A Note on the Jacobian Problem of Coifman, Lions, Meyer and Semmes

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 H1(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^1({\mathbb {R}}^n)$$\end{document}. 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.


Introduction
The real-variable Hardy space H 1 (R n ) often acts as a good substitute for L 1 (R n ), mirroring many of the ways in which BMO(R n ) substitutes L ∞ (R n ) [15,47].In order to define H 1 (R n ) we fix χ ∈ C ∞ c (R n ) with R n χ(x) dx = 0, denote χ t (x) := t −n χ(x/t) for every x ∈ R n and t > 0 and set We endow H 1 (R n ) with the norm f H 1 := sup t>0 |f * χ t (•)| L 1 .
Coifman & al. proceeded to ask whether these nonlinear quantities are surjections onto H 1 (R n ) [3, p. 258].The most famous open case is the following: (1.1) As a partial result, Coifman & al. showed that H 1 (R n ) is the smallest Banach space that contains the range of the Jacobian.More precisely, (1.2) . Further partial results were presented in [16,18,38].When the domain of definition of J is the inhomogeneous Sobolev space W 1,n (R n , R n ), the author proved non-surjectivity in [39].
The author was supported by the ERC Advanced Grant 834728. 1 In bounded domains, the Dirichlet problem Ju = f in Ω, u = id on ∂Ω 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 ∈ W 1,n  id (Ω, R n ) has Jacobian Ju ≥ 0 a.e., Müller's higher integrability result implies that Ju ∈ L log L(Ω).As an analogue of (1.1), Hogan & al. asked whether every non-negative f ∈ L log L(Ω) with mean 1 has a solution u ∈ W 1,n  id (Ω, 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 (Ω, R n ) with arbitrary boundary values in W 1−1/q,q (Ω, R n ) were constructed in [33].At any rate, compared to the Dirichlet problem, different ideas are needed in the case of R n due to the unboundedness of the domain and the absence of boundary conditions.
In [28], Iwaniec conjectured that for every n ≥ 2 and p ∈ [1, ∞) the Jacobian operator J : Ẇ 1,np (R n , R n ) → H p (R n ) not only is surjective but also has a continuous right inverse G : Hytönen proved in [27] the natural analogue of (1.2), so that L p (R n ) is, again, the minimal Banach space containing J( Ẇ 1,np (R n , R n )).We discuss Hytönen's contribution in §7.
We next briefly summarise some of the ideas presented in [28].Whenever f ∈ H p (R n ) and the equation Jv = f has a solution, it has a minimum norm solution u ∈ Ẇ 1,np (R n , R n ), that is, Ju = f and R n |Du| np = min Jv=f R n |Dv| np .Furthermore, the range of J is dense in H p (R n ).Iwaniec suggested a possible way of finding a continuous right inverse G: Strategy 1.1.When n ≥ 2 and 1 ≤ p < ∞, the following claims would yield a continuous right inverse of J : Ẇ 1,np (R n , R n ) → H p (R n ): (1) Every energy-minimal solution u satisfies Du n L np Ju H p .(2) For every f ∈ H p (R n ) there is a unique energy-minimal solution u f , modulo rotations.(3) There exist rotations R f ∈ SO(n) such that f → R f u f is a continuous right inverse of J.
Claim (1) would easily imply the surjectivity of J : Ẇ 1,np (R n , R n ) → H p (R n ) (see [38, p. 36]).In [16], (1) was shown to be, in fact, equivalent to the surjectivity of J : Ẇ 1,np (R n , R n ) → H p (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 ≤ p < ∞.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 : Ẇ 1,2 (R 2 , R 2 ) → H 1 (R 2 ).In §7, we discuss how to adapt the methods to the case n = 2, 1 < p < ∞.
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 In the plane, one can also deduce (1.2) from the two-sided commutator estimate ) and b ∈ BMO(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 ∈ {1, . . ., n} and T = R j is a Riesz transform, the upper bound estimate [b, [5,30,49] gives the weak factorisation (1.4) and the compactness of [b, R j ] immediately lead to the weak-to-weak * sequential continuity of the quadratic operator (ω, γ) → ωR j γ + γR j ω : L 2 (R n ) 2 → H 1 (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 1.2-1.3below, we axiomatise the above-mentioned properties of the Riesz transform R j and the quantity ωR j γ + γR j ω.We study the factorisation problem for quadratic weakly continuous quantities in real-variable function spaces such as H 1 (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 ).We address the following question: 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 ω → ω 2 − (Hω 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.
• Whenever f ∈ X * and the equation Qγ = f has a solution, it has a minimum norm solution ω ∈ H, that is, Qω = f and ω H = min Qγ=f γ H .
• We define an energy functional E : Almost  [16], for translation-invariant operators (such as the Jacobian) we have In particular, the surjectivity of Q : H → X * is equivalent to the statement that Q(H) is dense in X * and every minimum norm solution satisfies ω 2 H Qω X * .This gives a metamatematical justification for taking claim (1) as a goal.
• Given a minimum norm solution ω ∈ H of Qω = f , we may use calculus of variations to study ω via perturbed solutions ω ǫ ∈ H of Qω ǫ = f , ω ǫ − ω H → 0. However, it tends to be very hard to construct variations that satisfy the nonlinear constraint Qω ǫ = f .In particular, the first variation ω ǫ = ω + ǫϕ, ϕ ∈ H, is in general unavailable.
• The constraint Qω ǫ = f could be relaxed if we found a Lagrange multiplier, that is, b ∈ X * * satisfying If, furthermore, b X * * ≤ C uniformly in f , then setting ϕ = f in (1.5) would give the sought inequality E • X * .However, given an arbitrary minimum norm solution, the construction of a Lagrange multiplier b ∈ X * * is a formidable task-in particular, the standard Liusternik-Shnirelman method is not applicable.
• Nevertheless, many minimum norm solutions do possess a Lagrange multiplier.Indeed, (1.5) says that ω is a critical point of the functional • As every b ∈ X with T b H→H = 1 is a Lagrange multiplier, we use the norm of X given by and denote the resulting Banach spaces by X Q and X * Q .Thus every b ∈ S XQ is a Lagrange multiplier of some ω ∈ S H with Qω ∈ S X * Q .• As a consequence, the set 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]: 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, (ω, γ)) → (bω, −bγ) : R × R 2 → 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.

