Local pointwise second derivative estimates for strong solutions to the $\sigma_k$-Yamabe equation on Euclidean domains

We prove local pointwise second derivative estimates for positive $W^{2,p}$ solutions to the $\sigma_k$-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.

It is well-known that the equations (1.1 ± ) are elliptic. Furthermore, σ 1/k k is a concave function on the set of symmetric matrices with eigenvalues in Γ + k . The motivation behind (1.1 ± ) comes from conformal geometry: if g ij = u −2 δ ij is a metric conformal to the flat metric on a domain Ω ⊂ R n , then uA u is the (1, 1)-Schouten tensor of g, and the σ k -Yamabe equation in the so-called positive/negative (±) case is given by σ k (±uA u ) = 1, λ(±A u ) ∈ Γ + k , u > 0. (1. 2) The equations (1.2) and their counterparts on Riemannian manifolds were first studied by Viaclovsky in [63]. Since then, these equations have been addressed by various authorsfor a partial list of references, see [1-3, 8-12, 14, 16-19, 24, 27, 28, 31-33, 35, 39-41, 43, 44, 46, 47, 53, 54, 56, 64, 65] in the positive case and [13,23,25,29,30,42,45,55] in the negative case. When k = 1, these equations reduce to the original Yamabe equation. When k ≥ 2, they are fully nonlinear and elliptic at a solution (although, a priori, not necessarily uniformly elliptic). Fully nonlinear elliptic equations involving eigenvalues of the Hessian were first considered by Caffarelli, Nirenberg and Spruck in [6]. A priori local first and second derivative estimates play an important role in the study of the σ k -Yamabe equation, and were established in the positive case by Chen [14], Guan and Wang [27], Jin, Li and Li [39], Li and Li [40], Li [43] and Wang [65]. In the negative case, an a priori (global) C 1 estimate is proven by Gursky and Viaclovsky [30], but it is unknown whether a priori C 2 estimates hold. In this paper, we are concerned with the local regularity of positive W 2,p solutions to the equations (1.1 ± ). More precisely, for 2 ≤ k ≤ n we derive local pointwise boundedness of second derivatives, provided p > kn/2 in the positive case and p > (k + 1)n/2 in the negative case. To simplify the discussion, we do not include the case k = 1, in which the equations (1.1 ± ) are semilinear. We prove: Let Ω be a domain in R n (n ≥ 3) and let f ∈ C 1,1 loc (Ω × (0, ∞) × R n ) be a positive function. Suppose that 2 ≤ k ≤ n, p > kn/2 and u ∈ W 2,p loc (Ω) is a positive solution to (1.1 + ). Then u ∈ C 1,1 loc (Ω), and for any concentric balls B R ⊂ B 2R ⋐ Ω we have where C is a constant depending only on n, p, R, f and an upper bound for ln u W 2,p (B 2R ) .

