Uhlenbeck’s Decomposition in Sobolev and Morrey–Sobolev Spaces

We present a self-contained proof of Rivière’s theorem on the existence of Uhlenbeck’s decomposition for Ω∈Lp(Bn,so(m)⊗Λ1Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in L^p(\mathbb {B}^n,so(m)\otimes \Lambda ^1\mathbb {R}^n)$$\end{document} for p∈(1,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,n)$$\end{document}, with Sobolev type estimates in the case p∈[n/2,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in [n/2,n)$$\end{document} and Morrey–Sobolev type estimates in the case p∈(1,n/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,n/2)$$\end{document}. We also prove an analogous theorem in the case when Ω∈Lp(Bn,TCO+(m)⊗Λ1Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in L^p( \mathbb {B}^n, TCO_{+}(m) \otimes \Lambda ^1\mathbb {R}^n)$$\end{document}, which corresponds to Uhlenbeck’s decomposition with conformal gauge group.


Introduction
Throughout the paper, n ≥ 2.
Rivière [15] published a solution to the so-called Heinz-Hildebrandt conjecture on regularity of solutions to conformally invariant nonlinear systems of partial differential equations in dimension 2. The key tool he used was a theorem due to Karen Uhlenbeck, on the existence of the so-called Coulomb gauges, in which the connection of a line bundle takes particularly simple form; we quote the theorem below in somewhat imprecise terms, to avoid unnecessary technicalities. Theorem 1.1 (Uhlenbeck [21]). Let η be a vector bundle over the unit ball B n , n ≥ 2, with connection form Ω ∈ W 1,p , p ∈ [n/2, n). If the curvature field, F (Ω) = Ω ∧ Ω + dΩ, has sufficiently small L n/2 norm, then Ω is gaugeequivalent to a connectionΩ which is co-closed (i.e. d * Ω = 0), with estimates of the gauge and ofΩ given in terms of F (Ω) L p .
All the proofs of the above results are, up to details, adaptations of the original approach of Uhlenbeck and most of them refer the reader for certain parts of the reasoning to the original paper [21]. The latter, however, is written in the language of differential geometry (the result was used there in the context of the existence theory for Yang-Mills fields). Translating the results to Rivière's setting and filling all the sketched details was not trivial, which is probably why this extremely useful result went overlooked by the PDE community for over two decades.
All the proofs naturally split into two parts: • Proving the existence of the decomposition for any sufficiently small perturbation of a co-closed form * dζ, provided certain norm of dζ is small; • Proving that once we have Ω, P and ξ which satisfy the equation P −1 dP + P −1 ΩP = * dξ and additionally certain norms of Ω, dP and dξ are sufficiently small, the presumed estimates hold (the norms of P and ξ are bounded in terms of the norm of Ω). In the results mentioned above, two strategies of proving the existence of decomposition of Ω were used. The original strategy used by Karen Uhlenbeck was to solve the equation for P * d * (P −1 dP + P −1 ( * dζ + λ)P ) = 0 for a given perturbation λ of some fixed co-closed form. To do this, we look for P of the form P = e u , add some boundary condition on u (Neumann in the original result of Uhlenbeck, Cauchy in our proof) and define the nonlinear operator T (u, λ) = * d * (e −u de u + e −u ( * dζ + λ)e u ) acting on appropriate Banach spaces; in our case T (u, λ) : B × W 1,p → L p where B = W 2,p ∩W 1,p o . One can apply then the Implicit Function Theorem to show that T (u, λ) = 0 has a solution u λ continuously depending on λ. To do this, one has to show that the linearization of T at (u, λ) = (0, 0) with respect to the first argument, is an isomorphism B → L p . This strategy works in Sobolev spaces for p ∈ (n/2; n), but it fails when 1 < p < n/2.
Another strategy is used by Tao and Tian [20]. Again, one looks for P = e u and assumes u has zero boundary data. The equation * d * (P −1 dP + P −1 ( * dζ + λ)P ) = 0 is transformed into the form Δu = * d * du − e −u de u − e −u ( * dζ + λ)e u = * d * F (u, ζ, λ). Then, an iteration scheme is set to provide a solution: We use this strategy when 1 < p < n/2. Schikorra [17] gave an alternative, variational proof of the existence of Uhlenbeck's decomposition. His approach was inspired by a similar variational construction of a moving frame by Hélein [8]. Schikorra's methods, however, provided the gauge transformation P only in W 1,2 , even for Ω ∈ L p with p > 2 (while Uhlenbeck's and Rivière's approach gave P ∈ W 1,p ). On the other hand, his method was much simpler and allowed him to give alternative regularity proofs for systems studied by Rivière [15] and Rivière and Struwe [16].
