Interior Estimates for Monge-Amp\`ere Equation in Terms of Modulus of Continuity

We investigate the Monge-Amp\`ere equation subject to zero boundary value and with a positive right-hand side unction assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are obtained in terms of moduli of continuity. We explicate how the estimates depend on various quantities but have them independent of the solution's modulus of convexity. Our main theorem has many useful consequences. One of them is the nonlinear dependence between the H\"older seminorms of the solution and of the right-side function, which confirms the results of Figalli, Jhaveri and Mooney (J. Func. Anal. 2016). Our technique is in part inspired by Jian and Wang (SIAM J. Math. Anal. 2007) which includes using a sequence of so-called sections.


Introduction
In an open and bounded convex set Ω in R n with n ≥ 2, consider a strictly convex function v ∈ C 2 (Ω) ∩ C 0 (Ω) that satisfies the following Monge-Ampère equation ( 1 n . In the study of second order elliptic PDEs, ranging from the Laplacian equation ∆v = f to the fully nonlinear cases such as the MAE, a fundamental question is: can the solution v recover two derivatives of regularity from the right-hand side function f , at least over the interior of the domain? Caffarelli in [Caf90b] proved affirmative answer to this question for (1.1), (1.2) in the cases of Lebesgue integrable f and Hölder continuous f . These are regularity results without estimates. More than two decades later, Figalli et al. proved in [FJM16] interior estimates for MAE in terms of Hölder norms of D 2 v and f . In particular they proved that both upper and lower bounds of the solution's norm depend on the norm of f via nonlinear power laws as it grows larger. Remark 1.1 will say more about how our study examines the nature of this nonlinearity.
In this article, we prove estimates in terms of the modulus of continuity of f , denoted by ω f . Our main result covers a wide range of scenarios even those where f is discontinuous. We show how the moduli of continuity of Dv and D 2 v depend on ω f but does not depend on the modulus of convexity of v itself -the latter dependence is however required in [JW07] and [FJM16]. Although one could argue that the result [Caf90a, Cor. 2] provided some information on what the modulus of convexity in turn depends on, a detailed and integrated study as we conduct in this article is still necessary. In fact, due to the lack of such details in [JW07, Thm. 1(ii)], their Hölder estimate appears to be of linear dependence which is shown to be not the case in [FJM16].
The modulus of continuity of f is defined as It is non-decreasing for q ∈ (0, ∞). If q > diam(Ω), then ω f (q) = ω f diam(Ω) . Function f is called Dini continuous if, for some R > 0, which is a condition that is equivalent for any R > 0. Burch in [Bur78] studied a large family of linear elliptic PDEs using Dini continuity. See also [Kov97] by Kovats for early work on fully nonlinear elliptic PDE using ω f and note their assumption (*) is very close to Hölder continuity. The main theorem is as follows where the assumptions on the domain Ω and the lower bound of f can be easily generalised using John's Lemma 2.1 and affine invariance (3.3) of MAE. Our theorem includes both the result allowing R 0 ω f (q) dq q = ∞ and the result assuming R 0 ω f (q) dq q < ∞. In fact, the proofs for both cases share many common parts and will be presented in a unified way.
As usual, let f Ωρ denote the restriction of f onto Ω ρ so that ω f | Ωρ is defined accordingly.
Theorem 1.1 (Main Theorem). There exist positive constants C ω , β, β 1 ,C that only depend on dimension n so that the following is true. Consider a convex domain Ω satisfying B 1 ⊂ Ω ⊂ B n and a convex function v ∈ C 2 (Ω) ∩ C 0 (Ω) solving MAE (1.1), (1.2) with 1 ≤ f ≤ f max in Ω for constant f max . Fix ρ > 0 and constant q in satisfying q in ∈ (0, ρ] and Let Q in denote various (n, f max )-dependent positive powers of q −1 in times various (n, f max , ρ)-dependent positive constants. For any x, x ′ ∈ Ω ρ , letd := |x − x ′ | > 0 and Then we have If further f is Dini continuous or equivalently E 0 < ∞, then

5)
and (1.6) Finally, the theorem still holds if we let ω f | Ω ρ−q in replace every occurrence of ω f in (1.4) -(1.6), including those used in defining Ed and E 0 .
