p-harmonic coordinates for H\"older metrics and applications

We show that on any Riemannian manifold with H\"older continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^\alpha$ metric tensors is $C^{1+\alpha}$ regular, and that a manifold with $W^{1,n} \cap C^\alpha$ metric tensor and with vanishing Weyl tensor is locally conformally flat if $n \geq 4$. The results extend the works [LS14, LS15] from the case of $C^{1+\alpha}$ metrics to the H\"older continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.


Introduction
Various regularity results in Riemannian geometry can be achieved via the use of harmonic coordinates. If (M, g) is a Riemannian manifold, a local coordinate system x = (x 1 , . . . , x n ) is called harmonic if where ∆ g is the Laplace-Beltrami operator. The notion of harmonic coordinates goes back to Einstein [Ei16], and these coordinates have the property that the Ricci and Riemann curvature tensors become elliptic operators when written in harmonic coordinates [DK81]. The Ricci and Riemann tensors are invariant under diffeomorphisms, and fixing a harmonic coordinate system may be thought of as a gauge condition which makes these operators elliptic. Harmonic coordinates are useful when studying regularity of isometries or flatness of low regularity metrics [Ta06]. Also, the DeTurck trick for local wellposedness of the Ricci flow [CK04] is related to harmonic coordinates, see [GL91]. Harmonic coordinates are well suited to studying regularity results in Riemannian geometry, due to the fact that harmonic functions are preserved by isometries. In this paper we are interested in regularity results in conformal geometry. For this purpose it is useful to have a gauge condition which is preserved by conformal transformations. Such a condition was introduced in the recent work [LS14], which established the existence of p-harmonic coordinates on any smooth manifold with C s , s > 1, Riemannian metric. If p = n, the corresponding n-harmonic gauge condition is conformally invariant, and [LS14] applied n-harmonic coordinates to give a new proof of certain regularity results for conformal mappings. The subsequent article [LS15] showed that n-harmonic coordinates provide a gauge condition for ellipticity of conformal curvature tensors, and characterized local conformal flatness of manifolds of dimension ≥ 4 having C s , s > 1, metric tensor.
The aim of the present article is to extend the results of [LS14] to C s , s > 0, metric tensors and the local conformal flatness characterization of [LS15] to W 1,n ∩ C s , s > 0, metric tensors. Let us state the desired results in more detail.
In this paper M will be a smooth (= C ∞ ) manifold of dimension n ≥ 2, and g will be a Riemannian metric on M with finite regularity. If 1 < p < ∞, the p-Laplace operator on (M, g) is the operator u → −δ(|du| p−2 du) where d is the exterior derivative, δ is the codifferential and |du| is the g-norm of the cotangent vector du. This operator is given in local coordinates by −δ(|du| p−2 du) = |g| −1/2 ∂ j (|g| 1/2 g jk (g ab ∂ a u∂ b u) where (g jk ) is the metric in local coordinates, (g jk ) is the inverse matrix of (g jk ), and |g| = det(g jk ). Throughout the paper we use the Einstein summation convention, where a repeated index in upper and lower position is summed from 1 to n. If p = 2, (1.1) is the usual Laplace-Beltrami operator, whereas for p = 2 the p-Laplace operator is a quasilinear degenerate elliptic operator. We refer to [HKM93,Lin06] for more details on p-Laplace and more general A-harmonic operators.
We will state our results in terms of C s spaces for s ≥ 0. If s is a nonnegative integer, we understand that C s (R n ) is the space of functions in R n whose derivatives up to order s are bounded and continuous. If s = k +α where k ≥ 0 is an integer and 0 < α < 1, then C s = C k+α is the usual Hölder space with norm We will also consider the Zygmund spaces C s * (R n ) for s ∈ R, which are defined via the norm where (ψ j (ξ)) ∞ j=0 is a Littlewood-Paley partition of unity. One has C s = C s * , s > 0 not an integer, and C k ⊂ C k * if k is a nonnegative integer. Most often we will use local versions of the above spaces, and in local coordinate statements will write C s instead of C s loc etc. See [Ta96] for more details on the above spaces. The first main theorem states that whenever (M, g) is a Riemannian manifold with C s metric tensor where s > 0, then near any point of M there exist local coordinates all of whose coordinate functions are p-harmonic. This extends a result from [LS14] from the case s > 1 to s > 0.
Theorem 1.1 (p-harmonic coordinates). Let (M, g) be a Riemannian manifold whose metric is of class C s , s > 0, in a local coordinate chart about some point x 0 ∈ M . Let also 1 < p < ∞. There exists a local coordinate chart near x 0 whose coordinate functions are p-harmonic and have C s+1 * regularity, and given any ε > 0, one can arrange so that metric satisfies |g jk (x 0 ) − δ jk | < ε for j, k = 1, . . . , n. Moreover, all p-harmonic coordinates near x 0 have C s+1 The next result states that if φ is a 1-quasiregular map between two manifolds with C s metric tensors where s > 0 (in particular φ could be a C 1 conformal map), then φ is in fact a C s+1 local conformal diffeomorphism. This gives a new proof of results in [Re78,Iw82,Sh82] (the earlier work [Le76] considered C ∞ Riemannian metrics). We also mention the works [Ta06, CL15, CLO16, MT16] that give analogous results for Riemannian, subRiemannian and Finsler isometries and conformal maps.