Finally, one should mention the book of Wehrheim [23], who undertook the effort of clarifying and presenting in all detail the original result of K. Uhlenbeck.
The main result of the paper is a self-contained, complete proof of Rivière's theorem in the following settings.
First, with Sobolev type estimates, and such that Next, with Morrey-Sobolev type estimates, be an antisymmetric matrix of 1-differential forms on B n . Assume Ω ∈ L 2p,n−2p and dΩ ∈ L p,n−2p . There exists > 0 such that if Ω satisfies the smallness condition And finally, a version of Uhlenbeck's decomposition for a larger gauge group CO + (n) (i.e. conformal transformations), which gives the decomposition theorem for a larger class of matrix-valued differential forms.
and such that In view of Theorem 1.4, we may ask a natural question: Question. What is the largest Lie subgroup G of GL(n) that can be used in an analogue of Rivière's theorem: for any matrix of 1-differential forms Ω ∈ T Id G there exists a gauge transformation P ∈ G such that Ω P = P −1 dP + P −1 ΩP is co-closed and certain integrability estimates on P , Ω P and their derivatives, in terms of Ω, hold?
The paper is structured as follows. In Sect. 2, we recall Gaffney's inequality and discuss, how the boundary conditions on the decomposition components P and ξ allow us to estimate their Sobolev norms with only some of their derivatives.
Next, in Sect. 3 we recall the definitions and basic properties of Morrey and Morrey-Sobolev spaces we use.
Throughout the paper, wherever applicable, we use the operator norm |T | = sup |x|=1 |T x| of a linear operator T (thus, in particular, |P | = 1 almost everywhere)-this simplifies the estimates of compositions. A constant C may vary from line to line in calculations.

Sobolev Spaces and Differential Forms
Throughout the paper, we use differential forms with coefficients in Sobolev spaces, i.e. Sobolev differential forms. With this in mind, we write e.g.
which means that Ω is a W 1,p function with values in the vector space so(m)⊗ Λ 1 R n . This allows us to define the full Sobolev norm Ω W 1,p , which disregards the differential form aspect of it.
It is tempting to consider Sobolev spaces of differential forms using only the two derivatives that are natural in this setting: the differential d and the co-differential * d * , instead of the full derivative D. This is possible if we restrict to forms that satisfy certain boundary conditions, see e.g. [10]. It is important to realize, however, that for a general differential form ω, the differential dω and the co-differential * d * ω capture only some derivatives of the coefficients, and in general one should not expect to control the W 1,p -norm of ω by the L p -norms of ω, dω and/or * d * ω (we may skip the first star and use d * ω, since the Hodge star * is an isomorphism and * d * However, if the domain in which we consider our forms is smooth (as is in our case, where the only domains considered are n-dimensional balls) and the form ω satisfies certain boundary conditions (namely, ω has vanishing tangent or normal component on the domain's boundary), then Gaffney's inequality ( [5], see also [10,Theorem 4.8], [9,Theorem 10.4.1]) holds: where the constant C depends on the domain and the exponent p only. We refer the reader to the paper [10] and the book [9] for more details on Sobolev differential forms.
Let us now see how this applies to the components of Uhlenbeck's decomposition.
The orthogonal mapping P is a function (a 0-form), and thus its differential dP captures all the derivatives of its coefficients, i.e. ∇P L p = dP L p . Also, we assume that P restricted to the boundary of the ball B n equals (in the sense of traces) to the identity matrix Id, therefore dP = d(P − Id) has vanishing tangent component and Gaffney's inequality (2.1) or, since P is a function, standard elliptic estimates give us a comparison between dP W 1,p and dP L p + ΔP L p .
As for the (n − 2)-form ξ, we have strong assumptions given in (4.1): ξ vanishes on the boundary and it is co-closed. Thus both ξ and dξ have vanishing tangent components and again Gaffney's inequality (2.1) allows us to compare dξ W 1,p and dξ L p + Δξ L p , with Δξ = * d * dξ.