Main results. As a first largeness criterion on
) is contained in Theorem 5.1.We will find more refined information on Q(A ) by using the set-valued duality mapping.
Definition 1.7.The duality mapping D : S XQ → 2 By the Bishop-Phelps Theorem [12,Theorem 3.54 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.
The norm and relative weak * topologies also coincide in NA • 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 ∈ ext(S X * Q ).Then the minimum norm solution ω ∈ A of Qω = f is unique up to multiplication by c ∈ S K .
In view of (1.6) it is natural to ask whether 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:

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 ⊂ Z is open and g : U → Z is convex, where Z is a real Banach space.We say that g is Gâteaux differentiable at The operator L is then denoted by g ′ x and called the Gâteaux derivative of g at x.If the limit in (2.1) is uniform in h ∈ S X , then g is said to be Fréchet differentiable at x.
Note that the convex function g(x) = x X is Gâteaux (Fréchet) differentiable at x ∈ S X if and only if it is Gâteaux (Fréchet) differentiable at λx for every λ ∈ R \ {0}.We will use the following theorem of Asplund and Lindenstrauss (see [12,Theorem 8.21]) on the norm of X.
Recall that the duality mapping D : S X → 2 S X * is defined by We recall characterisations of the norm-to-norm upper semicontinuity of the duality mapping [13,24].Theorem 2.3.Let x ∈ S X .The following statements are equivalent.
(i) D is norm-to-norm upper semicontinuous at x and D(x) is norm compact.
(ii) For every sequence of points x * j in S X * such that x * j , x → 1, there exists a subsequence convergent to some x * ∈ D(x).(iii) The weak * and norm topologies agree on 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 • X is Fréchet differentiable at x ∈ 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 : Y → 2 Z \ {∅} 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 ∈ 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 ∈ 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.
Theorem 2.8.Let K be a non-empty subset of a Hausdorff locally convex space such that co(K) is compact.Then every extreme point of co(K) lies in K.

