Uhlenbeck's decomposition in Sobolev and Morrey-Sobolev spaces

We present a self-contained proof of Uhlenbeck's decomposition theorem for $\Omega\in L^p(\mathbb{B}^n,so(m)\otimes\Lambda^1\mathbb{R}^n)$ for $p\in (1,n)$ with Sobolev type estimates in the case $p \in[n/2,n)$ and Morrey-Sobolev type estimates in the case $p\in (1,n/2)$. We also prove an analogous theorem in the case when $\Omega\in L^p( \mathbb{B}^n, TCO_{+}(m) \otimes \Lambda^1\mathbb{R}^n)$, which corresponds to Uhlenbeck's theorem with conformal gauge group.


Introduction
In 2006 Tristan Rivière 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 [12]. The key tool he used was a theorem originally 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 [17]. One should mention that Coulomb gauges appeared earlier in the theory of geometrically motivated systems of PDE, namely in important papers of F. Hélein on regularity of harmonic mappings between manifolds ( [4,5]).
The power of Rivière's idea was in the fact that he used Uhlenbeck's theorem to antysymmetric differential forms, which a priori were not interpreted as connection forms, even if the problem had clear geometric motivation. Moreover, he reformulated the theorem in a language more suited for PDE applications. Simplifying, any antisymmetric matrix Ω of 1-forms on a ball Ω : B n → so(m) ⊗ Λ 1 R n with sufficiently small norm can be transformed by an orthogonal change of coordinates (gauge transformation) The original result of Uhlenbeck was stated for Ω ∈ W 1,p (B n ), with n/2 ≤ p < n. Since then, Uhlenbeck's decomposition appeared in numerous papers on nonlinear PDE's and variational problems, each time adapted to a specific system, function spaces and dimensions: • Rivière [12]: Ω ∈ L 2 , n = 2; • Rivière & Struwe [13]: Ω ∈ L 2,n−2 , n > 2; • Lamm & Rivière [7]: Ω ∈ W 1,2 , n = 4; • Meyer & Rivière [9] and Tao & Tian [16]: Ω ∈ W 1,2 ∩ L 4,n−4 , dΩ ∈ L 2,n−4 for n ≥ 4; • Müller & Schikorra [11]: Ω ∈ W 1,2 , n = 2; All the proofs of the above results (with the exception of [12]) 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 [17]. 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 a boundary condition on u 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 was used by T. Tao and G. Tian in [16]. Again, one looks for P = e u and assumes u has zero boudary data. The equation * d * (P −1 dP + P −1 ( * dζ + λ)P ) = 0 is transformed into the form Then, an iteration scheme is set to provide a solution: We apply this strategy when 1 < p < n/2.
In 2009 A. Schikorra gave an alternative, variational proof of the existence of Uhlenbeck's decomposition ( [14]). His approach was inspired by a similar variational construction of a moving frame by F. Hélein [5]. 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 [12] and Rivière & Struwe [13].
We begin the paper by recalling the definitions and basic properties of Morrey and Morrey-Sobolev spaces we use (Section 2). Next, we present a self-contained, complete proof of Uhlenbeck's decomposition in Rivière's form with both Sobolev-and Morrey-Sobolev type estimates, adapted to the abstract setting of PDE's. Namely, if we assume Ω ∈ W 1,p , then • for n/2 ≤ p < n: the smallness condition for Ω is in the L n norm and estimates are given in the Sobolev norms, P and ξ ∈ W 2,p (Section 3); • for 1 < p < n/2: the smallness condition is expressed in the suitable Morrey space and estimates for norms of P and ξ are given in Morrey-Sobolev-type spaces (Section 4).
We express the results and the proofs in the language of differential forms, which allows applications to higher-dimensional problems (c.f. [13]). Like in all the other existing proofs, we follow the original scheme of Uhlenbeck's proof; we adapt arguments used in various adaptations of Uhlenbeck's proof mentioned above, in particular in papers [13] and [16]. Finally, we study 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 Ω (Section 5).

Morrey spaces
Let Ω ⊂ R n be an open and bounded set.
Recall that the Morrey space L p,s (Ω) is a collection of all functions f ∈ L p (Ω) such that ||f || p L p,s = sup When s = 0, the Morrey space L p,0 (Ω) is the same as the usual Lebesgue L p (Ω) space. When s = n, the dimension of the ambient space, the Morrey space L p,n (Ω) is equivalent to L ∞ (Ω), due to the Lebesgue differentiation theorem. Morrey spaces are Banach spaces. For 1 ≤ p ≤ q < ∞ and s, σ ≥ 0 such that s−n p ≤ σ−n q we have L q,σ (Ω) ֒→ L p,s (Ω), in particular L q,n−q (Ω) ֒→ L p,n−p (Ω). (Ω) as Note that with this definition, f ∈ L p,n−kp k (Ω) does not imply Morrey estimates for f itself. However, the inclusion L p,n−kp k (Ω) ⊂ BM O(Ω) holds.
where f xo,r denotes the average of f on the set B r (x 0 ) ∩ Ω.