Morrey Spaces
In this section we recall the properties of Morrey spaces needed in our paper; for more information and for proofs of elementary properties of Morrey spaces we refer the reader e.g. to the monograph [1].
Throughout the paper we customarily use barred integral to denote the integral average and we write f E for the integral average of f over the set E: Let U ⊂ R n be an open and bounded set. Recall that the Morrey space When s = 0, the Morrey space L p,0 (U ) is the same as the usual Lebesgue L p (U ) space. When s = n, the dimension of the ambient space, it easily follows from the Lebesgue Differentiation Theorem that the Morrey space L p,n (U ) is equivalent to L ∞ (U ). Morrey spaces are Banach spaces.
For 1 ≤ p ≤ q < ∞ and s, σ ≥ 0 such that s−n p ≤ σ−n q we have in particular L q,n−q (U ) → L p,n−p (U ).
Note that with this definition, f ∈ L p,n−kp k (U ) does not imply Morrey estimates for f itself. Results Math Poincaré-Wirtinger's inequality on a ball B of radius r and Hölder's inequality immediately yield . We say that a domain U is of type (A) if there exists a constant C > 0 such that for any x 0 ∈ U and 0 < r < diam(U ) This excludes domains with outward cusps. For domains of type (A) we have the following generalization of Sobolev's embedding theorem (see [13]).

Proposition 3.3 (Morrey-Sobolev embedding). Let
Let us recall the BMO-Gagliardo-Nirenberg type result due to Adams and Frasier [2] (see also [19]): This result can be easily localized to functions in W 2,s ∩ BM O(B), for an arbitrary ball B ⊂ R n (see [22,Proposition 4.3]).

Proposition 3.5. For any ball
The proof goes exactly as in [22] (where p equals 2): we extend f to a functionf ∈ W 2,p ∩ BM O(R n ) and apply the estimate (3.2) tof .
As the consequence of the above estimates we obtain the following product estimate. Proof. First, let us observe that where the supremum is taken over all balls B r ⊂ B and the constant depends only on n and p. Indeed, for any such ball B r ⊂ B we have In particular, for α ∈ [0, p] we obtain (see also [18,Proposition 3.2])

Uhlenbeck's Decomposition, the Case n/2 ≤ p < n
In this section we prove Theorem 1.2. We first state it (as Lemma 4.1) and prove it in the case p ∈ (n/2, n); the case p = n/2 follows by approximation (see Corollary 4.7 at the end of this section).
and such that In what follows, we write, to keep the notation simple, We shall break the proof of Lemma 4.1 into several lemmata. Following Rivière, we introduce sets : Ω L n < and there exist P and ξ satisfying the system (4.1) and the estimate (4.2)}.
We show that for > 0 and sufficiently small the set U is closed and open in V , and since the latter is path connected (it is star-shaped in W 1,p ), it follows that U = V , which proves Lemma 4.1.
With every Ω k we associate P k and ξ k that satisfy (4.1): and the estimates (4.2) hold for P k , ξ k and Ω k , in particular The boundary condition on ξ k and boundedness of P k (recall that |P k | = 1) allow us to interpret (4.4) as boundedness of P k and ξ k in W 2,p , since the sequence (Ω k ), being convergent, is necessarily bounded in W 1,p . We can thus assume (after passing to subsequences) that P k and ξ k are weakly convergent in W 2,p to some P and ξ.
Both the boundary condition ξ| ∂B = 0 and the condition d * ξ = 0 are preserved when passing to the weak limit. Moreover, possibly after passing to a subsequence, we have and for any small δ > 0, Ω k , dP k and dξ k converge strongly (to Ω, P and ξ, respectively) in L s , with s = np n−p − δ, in particular in L n , since np/(n − p) > n. Also, P k are uniformly bounded in L ∞ and strongly convergent, by Sobolev embedding theorem, in L q for any q. This is enough to show the strong convergence of Ω k P k to ΩP in L s ; altogether, we may pass to the strong limit in L s in the system (4. 3), showing that the equation The estimates (4.2) for P and ξ are obvious.