Theorem 1.2.
Let Ω be a domain in R n (n ≥ 3) and let f ∈ C 1,1 loc (Ω × (0, ∞) × R n ) be a positive function. Suppose that 2 ≤ k ≤ n, p > (k + 1)n/2 and u ∈ W 2,p loc (Ω) is a positive solution to (1.1 − ). Then u ∈ C 1,1 loc (Ω), and for any concentric balls B R ⊂ B 2R ⋐ Ω we have where C is a constant depending only on n, p, R, f and an upper bound for ln u W 2,p (B 2R ) . Remark 1.3. As noted above, it is unknown whether a priori C 2 estimates hold for solutions to the σ k -Yamabe equation in the negative case. We also note that for the closely related σ k -Loewner-Nirenberg problem, there exist locally Lipschitz but non-differentiable viscosity solutions -see [45]. As far as the authors are aware, Theorem 1.2 currently provides the only available local second derivative estimate for solutions to the σ k -Yamabe equation in the negative case.
To put things in perspective, we note that our estimates in Theorem 1.1 are closely related to certain analytical aspects in the work of Chang, Gursky and Yang in [12]. In [12], under natural conformally invariant conditions on a Riemannian 4-manifold (M 4 , g 0 ), the authors established the existence of a metric in the conformal class [g 0 ] whose Schouten tensor has eigenvalues in Γ + 2 . An important part of the proof in [12] was to obtain W 2,s estimates for 4 < s < 5 on smooth solutions to a one-parameter family of regularised σ 2 -equations (see equation (A.1) in Appendix A) which are uniform with respect to the parameter. This was achieved by first obtaining a uniform W 1,4 estimate (see Theorem 3.1 in [12]), and subsequently carrying out an integrability improvement argument (see Sections 5 and 6 in [12]). With the W 2,s estimate in hand, the authors then applied a heat flow argument to obtain the desired conformal metric. Remark 1.4. A natural question to ask is whether the heat flow argument in [12] can be avoided by instead taking the regularisation parameter directly to zero. One application of Theorem 1.1 above and [46,Proposition 5.3] is that this can be achieved when (M 4 , g 0 ) is locally conformally flat. We refer the reader to Appendix A for the details.
Our work is also closely related to the work of Urbas in [60], where local pointwise second derivative estimates for W 2,p solutions to the k-Hessian equation were established on domains in R n . At the heart of Urbas' proof is also an integrability improvement argument, assuming an initial lower bound of p > kn/2 (see also [15,49,58,61,62]). By an application of Moser iteration, the C 1,1 loc estimate is then obtained. We note that Moser iteration has previously been utilised in the context of the σ k -Yamabe equation to establish local boundedness of solutions, see for instance [21,22,33].
We will in fact prove a more general version of Theorems 1.1 and 1.2, and consider an operator of the form ) for a given matrix-valued function H = H(x, z, ξ) ∈ C 1,1 loc (Ω × R × R n ; Sym n (R)), where Sym n (R) denotes the space of real symmetric n × n matrices. Rather than (1.1 ± ), we consider the equation Therefore, for the purpose of obtaining Theorems 1.1 and 1.2, it will suffice to consider the case that H is a multiple of the identity matrix: loc (Ω × R × R n ) a positive function and H ∈ C 1,1 loc (Ω × R × R n ; Sym n (R)). Suppose 2 ≤ k ≤ n, p ≥ 1 and u ∈ W 2,p loc (Ω) is a solution to (1.4), and that one of the following conditions holds: Then u ∈ C 1,1 loc (Ω), and for any concentric balls where C is a constant depending only on n, p, R, f, H and an upper bound for u W 2,p (B 2R ) . The matrix A H [u] introduced in (1.3) is sometimes referred to as an augmented Hessian of u. The corresponding augmented Hessian equations have been extensively studied in recent years -see [36][37][38] and the references therein. In this vein, it is therefore of interest to generalise Theorem 1.5 to arbitrary H ∈ C 1,1 loc . As we will see, the proof of Theorem 1.5 uses some favourable divergence structure in the case that H is a multiple of the identity matrix. However, when k = 2, a similar divergence structure holds for general H and we obtain the following: loc (Ω × R × R n ) a positive function and H ∈ C 1,1 loc (Ω × R × R n ; Sym n (R)). Suppose p > 3n 2 and u ∈ W 2,p loc (Ω) is a solution to (1.4) with k = 2. Then u ∈ C 1,1 loc (Ω), and for any concentric balls where C is a constant depending only on n, p, R, f, H and an upper bound for u W 2,p (B 2R ) . Remark 1.9. In [36][37][38] and the references therein, it is usually assumed that H satisfies a so-called co-dimension one convexity condition, which is known to be a necessary and sufficient condition to obtain C 1 estimates -see [50,51,57]. We point out that we do not assume a co-dimension one convexity condition in our treatment of second derivative estimates (the exception is Case 1 of Theorem 1.5, where we have convexity in ξ).
Under a stronger assumption on p, we will also obtain an extension of Theorem 1.8 to the case k ≥ 3 -see Section 6.
In adapting the methods of [60] to prove Theorems 1.5 and 1.8, we will need to deal with the term H[u] which, whilst being of lower order in the definition of A H [u], creates new higher order terms in our estimates. Roughly speaking, the two terms which are formally problematic consist of: with double difference quotients of H[u] ij (this arises as a result of taking difference quotients of (1.4) twice), and (ii) the divergence of F [u] ij multiplied by a term formally of third order in u (this arises after integrating by parts).
In [60], neither of these terms exist since F [u] ij is divergence-free when H ≡ 0. In the more general case that we are considering, it is unclear whether these third order terms have a favourable sign individually. However, we will estimate them so as to show that, when combined, they yield a cancellation phenomenon that ensures the overall higher order contribution is positive. For the estimates of the higher order terms arising from the divergence of F We close the introduction by noting that in Theorems 1.1 and 1.2, we do not know whether our lower bounds on p to obtain C 1,1 loc regularity are sharp, and it would be interesting to determine the sharp lower bounds. In the case of the k-Hessian equation for 3 ≤ k ≤ n, it is shown by Urbas in [59] that there exist W 2,p -strong solutions with p < k(k−1) 2 which fail to be C 1,α loc for any α > 1 − 2 k . Other lower bounds on p leading to C 1,1 loc regularity for k-Hessian equations have been studied in [15,49,58,61,62], for instance.