Note that whenever Ω supports the Poincaré inequality, e.g. if Ω has the interior cone property (see [8]), we have We say that a domain Ω is of type (A) if there exists a constant C > 0 such that for any x 0 ∈ Ω and 0 < r < diam(Ω) This excludes domains with infinitely sharp cusps. For domains of type (A) we have the following generalization of Sobolev's embedding theorem (see [10]).
The following Gagliardo-Nirenberg type estimate is a slight adaptation of [15,Proposition 3.2], with virtually no change in the proof.
Proposition 2.4. For any 1 < p ≤ n/2 there exists a constant C = C(n, p) such that for any α ∈ [0, p], any ball B ⊂ B n and any f ∈ L p,n−2p+2α In particular, for f ∈ L p,n−2p+2α (2.1) 3. Uhlenbeck's decomposition, the case n/2 ≤ p < n We first state and prove the theorem in the case p ∈ (n/2, n), recovering the original result of Uhlenbeck. The case p = n/2 follows by approximation (see Corollary 3.7 at the end of this section).
We show that for ǫ sufficiently small the set U ǫ is closed and open in V ǫ , and since the latter is path connected (it is star-shaped in W 1,p ), it follows that U ǫ = V ǫ , which proves Theorem 3.1.
Proof. Suppose (Ω k ) is a sequence in U ǫ , convergent in W 1,p to some Ω. With every Ω k we associate P k and ξ k that satisfy (3.1): and the estimates (3.2) 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 (3.4) 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 conserved 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 (3.3), showing that the equation is satisfied in the sense of distributions. The estimates (3.2) for P and ξ are obvious.
Remark. For any P and ξ in W 1,p that satisfy (3.3) 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 ǫ . In contrast with the previous lemma this is more delicate; we split the reasoning again into several lemmata.
Note that (3.5) 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 (3.5), we shall look for one of the form e u , where u ∈ B = W 2,p (B n , so(m))∩ W 1,p o (B n , so(m)) (this adds boundary restrictions on Q). We define the operator . 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. 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 ∆ −1 : L p → B is continuous, we get For κ sufficiently small we can have |||K||| B→B small and Id − K : and we can solve (3.6) 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 3.5. Suppose n/2 < p < n. There exists κ = κ(p, n) such that for any Ω ∈ V ǫ and P , ξ in W 2,p satisfying the system (3.1) and additionally the estimate Proof. The lemma follows from rather standard elliptic estimates, but we include them here for the sake of completeness. We have (3.9) 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: 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 ( [1]) 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 Putting (3.10) and (3.12) together we get and for κ < 1 C 1 this implies that the estimate (3.2a) holds. The above calculation is valid also for p = n, which yields the estimate (3.2b).
To show the estimate (3.2c), by taking * d * of both sides of (3.11), we see that with the constant C coming from the Sobolev embedding W 1,p ֒→ L np/(n−p) , thus dependent only on p and n.
Proof. Choose Ω o ∈ U ǫ and let P o and ξ o be the orthogonal transformation and antisymmetric (n − 2)-form form associated with Ω o , so that Theorem Setting P = P o Q we see that (3.15) reduces to * d * (P −1 dP + P −1 ΩP ) = 0.
Note that Q and P o ∈ W 2,p ∩ L ∞ imply that P ∈ W 2,p . 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 (3.16) is in W 1,p , which gives ξ ∈ W 2,p .
What remains to prove is that P , ξ and Ω satisfy the estimates (3.2). Observe that if Ω − Ω o W 1,p is small enough, then by continuity of the mapping λ → u λ so is P − P o W 1,p and ξ − ξ o W 1,p ; choosing η (which measures the distance Ω − Ω o W 1,p ) sufficiently small we may have We also know that Taking ǫ sufficiently small, we may ensure that with κ as in Lemma 3.5. Applying this lemma we show that the estimates (3.2) hold. Altogether, Ω ∈ U ǫ , which proves the openness of U ǫ .
Proof of Theorem 3.1. Since, by Lemmata 3.3 and 3.6, for ǫ sufficiently small the set U ǫ is closed and open in V ǫ , and since the latter is path connected (it is star-shaped in W 1,p ), it follows that U ǫ = V ǫ , which proves Theorem 3.1.