Remark. For any P and ξ in W 1,p that satisfy (4.3) we have that dξ ∈ W 1,p implies P ∈ W 2,p . Indeed, we have dP = P * dξ − ΩP , and for p > n/2 the right hand side is in W 1,p .
Now we proceed to prove the openness of U . In contrast with the previous lemma this is more delicate; we split the reasoning again into several lemmata.

Lemma 4.4. There exists a constant
Note that (4.5) implies, through Poincaré's lemma, that the term in parentheses is of the form * dζ, for some antisymmetric (n − 2)-formζ. The above Lemma should be understood as follows: Uhlenbeck's decomposition (i.e. Q andζ) exists if our matrix is a small (in W 1,p ) perturbation of a co-closed form * dζ, provided dζ L n is sufficiently small. Proof. Since we are interested in finding any Q ∈ W 2,p (B n , SO(m)) satisfying (4.5) and the boundary condition, we shall look for one of the form e u , where This is a well defined, smooth operator. Using Implicit Function Theorem, we prove that for any sufficiently small λ the equation T (u, λ) = 0 has a solution u λ , continuously depending on λ. To this end we linearize T at (u, λ) = (0, 0) with respect to the first argument: where the commutator [·, ·] denotes a commutator of two so(m) matrices. We have, by Hölder's inequality where the constant C S comes from the Sobolev embedding (recall that ψ ∈ B is a matrix-valued function, i.e. a 0-form, vanishing at the boundary). Therefore, for κ small, H is injective. Showing surjectivity of H amounts to showing that the system has a solution in B for arbitrary f ∈ L p . Let us consider an operator K : B → B, with K(ψ) defined as a solution to the system Using Hölder's and Sobolev's inequalities and the fact that the Newtonian potential Δ −1 : L p → B is continuous, we get For κ sufficiently small we can have |||K||| B→B small and If ψ is a solution to (4.6), then and we can solve (4.6) for any f ∈ L p by solving the above Poisson equation and applying to its solution the inverse mapping to Id − K.
Altogether, H : B → L p is an isomorphism, and we can apply the Implicit Function Theorem to get u λ as a continuous function of λ.
To end the proof of the lemma, we take Q(λ) = e u λ .
Lemma 4.5. Suppose n/2 < p < n. There exists κ = κ(p, n) such that for any Ω ∈ V and P , ξ in W 2,p satisfying the system (4.1) and additionally the estimate the estimates (4.2) hold.
Proof. The lemma follows from rather standard elliptic estimates, but we include them here for the sake of completeness. We have Note that for q = p/(p − 1), dξ L p is equivalent to where φ is a smooth, compactly supported (in particular with null boundary values) (n − 2)-form on B n . The inequality is obvious. Applying the Hodge decomposition to the (n−1)-form η, η = dφ+ψ with δψ = 0, dφ L q ≤ C q η L q , we get Denote byP the mean value of P over B n :P = B n P . For any φ as above, with dφ L q ≤ 1, with the constants C (possibly different in every line) dependent only on n and p. Note that, since P is an orthogonal matrix, |P | = |P −1 | = 1 and |dP −1 | = |dP |. In the estimate above we use, for the first summand, the Coifman-Lions-Meyer-Semmes div-curl inequality [3] and later the standard inclusion W 1,n → BM O; the second summand is estimated by Hölder's inequality.
On the other hand, taking L p norms of both sides of the equation [c.f. (4.1)] gives Putting (4.10) and (4.12) together we get and for κ < 1 C1 this implies that the estimate (4.2a) holds. The above calculation is valid also for p = n, which yields the estimate (4.2b).
To show the estimate (4.2c), by taking * d * of both sides of (4.11), we see that The constant C comes from Gaffney's inequality (2.1), standard elliptic estimates and the Sobolev embedding W 1,p → L np/(n−p) , thus it depends only on p and n. Similarly, using (4.9), (4.14) Note that, as pointed out in Sect. 2, the full Sobolev norms of dP and dξ can be estimated with the norms of Laplacians of P and ξ, thanks to the boundary conditions they satisfy. Composing (4.13) and (4.14) with the already proved estimate (4.2a) we get which, for κ sufficiently small, yields the estimate (4.2c). Take now Ω ∈ V close to Ω o in W 1,p : we ask that for Setting P = P o Q we see that (4.15) reduces to * d * (P −1 dP + P −1 ΩP ) = 0.