More information on constants C ω , β and β 1 can be found respectively in (4.3), (6.12) (also §6.4) and Lemma 2.7. They are independent of modulus of convexity of v. Technically, Lemma 5.1 shows this modulus of convexity depends on n, f max , ρ. This information is ultimately linked to various positive powers of q −1 in that is used to define the Q in notation. Subscript "in" stands for "initial" because q in determines the size of the first section of the sequence in the iterative proof.
Corollary 1.2. Consider a convex domain Ω satisfying B 1 ⊂ Ω ⊂ B n and a convex solution to (1.1), (1.2) in C 2,α (Ω) ∩ C 0 (Ω) for some α ∈ (0, 1). Suppose 1 ≤ f ≤ f max in Ω for constant f max . Fix ρ > 0. Let C stand for various (n, f max , ρ)-dependent positive constants, C for various (n, f max )-dependent positive constants and C for various n-dependent positive constants. Then, with H := |f | C α (Ω) , we have Remark. Thus, (1.8) confirms and consolidates Theorems 1 and 2 of [FJM16], at least for α away from 0 and 1; it is slightly stronger for α → 0 and slightly weaker for α → 1. In view of their Theorem 3 regarding the lower bound of D 2 v C α (Ωρ) , we see the sharpness of the power law (1.8).
In fact, a relatively simple examination of their §4.5 also reveals the sharpness of power law (1.7).
Remark. Although their Theorem 3 does not address boundary conditions, a small change to their proof can indeed make (1.2) satisfied in a domain of bounded eccentricity: while they defined in §4.4 a family of examples {u r } r→0 in domain B 1 , we can instead define them in a fixed section of u which is a convex function that they constructed to be independent of r. Also note by their Proof.
Remark. This result may seem weaker than that of [Caf91] which does not require the smallness of ω f (0+). However, we provide explicit dependence of the Hölder exponent (1 − γ) on ω f and shows that (1 − γ) can be approaching 1 if the size of discontinuities in f , measured by ω f (0+), approaches 0. Another route to making (1 − γ) approach 1 is to show v is in Sobolev space W 2,p for arbitrarily large but finite p ( [Caf90b]).
The proof is postponed to Appendix A. The same mollification and compactness argument there can be used to prove many more existence and regularity results as consequences of Theorem 1.1. We only mention two examples that involve logarithmic refinement of continuity function classes. The sketch proofs of them are also in Appendix A.
Example 1: under the same setup as Theorem 1.1, if Example 2: under the same setup as Theorem 1.1, if there exists constant a > 1 so that lim sup then there exists an (n, f max , ρ, ω f , a)-dependent upper bound for The paper is organised as follows. In §2, we collect some well-known results regarding MAE with constant or non-constant right-hand side. We also introduce normalised and quasi-normalised domains. In §3, we introduce sections of MAE solution and a notation F T that denotes a group action of the affine group on functions. We explain what it means for the MAE to be invariant under this group action -also known as affine invariance. Using these concepts, in Lemma 3.1, we study MAE with right-hand side sufficiently close to unit in the supremum norm. Building on this, we employ in §4 an iterative procedure to construct a sequence of shrinking sections of an MAE solution and obtain estimates that are linked to their "eccentricities". This is still a perturbative argument just like in §3 because the right-hand side of MAE, when restricted to the first and largest section in the sequence, must be close to unit. In order to kick-start the iteration for more general right-hand side, in §5, we prove the existence of small enough (first) section about any point inside the domain, and also obtain estimates on the geometry of this section, which will be ultimately linked to the scaling quantity q in in §6 where we complete the proof of the Main Theorem.
Remark. If a global C 1,1 (Ω) bound on v was already available, then our proofs would potentially be much simpler. We are not aware of such literature when one only assumes C 0 regularity of f and does not assume strict or uniform convexity of Ω, or certain regularity of ∂Ω. Consult, e.g., [GT01,§17] for global estimates and more detailed reference list on this topic; also see [FJM16, Remark 3.6] for references of early work.

Acknowledgement
Cheng is supported by the Leverhulme Trust (Award No. RPG-2017-098) and the EPSRC (Grant No. EP/R029628/1). O'Neill is supported by an EPSRC PhD studentship.

Well-known Results on Monge-Ampère Equation
Geometric techniques are crucial in the study of MAE. This is particularly due to its "affine invariant" nature, a term we use loosely to refer to how the Hessian of a function changes under affine transform on the independent variable, which makes the change of the determinant of the Hessian to be simply a constant scaling.