Theorem 1.2 (Regularity of conformal maps). Let (M, g) and (N, h) be Riemannian manifolds, n ≥ 3, where g, h ∈ C s for some s > 0, s = 1. Let φ : M → N be a non-constant mapping. Then the following are equivalent: φ is a Riemannian 1-quasiregular mapping, (1.2) φ is locally bi-Lipschitz and φ * h = c g a.e., φ is a local C s+1 diffeomorphism and φ * h = c g. (1.5) The third result states that the vanishing of the Weyl tensor on a manifold with W 1,n ∩C α metric tensor is equivalent with local conformal flatness. The Weyl tensor W (g) of (M, g) is the 4-tensor is the Riemann curvature tensor, and P ab is the Schouten tensor given by where R bc = g ad R abcd is the Ricci tensor and R = g rs R rs is the scalar curvature. If n = 3 the Weyl tensor is always zero, and if n ≥ 4 the Weyl tensor is conformally invariant in the sense that if c is a smooth positive function. It is a classical result that for C 3 metrics, the vanishing of the Weyl tensor is equivalent with the manifold being locally conformally flat in dimensions n ≥ 4 (see for instance [Au88,Chapter 4]).
The question of having a low regularity counterpart for this result has been raised in [IM01, Section 2.7] and [Ma14, Section 1]. Such a result was proved for C s metrics with s > 1 in [LS15]. Here we improve it to W 1,n ∩C s metrics with s > 0, where W k,p is the usual (local) Sobolev space of functions whose derivatives up to order k are (locally) in L p .
Theorem 1.3 (Local conformal flatness). Let M be a smooth n-dimensional manifold with n ≥ 4, and let g ∈ W 1,n ∩ C α in some local coordinates near x 0 ∈ M where 0 < α < 1. If W (g) = 0 near x 0 , then g ab = cδ ab for some positive W 1,n ∩ C α function c in some n-harmonic coordinates near x 0 .
In three dimensions, local conformal flatness of sufficiently regular metrics is characterized by the vanishing of the Cotton tensor. An analogue of Theorem 1.3 for n = 3 was proved in [LS15] for C s metrics with s > 2. It is likely that this regularity assumption could also be improved slightly.
We also remark that there are alternative gauge conditions for ellipticity of conformal curvature tensors, based on solving the Yamabe problem locally to achieve constant scalar curvature and using harmonic coordinates. This procedure may require more regularity than the method of n-harmonic coordinates presented in this paper, see [LS15] for further discussion.
This paper is organized as follows. Section 1 is the introduction. In Section 2 we prove Theorem 1.1, and more generally we show the existence of A-harmonic coordinates for any A satisfying certain structural assumptions including Hölder continuity in x. The main point in extending the result of [LS14] from s > 1 to s > 0 is that instead of working with strong solutions and nondivergence form equations, we keep the equations in divergence form and apply Campanato and Schauder estimates in order to show regularity of weak solutions. In Section 2 we also give an additional W 2,q ∩ C 1+α regularity result for the p-harmonic coordinates if g is W 1,q ∩ C α where 0 < α < 1 and q ≥ 2.
Section 3 considers the regularity of conformal mappings and establishes Theorem 1.2. Given Theorem 1.1, the argument is analogous to [LS14] and is based on representing the conformal map locally in n-harmonic coordinates in the target manifold. We also supply some details related to integer values of s that were missing in [LS14].
In Section 4 we discuss the Weyl tensor for low regularity metrics and prove Theorem 1.3. This uses the fact proved in [LS15] that the Weyl tensor becomes elliptic in n-harmonic coordinates after a conformal normalization. The vanishing of the Weyl tensor can be interpreted as a quasilinear overdetermined elliptic system involving a nonlinearity with quadratic growth, and to deal with this we will require regularity results for overdetermined elliptic systems.
There are many regularity results for square elliptic systems available in the literature, see for instance [Gi83,GM12]. In the case of sufficiently smooth coefficients, a linear overdetermined elliptic system P u = f can be reduced to the square system P * P u = P * f , and regularity results will follow from the corresponding results for square systems. Some results for overdetermined systems with nonsmooth coefficients (mostly in nondivergence form) are given in [Ta96,Ta00,LS15]. However, we need to deal with certain systems in divergence form which do not seem to be covered by these results. For this reason, in Appendix A we establish the basic Calderón-Zygmund and Schauder regularity results for linear overdetermined elliptic systems in divergence form. Complete proofs for these results are included, based on freezing coefficients and Fourier arguments.

p-harmonic coordinates
In this section we will prove Theorem 1.1. Since this theorem is a local statement, it is sufficient to consider a corresponding result in R n . Locally, p-harmonic functions in (M, g) are solutions of the A-harmonic equation 2) The next result shows the existence of A-harmonic coordinates for any A which satisfies the following structural assumptions for some M, δ > 0: Theorem 2.1 (A-harmonic coordinates). Let Ω be an open set in R n and let 1 < p < ∞ and 0 < α < 1. Suppose A : Ω × R n → R n is a vector field which satisfies (2.3)-(2.6). Given any point x 0 ∈ Ω, there is an C 1+α diffeomorphism U from some neighborhood of x 0 onto an open set in R n such that all coordinate functions of U are A-harmonic. Moreover, any C 1 diffeomorphism whose coordinate functions are A-harmonic has this regularity. Given any invertible matrix S and any ε > 0, there is U defined near We remark that (2.3)-(2.6) imply the conditions (for x ∈ Ω and ξ, h ∈ R n ) which appear in the C 1+β regularity results for the A-harmonic equation (2.1) [Ma86]. We follow the argument of [LS14] except for one part of the proof. In [LS14], we used the fact if A is C s in the x variable where s > 1, then an Aharmonic function whose gradient is nonzero must be C s+1 . The assumption s > 1 was used to first show that the A-harmonic function is W 2,2 by a difference quotient argument, and then the A-harmonic equation was brought into nondivergence form where Schauder estimates could be applied. In this paper, we will work under the assumption s > 0 and apply Campanato type estimates and a frozen coefficient argument to the divergence form equation instead.
