Global $W^{2,p}$ estimates for solutions to the linearized Monge–Ampère equations

Abstract

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge–Ampère equations under natural assumptions on the domain, Monge–Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin’s global $W^{2,p}$ estimates for the Monge–Ampère equations.

1 Introduction and statement of the main results

In this paper we consider the linearized Monge–Ampère equations and investigate global $L^p$ estimates for the second derivatives of their solutions. Let $\Omega \subset \mathbb{R }^n$ be a bounded convex domain and $\phi$ be a locally uniformly convex function on $\Omega$. The linearized Monge–Ampère equation corresponding to $\phi$ is

$$\begin{aligned} \mathcal L _{\phi } u:= \sum _{i, j=1}^{n} \Phi ^{ij} u_{ij}= f\quad \text{ in }\,\Omega , \end{aligned}$$

(1.1)

where $\Phi =\big (\Phi ^{ij}\big )_{1\le i, j\le n} := (\det D^2 \phi )~ (D^2\phi )^{-1}$ is the matrix of cofactors of the Hessian matrix $D^2\phi$. As the coefficient matrix $\Phi$ is positive semi-definite, $\mathcal L _{\phi }$ is a linear elliptic partial differential operator, possibly degenerate. The operator $\mathcal L _{\phi }$ appears in several contexts including affine maximal surface equation in affine geometry [30–33], Abreu’s equation in the context of existence of Kähler metric of constant scalar curvatures in complex geometry [10–13, 36], and semigeostrophic equations in fluid mechanics [1, 5, 25]. Solutions of many important problems in these contexts require a deep understanding of interior and boundary behaviors of solutions to (1.1).

The regularity theory for the linearized Monge–Ampère equation was initiated in the fundamental paper [4] by Caffarelli and Gutiérrez. They established an interior Harnack inequality for nonnegative solutions to the homogeneous equation $\mathcal L _{\phi }u=0$ in terms of the pinching of the Hessian determinant $\lambda \le \det D^{2} \phi \le \Lambda$. Their theory is an affine invariant version of the classical Harnack inequality for uniformly elliptic equations with measurable coefficients. This result played a crucial role in Trudinger–Wang’s resolution [31] of Chern’s conjecture in affine geometry concerning affine maximal hypersurfaces in $\mathbb{R }^3$ and in Donaldson’s interior estimates for Abreu’s equation in complex geometry [11]. Another contribution to the regularity theory comes from [18] where Gutiérrez and Tournier derived interior $W^{2,\delta }$ estimates for small $\delta$. The interior regularity for Eq. (1.1) was further developed by Gutiérrez and the second author in [16, 17] where the (sharp) interior $C^{1,\alpha }$ and $W^{2,p}$ estimates, respectively, were obtained.

Regarding the global regularity, by using Caffarelli–Gutiérrez’s interior Harnack estimates and Savin’s localization theorem, Savin and the first author [23] established boundary Hölder gradient estimates for solutions to the linearized Monge–Ampère equation. Furthermore, the first author [21] proved global Hölder estimates for solutions to (1.1) in uniformly convex domains, which are the global counterpart of Caffarelli–Gutiérrez’s interior Hölder estimates [4].

As mentioned above, Gutiérrez and the second author derived in [17] the interior $W^{2,p}$ estimates for solutions of (1.1) in terms of the $L^{q}$-norm of $f$ where $q>\max \{n, p\}$, the pinching of the Hessian determinant $\lambda \le \det D^{2} \phi \le \Lambda$ and the continuity of the Monge–Ampère measure $\det D^2 \phi$. The purpose of our paper is to establish global $W^{2,p}$ estimates for solutions to the linearized Monge–Ampère equation (1.1) under natural assumptions on the domain, Monge–Ampère measures and boundary data.

Our first main theorem is concerned with global $W^{2,p}$ estimates for the linearized equation (1.1) when the Monge–Ampère measure $\det D^2\phi$ is close to a constant.