First, we state a fundamental result on the geometry of convex sets.
If Ω ⊂ R n is a bounded, open convex set with nonempty interior and let E be the ellipsoid of minimum volume containing Ω. Then where n −1 E denotes the dilation of E with respect to its centre.
For such ellipsoid E, there exists an affine transform T x = A(x − ξ) for some ξ ∈ R n and real, invertible matrix A that maps n −1 E to the unit ball, which also maps Ω to a so-called normalised set T Ω. Apparently, matrix A can be chosen to be symmetric positive definite without "rotation" (otherwise apply polar factorisation on A). Although many results in literature are stated under the assumption of a normalised domain, we remark that the minimality of E is not necessary for their proofs to work. The essential assumption is that the domain can be affine transformed to one that contains B 1 and is contained in B n . We shall call such domain quasi-normalised.
Next onto properties of convex function u. A basic fact is that it is locally uniformly Lipschitz ( [Fig17,Cor. A.23]). In particular, if a convex function u ≤ 0 on Ω and it is differentiable at x ∈ Ω, then an elementary calculation ([Gut16, Lem. 3.2.1]) gives . Lemma 2.2. Consider a bounded and convex set Ω ⊂ R n and a convex function u ∈ C 2 (Ω) ∩ C 0 (Ω) with u = 0 on ∂Ω. Then, for constant C A , A useful consequence is that points at which the solution takes different values are well separated.
Corollary 2.3. Under the same assumptions and notations as Lemma 2.2, for any x 1 , x 2 ∈ Ω, Another type of well-known results is the comparison principle (e.g., [Gut16, Thm. 1.4.6], [Fig17, Thm. 2.10]). We again adapt it for C 2 functions in the next lemma.
Definition. For a bounded and strictly convex domain Ω, define w = Sol(Ω) to be the strictly convex solution to the boundary value problem For the existence and uniqueness of the (convex) Alexandrov solution w ∈ C 0 (Ω) to (2.3), we refer to [Gut16, Thm. 1.6.2] and [Fig17,Thm. 2.13] with the latter result actually not requiring strict or uniform convexity of Ω. The strict convexity of w inside Ω is due to (2.8). Also, see, e.g., [Fig17,Thm. 3.10] for the proof of w ∈ C ∞ (Ω).
Notations. All versions of w denote convex solutions to the MAE with unit right-hand side.
This means all (classical and generalised) solutions to MAE in this article are strictly convex in the interior of their respective domains, a fact that we will take for granted from now on. Moreover, in any occurrence of Sol(·) in this article, the strict convexity requirement for its definition (2.3) is always satisfied because we will make sure the domain for every instance of Sol(·) is always the section of another convex MAE solution in the interior of its domain, hence being strictly convex.
To finish this section, we state the following results on Monge-Ampère equation with constant right-hand side and sketch ideas behind their proofs.
Lemma 2.6. Let Ω ⊂ R n be strictly convex and satisfy B 1 ⊂ Ω ⊂ B n . Let w := Sol(Ω) and Consider any s ∈ (0, 1 2 n −2 ). If Ω 2s = ∅, then there exists positive constantc 2 (n, s) so that If Ω 4s = ∅, then there exist constantc 3 (n, s) so that (2.10) Proof. The existence and C ∞ (Ω) regularity of w is discussed below (2.3 Combining it with (2.11) and the fact that |w s | ≥ s in Ω 2s proves the upper bound of (2.9). Since all eigenvalues of D 2 w, counting multiplicity, multiply to 1, we prove the lower bound of (2.9). Now that we have shown w is uniformly elliptic on Ω 2s , we then can show the uniform ellipticity for the linearised version of det D 2 w = 1 as well as for the PDE of a given component of Dw that results from taking one derivative of det D 2 w = 1. Note that Corollary 2.3 gives lower bounds on dist(Ω 3s , ∂Ω 2s ) and dist(Ω 4s , ∂Ω 3s ). This means we can apply Evans-Krylov's Hölder interior estimates on D 2 w; see, e.g., [ The next lemma provides estimates on the difference of two solutions to MAE with constant right-hand sides. It requires information of two solutions in a shared domain, but does not require any information at the boundary.