We recall the following classical result by Campanato. To that aim we denote the mean value of u in a ball B(x 0 , r) by u x 0 ,r . The proof can be found in [Gi03].
We will also need the following standard algebraic lemma (see [Gi03,Lemma 7.3]).
Lemma 2.3. Let φ be a nonnegative and nondecreasing function on [0, r 0 ] such that where A, B, α, β are nonnegative constants such that β < α. Then for every γ ∈ (β, α) there exists a constant C such that it holds The next result concerns C 1+β regularity of A-harmonic functions, and requires the assumption that A is C α with respect to x for some α > 0. We recall that even in Euclidean space, solutions of the p-Laplace equation always have C 1+β regularity for some β > 0, but β could be very small and, in particular, solutions are in general not C 2 regular.
We write B r = B(0, r) for the open ball of radius r > 0 centered at the origin in R n , and in B(x 0 , r) and A satisfies (2.3)-(2.6). Then there are constants C > 0 and β ∈ (0, 1) only depending on n, p, α, δ and M such that u is C 1+β regular in B(x 0 , r) and Proof. We may assume that x 0 = 0 and u solves div A(x, ∇u) = 0 in B r . Define scaled coordinatesx = x/r and a scaled functionũ(x) = r −1 u(rx).
In particular, this implies that Undoing the scaling proves the result.
The following lemma will be useful for the Schauder estimate that we use later. We will apply it in the case where either the solution of the Aharmonic equation or the solution of the equation with frozen coefficient has nonvanishing gradient, and L 2 estimates will be natural in this context.
and u 0 is a weak solution of the equation with coefficients frozen at If either u 1 or u 0 satisfies the condition for some c 0 > 0, then we have the two estimates (2.9) The constants C and β depend on c 0 , n, p, α, δ and M .
Proof. Without loss of generality we may assume that x 0 = 0. We prove the result by assuming that u 0 satisfies (2.7) since in the other case the argument is similar.
Let us first prove (2.8). We use the assumptions on A and the fact that we may use u 1 − u 0 as a test function in the equations for u 1 and u 0 to conclude that Thus by (2.5) (2.10) Assume first that p ≥ 2. Then (2.7) and (2.10) imply Let us assume 1 < p < 2. We estimate the right hand side of (2.10) by Hölder inequality with exponents q = p 2−p and q = p 2p−2 and obtain Hence Therefore the the right hand side of (2.10) can be estimated by (2.11) and (2.12) To estimate the left hand side of (2.10) we use Proposition 2.4 and (2.12) show that on B r/2 Therefore (2.10) and (2.13) yield Br |∇u 0 | p and we again have (2.8).
It remains to prove (2.9) assuming (2.7) for u 0 . We first invoke Proposition 2.4 which shows that Using (2.12) and the fact that Combining this with (2.8) and increasing the constant slightly gives the two estimates n 2 +β and f = r β (∇u 1 − ∇u 0 )) gives This proves (2.9).
The next result shows that if A is C α with respect to x, then one can upgrade the C 1+β regularity of a weak solution u (recall that β could be very small) to C 1+α regularity in any open set where ∇u is nonvanishing.
Proof. Assume that ∇u is nonvanishing in U ⊂ B 1 . Our aim is to show that u is C 1+α near any x 0 ∈ U . Fix x 0 ∈ U and choose R > 0 so that B(x 0 , 8R) ⊂ U . We will show that for some small r 0 > 0 one has (2.14) whenever B(z, r) ⊂ B(x 0 , R) and r ≤ r 0 . This implies u ∈ C 1+α (B(x 0 , R)) by Proposition 2.2.
By Proposition 2.4, u is C 1+β regular in B 1 and thus there is a constant Fix z and r with B(z, r) ⊂ B(x 0 , R), and let v = v r ∈ W 1,p (B(z, 8r)) be the weak solution of the frozen coefficient equation By Lemma 2.5 we have (with C independent of z and r) (2.17) The bounds (2.15) and (2.17) imply that there is for every ρ ≤ r ≤ r 0 for some small r 0 . We use (2.16), (2.19) and the for every ρ ≤ r ≤ r 0 . The estimate (2.14) then follows from Lemma 2.3.