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 BMO functions and paracommutators.Specific examples are then given in §3.3.

Commutators of Calderón-Zygmund operators and BMO functions.
Recall that where the supremum is taken over cubes Coifman & al. showed in [5] that when T is a Calderón-Zygmund operator with a suitable smooth kernel Ω, we have [b, T ] L 2 →L 2 b BMO for all b ∈ BMO(R n ).Specifically, Ω was assumed to be homogeneous of degree zero with vanishing mean over S n−1 and to satisfy |Ω(x) − Ω(y)| < |x − y| for all x, y ∈ S n−1 .They also Uchiyama [49] and Janson [30] showed independently that in order to obtain b ∈ BMO(R n ), it suffices to show boundedness of [b, T ] in L 2 (R n ) for only one of the kernels Ω ≡ 0 in the result of [5].Uchiyama also showed that [b, T ] is compact if and only is The two-sided estimate (1.3) has been extended in numerous ways (to multiparameter 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 [b, T ] L 2 →L 2 ≃ b BMO under the very weak assumption that the kernel is "non-degenerate".We recall relevant definitions from [27].
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 In [31], Janson and Peetre gave conditions on the function A under which the boundedness of T b (A) : In order to state the result we need some definitions.We do not motivate them here but refer to [31] for more information.
for some σ-finite measure space (X, µ) and measurable functions α : Furthermore, M (U × V ) is a Banach algebra in the norm given by minimising the left hand side of (3.3) over all representations (3.2).
It is instructive to also consider the planar Jacobian in real variable notation.Define (3.5) Now T * b = −T b = 0 and thus T b is not self-adjoint.However, the operator , defined in Proposition 2.10, generates the Jacobian: ).This example serves to further motivate Assumption 1.3.
Example 3.6.On the real line, two quadratic H 1 -integrable quantities arise as follows.Isomorphically, H p (R) ∼ = H p (R, C) := {g + iHg : g, Hg ∈ L p (R)} for all p ∈ (0, ∞), where H is the Hilbert transform.Given ω ∈ L 2 (R) we set 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 , their product can be written as Each component of ΩΓ then belongs to H 1 (R n ) [51,Theorem 12.3.1].Note that when n = 2, the last term of (3.7) is a Jacobian in disguise: R Wu also studied more general combinations of Riesz transforms that arise via A(ξ, η) = 1 − (ξ • η) m /(|ξ| |η|) m , m ∈ N, and showed that each such A satisfies the assumptions of Theorem 3.4.As an example, where Λu = f .In this case, A(ξ, η) = 1 − η • ξ/(|η| |ξ|).Thus Wu's conjecture says that the Gâteaux derivative The planar Monge-Ampère equation arises as follows.Define we may write the Hessian as

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.Proof.Choose a minimising sequence so that Qω j = f for every j ∈ N and lim j→∞ ω j H = inf Qγ=f γ H . Since B H is sequentially weakly compact and Q is weak-to-weak * continuous, we have ω j ⇀ ω and Qω j * ⇀ Qω for a subsequence, and ω is the sought minimum norm solution.
We then show that surjectivity of Q : H → X * Q would follow from the weak * density of the range Q(H) in X * Q combined with a suitable a priori estimate.
Proof.Let f ∈ X * and choose minimum norm solutions ω j ∈ H with Qω j * ⇀ ω.For large enough j ∈ N we have ω j 2 H f X * + 1.After passing to a subsequence, a weak limit ω of (ω j ) ∞ j=1 satisfies Qω = 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 σ j ∈ H H and The example one should have in mind is σ j ω(x) = ω(x − je) and σ j f (x) := 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) Q(H) = X * Q and every minimum norm solution satisfies ω is a markedly less daunting task than disproving surjectivity [16].