What remains to prove is that P , ξ and Ω satisfy the estimates (4.2). Observe that if Ω − Ω o W 1,p is small enough, then by continuity of the mapping We also know that Taking sufficiently small, we may ensure that with κ as in Lemma 4.5. Applying this lemma we show that the estimates (4.2) hold.
Altogether, Ω ∈ U , which proves the openness of U .
Proof of Lemma 4.1. Since, by Lemmata 4.3 and 4.6, for sufficiently small the set U is closed and open in V , and since the latter is path connected (it is star-shaped in W 1,p ), it follows that U = V , which proves Lemma 4.1.
It is worth noting that for Ω ∈ W 1,n/2 , the proof of existence of the decomposition, i.e. Lemma 4.4, fails. However, we can proceed by a standard density argument: approximate Ω in W 1,n/2 with Ω k in W 1,p for p > n/2 and argue as in Lemma 4.3 (see also the proof of Theorem 1.3), obtaining Corollary 4.7. Let Ω ∈ W 1,n/2 . There exists > 0 such that if Ω is an antisymmetric matrix of 1-differential forms on B n such that Ω L n < then there exist P ∈ W 2,n/2 (B n , SO(m)) and ξ ∈ W 2,n/2 (B n , so(m)⊗Λ n−2 R n ) satisfying the system and such that dξ W 1,n/2 + dP W 1,n/2 ≤ C(n, m) Ω W 1,n/2 , dξ L n + dP L n ≤ C(n, m) Ω L n < C .

Uhlenbeck's Decomposition, the Case 1 < p < n/2
In this section we prove Theorem 1.3.
There exists > 0 such that if Ω satisfies the smallness condition Remark. Observe that for p = n/2, by the Sobolev Embedding Theorem, we have automatically Ω ∈ L n . The Morrey space L 2p,n−2p equals L n in this case and the smallness condition for the norm of Ω agrees with the one in Theorem 1.2.
As in Sect. 4, we shall break the proof of Theorem 1.3 into several lemmata. The proof of the existence of P and ξ (Lemma 4.4) cannot be adapted to the present situation. To avoid this difficulty, we first prove the theorem under more stringent regularity assumptions (see Lemma 5.1 below). To prove the existence of the elements of decomposition we follow the strategy of Tao and Tian [20]. At a certain moment of the proof (Lemma 5.3) we use the fact that due to the Morrey-Sobolev embedding (Proposition 3.3), for α > 0, This is not true for L p,n−p 1 . Also, as pointed out in [24], continuous functions are not dense in L p,s . Lemma 5.1. Let 1 < p < n/2. There exists > 0 such that for every α > 0 and for every Ω ∈ L 2p,n−2p+α such that dΩ ∈ L p,n−2p , Proof of the Lemma 5.1. As in the Sobolev case, for α, > 0 we introduce sets and there exist P and ξ satisfying the system (5.1) and estimates (5.3)} In Lemmata 5.2 and 5.6 below we show that for sufficiently small the set U α is closed and open in V α , and since the latter is path connected (it is starshaped), it follows that U α = V α . This (up to the proofs of these lemmata) completes the proof of the lemma.
Proof. Suppose (Ω k ) is a sequence in U α convergent in L p,n−2p 1 to some Ω. Observe that L p,n−2p 1 embeds continuously in W 1,p . Therefore the sequence (Ω k ) is convergent in W 1,p .
With every Ω k we have associated P k , ξ k that satisfy (5.1): We also have the estimates (5.3), in particular The inclusion of L p,n−p+α 1 in W 1,p , the boundary condition on ξ k and boundedness of P k (|P k | = 1) allow us to interpret the above as boundedness of P k and ξ k in W 2,p . We can thus assume (after passing to subsequences) that P k and ξ k are weakly convergent in W 2,p to some P and ξ.
Then, we argue as in Lemma 4.3: Both the boundary condition ξ| ∂B = 0 and the condition d * ξ = 0 are preserved when passing to the weak limit. Moreover, since n > 2p > p, after passing to a subsequence, and for any small δ > 0, Ω k , dP k and dξ k converge strongly (to Ω,P and ξ, respectively) in L s , for any s < np n−p . We also know that P k are uniformly bounded in L ∞ and strongly convergent, by the Sobolev embedding theorem, in L q for any q. This is enough to show the strong convergence of Ω k P k to ΩP in L s ; altogether, we may pass to the strong limit in L s in the system (4.3), showing that the equation is satisfied in the sense of distributions. The estimates (5.3) for P and ξ are then obvious.