The plan of the paper is as follows. We begin in Section 2 with an outline of the proof of Theorems 1.5 and 1.8. This prompts us to consider the divergence structure of the linearised operator, which we address in Section 3, and also motivate the estimates established from Section 4 onwards. In Section 4 we carry out the main body of our integral estimates. In Section 5, we use these estimates and the Moser iteration technique to obtain the desired C 1,1 loc estimates, completing the proofs of Theorems 1.5 and 1.8. In Section 6, we give the aforementioned extension of Theorem 1.8 to the case k ≥ 3.
2 Outline of the proofs of Theorems 1.5 and 1.8 Our proofs of Theorems 1.5 and 1.8 use an integrability improvement argument, from which the C 1,1 loc estimate is obtained by the Moser iteration technique. In Case 1 of Theorem 1.5, we will obtain, for a solution u ∈ W 2,q+k−1 loc (Ω) to (1.4) with q > kn 2 − k + 1, the estimate where ρ ∈ (0, R 3 ], β = kn kn−2k+2 and C 1 is a positive constant ensuring ∆u + C 1 ≥ 1 a.e. (see the paragraph after Remark 2.3 for the justification of the existence of C 1 ). Similarly, in Case 2 of Theorem 1.5 and in Theorem 1.8, we will obtain, for a solution u ∈ W 2,q+k now with β = (k+1)n (k+1)n−2(k+1)+2 . The estimates (2.1) and (2.2) then yield an improvement in integrability under the respective lower bounds on q, which can then be iterated to yield the desired C 1,1 loc estimates. 1 In the rest of this section we explain how the estimates (2.1) and (2.2) are obtained. Due to the lack of regularity, we derive our estimates through taking difference quotients of the equation (1.4). For an index l ∈ {1, . . . , n} and increment h ∈ R\{0}, we recall the first order difference quotient ∇ h l u(x) · · = h −1 (u(x + he l ) − u(x)) and the second order difference quotient The above expressions are well-defined for x ∈ Ω h · · = {y ∈ Ω : dist(y, ∂Ω) > |h|}.
It is well-known (see, for instance, [20,Lemma 7.23]) that We will also use the following fact -see Appendix B for a proof: Then v h → ∆u in L s loc (Ω) as h → 0. We assume now that both the increment h and our solution u are fixed, and write v as shorthand for v h . Taking difference quotients of the equation σ , ∇u(x)) and appealing to the concavity of σ 1/k k in Γ + k , we will derive (at the start of Section 4) the pointwise estimate Remark 2.2. In (2.6), and from this point onwards, summation notation is employed only over repeated indices which appear in both upper and lower positions. Positioning of indices is purely to indicate whether summation convention is being utilised; since we are working with the Euclidean metric, we are free to raise and lower indices at will. For instance, A ij , A i j , A j i and A ij all denote the (i, j)-entry of a symmetric matrix A. Similarly, we do not distinguish between the derivatives ∇ i and ∇ i when using index notation. The estimates (2.1) and (2.2) are derived by testing (2.6) against suitable test functions. First fix a ball B 2R ⋐ Ω h . Since λ(A H ) ∈ Γ + 2 is equivalent to tr(A H ) = ∆u − tr(H) > 0 and σ 2 (A H ) > 0, there exists a constant C 1 ≥ 0 (depending on an upper bound for H C 0 (Σ) -see Remark 1.6) for which ∆u + C 1 ≥ 1 and |∇ 2 u| ≤ ∆u + C 1 a.e. in B 2R . We definẽ v · · = v + C 1 , and for a small parameter δ > 0 (that we eventually take to zero) we denote for all q > 1 and u ∈ W 2,q+k−1 loc (Ω) ∩ W 1,∞ loc (Ω) solving (1.4). For ease of outlining our argument, let us suppose that f = f (x, z) (the general case f = f (x, z, ξ) will only require minor changes -see Section 5.3). Then the integrand on the left hand side (LHS) of (2.7) is a lower order term, whereas the integrands on the RHS of (2.7) involve higher order terms, formally of fourth and third order in the limit h → 0, and thus need to be treated.
In Section 4, we integrate by parts in the first integral on the RHS of (2.7), using a result of Section 3 that tells us ∇ i F [u] ij is a regular distribution belonging to L . After taking δ → 0 and carrying out some further calculations (see Lemmas 4.2 and 4.3), we will obtain the estimate where C is a constant independent of h, q and ρ. Whilst the first integral on the LHS of (2.8) is a favourable positive higher order term, the other two integrals on the LHS (which we denote by (I 2 ) h and (I 3 ) h , respectively) involve higher order terms which are, a priori, of unknown sign. Treating (I 2 ) h and (I 3 ) h is the most technical part of our proof. Now, if we momentarily assume sufficiently high regularity on u, say u ∈ W 2,q+2k−1 loc (Ω) ∩ W 1,∞ loc (Ω) (q > 1), the issue of dealing with (I 2 ) h and (I 3 ) h is largely simplified. As will be detailed in the proof of Theorem 6.1, one may apply the Cauchy inequality to each of the integrands and absorb the resulting third order terms into the positive term on the LHS of (2.8). Under the stated integrability assumption, this crude estimation is sufficient to show An estimate analogous to (2.1) and (2.2) can then be obtained, assuming q > kn − 2k + 1.
The difficulty is to therefore deal with (I 2 ) h and (I 3 ) h under the weaker integrability assumptions of Theorems 1.5 and 1.8. At this point, we make the distinction between the various cases. In each case, we estimate (I 2 ) h and (I 3 ) h so as to produce a cancellation phenomenon when combined, leaving only lower order terms; see Lemmas 4.4 and 4.5 for the estimates on (I 2 ) h , Lemma 4.10 for the estimates on (I 3 ) h , and Corollaries 4.12, 4.13 and 4.14 for the resulting cancellations. It will then follow from (2.8) that, in Case 1 of Theorem 1.5 with the relaxed assumption u ∈ W 2,q+k−1 Similarly, in the remaining cases with u ∈ W 2,q+k To obtain (2.1) from (2.9) (resp. (2.2) from (2.10)), we proceed as follows (the details can be found in Section 5). We first obtain an integral estimate for ∇ (ṽ + ) q/2 2 , to which we can apply the Sobolev inequality. We then justify taking the limit h → 0 and impose the lower bound q + k − 1 > kn 2 (resp. q + k > (k+1)n 2 ), from which we obtain (2.1) (resp. (2.2)).