Definition and existence of Lagrange multipliers.
When ω ∈ H is a minimum norm solution, it is natural to look for a Lagrange multiplier of ω, as discussed in §1.3.The Lagrange multiplier condition (1.5) can be written more concisely as follows: 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 ′ ω maps H onto X * Q , then ω 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 Proof.Choose a maximizing sequence: T b ω j , ω j H → 1.For a subsequence, ω j ⇀ ω in H, and so (iii) ω ∈ ker(I − T b ).If (i)-(iii) hold, then ω is a minimum norm solution and belongs to A .
Proof.We first prove (i) ⇔ (ii).Suppose (ii) holds and fix ϕ ∈ H and small δ > 0. The function is maximised at ǫ = 0, and therefore k − 2Re ω, ϕ H = 0, giving (i).The direction (i) ⇒ (ii) is proved by setting ϕ = ω.We then prove (ii) ⇔ (iii).First, (ii) gives ω, T b ω H = 1.Since T b H→H = 1 and H is strictly convex, we conclude that (iii) holds.On the other hand, if (iii) holds, then b, Qω X 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 z = ω as before, (i)-(iii) are equivalent to 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 S X * Q , and Lemma 5.1 then gives f ∈ Q(A ).We also characterise elements of S H that possess a Lagrange multiplier.Proposition 4.12.Let ω ∈ S H .The following conditions are equivalent.
Using the results above we collect many equivalent formulations of Question 1.6.
) Every minimum norm solution has a Lagrange multiplier in S X * * Q .

The proof of Theorem 1.10
We get the proofs of Theorems 1.9 and 1.10 underway by proving (1.6).
Claim 6.4.The norm and relative weak * topologies coincide in Q(A ).
Proof.The result follows directly from Claim (v) and Theorem 2.3.

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 : Ẇ 1,2p (R 2 , R 2 ) → L p (R 2 ) when 1 < p < ∞.In this section we outline such an approach and list those results whose proof extends to this new setting in a straightforward manner.
We Hytönen proved (7.1) rather recently in [27].More generally, on R n , Hytönen showed that the commutator of a degenerate Calderón-Zygmund operator with b ∈ L 1 loc (R n ) defines a bounded operator from L q1 (R n ) into L q2 (R n ), q 1 > q 2 > 1, if and only if b = a + c with a ∈ L r (R n ), 1/r = 1/q 2 − 1/q 1 and c ∈ R. In notable contrast to the case of [b, T ] : L 2 (R n ) → L 2 (R n ), the possible cancellations of b do not play an important role; recall, for instance, that there exist b / ∈ BMO(R n ) such that |b| ∈ BMO(R n ).
As a corollary of (7.1), Hytönen obtained an analogue of (1. f 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 ] : L 2p (R 2 , R 2 ) → L (2p) ′ (R 2 , R 2 ) for all b ∈ L p ′ (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 < ∞ 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) condition (i) of Definition 3.2 with the Dini condition

Assumption 1 . 3 .∞ j=1 ω j 2 H
The bilinear mapping (b, ω) → T b ω satisfies the following conditions for every b ∈ X * * :(i) T b is self-adjoint.(ii) T b H→H = sup f H =1 T b f, f .Definition 1.4.Given X, H and (b, ω) → T b f we define a norm-to-norm and weak-to-weak * sequentially continuous mapping Q : H → X * by b, Qω X−X * := T b ω, ω 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 * = { ∞ j=1 Qω j : < ∞}.Whenever the map (b, ω) → T b ω satisfies Assumption 1.2, the modified operator (b, (ω, γ)) → Tb (ω, γ) := (T * b γ, T b ω) : X * * × (H × H) → H × H satisfies Assumptions 1.2-1.3(see Proposition 2.10).Examples 3.5-3.6illustrate the role of Assumption 1.3 further.The planar Jacobian arises as follows (see Example 3.5): defining T b : L 2 (C, C) → L 2 (C, C) by T b ω := (Sb − bS)Sω, where S is the Beurling transform, we can write Question 1.6 reduces to the question whether D(b) ∩ QA = D(b) for every b ∈ S XQ .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 A (see Remark 5.6).