It is worth noting that, for Ω ∈ W 1,n/2 , the proof of existence of the decomposition, i.e. Lemma 3.4, 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 3.3 (see also the proof of Theorem 4.1), obtaining Corollary 3.7. Let Ω ∈ W 1,n/2 . There exists ǫ > 0 such that if Ω is an antisymmetric matrix of 1-differential forms on B n such that Ω L n < ǫ 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 ) 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 < Cǫ.
As previously, we shall break the proof of Theorem 4.1 into several lemmata. The proof of existence of P and ξ (Lemma 3.4) cannot be adapted to the present situation. To avoid this difficulty we first prove the Uhlenbeck result under more stringent regularity assumptions (see Lemma 4.2 below). To prove the existence of the elements of decomposition we follow the strategy of Tao and Tian [16]. At a certain moment of the proof (Lemma 4.4) we use the fact that due to the Morrey-Sobolev embedding (Proposition 2.3), for α > 0, L p,n−p+α This is not true for L p,n−p
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 3.3: Both the boundary condition ξ| ∂B = 0 and the condition d * ξ = 0 are conserved 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 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   Note that (4.4) implies, through Poincaré's lemma, that the term in parentheses is of the form * dζ, for some antisymmetric (n − 2)-formζ.
Proof. Since we are interested in finding any Q satisfying (4.4), we shall look for one of the form e u , where u : B n → so(m), u ∈ L p,n−2p+2α 2 This adds boundary restrictions on Q -since Q(x) ∈ SO(m), we may assume it is close to identity at the boundary of B n , so u has zero boundary values.
The equation (4.4), together with the boundary condition, can be rewritten as ∆u = * d * du − e −u de u − e −u ( * dζ + λ)e u , u = 0 on ∂B n . (4.5) We follow the proof of Tao and Tian [16], 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 recurence if u k L p,n−2p+2α 2 ≤ δ then u k+1 L p,n−2p+2α 2 ≤ δ.
We will show first that the following pointwise estimates hold Indeed, applying the Campbell-Hausdorff-Baker formula we obtain Therefore, taking the structure of the right hand side of (4.6) into account (see (4.7)), we obtain * d * F (u k , ζ, λ) We easily see that Since (4.11) holds, we obtain which proves the pointwise estimates (4.12) hold true, i.e.
Passing from pointwise to L p,n−2p+2α estimates and using (4.10) we obtain * d * F (u k , ζ, λ) L p,n−2p+2α where the last inequality follows from (4.8), with u k in place of f . Regularity estimates for linear elliptic systems yield W.l.o.g. we may assume C 3 > 1. Let us choose η and κ such that Now, let us apply the same scheme to diferences of u k . Fix a constant β ∈ (0, 1/2) (it will be specified at the end). We adjust δ if necessary, i.e. we take (4.14) where C M is the constant in (4.8). Then, we assume that u, v ∈ L p,n−2p+2α 2 , u L p,n−2p+2α 2 < δ, v L p,n−2p+2α 2 < δ and u, v = 0 on ∂B n . This, in particular, implies (by (4.9)), that u L ∞ and v L ∞ are at most β.
We have By (4.13), we have Likewise, S 2 can be rewritten as Next, we want to estimate (pointwise) * d * S 1 . Keeping in mind that an ℓ- tuple commutator [a 1 , [a 2 , . . . [a ℓ−1 , a ℓ ] . . .]] stands for a sum of 2 ℓ−1 products of the form a i 1 a i 2 · · · a i ℓ , we obtain Using the fact that |u|, |v| < β and 2β < 1, we estimate further where C is a universal constant.
To estimate | * d * S 2 |, note that (4.16) Altogether, adding up the estimates (4.15) and (4.16), we obtain the following pointwise estimate with C a universal constant.
If we denote H(u k ) = u k+1 , where u k+1 is a solution to (4.6), then where C E is an absolute constant from eliptic estimates (see [3]). 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 (4.6) converges and we obtain the desired solution of the system (4.5).
The rest of the proof mimics the proof in the Sobolev case, and the Lemmata 4.5 and 4.7 are direct counterparts of Lemmata 3.5 and 3.6 from Section 3.
Lemma 4.5. Suppose p < n/2. There exists κ = κ(p, n) with the following property: suppose that for Ω ∈ V α ǫ there exist P and ξ in L p,n−2p+2α 2 (B n ) satisfying the system (4.1) and additionally the estimate Then the estimates (4.3) hold.
