonormal basis of
$\ker(I-T_{b})$ and $Q\omega\in D(b)\cap Q(\mathscr{A})$, we may write
$Q\omega\in\operatorname{span}\cup_{j,k=1}^{n}\\{Q^{\prime}_{\omega_{j}}\omega_{k}\\}$
(when $\mathbb{K}=\mathbb{R}$) or
$Q\omega\in\operatorname{span}\cup_{j,k=1}^{n}\\{\mathbf{Re}(Q^{\prime}_{\omega_{j}}\omega_{k}),\mathbf{Re}(Q^{\prime}_{\omega_{j}}(i\omega_{k}))\\}$
(when $\mathbb{K}=\mathbb{C}$). ∎

######  Claim 6.4.

The norm and relative weak∗ topologies coincide in $Q(\mathscr{A})$.

###### Proof.

Suppose $\omega_{j},\omega\in\mathscr{A}$ and
$Q\omega_{j}\overset{*}{\rightharpoonup}Q\omega$. Seeking a contradiction,
suppose that for a subsequence, $\liminf_{j\to\infty}\left\lVert
Q\omega_{j}-Q\omega\right\rVert_{X_{Q}^{*}}>0$. By passing to a further
subsequence, $\omega_{j}\rightharpoonup\gamma\in\mathbb{B}_{H}$. Now
$Q\omega_{j}\overset{*}{\rightharpoonup}Q\gamma$ and thus $Q\gamma=Q\omega$.
This implies that $1=\left\lVert
Q\gamma\right\rVert_{X_{Q}^{*}}\leq\left\lVert\gamma\right\rVert_{H}\leq 1$.
Thus $\omega_{j}\rightharpoonup\gamma$ and
$\left\lVert\omega_{j}\right\rVert_{H}\to\left\lVert\gamma\right\rVert_{H}$,
giving $\left\lVert\omega_{j}-\gamma\right\rVert_{H}\to 0$ and so $\left\lVert
Q\omega_{j}-Q\gamma\right\rVert_{H}\to 0$, and the latter yields the sought
contradiction. ∎

###  6.2. Claims (iv)–(vi)

We recall and prove Claims (iv)–(vi) below. In this subsection, we assume that
Assumptions 1.2, 1.3 and 1.8 hold.

######  Claim 6.5.

For every $b$ in a relatively open dense subset of $\mathbb{S}_{X_{Q}}$,
$D(b)=\\{Q\omega\\}$ for some $\omega\in\mathscr{A}$ and
$\left\lVert\cdot\right\rVert_{X_{Q}}$ is Fréchet differentiable at $b$.

###### Proof.

By Theorem 2.1, $\left\lVert\cdot\right\rVert_{X_{Q}}$ is Fréchet
differentiable in a dense subset of $\mathbb{S}_{X_{Q}}$. Seeking a
contradiction, suppose $b$ is a point of Fréchet differentiability but the
points $b_{j}\to b$, $b_{j}\in\mathbb{S}_{X_{Q}}$, are not. Thus the subspaces
$\textup{ker}(I-T_{b_{j}})$ are at least two-dimensional but, by Corollary
5.3, $\textup{dim}(\textup{ker}(I-T_{b}))=1$.

For every $j\in\mathbb{N}$ choose
$\omega_{j},\gamma_{j}\in\ker(I-T_{b})\cap\mathbb{S}_{H}$ such that
$\langle\omega_{j},\gamma_{j}\rangle_{H}=0$. Since $\mathbb{B}_{H}$ is weakly
compact, we may assume that $\omega_{j}\rightharpoonup\omega$ and
$\gamma_{j}\rightharpoonup\gamma$ in $H$. Then
$Q\omega_{j}\overset{*}{\rightharpoonup}Q\omega$ and
$Q\gamma_{j}\overset{*}{\rightharpoonup}Q\gamma$ in $X_{Q}^{*}$.