Divergence structure of the linearised operator F [u] ij
In this section we derive a divergence formula for the linearised operator F [u] ij (defined in (1.6)), which we will use at various stages of our proof. We note that in the case that In the former case, F [u] ij is divergence-free with respect to the flat metric (see [52]), and in the latter case, u 1−k F [u] ij is divergence-free with respect to the conformal metric g ij = u −2 δ ij (see [63]). For related discussions, see also [4,5,26,34,53].
For A ∈ Sym n (R) and 1 ≤ k ≤ n, define the k'th Newton tensor of A inductively by It is well-known (see [52]) that and and moreover T k−1 (A) ij is positive definite when λ(A) ∈ Γ + k (see [6]). In particular, by (1.6) and Let Ω ⊂ R n be a domain and u ∈ C 3 (Ω). Then for H ∈ C 1 (Ω × R × R n ; Sym n (R)) and 2 ≤ k ≤ n, Proof. The identity (3.4) will follow once we show that for 1 ≤ k ≤ n − 1, Similarly, (3.5) will follow once we show that for 1 ≤ k ≤ n − 1 and H(x, z, ξ) = H 2 (x, z, ξ)I, To this end, we take the divergence of both sides in (3.1), which yields Then (3.6) is readily seen by applying (3.8) iteratively. We now turn to (3.7), for which we apply an induction argument on k using (3.8). The base case k = 1 is clear. We suppose that for some k ≥ 2 we have the identity and we show that (3.7) then follows. First observe that, by (3.9) and the fact H ij = H 2 δ ij , (3.8) simplifies to After substituting (3.1) and (3.3) into the last term and the penultimate term in (3.10), respectively, we arrive at (3.7).
Note that V [u] j (defined in (3.4)) contains at most second order derivatives of u. As a consequence, . More precisely, we have: Let Ω ⊂ R n be a domain and u ∈ W 2,q+k−1 loc (Ω) ∩ W 1,∞ loc (Ω) with q > 1 and 2 ≤ k ≤ n. Then for H ∈ C 0,1 loc (Ω × R × R n ; Sym n (R)) and ϕ ∈ W 1,s 0 (Ω; R n ), s · · = q+k−1 q , we have (Ω) and where C is a constant depending on an upper bound for (Ω). In particular, we can take m → ∞ in (3.13) to get (3.11). The estimate (3.12) follows from the definition of V [u] j .