Proof. Since we are interested in finding any Q satisfying (5.5), we shall look for one of the form e u , where u : B n → so(m), u ∈ L p,n−2p+α 2 Also, we need the boundary condition on Q to hold, so we ask that u has zero boundary values. The equation (5.5), together with the boundary condition, can be rewritten as We follow the proof of Tao and Tian [20], setting up the iteration scheme Some calculations need more subtle justification though, since we work in noncommutative setting. We will show that there exists δ > 0 such that in each step of the recurrence We start with an easy observation. Since Although the value of β will be fixed later, we may assume already that β < 1 2 . We will show first that the following pointwise estimates hold  and Passing from pointwise to L p,n−2p+α estimates, using Lemma 3.6 we obtain * d * F (u k , ζ, λ) L p,n−2p+α The smallness conditions (5.4), (5.9) and (5.10) then imply Regularity estimates for linear elliptic systems (see [6]) yield W.l.o.g. we may assume C 3 > 1. Let us choose β, η and κ such that

Now we set
Then, if Now, let us apply the same scheme to the differences of u k . We assume that u, v ∈ L p,n−2p+α most β and although β is to be specified later, we assume as before that it is less than 1 2 . We have Next, to estimate | * d * F (u, ζ, λ) − F (v, ζ, λ)|, we shall estimate separately | * d * S 1 | and | * d * S 2 |, to avoid multi-line calculations.
Using as before boundedness and Lipschitz continuity of E, DE, exp and D exp on {|u| ≤ β} and keeping in mind that * d * ( * dζ) = 0 we get (5.12) and Altogether, adding up the estimates (5.12) and (5.13), we obtain the following pointwise estimate (5.14) with C a universal constant. Passing from the pointwise to L p,n−2p+α estimates, using repeatedly (5.9), Hölder's inequality, Lemma 3.6 and keeping in mind all smallness conditions, i.e.
If we denote H(u k ) = u k+1 , where u k+1 is a solution to (5.7), then where C E is an absolute constant from elliptic estimates. Now, in order to show that H is a contraction, we choose β and κ and η sufficiently small. The choice of β and η results in the choice of δ. Therefore, by the Banach fixed point theorem, the iteration scheme (5.7) converges and we obtain the desired solution of the system (5.6).
The rest of the proof mimics the proof in the Sobolev case, and the Lemmata 5.4 and 5.6 are direct counterparts of Lemmata 4.5 and 4.6 from Sect. 4. Lemma 5.4. Suppose p < n/2. There exists κ = κ(p, n) with the following property: suppose that for Ω ∈ V α there exist P and ξ in L p,n−2p+α 2 (B n ) satisfying the system (5.1) and additionally the estimate dP L 2p,n−2p (B n ) + dξ L 2p,n−2p (B n ) < κ. (5.15) Then the estimates (5.3) hold.
Proof of Lemma 5.4. We have in the ball B n where φ is a smooth, compactly supported (in particular with null boundary values) (n − 2)-form on B ∩ B n (c.f. the proof of Lemma 4.5). Denote byP the mean value of P over B n :P = B n P . For any φ as above, with the constants C (possibly different in every line) dependent only on n and p. Note that since P is an orthogonal matrix, |P | = |P −1 | = 1, |dP −1 | = |dP |. In the estimate above we use the Coifman-Lions-Meyer-Semmes div-curl inequality [3] and later the inclusion We have then due to the smallness assumption (5.15). Therefore On the other hand, taking L p norms of both sides of the equation and thus Putting (5.19) and (5.21) together we get, for α ≥ 0, Taking κ small enough we conclude the estimates (5.3a), (5.3b) hold, i.e.