Since $b_{j}\to b$ in $X_{Q}$ and
$Q\omega_{j}\overset{*}{\rightharpoonup}Q\omega$ and
$Q\gamma_{j}\overset{*}{\rightharpoonup}Q\gamma$ in $X_{Q}^{*}$, we conclude
that

$\displaystyle\langle
Q\omega,b\rangle_{X_{Q}^{*}-X_{Q}}=\lim_{j\to\infty}\langle
Q\omega_{j},b_{j}\rangle_{X_{Q}^{*}-X_{Q}}=1,$ $\displaystyle\langle
Q\gamma,b\rangle_{X_{Q}^{*}-X_{Q}}=\lim_{j\to\infty}\langle
Q\gamma_{j},b_{j}\rangle_{X_{Q}^{*}-X_{Q}}=1.$

Since $D(b)$ is a singleton by assumption, we obtain $Q\omega=Q\gamma$. By
Corollary 5.3,

(6.1) $\gamma=c\omega\quad\text{for some }c\in\mathbb{S}_{\mathbb{K}}.$

Furthermore,

$1=\lim_{j\to\infty}\left\lVert\omega_{j}\right\rVert_{H}^{2}\geq\left\lVert\omega\right\rVert_{H}^{2}\geq\|Q\omega\|_{X_{Q}^{*}}\geq\langle
Q\omega,b\rangle_{X_{Q}^{*}-X_{Q}}=1.$

We thus have $\omega_{j}\rightharpoonup\omega$ and
$\|\omega_{j}\|_{H}\to\|\omega\|_{H}$, whereby $\|\omega_{j}-\omega\|_{H}\to
0$. Similarly, $\|\gamma_{j}-\gamma\|_{H}\to 0$. We conclude, via (6.1), that

$0=\langle\omega_{j},\gamma_{j}\rangle\to\langle\omega,\gamma\rangle=\langle\omega,c\omega\rangle=\overline{c},$

which yields a contradiction. ∎

######  Remark 6.6.

Note that a straightforward modification of the proof above proves the
relative openness in $\mathbb{S}_{X_{Q}}$ of
$\\{b\in\mathbb{S}_{X_{Q}}\colon\dim(\ker(I-K_{b}))\leq n\\}$ for every
$n\in\mathbb{N}$.

######  Claim 6.7.

$D\colon\mathbb{S}_{X_{Q}}\to 2^{\mathbb{S}_{X_{Q}^{*}}}$ is a cusco map.

###### Proof.

The duality mapping $D\colon S_{X_{Q}}\to S_{X_{Q}^{*}}$ is point-to-compact
since for every $b\in\mathbb{S}_{X_{Q}}$, the closed, bounded set $D(b)$ is
contained in a finite-dimensional subspace of $X_{Q}^{*}$. We next intend to
show that $D$ is norm-to-norm upper semicontinuous. For this, suppose
$b\in\mathbb{S}_{X_{Q}}$ and $f_{j}\in\mathbb{S}_{X_{Q}^{*}}$ satisfy
$\lim_{j\to\infty}\langle f_{j},b\rangle_{X_{Q}^{*}-X_{Q}}=1$. By Theorem 2.3
it suffices to show that $f_{j}\to f\in D(b)$ for a subsequence.

Denote $\operatorname{dim}(\ker(I-T_{b}))=n\in\mathbb{N}$. Then
$\operatorname{dim}(\ker(I-T_{b_{j}}))\leq n$ from some index on by Remark
6.6. Thus we may write every $f_{j}$ as a convex combination
$f_{j}=\sum_{k=1}^{N}\lambda_{j}^{k}Q\omega_{j}^{k}$. By passing to a
subsequence, for every $k\in\\{1,\ldots,N\\}$ we have
$\lambda_{j}^{k}\to\lambda^{k}\in[0,1]$ and
$\omega_{j}^{k}\rightharpoonup\omega^{k}\in\mathbb{B}_{H}$. In particular,
$\sum_{k=1}^{N}\lambda^{k}=1$.

Suppose now $\lambda^{k}>0$. Then necessarily $\langle
Q\omega_{j}^{k},b\rangle_{X_{Q}^{*}-X_{Q}}\to 1$ and so
$\|\omega_{j}^{k}\|_{H}\to 1$, giving $\|\omega_{j}^{k}-\omega^{k}\|_{H}\to 0$
and $\|Q\omega_{j}^{k}-Q\omega^{k}\|_{X_{Q}^{*}}\to 0$. We conclude that
$f_{j}\to\sum_{k=1}^{N}\lambda^{k}Q\omega^{k}\in D(b)$. ∎

######  Claim 6.8.

The norm and relative weak∗ topologies also coincide in
$\textit{NA}_{\left\lVert\cdot\right\rVert_{X_{Q}}}$.