Theorem 1.1

Let $\Omega$ be a bounded, uniformly convex domain with $\partial \Omega \in C^3$, and let $\phi \in C(\overline{\Omega })$ be a convex function satisfying $\phi =0$ on $\partial \Omega$. Let $u\in C(\overline{\Omega })\cap W^{2,n}_{loc}(\Omega )$ be the solution to the linearized Monge–Ampère equation

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \mathcal{L }_\phi u =f &{} \text{ in }\, \Omega ,\\ u =0 &{} \text{ on }\, \partial \Omega , \end{array}\right. \end{aligned}$$

where $f\in L^{q}(\Omega )$ with $n<q<\infty$. Then, for any $p\in (1, q)$, there exist $0<\epsilon <1$ and $C>0$ depending only on $n, p, q$ and $\Omega$ such that

$$\begin{aligned} \Vert u\Vert _{W^{2,p}(\Omega )}\le C\Vert f\Vert _{L^{q}(\Omega )} \end{aligned}$$

provided that the Monge–Ampère measure of $\phi$ satisfies

$$\begin{aligned} 1-\epsilon \le \det D^2\phi \le 1 + \epsilon \quad \text{ in }\,\Omega .\end{aligned}$$

As a corollary of our method of the proof of Theorem 1.1, we obtain global $W^{2,p}$ estimates for Eq. (1.1) when the Monge–Ampère measure $\det D^2\phi$ is continuous. Our second main theorem states as follows.

Theorem 1.2

Let $\Omega$ be a bounded, uniformly convex domain with $\partial \Omega \in C^3$, and let $\phi \in C(\overline{\Omega })$ be the convex solution to the Monge–Ampère equation

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \det D^2 \phi =g &{} \text{ in }\, \Omega ,\\ \phi =0&{} \text{ on }\, \partial \Omega , \end{array}\right. \end{aligned}$$

where $g\in C(\overline{\Omega })$ is a continuous function satisfying $0<\lambda \le g(x)\le \Lambda$ in $\Omega$. Let $u\in C(\overline{\Omega })\cap W^{2,n}_{loc}(\Omega )$ be the solution to the linearized Monge–Ampère equation

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \mathcal{L }_\phi u =f &{} \text{ in }\, \Omega ,\\ u =\varphi &{} \text{ on }\, \partial \Omega , \end{array}\right. \end{aligned}$$

where $\varphi \in W^{2, s}(\Omega ),\; f\in L^{q}(\Omega )$ with $n<q<s<\infty$. Then, for any $p\in (1, q)$, there exists $C>0$ depending only on $\lambda , \Lambda , n, p, q,s, \Omega$ and the modulus of continuity of $g$ such that

$$\begin{aligned} \Vert u\Vert _{W^{2,p}(\Omega )}\le C\left( \Vert \varphi \Vert _{W^{2, s}(\Omega )} + \Vert f\Vert _{L^{q}(\Omega )}\right) \!. \end{aligned}$$

Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter [35] for fully nonlinear, uniformly elliptic equations, and are also linearized counterparts of Savin’s global $W^{2,p}$ estimates for the Monge–Ampère equation [28]. We note that the continuity condition on the Monge–Ampère measure in Theorem 1.2 is sharp in view of Wang’s counterexample [34] for solutions to the Monge–Ampère equation and the fact that $\mathcal{L }_\phi \phi = n \det D^2\phi =ng$. The global second derivative estimates in Theorems 1.1 and 1.2 depend only on the bounds on the Hessian determinant $\det D^{2} \phi$ and its continuity or closeness to a constant, the geometry of $\Omega$ and the quadratic separation of $\phi$ from its tangent planes on the boundary $\partial \Omega$. This quadratic separation is guaranteed by the $C^{3}$ character of boundary domain $\partial \Omega$, data $\phi \mid _{\partial \Omega }$ and the uniform convexity of $\Omega$ (see Proposition 2.4). Under the assumptions in the main theorems, the linearized Monge–Ampère operator $\mathcal L _{\phi }$ is not uniformly elliptic, i.e., the eigenvalues of $\Phi = (\Phi ^{ij})$ are not necessarily bounded away from $0$ and $\infty .$ Moreover, $\mathcal L _{\phi }$ can be possibly singular near the boundary. The degeneracy and singularity of $\mathcal L _{\phi }$ are the main difficulties in establishing our boundary regularity results. We handle the degeneracy of $\mathcal L _{\phi }$ by working as in [4, 16, 17, 21, 23] with sections of solutions to the Monge–Ampère equations. These sections have the same role as Euclidean balls have in the classical theory. To overcome the singularity of $\mathcal L _{\phi }$ near the boundary, we use a Localization Theorem at the boundary for solutions to the Monge–Ampère equations which was obtained by Savin [26, 27]. In order to obtain the desired global second derivative estimates for solutions $u$ of $\mathcal L _{\phi }$, we need to have good global decay estimates for the distribution function of the second derivatives of $u$. To this end, we approximate $u$ by solutions of $\mathcal L _{w}$ where $w$ solves the standard Monge–Ampère equation $\det D^2 w=1$ with appropriate boundary conditions, and use fine geometric properties of boundary sections for solutions to the Monge–Ampère equation which were obtained recently in [22].

Though the statements of our main theorems are rather succinct, their proofs are quite delicate. There are essentially two main steps for the proof of the main estimates:

Step 1: We consider the quasi distance $d(x,\bar{x})$ induced by the solution $\phi$ to the Monge–Ampère equation and is defined by $d(x,\bar{x})^2 := \phi (x) -\phi (\bar{x}) - \nabla \phi (\bar{x})\cdot (x -\bar{x})$. We then bound the distribution function of the second derivative $D^{2} u$ by the Lebesgue measures of the “bad” sets on whose complements the quasi distance $d(x, \bar{x})$ is comparable to the Euclidean distance $|x-\bar{x}|$ in a controllable manner and the graph of $u$ is touched from above and below by “quasi paraboloids” generated by the quasi distance. Intuitively, the better the regularity of $\phi$ is, the faster these decay estimates can be expected. When $\phi (x) = \left| x\right| ^2/2$, the Monge–Ampère measure $\det D^2\phi$ is the usual Lebesgue measure and $d(x, \bar{x})$ corresponds to the Euclidean distance. In this step, we establish preliminary power decay estimates for the bad sets under natural assumptions on the domain $\Omega$ and the boundary data of $\phi$. As a result, we obtain global $W^{2,\delta }(\Omega )$ estimates for $u$ where $\delta >0$ is small under these natural assumptions provided that the Monge–Ampère measure $\det D^2\phi$ is close to a constant. We also give a more direct proof of global $W^{2,\delta }$ estimates for solutions to the linearized Monge–Ampère equations when the Monge–Ampère measure is only assumed to be bounded away from $0$ and $\infty$. This direct proof is based on interior estimates without resorting to decay estimates of the distribution function of the second derivatives. These estimates, that are of independent interest, are global counterparts of Gutiérrez-Tournier’s interior $W^{2,\delta }$ estimates for solutions to the linearized equation (1.1). Our idea, which is similar to Savin’s arguments in [28], is rather simple but useful for the second step and can be roughly described as follows:

$$\begin{aligned} \text {local estimates} + \text {appropriate covering results}\Longrightarrow \text {global estimates}. \end{aligned}$$

Step 2: We improve the power decay estimates obtained in Step 1 assuming in addition that $\det D^2\phi$ is sufficiently close to $1$. This will involve two main auxiliary results:

• (1) a global stability of cofactor matrices: we prove that the cofactor matrices of the Hessian matrices of two convex functions defined on the same domain are close if their Monge–Ampère measures and boundary values are close in the $L^{\infty }$ norm;

• (2) a global approximation result: we approximate the solution $u$ by smooth solutions of linearized Monge–Ampère equations associated with convex functions whose Monge–Ampère measures and boundary data are close to those of $\phi$.