We continue to study the regularity properties of A-harmonic functions and prove a Calderón-Zygmund type result which will be needed later in Section 4. We denote the partial derivatives of x → A(x, ξ) by ∂ x i A(x, ξ) and of ξ → A(x, ξ) by ∂ ξ i A(x, ξ). Let us assume that in addition to the structural assumptions (2.3)-(2.6) the vector field A satisfies Proof. Assume that ∇u is nonvanishing in B(x 0 , 2r) ⊂ B 1 . Without loss of generality we assume that x 0 = 0. By the previous theorem we have that u ∈ C 1,α (B 3r 2 ). We show first that u ∈ W 2,2 (B r ). Let us fix a direction e i and choose a cutoff function ζ ∈ C ∞ 0 (B 4r 3 ) such that 0 ≤ ζ ≤ 1 and ζ ≡ 1 in B r . For a small h > 0 denote the difference quotient by ∆ h u(x) = 1 h (u(x + he i ) − u(x)). We use the test function ϕ = ∆ h u(x)ζ 2 and integrate against the equations −div A(x, ∇u(x)) = 0 and − div A(x + he i , ∇u(x + he i )) = 0.
By rearranging terms, this gives that (2.22) We use the assumption (2.6), the C 1,α regularity of u and the assumption |∇u| ≥ c > 0 to estimate the left hand side of (2.22) for some constant c > 0. Next we use (2.20) and Young's inequality to estimate the first term on the right hand side of (2.22) (2.24) (2.25) Moreover by the assumption (2.4), the C 1,α regularity of u and |∇u| ≥ c > 0 we have that (2.26) We divide (2.22) by h and deduce from (2.23), (2.24), (2.25) and (2.26) that By letting h → 0 we obtain that u ∈ W 2,2 (B r ).
We may thus differentiate the equation −div A(x, ∇u(x)) = 0 with respect to x i and conclude that v = ∂ x i u is a solution of −div (∂ ξ i A(x, ∇u(x))∇v) = div (∂ x i A(x, ∇u(x))) . (2.27) By the assumption (2.6) the linear operator on the left hand side of (2.27) is elliptic. Moreover by (2.21) x → ∂ ξ i A(x, ∇u(x)) is continuous. On the other hand the vector field x → ∂ x i A(x, ∇u(x)) on the right hand side of (2.27) is L q integrable and therefore by the classical Calderón-Zygmund estimate (also contained in Proposition A.1(a)) we have that v ∈ W 1,q . This implies u ∈ W 2,q (B r ).
Proof of Theorem 2.1. We may assume that x 0 = 0, B 1 ⊂ Ω, and S is the identity matrix (otherwise consider (Sx) j instead of x j below). The Aharmonic coordinates will be obtained as the map U = (u 1 , . . . , u n ), where each u j = u j r solves for r small the Dirichlet problem div A(x, ∇u j ) = 0 in B r , u j | ∂Br = x j .
Let us fix j ∈ {1, . . . , n}. Then v j (x) = x j is a solution of div A(0, ∇v j ) = 0 in B r and |∇v j | = 1 in B r .
We use Lemma 2.5 to conclude Thus there is r > 0 such that |∇u j − ∇v j | ≤ 1 100 in B r/4 and, in particular, ∇u j is nonvanishing in B r/4 . Proposition 2.6 shows that u j ∈ C 1+α (B r/8 ). Now each u j is C 1+α near 0 and the Jacobian matrix of U = (u 1 , . . . , u n ) is invertible at 0. The inverse function theorem shows that U is a C 1 diffeomorphism near 0. Moreover U is C 1+α and so is U −1 since (2.28) The fact that any C 1 diffeomorphism whose coordinate functions are Aharmonic is a C 1+α diffeomorphism follows from Proposition 2.6.
Proof of Theorem 1.1. As discussed above, in any local coordinate system where the metric has C α regularity, the p-harmonic equation is of the form div A(x, ∇u) = 0 where A is given by (2.2) and satisfies (2.3)-(2.6). The existence and regularity of p-harmonic coordinates is then a consequence of Theorem 2.1 for s < 1. The case s > 1 is contained in [LS14]. For s = 1, we have g ∈ W 1,q for all q < ∞ and thus by Proposition 2.7 the p-harmonic coordinates are W 2,q regular. This regularity is enough for us to write the p-harmonic equation for each coordinate function u = u j in nondivergence form as a jk ∂ jk u = f where the coefficients a jk and the function f are defined by Now the linear equation a jk ∂ jk u = f is elliptic since ∇u is nonvanishing. Since u ∈ W 2,q for all q < ∞, we have ∇u ∈ C γ for any γ < 1 by the Sobolev embedding. Consequently a jk ∈ C γ for γ < 1 and f ∈ C 0 . Thus u ∈ C 2 * , see e.g. [Ta96, Theorem 14.4.2]. That the inverse of the p-harmonic coordinates are also of this regularity follows from the formula (2.28). In deriving this conclusion, note that Zygmund spaces are closed under pointwise multiplication [Tr83].
Suppose that the metric in original coordinates at x 0 is represented by the matrix G 0 , and let V be the corresponding A-harmonic coordinates. The metric in p-harmonic coordinates at x 0 is given by Choosing V so that DV (0) is very close to G 1/2 0 , we can arrange that the metric in p-harmonic coordinates is arbitrarily close to identity at x 0 .
Corollary 2.8. Let (M, g) be a Riemannian manifold with g ∈ W 1,q , q > n, in a local coordinate chart about some point x 0 ∈ M . There exists a p-harmonic coordinate system on a neighborhood of x 0 that is a local W 2,q diffeomorphism. Moreover, the Riemannian metric is W 1,q in these p-harmonic coordinates.