we apply
Jacobi's formula on the t derivative, followed by replacing D 2 (w − w ′ ) = 2 b+b ′ D 2 u, to find, where "adj" denotes matrix adjugate so that adj(D 2 w t ) = (D 2 w t ) −1 det D 2 w t . Since D 2 w t is a convex combination of D 2 w and D 2 w ′ , we apply the Minskowski determinant theorem to obtain a lower bound for det D 2 w t and also use Λ n as the upper bound for det D 2 w t to have so we can view (2.14) as a uniformly elliptic, 2nd order linear PDE for u. We then prove the ℓ = 2 case of (2.12) via combining (2.13) and Schauder's interior estimates applied on (2.14), a process that will incur a factor at the order of the distance-weighted C 0,α (Ω) norm of the coefficients of D 2 u in (2.14) for which there exists an upper bound at the order of adj( Here, an entry of adj(D 2 w t ) equals a sum of products of entries of D 2 w t . Also, consult the proofs of [GT01, Lem. 6.1, Thm. 6.2] to see that the final estimates depend linearly on adj(D 2 w t ) C α and also depend on λ, Λ via power laws (which has been simplified in our case since λ ≤ 1 ≤ Λ). The ℓ = 3 case of (2.12) can be proven by mimicking the above steps, starting with taking an arbitrary directional derivative of (2.14).
Remark. By the definition of u and the assumed C 2 upper bound on w, w ′ , we can also bound Convention on constants. From now on, constants are always strictly positive and, unless their dependence on other parameters is explicitly given, constants only depend on dimension n.

Sections of Solutions to Monge-Ampère Equation
The use of sections in studying the solution regularity of MAE has seen success since Caffarelli's [Caf90b,Caf91,Caf90a]. Here, we define it only for C 1 function v, and in the following form as a mapping from a triplet of arguments to a subdomain, We say this section is about point x. Let S(v, h) := S(v, h, 0).
To describe the effects of affine transforms, we introduce the following notations. For any member of the affine group on R n , i.e., an invertible affine transform T , define For function v, define Then F T is a left group action of the affine group on functions, namely, it satisfies We also have the following "commutation property" between T, F T and the sections which is straightforward to prove when v is C 1 . Next, we define the norm for a square matrix A to be induced from the vector 2-norm, namely and define the norm of affine transform T as T := sup Derivatives under affine transforms are governed by the following elementary properties, We will refer to (3.3) as the "affine invariance" of MAE from here on. Next, for invertible affine transform T and matrix A, we introduce their rescaled versions as so that detT = 1 = detȂ. As an attempt to increase readability, we use [ ] to denote the matrix/transform norm when the argument is known to have unit determinant, namely, ]. Since A 1 has n real eigenvalues (counting multiplicity) whose product equals unit, and among them, the smallest absolute value equals [A −1 1 ] −1 and the greatest absolute value In the rescaled notations, the transform estimates (3.2) amount to ∀ v ∈ C ℓ,α (Ω), α ∈ [0, 1] and integer ℓ ≥ 0. (3.5) The next lemma is on MAE with right-hand side sufficiently close to unit, and will serve different technical purposes in the proofs of Lemmas 4.1 and 4.2. We are motivated by [Caf90b, Lem. 7, Cor. 2, 3] while endeavouring to provide enough details that will be useful in proving the Main Theorem. For example, for the iterative argument that we will employ in §4, estimates (3.15) are concerned with F T w ♯ which will effectively plays the role of w in the next step of iteration, and the right-hand side of (3.15) shows the distance between identity and its Hessian -crucially the square root of this Hessian is used to define the next affine transform in the iterative proof of §4. (3.10) Further, there exist positive constants C 4 ≤ 1 and h c ≤ 1 5 so that if 0 < h ≤ h c , (3.11) 0 ≤ δ ≤ min C 4 h , 1 5 n −2 , (3.12) then S(u, h) ⊂ S(u, L/2) and the following hold.
(i) The affine transform where the hidden constant coefficients in the O( ) notations only depend on n, c 2 , C 2 , C 4 .

   (3.22)
Yet again by (3.18), Perform Taylor expansion on w(x) − w(0) with a 3rd derivative remainder that is bounded using (3.20) to find By (3.10) and (3.22), the right-hand side is bounded by which proves the B r − ⊂ T S(u, h) ⊂ B r + part of (3.13) with r ± satisfying (3.14). By choosing a suitable constant h c as the upper bound for h and a suitable constant C 4 as the upper bound for δ/h, we show the first and last inclusions of (3.13).