By Poincaré's lemma this implies that P −1 dP + P −1 ΩP is a coexact form, i.e. there exists an antisymmetric (n − 2)-form ξ such that * dξ = P −1 dP + P −1 ΩP, (5.23) thus P and ξ give Uhlenbeck's decomposition of Ω. Note that Q and P o ∈ L p,n−2p+α 2 ∩ L ∞ imply that P ∈ L p,n−2p+α 2 . By the Hodge decomposition theorem we can choose ξ to be coclosed (d * ξ = 0 on B n ) and to have zero boundary values (ξ| ∂B n = 0). Finally, the right hand side of (5.23) is in L p,n−2p+α 1 , which gives ξ ∈ L p,n−2p+α What remains to prove is that P , ξ and Ω satisfy the estimates (5. ) sufficiently small we get We also know that Taking sufficiently small we may ensure that , nor L 2p,n−2p+α does into L 2p,n−2p . However (cf. [16], proof of Lemma 3.1), one can easily prove that if (φ r ) is a standard mollifier and f ∈ L q,n−q (B), q ≥ 1, then on any ball B = B(x, ρ) such that 2B = B(x, 2ρ) ⊂ B n we have, for r < ρ, that f * φ r L q,n−q (B) ≤ f L q,n−q (2B) . Reasoning like in the proof of Meyers-Serrin's theorem and using a suitable decomposition of unity we can show then that there exists a sequence f k ∈ C ∞ (B) convergent to f in L q (and in any other appropriate Lebesgue and Sobolev norm) such that f k L q,n−q (B) ≤ C(n) f L q,n−q (B) . We thus proceed as follows: we approximate Ω in W 1,p by a sequence of smooth Ω k such that for all k Assuming that in the condition Ω L 2p,n−2p < is taken small enough we can ensure, through (5.24), that all Ω k satisfy the analogous smallness condition in Lemma 5.1. This provides us with sequences of P k and ξ k that give the Uhlenbeck decomposition for Ω k , together with the uniform estimate dξ k L p,n−p + dP k L p,n−p ≤ C(n, m) Ω L 2p,n−2p < C .
Then we proceed as in the proof of closedness of U α in Lemma 5.2, obtaining convergent subsequences of P k and ξ k . As Ω k are smooth, they satisfy the assumptions of Lemma 5.1, which gives us the estimates (5.3) dξ k L p,n−p + dP k L p,n−p ≤ C(n, m) Ω k L 2p,n−2p , Δξ k L p,n−2p + ΔP k L p,n−2p ≤ C(n, m) ( Ω k L 2p,n−2p + ∇Ω k L p,n−2p ) .
The sequences P k and ξ k converge in L p,n−2p 2 to appropriate elements of a decomposition of Ω (this follows from the equation they satisfy). Thus, the above estimates, together with (5.24), yield the desired estimates (5.3) for Ω. This completes the proof of Theorem 1.3.

Uhlenbeck's Decomposition and Conformal Matrices
A natural extension of the orthogonal gauge group SO(m) is the conformal group CO + (m). The interest in this group has deep roots in complex analysis, in particular in the studies related to Liouville's Theorem (see [9] for a detailed exposition). This is a non-compact group, defined as CO + (m) = {λP : λ ∈ R + , P ∈ SO(m)}. Clearly, S ∈ CO + (m) iff SS T = λ 2 Id, where by Id we denote the m × m identity matrix.
The tangent space at Id to CO + (m), which we denote by T CO + (m), is given as or, equivalently, T CO + = {A + μId : A ∈ so(m), μ ∈ R}, see e.g. [4]. Our objective is to prove an analogue of Theorem 1.2 for the conformal gauge group, i.e. Theorem 1.4.
Theorem. Let n 2 < p < n. There exists > 0 such that for any Ω ∈ W 1,p (B n , T CO + (m) ⊗ Λ 1 R n ) such that Ω L n < there exist S : B n → CO + (m) satisfying ln |S| ∈ W 2,p (B n ), S/|S| ∈ W 2,p (B n , SO(m)) and ζ ∈ W 2,p (B n , T CO + (m) ⊗ Λ n−2 R n ) such that ⎧ ⎨ The integrability conditions on ln |S| should be understood as (rather weak) integrability conditions both on S and S −1 . We should also note that if S satisfies the above theorem, so does tS for any non-zero constant t.
The above theorem is rather simple, but it provides a new interpretation to gradient-like terms df ⊗ Id in nonlinear systems-we can incorporate them in antisymmetric expressions and perform Uhlenbeck's decomposition on the resulting T CO + matrix of differential forms instead of dealing with both kinds of terms separately.