Proof. Since W 1,q ⊂ C α for some α > 0, there exists a C 1+α regular pharmonic coordinate system (v i ) on a neighborhood of x 0 by Theorem 1.1. Let us denote V = (v 1 , . . . , v n ). For g ∈ W 1,q , the function A defining p-harmonic functions satisfies the conditions (2.20) and (2.21), and thus by Proposition 2.7, we have that V ∈ W 2,q . By the formula it is easy to see (by using integration by substitution and the standard cofactor formula for an inverse matrix) that the same holds for the inverse of V . Note that W 1,q is an algebra, see Section 4.
That the Riemannian metric is W 1,q in p-harmonic coordinates follows from the coordinate transformation rule for the Riemannian metric Here g andg are the metrics in the original and in the p-harmonic coordinates respectively.

Regularity of conformal and 1-quasiregular mappings
The first application of n-harmonic coordinates is to the regularity of C 1 conformal mappings, or W 1,n 1-quasiregular mappings, between Riemannian manifolds with C s , s > 0, s = 1, metric tensors. This gives a new proof of results in [Re78,Iw82,Sh82].
The proofs are essentially the same as in [LS14] and are only sketched. We however cover a slight lapse in the proof of [LS14, Theorem 3.1] concerning the integer values s = 2, 3, 4, . . . of the regularity parameter s. We also comment on the case s = 1 after the theorem.
Theorem 3.1. Let (U, g) and (V, h) be Riemannian manifolds such that g, h ∈ C s , s > 0, s = 1. If φ : U → V is a C 1 diffeomorphism that is conformal in the sense that φ * h = cg for some continuous positive function c, then φ is a C s+1 diffeomorphism from U onto V .
Proof. Let first g, h ∈ C s with s > 0. Fix x 0 ∈ U , and let v be n-harmonic coordinates near φ(x 0 ). Since h ∈ C s , Theorem 1.1 ensures that v ∈ C s+1 * . Let u = v • φ. Since φ is a C 1 conformal diffeomorphism, a straightforward calculation [LS14, Theorem 3.1] shows that u is an n-harmonic coordinate system near x 0 . Now g is C s , which implies that u is C s+1 * by Theorem 1.1.
Assume now that s > 0 is not an integer. Then u, v and their inverses are C s+1 , and thus φ and also φ −1 are C s+1 (the spaces C s+1 = C s+1 * are closed under composition for s > 0 noninteger). Assume next that s ∈ {2, 3, 4, . . .}. We first show that the conformal factor c in φ * h = cg is a C s function (this part of the proof is missing in [LS14, Theorem 3.1]). The argument above yields that φ ∈ C s+1−ε and c ∈ C s−ε for any ε > 0. Let us recall the transformation rules for the Ricci tensor and the scalar curvature under conformal scaling. Ifg = e 2f g, we have if f and g are sufficiently smooth, see e.g. [Be87]. Since φ is at least C 3−ε and g, h are at least C 2 , we have This, together with the assumption that g, h ∈ C 2 , yield that the identities (3.1) hold in the distributional sense as described later in Section 4. In addition to the identities above, the tensoriality identities also hold in the distributional sense.
Let us now apply the scalar curvature to both sides of the equation The identities above yield φ * R(h) = e −2f (R(g) + 2(n − 1)∆f − (n − 2)(n − 1)|df | 2 g ). Since g, h ∈ C s and φ ∈ C s+1−ε , f ∈ C s−ε , we may solve for ∆f which implies that ∆f ∈ C s−2 . Now applying the Ricci tensor to φ * h = e 2f g, using the identities above and using that ∆f ∈ C s−2 , gives that (3.2) Thus f , and consequently c, is a C s function. The rest of the proof of the theorem now follows by expressing the second derivatives of φ by the coordinate transformation formula [CH70,Ta06]). Since c ∈ C s , the right hand side of this equation is in C s−1 completing the proof.
The reason we could not prove the theorem for s = 1 is that in this case the relation (3.2) for the second derivatives of the conformal factor would involve terms containing second distributional derivatives of C 1 functions. We could not verify that the conformal factor is in C 1 in this case.
The next result concerns the regularity of Riemannian 1-quasiregular mappings and the regularity of conformal bi-Lipschitz mappings. We refer to [LS14,Li13] for details about Riemannian quasiregular mappings. The proof is the same as the proof of Theorem 4.4 in [LS14], except for the additional details for s = 2, 3, 4, . . . given above. We omit the proof.
Theorem 3.2. Let (M, g) and (N, h) be Riemannian manifolds, n ≥ 3, with g, h ∈ C s , s > 0, s = 1. Let φ : M → N be a non-constant mapping. Then the following are equivalent: φ is a Riemannian 1-quasiregular mapping, φ is locally bi-Lipschitz and φ * h = c g a.e., φ is a local C 1 diffeomorphism and φ * h = c g, φ is a local C s+1 diffeomorphism and φ * h = c g.
Finally, we remark that the theorem holds also for n = 2 if we a priori assume φ to be a local homeomorphism. This gives an extension of [LS14,Theorem 4.5] to Hölder exponents s > 0, s = 1.