Main estimates
In this section we prove our main estimates, which will then be used in the proof of our main results in Section 5. Largely, our estimates will be concerned with terms involving the contraction of the linearised operator F = (F ij ) and its divergence with various other tensors, such as ∇ 2ṽ , ∇ṽ and (∆ h ll H[u] ij ).

Initial integral estimates: isolating higher order terms
The following lemma provides the starting point for our integral estimates: Proof. The proof follows [60], with some adjustments. For A ∈ Sym n (R), let G ij (A) = ∂σ Adding the two equations in (4.2), dividing through by h 2 and summing over l, we have Substituting the equation σ 1/k k (A H ) = f into the LHS of (4.4), and commuting difference quotients with derivatives on the RHS of (4.4), we arrive at (4.1).
As outlined in Section 2, we proceed to derive a series of integral estimates by multiplying (4.1) by suitable test functions and integrating by parts using the divergence structure proved in Lemma 3.2. Recall that for a fixed increment h > 0, we defined v(x) = l ∆ h ll u(x), and that we fixed a ball B 2R ⋐ Ω h and a constant C 1 (depending on an upper bound for H C 0 (Σ) ) such that ∆u + C 1 ≥ 1 and |∇ 2 u| ≤ ∆u + C 1 a.e. in B 2R . The existence of such a constant is guaranteed by the assumption λ(A H ) ∈ Γ + 2 . We then definedṽ = v + C 1 , and for a small parameter δ > 0 (that we eventually take to zero) we defined which is just the estimate (2.7) in Section 2, repeated here for convenience.
We are now in a position to prove our first integral estimate. In what follows, let Roughly speaking, if u ∈ W 2,s loc (Ω) then J (s) h should be interpreted as a lower order term, and terms bounded by J (s) h are consequently considered 'good terms'. We will first address the case f = f (x, z) for simplicity and postpone the more general case until Section 5.3. The relevant equation is therefore (4.6) Throughout Section 4, unless otherwise stated, C will denote a generic positive constant which may vary from line to line, depending only on n, R, f, H and an upper bound for u W 1,∞ (B 2R ) . In particular, C is independent of h, q and ρ, and any norm of ∇ 2 u. In addition, we will often use the inequalities ∆u + C 1 ≥ 1 and |∇ 2 u| ≤ ∆u + C 1 without explicit reference.
and |h| sufficiently small, we have Proof of Lemma 4.2. Appealing to Lemma 3.2 with ϕ j = ηQ q−1 δ ∇ jṽ , and noting that Rearranging (4.8) to get the desired integration by parts formula for B R+2ρ ηQ q−1 δ F ij ∇ i ∇ jṽ , and substituting this back into (4.5), we obtain We now take δ → 0 in (4.9), using Fatou's lemma for the first integral (which is positive) and the dominated convergence theorem elsewhere (which is justified since q > 1). This yields To conclude the proof of Lemma 4.2, we must bound the RHS of (4.10) from above by . We begin with the first integral on the RHS of (4.10). Appealing again to Lemma 3.2, now with ϕ j = 1 q (ṽ + ) q ∇ j η and Therefore, where F = (F ij ). Recalling |F | ≤ C(∆u + C 1 ) k−1 and applying Hölder's inequality to the penultimate integral in (4.11), we see that . The final integral in (4.11) satisfies the same estimate, since | div F [u]| ≤ C(∆u + C 1 ) k−1 by (3.12).
It remains to estimate the second term on the RHS of (4.10). Keeping in mind that f = f (x, z) ∈ C 1,1 loc (Ω × R), we apply Hölder's inequality followed by (2.5) to obtain This concludes the proof.
To clear up notation, we denote the three integrals on the LHS of (4.7) involving higher order terms by The terms (I 1 ) h , (I 2 ) h and (I 3 ) h will be considered in turn. In Section 4.2, we prove an estimate for (I 1 ) h . In Section 4.3.1, we estimate (I 2 ) h in the case that H is a multiple of the identity, and in Section 4.3.2 we estimate (I 2 ) h for general H when k = 2. The estimate for (I 3 ) h in the general case is slightly involved, so for illustrative purposes we first address the simpler case when H(x, z, ξ) = H 1 (x, z)|ξ| 2 I with H 1 ≥ 0, which includes the σ k -Yamabe equation in the positive case. This is done in Section 4.4.1. The estimate for (I 3 ) h in the general case is proved in Section 4.4.2. In the process, we will prove the cancellation phenomenon between (I 2 ) h and (I 3 ) h alluded to earlier -see Corollaries 4.12, 4.13 and 4.14.

A pointwise lower bound for
The term F ij ∇ iṽ ∇ jṽ in (I 1 ) h can be bounded in the same way as in [60] (see equation (3.6) therein). We reproduce the argument here for the reader's convenience.

Integral estimates for
In this section we obtain estimates for the term ( The case in which H is a multiple of the identity matrix will be dealt with first, in Section 4.3.1. The case for general H when k = 2 will then be addressed in Section 4.3.2.
Remark 4.7. The first term on the RHS of (4.17) and (4.18) will later be shown to cancel with a term arising from our estimate for (I 3 ) h .
Proof of Lemmas 4.4 and 4.5. The proof consists of three steps. In Step 1, we prove a preliminary estimate assuming only u ∈ W 2,q+k−1 loc (Ω) ∩ W 1,∞ loc (Ω) and H = H 2 (x, z, ξ)I, but we do not assume at this point that u necessarily solves (4.6). Only in Steps 2 and 3 will we appeal to the specific hypotheses of Lemmas 4.4 and 4.5.
Our starting point is the following expression for (I 2 ) h , which follows from (3.5): Step 1: In this step, we show that for every u ∈ W 2,q+k−1 loc (Ω) ∩ W 1,∞ loc (Ω), Note that the first integral on the RHS of (4.19) is the desired term seen in (4.17) and (4.18). First observe that by the chain rule, Denote the top two lines of the RHS of (4.20) collectively by L 1 , and the bottom line by L 2 .
Recalling that ∇ j ∇ a u = H 2 δ ja + (A H ) ja and, in view of (3.1) and (3.3), that we have Substituting this identity for L 2 into (4.20) yields We claim that the terms on the top line of the RHS of (4.22) are bounded from below by −Cρ −1 J Thus, after integrating by parts using Lemma 3.2 and applying Hölder's inequality, the lower bound for these terms follows.
To estimate the penultimate integral in (4.22), we integrate by parts using Lemma 3.2 and apply the identity After an application of Hölder's inequality, this gives from which (4.19) follows.
Step 3: In this step we prove Lemma 4.4. Since we assume in this case that H 2 (x, z, ξ) = H 1 (x, z)|ξ| 2 and that u solves (4.6), rather than estimating as in Step 2 we observe Substituting (4.23) into the second integral in (4.19), we arrive at (4.17).