For the final part of the proof, first, since det 1 n T = 1/ √ h due to det D 2 w = 1, combining (3.18) and transform estimates (3.5) proves We then validate assumptions (3.6), (3.12) on F T u and g • T −1 in the domain T S(u, h). Since T S(u, h) is quasi-normalised due to (3.13), we apply (3.7) on F T (u + L − h) and F T w ♯ to find Also, considering F T w ♯ in domain T S(u, h), we apply (3.8), (3.20) to obtain C 1,α 0 estimates of D 2 (F T w ♯ ) in T S(u, h/2). Combining the original form of (3.8), (3.20) on w with transform estimate (3.5) also gives the corresponding C 1,α 0 estimates of D 2 F T w in T S(u, L/2) ⊃ T S(u, h/2). Noting that only the ℓ = 2, 3 cases of (3.5) was used and noting det 1 n T = 1/ √ h again, we find these (upper) bounds depend on non-negative powers of h and hence can be relaxed to be independent of h. Finally, since the convexity of u implies T S(u, h/2) ⊃ 1 2 T S(u, h), we use (3.13) to have T S(u, h/2) ⊃ B 1/2 . Therefore, we apply Lemma 2.7 to F T w, F T w ♯ in the domain B 1/2 together with (3.24) and (3.25), noting that D 2 (F T w)(0) = I by the chain rule, to prove (3.15), (3.16).

Iterative Steps
By iterating what was done in Lemma 3.1 going from a section to a smaller section about the same point, we construct a sequence of shrinking sections of the MAE solution and obtain bounds on the eccentricity of the affine transforms that quasi-normalise them.
Assumptions. Suppose V is strictly convex and C 2 in a closed section S(V, h 0 ) for h 0 > 0, and for some ξ ∈ R n . Definitions. For constant h c from Lemma 3.1, define w 0 := Sol(S(V, h 0 )) and Then, iteratively define transforms P 0 , T k and P k+1 for all k = 0, 1, 2, . . . as follows: Apparently, for all k ≥ 0, (4.6) Finally, define so that, by (4.4) and Lemma 4.1. Under the set-up so far in § 4 and the same constants as in Lemma 3.1, we have and We also have what will be referred to as "compound estimates" where the sum is understood as 0 if k = 1, making M 1 = 1. Finally, there exists a constant C 7 that only depends on n so that, for all k ≥ 0, Proof. We prove (4.9) by induction. Fixing any k ≥ 0, suppose we have established the inductive hypothesis -note when k = 0, the inductive hypothesis is just (4.1). Invoke Lemma 3.1 where the role of S(u, L) is played by S(V, h 0 ) when k = 0 and played by P k S(V, h k c ) when k ≥ 1, and the roles of h , δ , g , u (4.14) Then, the definitions of w k , T k means that the roles of w , T are played by All assumptions in Lemma 3.1 are verified as follows. The quasi-normalisation assumption is satisfied due to the inductive hypothesis whereas assumption (3.6) is satisfied due to (4.2) and the roles of u, g as assigned above. The definition of h c makes (3.11) satisfied whereas (4.8) and (4.3) make (3.12) satisfied. With all assumptions verified and noting (4.16) (the equality is due to property (3.1) and determinant (4.6)), we invoke inclusions (3.13) on the above set to prove (4.9) for P k+1 S(V, h k+1 c ). The inductive step is complete. To prove the rest of the Lemma, let us use the same notations as within the above inductive step. Then, (4.16) with the definition of w k+1 implies F P k w k+1 plays the role of Sol(S(u, h)), namely, w ♯ in Lemma 3.1, hence F P k+1 (w k+1 − w k ) plays the role of F T (w ♯ − w). Then, we apply (3.16) to show estimate (4.13) and apply (3.15) to show (4.17) Recalling definition (4.5) but for T k+1 and using a straightforward eigenvalue analysis on (4.17), we have T k+1 − I ≤ 1 2 C 5 δ k /h c which is further bounded by 1 6 due to (4.8) and (4.3). Thus, Combining it with h c ≤ 1 5 from Lemma 3.1, we show (4.10). Also, by Taylor expansion, we have ln [T k+1 ] 2 ≤ 2 ln(1 + 1 , this proves (4.11). To prove (4.12), recall estimates (4.17) with its right-hand side further bounded by Then, substituting it into (4.17), applying transform estimates (3.5) and combining the result with (4.11) applied on P k+1 proves (4.12).