Weyl tensor
We now consider low regularity Riemannian metrics and establish an elliptic regularity result for the Weyl tensor. We first assume that g jk ∈ W 1,2 ∩ L ∞ in some system of local coordinates near a point. (In this section all function spaces are assumed to be of the local variety near a point, that is, we write W 1,2 instead of W 1,2 loc (U ) etc.) This seems to be a minimal assumption for defining the Weyl tensor: the set W 1,2 ∩ L ∞ is an algebra under pointwise multiplication [KP88], [BBR12], and therefore g jk , |g| ∈ W 1,2 ∩ L ∞ . We see that abc ∈ W −1,2 + L 1 . We write this schematically as where O(g −1 ∂g) denotes a finite linear combination of terms of the form g pq ∂ j g rs , etc. Now, since multiplication by an W 1,2 ∩ L ∞ function maps W 1,q to W 1,2 for any q > n, it also maps W −1,2 to W −1,q , and we have R abcd = O g∂(g −1 ∂g) + g(g −1 ∂g) 2 ∈ W −1,q + L 1 . Since where R = g rs R rs is the scalar curvature, we have W abcd = R abcd + P ac g bd − P bc g ad + P bd g ac − P ad g bc = (R abcd + R ac g bd − R bc g ad ) + 1 n − 2 − 1 (R ac g bd − R bc g ad ) This shows that if g jk ∈ W 1,2 ∩L ∞ in some system of local coordinates, then one can make sense of the components W abcd of the Weyl tensor as elements in W −1,q + L 1 in these coordinates for any q > n. It is easy to see that the identity W (cg) = cW (g) remains true if c, g jk ∈ W 1,2 ∩ L ∞ .
We wish to understand the equation W abcd (g) = 0 as a quasilinear divergence form system for the matrix elements g jk . Below we write D = −i∇.
Lemma 4.1. Let g jk ∈ W 1,2 ∩L ∞ in some local coordinates. If W abcd (g) = 0 in these coordinates and if u ∈ W 1,2 ∩ L ∞ is the column vector consisting of the matrix elements g jk , then u is a distributional solution of Du) where B is given by and each A lm (resp. B lm a ) is an M × N (resp. N × N ) matrix function whose entries are rational functions of the components of u. Moreover, each A lm and B lm a is smooth in the set of vectors u corresponding to positive definite symmetric matrices. One can choose M = n 4 and N = n(n+1) 2 .
Proof. Follows from (4.1) after using the Leibniz rule: The fact that A lm and B lm a are rational functions follows by using Cramer's rule for the inverse matrix (g jk ).
The main point, as observed in [LS15], is that the equation W abcd (g) = 0 becomes elliptic if it is written in n-harmonic coordinates and the determinant is normalized to one. This is expressed by the next result.
Lemma 4.2. Let g jk ∈ W 1,q ∩ C α in some local coordinates, where q ≥ 2 and 0 < α < 1. If g ab is the representation of g in any n-harmonic coordinate system, and ifĝ ab = |g| −1/n g ab , thenĝ ab ∈ W 1,q ∩ C α . Ifû is the column vector consisting of the matrix elementsĝ ab , and ifÂ lm (x) = A lm (û(x)), then the linear operator is overdetermined elliptic (i.e. its principal symbol is injective).
Proof. Since g jk ∈ W 1,q ∩ C α , by Theorem 1.1 and Proposition 2.7 we know that any n-harmonic coordinate system is W 2,q ∩ C 1+α . Now W 1,q ∩ C α is an algebra under pointwise multiplication, which shows that g ab andĝ ab are in W 1,q ∩ C α .
To prove ellipticity, we need to show that the matrix valued symbol A lm ξ l ξ m is injective for any ξ ∈ R n \ {0}. In order to simplify notation we will omit all hats in the rest of this proof and write g ab instead ofĝ ab etc. Assuming for the moment that g is C ∞ , we note the following consequences of Bianchi identities, where M abcd = R abcd + R ac g bd − R bc g ad , N abcd = R ab g cd − 1 2 Rg ab g cd .
The left hand sides of these Bianchi type identities may be viewed as third order nonlinear operators acting on g. After linearization we obtain the following symbol versions of the previous identities, Here σ ξ (P ) = σ ξ (P )(g)h is the principal symbol of the linearization of an operator P , linearized at g = g ab and acting on some h = h ab . Now (4.2) are algebraic equations for the principal symbols σ ξ (M abcd ) and σ ξ (N abcd ) when g ∈ C ∞ , but the since these principal symbols can also be computed for W 1,q ∩ C α metrics and their algebraic expressions remain identical, it follows that (4.2) is valid for g ab ∈ W 1,q ∩ C α . By (4.1) we have Rg ac g bd + 1 2(n − 2) (Rg bd g ac − Rg ad g bc ) − R (n − 1)(n − 2) (g ac g bd − g bc g ad ).
Let us now assume that σ ξ (W abcd )h = 0, where h = (h ij ). Contracting this equation by ξ a ξ d and using (4.2) yields The symbol on the right hand side is now (a scalar multiple of) the symbol of the Bach tensor, see [LS15, Lemma 3.2]. The W 1,q ∩ C α regularity is also enough for the n-harmonic coordinate gauge condition in [LS15] for the contracted Christoffel symbols (with g having determinant one) to be written as Γ k (g) = n − 2 2 g ka ∂ a log g kk .