###### Proof.

The result follows directly from Claim (v) and Theorem 2.3. ∎

##  7\. The Jacobian equation with $L^{p}$ data

Many of the ideas of this paper can be adapted to study the range of the
operator $J\colon\dot{W}^{1,2p}(\mathbb{R}^{2},\mathbb{R}^{2})\to
L^{p}(\mathbb{R}^{2})$ when $1<p<\infty$. In this section we outline such an
approach and list those results whose proof extends to this new setting in a
straightforward manner.

We can again use basic properties of the Beurling transform to write

$\langle b,Q\omega\rangle_{L^{p^{\prime}}-L^{p}}=\langle
T_{b}\omega,\omega\rangle_{L^{(2p)^{\prime}}-L^{2p}},\quad
T_{b}\omega\mathrel{:\mkern-0.25mu=}\overline{({\mathcal{S}}b-b{\mathcal{S}})\overline{{\mathcal{S}}\omega}},\quad
Q\omega\mathrel{:\mkern-0.25mu=}\left\lvert{\mathcal{S}}\omega\right\rvert^{2}-\left\lvert\omega\right\rvert^{2}.$

Note that $\left\lVert T_{b}\right\rVert_{L^{2p}\to
L^{(2p)^{\prime}}}\lesssim_{p}\left\lVert b\right\rVert_{L^{p^{\prime}}}$ for
all $b\in L^{p^{\prime}}(\mathbb{C})$ by a simple application of Hölder’s
inequality and the boundedness of ${\mathcal{S}}\colon
L^{p}(\mathbb{C},\mathbb{C})\to L^{p}(\mathbb{C},\mathbb{C})$. The lower bound
estimate

(7.1) $\left\lVert T_{b}\right\rVert_{L^{2p}\to
L^{(2p)^{\prime}}}\gtrsim_{p}\left\lVert b\right\rVert_{L^{p^{\prime}}},$

however, is far from trivial and cannot be proved by simply following the
proof of estimates (1.3).

Hytönen proved (7.1) rather recently in [27]. More generally, on
$\mathbb{R}^{n}$, Hytönen showed that the commutator of a degenerate Calderón-
Zygmund operator with $b\in L^{1}_{loc}(\mathbb{R}^{n})$ defines a bounded
operator from $L^{q_{1}}(\mathbb{R}^{n})$ into $L^{q_{2}}(\mathbb{R}^{n})$,
$q_{1}>q_{2}>1$, if and only if $b=a+c$ with $a\in L^{r}(\mathbb{R}^{n})$,
$1/r=1/{q_{2}}-1/{q_{1}}$ and $c\in\mathbb{R}$. In notable contrast to the
case of $[b,T]\colon L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$, the
possible cancellations of $b$ do not play an important role; recall, for
instance, that there exist $b\notin\textup{BMO}(\mathbb{R}^{n})$ such that
$\left\lvert b\right\rvert\in\textup{BMO}(\mathbb{R}^{n})$.

As a corollary of (7.1), Hytönen obtained an analogue of (1.2): for every
$f\in L^{p}(\mathbb{C})$ there exist
$u_{j}\in\dot{W}^{1,2p}(\mathbb{C},\mathbb{C})$ such that
$f=\sum_{j=1}^{\infty}Ju_{j}$ and $\sum_{j=1}^{\infty}\left\lVert
Du_{j}\right\rVert_{L^{2p}}^{2}\lesssim\left\lVert f\right\rVert_{L^{p}}$
[27]. Hytönen also showed an analogous result in higher dimensions by
different methods.