Consequences of Dini continuity.
When f is Dini continuous, estimate (4.12) allows us to approximate D 2 V (0) using sequence {D 2 w k (0)}, leading to an explicit upper bound on D 2 V (0). For conciseness however, we devise the next lemma so that it requires an assumption, namely (4.19), that is not exactly Dini continuity but rather a consequence of it. This latter link will be established in §6. then Proof. We will use without reference the ℓ = 2, α = 0 version of (3.5).
Since {M k } is an increasing sequence, estimate (4.12) and assumption (4.19) imply D 2 w k (0) is a Cauchy sequence and it remains to show the limit of this Cauchy sequence is D 2 V (0).
In fact, since diam(S(V, h k c )) ≤ 2 P −1 k due to (4.9), we use (4.10) to deduce Then, by the C 2 regularity of V , the oscillation of D 2 V in the domain S(V, h k c ) vanishes as k → ∞. Thus, in view of (4.19) and (4.11), we have as k → ∞. (4.23) Second, by (4.7) and (4.22), we have δ k → 0 which means, for any integer i ≫ 1 we can find large enough integer K i so that (4.24) Using K = K i for brevity, we invoke Lemma 3.1 with the roles of S(u, L) , h , δ , u , w , played by Most assumptions of Lemma 3.1 are validated in the same way as in the proof of Lemma 4.1 except that it is (4.24) that implies δ = δ K satisfies assumption (3.12). The role of T of Lemma 3.1 is played by h 2i+1 c − 1 2 D 2 (P K w K )(0) = h −i c T K (recall definition (4.5) for T K ). Then, recalling (3.1) and (4.6), we have that the role of T S(u, h) is played by Then, by inclusions (3.13) and estimates (3.14), we obtain due to (4.24) and the role of h assigned above. Therefore, for any unit vector e ∈ R n , there exists a scalar r ′ so that (r ′ ) 2 = 2 + O(h i c ) and r ′ h i c e is on the boundary of the above middle set. Then, by Taylor expansion and (4.23), recalling K = K i , we find Since this holds for any unit vector e, we have Then, in view of (4.17) with δ k → 0 and the uniform bound (4.19), we prove that the Cauchy sequence D 2 w k (0) indeed tends to D 2 V (0), hence completing the proof of (4.20). Then, combining the k = 0 case of (4.20) and the upper bound in (3.8) applied to w 0 proves the C 2 estimate (4.21).

Estimates on the First Section
Since the Iteration Lemma 4.1 requires in (4.4) small oscillation of the right-hand side function over the first section in the sequence, we now focus on the existence of such first section in the interior of the domain. We also prove estimates on the affine transform that quasi-normalises the first section, which will be needed in piecing together the proof of the Main Theorem (see (6.6)). Caffarelli's strict convexity result ( [Caf90a]) lays the foundation of the following proof.
Proof. We first prove the j = 0 case of (5.1). Suppose not. Then there exist a sequence of convex domains {Ω i } for large integers i. Each domain is sandwiched by B 1 and B n , and there exists a C 2 convex function v i defined on Ω i that satisfies 1 ≤ det On the other hand, {x i }, {z i } are apparently compact. Also, by the Lipschitz bound (2.1), we have that {∇v i (x i )} are compact with a subsequence converging to p ∞ . Then, taking the limit of (5.2) shows exists a point x ∞ ∈ (Ω ∞ ) ρ and a point z which contradicts the strict convexity of v ∞ . Therefore, we have proven the j = 0 case of (5.1).

(5.3)
In fact, the ratio on the left-hand side is apparently invariant under invertible affine transforms. Affine invariance of MAE means that the assumptions on x, y 1 , y 1 2 in (5.3) as well as the range f ∈ [1, f max ] also remain unchanged up to a harmless redefinition of h. So, it suffices to prove (5.3) in a normalised domain Ω ′ = S(v, h, x) with v ∂Ω ′ = 0. Then, by maximum principle Lemma 2.2, Next, the comparison principle Lemma 2.4 implies 1 2 (|y| 2 − n 2 )f max ≤ v(y) for any y ∈ Ω ′ , and thus noting v(x) = −h gives |x| 2 ≤ n 2 − 2h/f max , which implies By Lemma 2.4, we also have 1 2 (|y| 2 − 1) ≥ v(y) for any y ∈ Ω ′ , and thus h = − min Ω ′ v ≥ 1 2 . Therefore, gathering all the estimates so far gives Hence, by co-linearity of x, y 1 , y 1 2 and that y 1 2 is between x, y 1 , we have |x − y 1 2 |/|x − y 1 | bounded from above by θ = θ n, f max , whence proving (5.3).