The case k = 2 for general H
In this section we obtain an estimate in the case k = 2 analogous to (4.17) and (4.18). We do not assume that H is a multiple of the identity and, as in Lemma 4.5, we do not assume that u solves (4.6): Lemma 4.8. Suppose H ∈ C 1,1 loc (Ω×R×R n ; Sym n (R)), k = 2 and u ∈ W 2,q+2 and |h| sufficiently small, we have

(4.24)
Remark 4.9. The first two terms on the RHS of (4.24) will later be shown to cancel with a term arising from our estimate for (I 3 ) h (cf. Remark 4.7).
Proof of Lemma 4.8. As k = 2, we have (3.4)) and ∇ j ∇ a u = tr(A H )δ ja − F ja − H ja . It follows that The integral on the last two lines of (4.25) can be bounded from below by −Cρ −1 J

Integral estimates for F [u] ij ∆ h ll H[u] ij
In this section we obtain estimates for the quantity ( ij . More precisely, we will prove the following lemma: , then for |h| sufficiently small, we have (Ω) ∩ W 1,∞ loc (Ω) (q > 1) and H(x, z, ξ) = H 1 (x, z)|ξ| 2 I with H 1 ≥ 0, then for |h| sufficiently small, we have Remark 4.11. Neither estimate in Lemma 4.10 requires u to be a solution to (4.6).
A similar cancellation also holds in the setting of Theorem 1.8, although this requires a little more work: Corollary 4.14. Suppose H ∈ C 1,1 loc (Ω × R × R n ; Sym n (R)), k = 2 and u ∈ W 2,q+2 loc (Ω) ∩ W 1,∞ loc (Ω) (q > 1). Then for |h| sufficiently small, we have If, in addition, u solves (4.6) for some positive f ∈ C 1,1 loc (Ω × R), then Proof. The estimate (4.33) will immediately follow once (4.32) is established, by substituting (4.14) and (4.32) into (4.7). Taking k = 2 in Lemma 4.10 a) and using Now, the first term on the RHS of (4.34) cancels with the second term on the RHS of (4.24), and the first term on the last line of (4.34) can be estimated by −Cρ −1 J (q+2) h , after integrating by parts and applying Hölder's inequality. Therefore, combining (4.24) and (4.34), we obtain Now, if u were to have enough regularity, we could integrate by parts here, observe that the third derivatives of u cancel, and obtain (4.32) by estimating the remaining terms in the usual way. To circumvent the lack of regularity, we instead apply the following lemma: Lemma 4.15. Let U ⊂ R n be a smooth bounded domain and let B ∈ L ∞ (U; R n×n ) be an antisymmetric matrix with supp(B) ⋐ U. For 1 ≤ p < ∞ and p ′ · · = p p−1 , consider the bilinear form B : If div B ∈ L q (U; R n ) with 1 p + 1 q = 1 − 1 r for some 1 ≤ q, r ≤ ∞, then we have the estimate for all g ∈ W 1,p (U) and h ∈ W 1,p ′ (U) ∩ L r (U).
Before proving Lemma 4.15, we use it to complete the proof of (4.32): for each i ∈ {1, . . . , n}, taking B a j = η It remains to prove Lemma 4.15. By a standard approximation argument, it suffices to prove (4.36) for g, h ∈ C ∞ (U). We are then justified in integrating by parts in (4.35), giving where we have used antisymmetry of B to assert that B a j ∇ a ∇ j g = 0.