In order to adapt the methods of this paper to the $L^{p}$ case, another
crucial ingredient is the compactness of $[b,T]\colon
L^{2p}(\mathbb{R}^{2},\mathbb{R}^{2})\to
L^{(2p)^{\prime}}(\mathbb{R}^{2},\mathbb{R}^{2})$ for all $b\in
L^{p^{\prime}}(\mathbb{R}^{2})$. Hytönen & al. recently proved this property
for a large class of degenerate Calderón-Zygmund kernels in [25]. We formulate
their two main results.

######  Theorem 7.1.

Let $1<q_{2}<q_{1}<\infty$ and $1/r=1/{q_{2}}-1/{q_{1}}$. Suppose $T$ is a
non-degenerate Calderón-Zygmund operator which satisfies one of the following
two:

  1. (1)

condition (i) of Definition 3.2 with the Dini condition
$\int_{0}^{1}t^{-1}\omega(t)\textup{d}t<\infty$,

  2. (2)

condition (ii) of Definition 3.2 with $\Omega\in L^{\nu}(S^{n-1})$ for some
$\nu\in(1,\infty]$.

Then $[b,T]$ is compact from $L^{q_{1}}(\mathbb{R}^{n})$ to
$L^{q_{2}}(\mathbb{R}^{n})$ for all $b\in L^{r}(\mathbb{R}^{n})$.

In the next subsection we put the operator
$Q=\left\lvert{\mathcal{S}}\cdot\right\rvert^{2}-\left\lvert\cdot\right\rvert^{2}\colon
L^{2p}(\mathbb{C},\mathbb{C})\to L^{p}(\mathbb{C})$ into a general Banach-
space geometric framework.

###  7.1. Mathematical setting

We assume that $X$ is a separable, reflexive Banach space and that $Y$ is a
separable, reflexive, smooth, strictly convex Banach space with the Kadec-Klee
property. Assumptions 1.2–1.3, Definition 1.4 and Proposition 2.10 have
natural analogues:

######  Assumption 7.2.

A bilinear mapping

$(b,f)\mapsto T_{b}f\colon X\times Y\to Y^{*}$

satisfies the following conditions:

  1. (i)

$c\left\lVert b\right\rVert_{X}\leq\left\lVert T_{b}\right\rVert_{Y\to
Y^{*}}\leq C\|b\|_{X}$ for every $b\in X$,

  2. (ii)

$T_{b}$ is compact for every $b\in X$.

######  Assumption 7.3.

The following conditions hold for every $b\in X$:

  1. (1)

$T_{b}^{*}=T_{b}$.

  2. (2)

$\left\lVert T_{b}\right\rVert_{Y\to
Y^{*}}=\sup_{\left\lVert\omega\right\rVert_{Y}=1}\langle
T_{b}\omega,\omega\rangle_{Y^{*}-Y}$.

######  Definition 7.4.

Suppose Assumptions 7.2–7.3 hold. Define the norm-to-norm and weak-to-weak
sequentially continuous map $Q\colon Y\to X^{*}$ by

$\langle b,Qf\rangle_{X-X^{*}}\mathrel{:\mkern-0.25mu=}\langle
T_{b}f,f\rangle_{Y^{*}-Y}.$

######  Proposition 7.5.

If $T$ satisfies Assumption 7.2, then $\tilde{T}_{b}\colon Y\times Y\to
Y^{*}\times Y^{*}$,
$\tilde{T}_{b}(\omega,\gamma)\mathrel{:\mkern-0.25mu=}(T_{b}^{*}\gamma,T_{b}\omega)$,
satisfies Assumptions 7.2–7.3.

In Proposition 7.5 we have endowed $Y\times Y$ with the norm
$\left\lVert(\omega,\gamma)\right\rVert_{Y\times
Y}\mathrel{:\mkern-0.25mu=}(\left\lVert\omega\right\rVert_{Y}^{2}+\left\lVert\gamma\right\rVert_{Y}^{2})^{1/2}$
and similarly for $Y^{*}\times Y^{*}$, so that $Y\times Y$ and $Y^{*}\times
Y^{*}$ are smooth and strictly convex since $Y$ and $Y^{*}$ are. Denoting
$D(\omega)=\\{\omega^{*}\\}$ for all $\omega\in\mathbb{S}_{Y}$ we have
$D(\omega,\gamma)=\\{(\omega^{*},\gamma^{*})\\}$ for all
$\omega,\gamma\in\mathbb{S}_{Y}$.