Since the argument below (5.3) shows that it also applies to the un-normalised section S(v, 2 −j H, x), we prove (5.1) by combinging (5.3) with the previously proven S(v, h, x) ⊂ B ρ (x).
Remark. The above lemma is of interior estimates. If point x were allowed to be arbitrarily close to the boundary, then in the first half of the proof, x ∞ , z ∞ could possibly be situated on ∂Ω ∞ and since there is no a priori information on the curvature of the boundary, the part of ∂Ω ∞ between them could be possibly a straight segment, which would make the contradiction (to strict convexity) approach fail to work.
Lemma 5.2. Consider convex domain Ω ⊂ R n and let v ∈ C 2 (Ω) ∩ C 0 (Ω) be a convex solution to det 1 n D 2 v = f over Ω. Consider a section S(v, h, x) ⊂ Ω and suppose 1 ≤ f ≤ f max in S(v, h, x) for constant f max . Suppose an affine transform T satisfies for some 0 < r 1 ≤ r 2 and ξ 1 , ξ 2 ∈ R n . Then, Next, applying T −1 on the three sets in (5.4) will stretch B r 2 (ξ 2 ) along any direction by a factor of at most T −1 and stretch B r 1 (ξ 1 ) along a certain direction by a factor of exactly T −1 . Therefore proving (5.7). Since T −1 = [T −1 ](det T ) − 1 n , combining the above with (5.5) proves (5.6).
6. Proof of Theorem 1.1 Recall 1 ≤ f ≤ f max in Ω. For a given 0 < ρ < 1 2 diam(Ω), recall the positive θ = θ(n, f max ) < 1 from Lemma 5.1. Recall q in from (1.3) with C ω given in (4.3). Define integer J := ln(ρ/q in )/ln θ −1 which is non-negative due to q in ≤ ρ and θ < 1. Then, Lemma 5.1 together with θ J ≤ q in /ρ implies for constant H = H(n, f max , ρ), We fix this point x from now on and define Let A be the matrix of the affine transform that normalises S(v, 2 −J H, x). Define transform τ which makes τ x = 0 and τ −1 0 = x. By commutation property (3.1), there exists constant h 0 > 0 (the value of which is irrelevant) so that and section S(V, h 0 ) is quasi-normalised modulo a translation. Just to be clear, we have Then, assumptions (4.1), (4.2) are validated. Also, assumption (4.4) is validated due to (1.3), (6.1) and b ≥ 1.
In estimates (5.5) and (5.6) of Lemma 5.2, let the role of h be played by 2 −J H(n, f max , ρ) with 2 J−1 < 2 ln(ρ/q in )/ln(θ −1 ) = (ρ/q in ) ln 2/ln(θ −1 ) ≤ 2 J , so that combining it with Convention. Let Q in denote various values that satisfy dependence (6.6), which is consistent with its definition in Theorem 1.1. A useful estimate: since by (6.1), (6.3) and S(V, h c ) ⊂ S(V, h 0 ), we have τ −1 S(V, h c ) ⊂ B q in (x), and since (4.9) shows P 1 S(V, h c ) = P 1 τ (τ −1 S(V, h c )) ⊃ B 1 , we apply (5.7) to obtain (6.7) Now consider x, x ′ ∈ Ω ρ withd := |x − x ′ | > 0. The Q ind ≥ 1 case of (1.4) follows directly from C 1 estimate (2.1) and maximum principle Lemma 2.2. The Q ind ≥ 1 case of (1.6) follows directly from the C 2 bound (1.5). So it suffices to consider the case of Q ind < 1 from here on. By (4.11) and (6.6), we can always choose a suitable Q in in the condition Q ind < 1 so thatd < 1 2 P 1 τ −1 . Also, (4.10) implies P k τ k≥1 is strictly increasing. Therefore, we obtain that 1 2 P k+1 τ −1 ≤d < 1 2 P k τ −1 for unique integer k ≥ 1, (6.8) and fix this k ≥ 1 from now on. Next, we establish some useful estimates in terms ofd. Using where the upper bound was due to (4.11) and (6.6), we can obtain an upper bound for τ −1 P −1 k by considering (6.8) and P k+1 τ ≤ 7 6 P k τ / √ h c (due to (4.18)). Apparently, we