Proof of Theorem 1.3. The assumption that g ∈ W 1,n ∩C α in some local coordinates together with Theorem 1.1 implies that there exists an n-harmonic coordinate system with C 1+α regularity. By Lemma 4.2 we know that in any n-harmonic coordinates, g ab andĝ ab = |g| −1/n g ab are in W 1,n ∩ C α . We will prove thatĝ Since the Weyl tensor ofĝ ab vanishes, we get thatĝ ab = c 0 δ ab where c 0 ∈ C ∞ . Thus g ab = |g| 1/n c 0 δ ab where c = |g| 1/n c 0 is a positive function in W 1,n ∩ C α as required. It remains to prove (4.3). In the rest of the proof, we will omit hats and write g ab and u instead ofĝ ab andû etc. Since the Weyl tensor of g vanishes, Lemmas 4.1-4.2 imply that u solves the overdetermined elliptic system where u ∈ W 1,n ∩ C α is the column vector consisting of the matrix elements of g ab . We claim that for any t ≥ n, one has where δ = α 1−α n 4 . If (4.4) is known, then starting from u ∈ W 1,n ∩ C α and iterating (4.4) gives u ∈ W 1,q for any q < ∞. Then by Sobolev embedding u ∈ C β for any β < 1, and u solves where A lm (u) ∈ C β and B(u, Du) ∈ L q for all β < 1 and q < ∞. The Schauder estimates in Proposition A.1(c) give that u ∈ C 1+β for all β < 1, and iterating the higher order Schauder estimates in Proposition A.1(d) gives that u ∈ C ∞ . This shows (4.3).
To prove (4.4), note that if u ∈ W 1,t ∩ C α and t ≥ n, then x → A lm (u(x)) and x → B lm a (u(x)) are in W 1,t ∩ C α (this set is an algebra under pointwise multiplication). Thus, if we write f (x) = B(u(x), Du(x)), we see that u solves with A lm (u) ∈ W 1,t ∩ C α and f ∈ L t/2 . The Calderón-Zygmund estimate in Proposition A.1(b) implies that u ∈ W 2,t/2 . But we also know that u ∈ C α , and C α ⊂ W ε,q whenever ε < α and q < ∞. Under these conditions we have u ∈ W 2,t/2 ∩ W ε,q , and by complex interpolation u ∈ [W 2,t/2 , W ε,q ] θ for 0 < θ < 1. We choose θ so that (1 − θ) · 2 + θ · ε = 1, which gives θ = 1 2−ε . Thus u ∈ W 1,p where 1 Since q < ∞ was arbitrary, we get u ∈ W 1,p whenever p < 2−ε 1−ε t 2 and ε < α. In particular this gives (4.4). We will need some standard facts about function spaces, see [BL76], [Ta96] for more details. If s ∈ R and 1 < p < ∞, then W s,p = W s,p (R n ) is the Banach space with norm Here, ξ = (1 + |ξ| 2 ) 1/2 and D s is the Fourier multiplier corresponding to ξ s . In general, if m(ξ) is a C ∞ function which is polynomially bounded in R n together with its derivatives, the corresponding Fourier multiplier m(D) is the map on S (R n ) (the space of tempered distributions) defined by If k ∈ {0, 1, 2, . . .} and 0 < α < 1, then C k+α = C k+α (R n ) is the Banach space with norm One has C k+α = C k+α * = B k+α ∞∞ where C s * are the Zygmund spaces and B s pq the Besov spaces in R n . We write C −1+α = C −1+α * for 0 < α < 1. The operator D t is an isomorphism W s,p → W s−t,p and C s * → C s−t * , and D l is a bounded map W s,p → W s−1,p and C s * → C s−1 * for any s, t ∈ R and 1 < p < ∞ [Ta96, Section 13]. We will also use the Sobolev embeddings where p * = np n−p for 1 ≤ p < n. The first step in the proof of Proposition A.1 is a parametrix construction for solutions of P x 0 v = f in R n , where P x 0 the frozen coefficient operator We fix a cutoff function ψ ∈ C ∞ c (R n ) with ψ = 1 for |ξ| ≤ 1/2 and ψ = 0 for |ξ| ≥ 1.
Lemma A.2. Fix x 0 ∈ Ω. There is an operator E x 0 = e x 0 (D) such that for where e x 0 : R n → R N ×M is the bounded smooth matrix valued function The operator E x 0 is a bounded map W s,p (R n , R M ) → W s+2,p (R n , R N ) and C s * (R n , R M ) → C s+2 * (R n , R N ) whenever s ∈ R and 1 < p < ∞. Proof. The operator P x 0 has full symbol p(x 0 , ξ) = A lm (x 0 )ξ l ξ m . Then The ellipticity condition (A.3) implies that det (p(x 0 , ξ) * p(x 0 , ξ)) = 0, ξ ∈ R n \ {0}.
Since p(x 0 , ξ) * p(x 0 , ξ) is homogeneous of degree 4 in ξ, the function e x 0 defined in the statement of the lemma is smooth in ξ and homogeneous of degree −2 for |ξ| large. If implies v = E x 0 P x 0 v + ψ(D)v by taking inverse Fourier transforms. Now e x 0 (ξ) is a matrix valued classical symbol in the class S −2 1,0 , and the mapping properties of the corresponding pseudodifferential operator [Ta96, Section 13] imply the mapping properties for E x 0 . (Alternatively, one could apply the vector valued Mihlin multiplier theorem [BL76, Theorem 6.1.6] and its version on Besov spaces to m x 0 = ξ 2 e x 0 .) Next we give a local representation formula for solutions of the equation P (x, D)u = f . To prove the Schauder estimates it is convenient to do a rescaling: if x 0 ∈ Ω and r > 0 is small, we writẽ A lm (y) = A lm (x 0 + ry),ũ(y) = u(x 0 + ry),f (y) = f (x 0 + ry).