######  Remark 7.6.

Assumption 7.3 reveals an important difference to the case of
$\mathscr{H}^{1}$ data. In the case of
$J\colon\dot{W}^{1,2}(\mathbb{C},\mathbb{C})\to\mathscr{H}^{1}(\mathbb{C})$,
the associated linear operators $T_{b}\colon L^{2}(\mathbb{C},\mathbb{C})\to
L^{2}(\mathbb{C},\mathbb{C})$ are self-adjoint and therefore the operator norm
and numerical radius of $T_{b}$ coincide, $\left\lVert
T_{b}\right\rVert_{L^{2}\to
L^{2}}=\sup_{\left\lVert\omega\right\rVert_{L^{2}}=1}\left\lvert\langle
T_{b}f,f\rangle\right\rvert$. The standard proof (see e.g. [6, p. 34]) employs
the fact that as a Hilbert space, $L^{2}(\mathbb{C},\mathbb{C})$ satisfies the
parallelogram law
$\left\lVert\omega+\gamma\right\rVert_{L^{2}}^{2}+\left\lVert\omega-\gamma\right\rVert_{L^{2}}^{2}=2(\left\lVert\omega\right\rVert_{L^{2}}^{2}+\left\lVert\gamma\right\rVert_{L^{2}}^{2})$.
The analogue $\left\lVert T_{b}\right\rVert_{L^{2p}\to
L^{(2p)^{\prime}}}=\sup_{\left\lVert\omega\right\rVert_{L^{2p}}=1}\left\lvert\langle
T_{b}\omega,\omega\rangle_{L^{(2p)^{\prime}}-L^{2p}}\right\rvert$ seems to
fail (even though the left and right hand sides are comparable), owing to the
fact that the inequality
$\left\lVert\omega+\gamma\right\rVert_{L^{2p}}^{2}+\left\lVert\omega+\gamma\right\rVert_{L^{(2p)^{\prime}}}^{2}\leq
2(\left\lVert\omega\right\rVert_{L^{2p}}^{2}+\left\lVert\gamma\right\rVert_{L^{2p}}^{2})$
fails (see [32, Theorem 2]).

In order to study the planar Jacobian via our operator theoretic framework,
one then needs to make the concession that
$Qf\mathrel{:\mkern-0.25mu=}\left\lvert{\mathcal{S}}f\right\rvert^{2}-\left\lvert
f\right\rvert^{2}$ is replaced by
$\tilde{Q}(f,g)=Q^{\prime}_{f}g=J(u_{1},v_{2})+J(v_{1},u_{2})$, where
$u=u_{1}+iu_{2}=\mathcal{C}f$ and $v=v_{1}+iv_{2}=\mathcal{C}g$. It is of
course equivalent to studying the range of $(u,v)\mapsto
Ju+Jv\colon\dot{W}^{1,2p}(\mathbb{C},\mathbb{C})^{2}\to L^{p}(\mathbb{C})$.

We define norms on $X$ and $X^{*}$ by $\left\lVert
b\right\rVert_{X_{Q}}\mathrel{:\mkern-0.25mu=}\sup_{\left\lVert\omega\right\rVert_{X}=1}\langle
b,Q\omega\rangle_{X-X^{*}}=\left\lVert T_{b}\right\rVert_{Y\to Y^{*}}$ and
$\left\lVert
f\right\rVert_{X_{Q}^{*}}\mathrel{:\mkern-0.25mu=}\sup_{\left\lVert
b\right\rVert_{X_{Q}}=1}\langle f,b\rangle_{X^{*}-X}$. We also denote
$\mathscr{A}\mathrel{:\mkern-0.25mu=}\\{\omega\in\mathbb{S}_{Y}\colon
Q\omega\in\mathbb{S}_{X_{Q}^{*}}\\}$.