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 3.5 ). 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 ( [1]) and later the inclusion We have then due to the smallness assumption (4.18). Therefore Combining (4.20) and (4.21) we conclude with On the other hand, taking L p norms of both sides of the equation and thus (4.24) dP L p,n−p+α (B n ) ≤ dξ L p,n−p+α (B n ) + Ω L p,n−p+α (B n ) .
Observe that the the above inequality is a consequence of the equation (4.23) and the Hölder inequality only, so the estimate holds in any Morrey space L p,γ with γ > n − 2p in which both sides of the inequality are finite. Take now Ω ∈ V α ǫ close to Ω o in L 2p,n−2p+2α ∩ L p,n−2p+2α 1 : we ask that ). Applying Lemma 4.4 with ζ = ξ o we find Q ∈ L p,n−2p+2α 2 (B n , SO(m)) such that Setting P = P o Q we see that (4.25) reduces to * d * (P −1 dP + P −1 ΩP ) = 0.
What remains to prove is that P , ξ and Ω satisfy the estimates (4.3). Observe that if Ω − Ω o L p,n−2p+2α 1 is small enough, then by continuity of the mapping λ → u λ so is P − P o L p,n−2p+2α 1 and ξ − ξ o L p,n−2p+2α 1 ; choosing η (which measures the distance Ω − Ω o L p,n−2p+2α 1 ) sufficiently small we get We also know that Taking ǫ sufficiently small we may ensure that with κ as in Lemma 4.5. Applying this lemma we show that the estimates (4.3) hold. Altogether, Ω ∈ U α ǫ , which proves the openness of U α ǫ .
Proof of Theorem 4.1. The proof mimics, in a way, the passage from Theorem 3.1 to Corollary 3.7, i.e. from the Uhlenbeck decomposition in W 1,p for p > n/2 to the decomposition for p = n/2. There we could simply argue by approximation. In the Morrey space setting, however, neither L p,n−p+α 1 embeds densely in L p,n−p 1 , nor L 2p,n−2p+2α does into L 2p,n−2p . However (cf. [13], 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 ǫ in the condition Ω L 2p,n−2p < ǫ is taken small enough we can ensure, through (4.27), that all Ω k satisfy the analogous smallness condition in Lemma 4.2. 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 < Cǫ.
Then we proceed as in the proof of closedness of U α ǫ in Lemma 4.3, obtaining convergent subsequences of P k and ξ k . As Ω k are smooth, they satisfy the assumptions of Lemma 4.2, which gives us the estimates (4.3) dξ k L p,n−p + dP k L p,n−p ≤ C(n, m) Ω k L 2p,n−2p , ∆ξ k L p,n−2p + ∆P 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 (4.27), yield the desired estimates (4.3) for Ω.

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 study related to Liouville Theorem (see [6] 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 I, where by I we denote the m × m identity matrix.
The tangent space at I to CO + (m), which we denote by T CO + (m), is given as T CO + = {K ∈ M m×m : K + K T = 2T r(K) m ⊗ I}, or, equivalently, T CO + = {A + µI : A ∈ so(m), µ ∈ R}, see e.g. [2]. Our objective is the following analogue of Theorem 3.1: Theorem 5.1. Let n 2 < p < n. There exists ǫ > 0 such that for any Ω ∈ W 1,p (B n , T CO + (m) ⊗ Λ 1 R n ) such that Ω L n < ǫ 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 and such that dζ W 1,p + d(S/|S|) W 1,p + d ln |S| W 1,p ≤ C(n, m) Ω W 1,p (5.2a) dζ L p + d(S/|S|) L p + d ln |S| L p ≤ C(n, m) Ω L p , (5.2b) dζ L n + d(S/|S|) L n + d ln |S| L n ≤ C(n, m) Ω L n . (5.2c) 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.
Proof. We shall construct S : B n → CO + (m) satisfying the above conditions. Let us first fix some notation: We shall write S = λP , where λ = |S| ∈ R + and P = S/|S| ∈ SO(m), we also decompose Ω into its antisymmetric and diagonal part: with A ∈ W 1,p (B n , so(m) ⊗ Λ 1 R n ).
The above theorem is rather simple, but it provides a new interpretation to gradient-like terms df ⊗ I 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. We also can ask a natural question.
Question. What is the largest Lie subgroup G of GL(n) that can be used in an analogue of Uhlenbeck's theorem: for any matrix of 1-differential forms Ω ∈ T I G there exists a gauge transformation P ∈ G such that Ω P = P −1 dP + P −1 ΩP is coclosed and certain integrability estimates on P , Ω P and their derivatives in terms of Ω hold?