Proof of Lemma 4.10 b)
where the error terms satisfy (4.42) To keep notation succinct, we denote x ± = x ± he l and F = (F ij ) in what follows.
Step 1: We first prove a lower bound for F ij (x)∆ h ll (H[u](x)) ij , identifying the error terms in (4.41). Observe that by (4.40) and the fact that F ij is positive definite in Γ + k , we have where to obtain the second inequality we have estimated Recalling the definition of ∆ h ll (H[u](x)) ij , we therefore see that for a.e. x ∈ B R+2ρ , Step 2: To prove (4.26), we need to show that the error terms in last two lines of (4.43) satisfy (4.42). Formally, these terms behave like |F |(|∇ 2 u| 2 + |∇ 2 u|), and so by the estimate |F | ≤ C(∆u + C 1 ) k−1 , the bound (4.42) is then conceivable. We now give the details. Denote the terms on the penultimate line of (4.43) collectively by E 1 , and the terms on the last line of (4.43) collectively by E 2 . The error terms in E 2 are easier to deal with. Indeed, by the bound |F | ≤ C(∆u + C 1 ) k−1 , Hölder's inequality and (2.4), we have In exactly the same way, one can show B R+2ρ η(ṽ + ) q−1 |F ||∇ ±h l ∇u| ≤ CJ (q+k−1) h , and combining these estimates we obtain B R+2ρ η(ṽ + ) q−1 |E 2 | ≤ CJ (q+k) h . We now treat the error terms in E 1 . We first observe that by the fundamental theorem of calculus followed by the chain rule, we have the identities and therefore Now, by the C 1,1 loc regularity of H and the Lipschitz regularity of the mapping (x, z, p) → ∂H ij ∂z (x, z, ξ)p l for fixed ξ and each l ∈ {1, . . . , n}, we can estimate the last line of (4.45) from above by Ch 2 and the middle line of (4.45) from above by Applying these estimates in (4.45) and taking ξ = ∇u(x), we therefore see that Using (4.46), one readily obtains the estimate B R+2ρ η(ṽ + ) q−1 |E 1 | ≤ CJ (q+k−1) h , applying the same line of argument as seen above for E 2 . For example, by Fubini's theorem and Young's inequality, we have This completes the proof of Lemma 4.10 a).
Step 3: It remains to prove Lemma 4.10 b) (see Section 4.4.1 for an alternative proof which is independent of calculations in Steps 1 and 2 above). Note that in this case, we may take C Σ = 0 in (4.43) and so the error terms on the last two lines of (4.43) formally behave like |F ||∇ 2 u|. By the same argument as in Step 2, the error terms E 1 and E 2 considered in Step 2 therefore satisfy B R+2ρ η(ṽ , and the conclusion follows.

Proof of main results
In this section we use Corollaries 4.12, 4.13 and 4.14 to prove Theorems 1.5 and 1.8, as outlined at the end of Section 2. We will first give a detailed proof of Case 1 of Theorem 1.5 when f = f (x, z) in Section 5.1, and then indicate the necessary adjustments for remaining cases, still when f = f (x, z), in Section 5.2. In Section 5.3, we extend these results to the case f = f (x, z, ξ), completing the proofs of Theorems 1.5 and 1.8.
We now carry out the Moser iteration argument. Let p > kn 2 be as in the statement of Theorem 1.5, and define a sequence q j inductively by q 0 = p − k + 1, q j = βq j−1 − k + 1 for j ≥ 1.
Applying (5.6) iteratively with q = q j and ρ = 3 −j−1 R, we have for each j ≥ 0 Letting j → ∞ and appealing once again to (5.7), we arrive at which implies the desired bound on ∇ 2 u L ∞ (B R ) by the choice of C 1 . In these cases, we recall that by Corollaries 4.13 and 4.14 we have the estimate