Proof. Note thatũ satisfies the equation If v = χũ, we compute Note that the left hand side operator frozen at 0 is just P x 0 . Thus where Q x 0 is as stated. It follows that The result follows from Lemma A.2.
Proof of Proposition A.1. Let u ∈ W 1,q 0 loc (Ω, R N ) solve P (x, D)u = f in Ω, where q 0 > 1. Fix x 0 ∈ Ω, and assume that B(x 0 , 2r) ⊂ Ω. By Lemma A.3, the function v = χũ satisfies Assume now the conditions in (a). We may assume q 0 < n, so W 1,q 0 ⊂ L q * 0 by Sobolev embedding. We may also assume q 0 < q (otherwise there is nothing to prove). Fix p 0 > 2 so that q 0 , q ∈ [p 0 , p 0 ]. We claim that if r is chosen small enough, then T is invertible on W 1,p (R n , R N ) for p ∈ [p 0 , p 0 ], and (A.5) G ∈ W 1,p for p ∈ [q 0 , min{q * 0 , q}]. (A.6) Assuming these two claims, it follows that v = T −1 (G) ∈ W 1,min{q * 0 ,q} . If q ≤ q * 0 we have v ∈ W 1,q and we are done. Otherwise we get v ∈ W 1,q * 0 , and repeating the argument with q 0 replaced by q * 0 we see that G and thus also v are in W 1,min{q * * 0 ,q} . Iterating this procedure finitely many times gives v ∈ W 1,q , which proves u ∈ W 1,q loc upon varying x 0 . To see (A.5), note that E x 0 w W 1,p ≤ C w W −1,p , w ∈ W −1,p , p ∈ [p 0 , p 0 ].
Since A lm are continuous, we can choose r small enough so that This implies (A.5) by Neumann series. To see (A.6), note that our assumptions and Sobolev embedding imply Since all the terms are compactly supported, we get F ∈ W −1,p , p ∈ (1, min{q * 0 , q}]. The mapping properties of E x 0 in Lemma A.2 and the fact that ψ(D)χ 2 v is Schwartz imply (A.6).
Assume next the conditions in (b). We claim that for r > 0 small, T is invertible on W 2,q (R n , R N ), and (A.7) G ∈ W 2,q . (A.8) Since also the conditions in (a) hold, we have v = T −1 (G) and thus (b) follows immediately from these claims upon varying x 0 . To prove (A.7), note that Lemma A.2 gives Combining this with the pointwise multiplier property au W 1,q ≤ au L q + ∇(au) L q ≤ C( a L ∞ + ∇a L t ) u W 1,q by using the assumption on t (if p = n this holds for a supported in B(0, 2)), we get χ 2 (Ã lm −Ã lm (0))D m w W 1,q ≤ Co r→0 (1) w W 2,q using the fact that A lm ∈ C 0 loc ∩ W 1,t loc . This shows (A.7) by Neumann series. To prove (A.8), consider first the case 1 < q < n. Then L q ⊂ W −1,q * , so u ∈ W 1,q * loc by (a). Since L n , L q * ⊂ L q , we get F ∈ L q and therefore G ∈ W 2,q by the mapping properties of E x 0 . In the case n ≤ q < ∞ we get L q loc ⊂ W −1,p loc for all p < ∞, thus u ∈ W 1,p loc for all p < ∞ by (a), and u ∈ L ∞ by Sobolev embedding. The assumption A lm ∈ C 0 loc ∩ W 1,t loc gives F ∈ L q and G ∈ W 2,q . This proves (A.8) and also (b).
Assume next the conditions in (c). It is enough to prove that for r > 0 small, T is invertible on C 1+α (R n , R N ), and (A.9) G ∈ C 1+α . (A.10) The main point is that |A lm (y) − A lm (z)| ≤ M |y − z| α , so we have χ 2 (Ã lm −Ã lm (0)) C α ≤ Cr α and consequently This proves (A.9) by Neumann series if r > 0 is small enough. To prove (A.10) one first notes that f ∈ W −1,q loc for all q, hence u ∈ W 1,q loc for all q by (a) and u ∈ C β loc for 0 < β < 1 by Sobolev embedding. Thus F ∈ C −1+α , so that G ∈ C 1+α by the mapping properties of E x 0 .
Finally, assume the conditions in (d). We do the proof by induction on k. If k = 0 the result follows from (c). Assume the result holds smoothness indices ≤ k − 1 where k ≥ 1, and assume that A lm ∈ C k+α loc , f ∈ C k−1+α loc .
By induction hypothesis, u ∈ C k+β loc for β < 1. Also, since k ≥ 1, part (b) implies that u ∈ W 2,q loc for any q < ∞. Hence it is possible to differentiate the equation once to get D l (A lm D m (D j u)) = D j f − D l ((D j A lm )D m u).
The right hand side is in C k−2+α loc , hence the induction hypothesis gives D j u ∈ C k+α loc for all j. This proves that u ∈ C k+1+α loc .