##  8\. Results

In the current setting, the Lagrange multiplier condition

$\left.\frac{d}{d\epsilon}(\langle
b,Q(\omega+\epsilon\varphi)\rangle_{X_{Q}-X_{Q}^{*}}-\left\lVert\omega+\epsilon\varphi\right\rVert_{Y}^{2})\right|_{\epsilon=0}=0\qquad\text{for
every }\varphi\in Y$

is written concisely as $T_{b}\omega=\omega^{*}$, where $\omega^{*}$ is the
unique element of $Y^{*}$ such that
$\langle\omega,\omega^{*}\rangle_{Y-Y^{*}}=\left\lVert\omega\right\rVert_{Y}\left\lVert\omega^{*}\right\rVert_{Y^{*}}$.
(In particular, $D(\omega)=\\{\omega^{*}\\}$ for all
$\omega\in\mathbb{S}_{Y}$.) This is seen by following the proof of Proposition
4.9.

We list results of this paper that allow a straightforward adaptation to the
current setting: Theorem 4.3; Corollaries 4.7, 4.8, 4.13 and 5.3; Propositions
2.10, 4.1, 4.2, 4.5, 4.9 and 4.12; Lemmas 2.9, 4.10 and 5.1.

Theorem 1.9 (i) has at least the weaker variant that
$D\colon\mathbb{S}_{X_{Q}}\to\mathbb{S}_{X_{Q}^{*}}$ is point-to-compact. It
is natural to ask whether, again, $\textup{co}(\textup{ext}(D(b)))=D(b)$ for
all $b\in\mathbb{S}_{X_{Q}}$. Since
$\textup{ext}(\mathbb{B}_{X_{Q}^{*}})\subset Q\mathscr{A}$, we could then
write each $f\in\mathbb{S}_{X_{Q}^{*}}$ as a convex combination of elements
$Q\omega$, $\omega\in\mathscr{A}$. By a standard Baire category argument, one
would then obtain an upper bound for the number of terms in these sums [16]:

######  Proposition 8.1.

Suppose $\textup{co}(\textup{ext}(D(b)))=D(b)$ for all
$b\in\mathbb{S}_{X_{Q}}$. Then there exists $N\in\mathbb{N}$ such that every
$f\in X_{Q}^{*}$ can be written as

$f=\sum_{j=1}^{N}Q\omega_{j},\qquad\sum_{j=1}^{N}\left\lVert\omega_{j}\right\rVert_{Y}^{2}\lesssim\left\lVert
f\right\rVert_{X_{Q}^{*}}.$

##  Appendix A A proof of (3.6)

While formula (3.6) is well-known and classical, the author has been unable to
find a proof in the literature, and therefore one is sketched in this
appendix. Note that (3.6) is equivalent to the following proposition.

######  Proposition A.1.

Every $f\in\mathcal{H}^{1}(\mathbb{R})$ can be written as

$f=\omega\gamma-H\omega H\gamma$

for some $\omega,\gamma\in L^{2}(\mathbb{R})$. However, there exists
$f\in\mathcal{H}^{1}(\mathbb{R})$ that cannot be written as
$f=\omega^{2}-(H\omega)^{2}$ for any $\omega\in L^{2}(\mathbb{R})$.

The relevant background can be found e.g. in [10] or [40]. Whenever
$0<p<\infty$, we denote

$\left\lVert
U\right\rVert_{p}\mathrel{:\mkern-0.25mu=}\sup_{0<y<\infty}\left(\int_{-\infty}^{\infty}\left\lvert
U(x+iy)\right\rvert^{p}dx\right)^{\frac{1}{p}}.$

The analytic Hardy space $\mathcal{H}^{p}(\mathbb{C}_{+})$ consists of
analytic functions $U\colon\mathbb{C}_{+}\to\mathbb{C}$ with $\left\lVert
U\right\rVert_{p}<\infty$.