can also use (6.8) to obtain a lower bound. In short, (6.9) Also, combining (4.11), (6.6) and (6.8) shows (6.10) We move on to δ k and M k . Since (4.9) implies τ −1 S(V, h k c ) ⊂ B 2 τ −1 P −1 k , we combine this with definition (4.7) of δ k and definition (6.5) of Φ, noting bound (4.8), to have Then, ln M 2 1.12 C 5 /hc ≤ C ω , and for k ≥ 2, we use monotonicity of ω f and (4.10) to find ln M k+1 whereCd was due to (6.9) and (4.10), and Cq in was due to (6.7). Then, by defining Also, in view of (6.9), (6.11) and (6.13), we have δ k ≤ ω f (Q in Edd). (6.15) In the final stage of the proof, we introduce w v to be the "backward transform" of w k from domain S(V, h k c ) onto τ −1 S(V, h k c ) while also "adding back" the information of the gradient of v, w v (z) τ −1 S(V,h k c ) := F τ −1 w k S(V,h k c ) + b −1 Dv(x) · (z − x). (6.16) Then, in view of (6.4), we have Recall definition w k = Sol(S(V, h k c )). By the quasi-normalisation (4.9) and estimate (4.8) on δ k , the assumptions for Lemma 3.1 are verified with the roles of u, w played according to (4.14) and (4.15), so that we can invoke (3.7) -(3.10) followed by transforming the domain from P k S(V, h k c ) back to τ −1 S(V, h k c ) ⊃ τ −1 P −1 k B 1 (which should be combined with S(V, h k c /2) ⊃ 1 2 S(V, h k c ) by convexity of v) and applying transform estimate (3.2) or (3.5) together with using (6.8) to relax factors of P k τ and using (6.10), (6.13) to relax factors of det where in the last estimate we also used DV (0) = 0.
Define v ′ in the same fashion as (6.3) but in a neighbourhood of x ′ , and define w ′ v in the same fashion as (6.16) but in terms of v ′ , x ′ and constant b ′ = f (x ′ ). Let x m := 1 2 (x + x ′ ). In the rest of §6, we will use transform estimates (3.5) without referencing.
6.1. Second derivative estimate. Suppose f is Dini continuous in B q in (x) and B q in (x ′ ).
Split the difference in the values of D 2 v at x, x ′ so that, by symmetry and without loss of generality, (6.22) We start with estimating the last term. By definition of x m , we have Bd /2 (x m ) ⊂ Bd(x) and Bd /2 (x m ) ⊂ Bd(x ′ ). By symmetry and without loss of generality, we can have w ′ v satisfy (6.18) -(6.20) in the same domain Bd /2 (x m ) except that b should be replaced by b ′ . Then, by (6.18), (6.23) . Therefore, by invoking Lemma 2.7 with the role of δ played by Q in Edd 2 δ k and r, r 0 played by 1 2d , 1 4d , and combining the result with (2.15), (6.15), (6.19), (6.20), we show in Bd /4 (x m ). (6.24) To estimate the second last term of (6.22), we combine (4.13), (6.6) and (6.13) to find Since (6.8) implies Bd /2 (x) ⊂ τ −1 P −1 j+1 B 1/4 for all j ∈ [1, k − 1], summing the above estimates and also using (3.9) to bound D 3 (F τ −1 w 1 ) in τ −1 P −1 1 B 1/4 , we find Then, by a similar argument to the one between (6.11) and (6.12), we show Cd ω f (q) dq q 2 . (6.25) The first term on the right-hand side of (6.22) and also |D 2 v(x)| are estimated as follows. By (6.14), we verify assumption (4.19) of Lemma 4.2. Thus, (4.20) and (4.21) follows, the latter of which together with (6.4), (6.14) shows |D 2 v(x)| ≤ Q in E 0 in terms of v, proving the C 2 bound (1.5). By factoring out M k+1 on the right-hand side of (4.20) and estimating the remaining factor using simply inequality e z − 1 ≤ e z z for z ≥ 0, we have Consider a sequence of MAEs, det 1 n D 2 v i = f i in Ω, v i = 0 on ∂Ω.