5.3
Proof of Theorems 1.5 and 1.8 for f = f (x, z, ξ) In this section we explain how the preceding arguments may be adjusted to treat the general case f = f (x, z, ξ), thus completing the proofs of Theorems 1.5 and 1.8: Proof of Theorems 1.5 and 1.8. The arguments up until (4.12) remain valid for f = f (x, z, ξ), but the last term in (4.10) can no longer be estimated as in (4.12). Consequently, under otherwise the same hypotheses, the conclusion of Lemma 4.2 now reads where (I 1 ) h , (I 2 ) h and (I 3 ) h are as before and The estimates for (I 1 ) h , (I 2 ) h and (I 3 ) h are unchanged (see Lemmas 4.3, 4.4, 4.5, 4.8 and 4.10), since they do not involve differentiating f . The integrand of (I 4 ) h was previously a lower order term, but is now formally of third order in u. However, this can be treated using some of the ideas already seen in the proof of Lemma 4.10. Indeed, by the same argument leading to (4.43), we have for each l ∈ {1, . . . , n} and a.e. x ∈ B R+2ρ the estimate As before, the constant C Σ > 0 is such that the mapping ξ −→ f (x, z, ξ) + C Σ |ξ| 2 is convex for all (x, z, ξ) ∈ Σ. Denoting all but the first term on the RHS of (5.10) as error terms, it follows from (5.10) that Now, in the same way that we dealt with the error terms in Step 2 of the proof of Lemma 4.10, one readily obtains B R+2ρ kη(ṽ . For the first integral on the RHS of (5.11), we integrate by parts and apply Hölder's inequality to obtain Returning to (5.11), we therefore obtain ( . As a consequence, the estimates (5.1) and (5.8) hold, and the arguments of Section 5 therefore apply without any changes.
6 The case k ≥ 3 for general H In this final section we consider a minor extension of Theorem 1.8. Recall that our proof of Theorems 1.5 and 1.8 exploited a cancellation phenomenon between higher order terms arising from (I 2 ) h and (I 3 ) h , where the divergence structure of F ij played a role in estimating (I 2 ) h . When 3 ≤ k ≤ n and H is not necessarily a multiple of the identity, the divergence structure given in (3.4) is more involved and the resulting arguments fall outside the scope of the present paper. That said, if one assumes higher integrability on ∇ 2 u from the outset, the terms (I 2 ) h and (I 3 ) h may be estimated by using Cauchy's inequality and absorbing the resulting negative higher order terms into the positive term (I 1 ) h . This avoids the need to prove any cancellation between (I 2 ) h and (I 3 ) h . We establish: Let Ω be a domain in R n (n ≥ 3), f = f (x, z, ξ) ∈ C 1,1 loc (Ω×R×R n ) a positive function and H ∈ C 1,1 loc (Ω × R × R n ; Sym n (R)). Suppose 3 ≤ k ≤ n, p > kn and u ∈ W 2,p loc (Ω) is a solution to (1.4). Then u ∈ C 1,1 loc (Ω), and for any concentric balls where C is a constant depending only on n, p, R, f, H and an upper bound for u W 2,p (B 2R ) .
Proof. Following the proof of Theorem 1.8 in Section 5.3 but leaving the terms (I 2 ) h and (I 3 ) h untreated, we have for u ∈ W 2,q+k−1 We now suppose further that ∇ 2 u ∈ L q+2k−1
A A remark on the regularity of solutions to the σ 2 -Yamabe equation obtained by vanishing viscosity Let (M 4 , g 0 ) be a 4-manifold with scalar curvature R 0 > 0 and Schouten tensor A 0 satisfying M 4 σ 2 (A 0 ) dv 0 > 0. In [12], the existence of smooth solutions g w δ = e 2w δ g 0 with positive scalar curvature to the fourth order equation is established for each δ ∈ (0, 1], where η is any fixed non-vanishing (0, 2)-tensor and γ 1 < 0 is the conformal invariant obtained by integrating both sides of (A.1). Moreover, solutions are shown to satisfy the uniform estimates w δ W 2,s (M 4 ,g 0 ) ≤ C for all δ ∈ (0, 1], 1 ≤ s < 5, (A.2) where the constant C = C(s) is independent of δ. A heat flow argument is then applied to obtain a conformal metric g with λ(A g ) ∈ Γ + 2 . In this appendix, we show that in the case that (M 4 , g 0 ) is locally conformally flat, we may take the limit δ → 0 more directly in (A.1) to obtain the desired conformal metric with λ(A g ) ∈ Γ + 2 . More precisely, using Theorem 1.1 and a result of [46], we show that, along a subsequence, the solutions w δ converge weakly to a smooth solution of the equation σ 2 (A gw δ ) = −2γ 1 |η| 2 gw δ > 0. To this end, fix 4 < s < 5. We first observe that by (A.2), we can find a sequence δ i → 0 for which w i · · = w δ i converges weakly in W 2,s (M 4 , g 0 ), say to w ∈ W 2,s (M 4 , g 0 ). By the Morrey embedding W 2,s (M 4 , g 0 ) ֒→ C 1,1− 4 s (M 4 , g 0 ), we may assume w i → w in C 1,α (M 4 , g 0 ) for some α > 0. It then follows from [ Moreover, as R gw i > 0 for each i, it follows that R gw ≥ 0, and by (A.4) we therefore have R gw > 0 a.e. If (M 4 , g 0 ) is locally conformally flat, we therefore obtain from Theorem 1.1 that u · · = e −w ∈ C 1,1 (M 4 , g 0 ), and consequently (A.4) is uniformly elliptic at w.
At this point, we apply the Evans-Krylov theorem to obtain u ∈ C 2,α (M 4 , g 0 ). Indeed, by the proof of [7,Theorem 6.6], it suffices to observe that, by Lemmas 4.1 and 4.16, v = l ∆ h ll u is a subsolution to a uniformly elliptic linear equation, namely where F ij is uniformly elliptic and F ij , B i and C are bounded. Furthermore, since f (x, u) · · = −2γ 1 |η(x)| 2 u −2 g 0 = −2γ 1 u 4 |η(x)| 2 g 0 is smooth, standard elliptic regularity ensures that u (and hence w) belongs to C ∞ (M 4 , g 0 ).

B Proof of Lemma 2.1
The proof is a standard argument using Taylor's theorem. We claim that for all u ∈ W 2,s (Ω) and Ω ′ ⋐ Ω satisfying |h| < dist(Ω ′ , ∂Ω), from which the conclusion follows by the continuity of the translation operator in L s (Ω). By density it suffices to prove (B.1) for u ∈ C 2 (Ω). Let Ω ′ be as above. Then for each x ∈ Ω ′ and l ∈ {1, . . . , n}, we have by Taylor's theorem u(x ± he l ) = u(x) ± h∇ l u(x) + h 2 Let s ′ be such that 1 s + 1 s ′ = 1. It follows from (B.2) and Hölder's inequality that for all g ∈ L s ′ (Ω ′ ) satisfying g L s ′ (Ω ′ ) ≤ 1, we have