###### Proof of Proposition A.1.

Let $f\in\mathcal{H}^{1}(\mathbb{R})$; then the Hilbert transform $Hf\in
L^{1}(\mathbb{R})$. We extend $f+iHf$ analytically into $\mathbb{C}_{+}$ by
using the Poisson kernel $P_{y}$ and the conjugate Poisson kernel $Q_{y}$,

$P_{y}(x)\mathrel{:\mkern-0.25mu=}\frac{1}{\pi}\frac{y}{x^{2}+y^{2}},\quad
Q_{y}(x)\mathrel{:\mkern-0.25mu=}\frac{1}{\pi}\frac{x}{x^{2}+y^{2}}.$

We denote

$U_{1}(x+iy)+iU_{2}(x+iy)\mathrel{:\mkern-0.25mu=}f*P_{y}(x)+i\,f*Q_{y}(x);$

then $U=U_{1}+iU_{2}$ belongs to $\mathcal{H}^{1}(\mathbb{C}_{+})$ and,
furthermore, $\lim_{y\searrow 0}\left\lVert
U_{1}(\cdot,y)-f\right\rVert_{L^{1}}=0$ and $\lim_{y\searrow 0}\left\lVert
U_{2}(\cdot,y)-Hf\right\rVert_{L^{1}}=0$.

We form the Blaschke product $B$ of $U$ and write $U=B\Phi$, where
$\Phi\in\mathcal{H}^{1}(\mathbb{C}_{+})$ is zero-free. Since $\mathbb{C}_{+}$
is simply connected, $\Phi$ has an analytic square root $\Phi^{1/2}$. Set

$V\mathrel{:\mkern-0.25mu=}B\Phi^{1/2}\in\mathcal{H}^{2}(\mathbb{C}_{+})\quad\text{and}\quad
W\mathrel{:\mkern-0.25mu=}\Phi^{1/2}\in\mathcal{H}^{2}(\mathbb{C}_{+}).$

Then there exist $\omega,\gamma\in L^{2}(\mathbb{R})$ such that

$V_{1}(\cdot+iy)+iV_{2}(\cdot+iy)\to\omega+iH\omega,$
$W_{1}(\cdot+iy)+iW_{2}(\cdot+iy)\to\gamma+iH\gamma$

in $L^{2}(\mathbb{R})$ as $y\searrow 0$. As a consequence,

$\mathbf{Re}[VW]=V_{1}W_{1}-V_{2}W_{2}\to\omega\gamma-H\omega H\gamma$

in $L^{1}(\mathbb{R})$ as $y\searrow 0$. On the other hand,

$\mathbf{Re}[VW]=\mathbf{Re}\,U\to f$

in $L^{1}(\mathbb{R})$ as $y\searrow 0$. Thus $f=w\gamma-H\omega H\gamma$.

We then find $f\in\mathcal{H}^{1}(\mathbb{R})$ that cannot be written as
$f=\omega^{2}-(H\omega)^{2}$ for any $\omega\in L^{2}(\mathbb{R})$. Choose
$U\in\mathcal{H}^{1}(\mathbb{C}_{+})$ that has at least one zero of odd order.
(Given $W\in\mathcal{H}^{1}(\mathbb{C}_{+})$ one may set
$U(z)=[(z-z_{0})/(z-\bar{z_{0}})]W(z)$, where $z_{0}\in\mathbb{C}_{+}$ and
$W(z_{0})\neq 0$.) Then $U$ is not the square of any analytic function. We
write $f\mathrel{:\mkern-0.25mu=}\lim_{y\searrow
0}U_{1}\in\mathcal{H}^{1}(\mathbb{R})$ where the limit holds in
$L^{1}(\mathbb{R})$.

Seeking a contradiction, assume that $f=\omega^{2}-(H\omega)^{2}$ for some
$\omega\in L^{2}(\mathbb{R})$. Then there exists
$V=V_{1}+iV_{2}\in\mathcal{H}^{2}(\mathbb{C}_{+})$ such that $\lim_{y\searrow
0}\left\lVert V_{1}(\cdot+iy)-\omega\right\rVert_{L^{2}}=0$. Now
$V^{2}\in\mathcal{H}^{1}(\mathbb{C}_{+})$ and

$\mathbf{Re}[V^{2}]=V_{1}^{2}-V_{2}^{2}\to\omega^{2}-(H\omega)^{2}=f$

in $L^{1}(\mathbb{R})$ as $y\searrow 0$. Thus
$U-V^{2}\in\mathcal{H}^{1}(\mathbb{C}_{+})$ has vanishing boundary values at
$y=0$, and so $U=V^{2}$, which yields a contradiction. ∎

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