Theorem 1 . 9 .
Under Assumptions 1.2-1.3, the following statements hold: (i) For every b ∈ S XQ , the convex set D(b) has finite affine dimension.(ii) For every b ∈ S XQ , D(b) ∩ Q(A ) contains ext(D(b)) and is path-connected.(iii) The norm and relative weak * topologies coincide in Q(A ).Under the further Assumption 1.8, the following statements hold: (iv) For every b in a dense, relatively open subset of S XQ , D(b) = {Qω} for some ω ∈ A and • XQ is Fréchet differentiable at b.

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 : H → H is K-linear.Then there exists a unique selfadjoint K-linear operator B : H × H → H × H such that B(x, y), (x, y) H×H = 2Re Ax, y H for all x, y ∈ H.The operator B is of the form B(x, y) = (A * y, Ax) and satisfies B H×H→H×H = A H→H .Proposition 2.10.If a bilinear mapping T : X * * × H → H satisfies Assumption 1.2, then the modified operator where A : R n × R n → R and ω : R n → R are functions and b : R n → R is called the symbol of T b (A).As the most basic example, A(ξ, η) ≡ 1 yields (under suitable integrability assumptions) the multiplication operator T b ω = bω.The commutators T b (A) = [b, R j ] arise via A(ξ, η) = ξ j / |ξ| − η j / |η|.More generally, if T is a Calderón-Zygmund singular integral operator with Fourier symbol m and A(ξ, η) = m(ξ) − m(η), then T b (A) = [b, T ].Several further examples are presented in [31, pp.469-473] and [45, pp.513-519].

4. 1 .Proposition 4 . 1 .
Minimum norm solutions.We begin by noting that if the equation Qω = f has a solution, then the direct method gives a minimum norm solution.Suppose f ∈ X * Q , γ ∈ H and Qγ = f .Then there exists ω ∈ H with Qω = f and ω H = min Qγ=f γ H .
Hilbert space onto X * .Proposition 4.5.Every b ∈ S XQ is a Lagrange multiplier of some ω ∈ A.

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.

Claim 5 . 2 .
Suppose Assumptions 1.2-1.3 and 1.8 hold, and let f ∈ ext(B X * Q ).Then the minimum norm solution ω ∈ A of Qω = f is unique up to multiplication by c ∈ S K .

Claim 6 . 5 .
For every b in a relatively open dense subset of S XQ , D(b) = {Qω} for some ω ∈ A and • XQ is Fréchet differentiable at b. Proof.By Theorem 2.1, • XQ is Fréchet differentiable in a dense subset of S XQ .Seeking a contradiction, suppose b is a point of Fréchet differentiability but the points b j → b, b j ∈ S XQ , are not.Thus the subspaces ker(I − T bj ) are at least two-dimensional but, by Corollary 5.3, dim(ker(I − T b )) = 1.

Remark 6 . 6 .
Note that a straightforward modification of the proof above proves the relative openness in S XQ of {b ∈ S XQ : dim(ker(I − K b )) ≤ n} for every n ∈ N. Claim 6.7.D : S XQ → 2 S X * Q is a cusco map.Proof.The duality mapping D : S XQ → S X * Q is point-to-compact since for every b ∈ S XQ , 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 ∈ S XQ and f j ∈ S X * Q satisfy lim j→∞ f j , b X * Q −XQ = 1.By Theorem 2.3 it suffices to show that f j → f ∈ D(b) for a subsequence.Denote dim(ker(I − T b )) = n ∈ N. Then dim(ker(I − T bj )) can again use basic properties of the Beurling transform to write b, Qω L p ′ −L p = T b ω, ω L (2p) ′ −L 2p , T b ω := (Sb − bS)Sω, Qω := |Sω| 2 − |ω| 2 .Note that T b L 2p →L (2p) ′ p b L p ′ for all b ∈ L p ′ (C)by a simple application of Hölder's inequality and the boundedness of S : L p (C, C) → L p (C, C).The lower bound estimate (7.1)T b L 2p →L (2p) ′ p b L p ′ ,however, is far from trivial and cannot be proved by simply following the proof of estimates (1.3).
By classical results by Fefferman and by Coifman and Weiss, respectively, we have the dualities [H 1