The main estimates will then follow from a covering theorem for boundary sections and a strong-type $p-p$ estimate for the maximal function corresponding to boundary sections.

Without going into details, we now indicate key technical points that entail for getting global $W^{2,p}$ estimates. First, we show that the distribution function $|\{x: |D^2 u| >\beta \}|$ of the second derivatives of the solution $u$ to $\mathcal L _{\phi } u=f$ has some decay of the form $C\beta ^{-\tau }$ with $\tau >0$ small and $C>0$ depending only on the structural constants in our equation; see Propositions 3.6 and  3.7. In the next step, we refine these decay estimates by working in very small regions of the domain and by rescaling our equation and domain. In this rescaled setting, the constant $C$ above can be improved, roughly by a factor of ; see Lemma 5.1. Here $W$ is the matrix of the cofactors of $D^2 w$ where $w$ is the solution to the standard Monge–Ampère equation $\det D^2 w=1$ having the same boundary values as $\phi$ in small regions. When $\det D^2\phi$ is close to $1$, the term $\Vert \Phi -W\Vert _{L^{n}}$ can be made as small as we want thanks to the stability of cofactor matrices in Proposition 3.14. The term is invariant under a rescaling of our equation that almost preserves the $L^{\infty }$-norm of the second derivative $D^2u.$ There are two natural rescalings of our equation to be explained in Sect. 2 but the aforementioned rescaling is the most crucial. As a consequence, can be made as small as we want provided that $f$ has higher integrability than $L^{n}$, but this is the assumption in our main theorems.

The rest of the paper is organized as follows. In Sect. 2, we recall the main tool used in our proof: the Localization Theorem at the boundary for solutions to the Monge–Ampère equation, and state relevant results on the geometry of their sections. We also discuss properties of solutions to the Monge–Ampère equation and its linearization under suitable rescalings using the Localization Theorem. In addition, we establish boundary $C^{2,\alpha }$ estimates for solutions to the standard Monge–Ampère equations $\det D^2 w=1$ having the same boundary values as $\phi$ on its rescaled sections at the boundary. In Sect. 3, we derive preliminary power decay estimates for the distribution function of the second derivatives of solutions to the linearized Monge–Ampère equations (1.1). We also establish the global $W^{2,\delta }$ estimates for solutions to (1.1), paving the way for proving the global stability of cofactor matrices in Sect. 3.4. Moreover, applying the global stability of cofactor matrices, we obtain in Sect. 3.5 global $W^{2, 1+\epsilon }$ estimates for convex solutions to the linearized Monge–Ampère equations when the Monge–Ampère measure is only assumed to be bounded away from zero and infinity. These estimates can be viewed as affine invariant versions of results obtained by De Phillipis–Figalli–Savin and Schmidt. In Sect. 4.1, we prove the global Hölder continuity property of solutions to (1.1). This property together with the boundary $C^{2, \alpha }$ estimates in Sect. 2 will be instrumental in the global approximation lemmas in Sect. 4.2. In the last section, Sect. 5, by combining these approximation lemmas with the preliminary power decay estimates, we obtain density estimates, which improve the power decay estimates in Sect. 3, when the Monge–Ampère measure $\det D^2\phi$ is close to a constant. The proofs of the main results will be given at the end of this section using these density estimates, a covering theorem and a strong-type $p-p$ estimate for the maximal function with respect to sections.

2 The localization theorem and geometry of the Monge–Ampère equation

The results in this section hold under the following global information on the convex domain $\Omega$ and the convex function $\phi$. We assume there exists $\rho >0$ such that

$$\begin{aligned} \Omega \!\subset \! B_{1/\rho }, ~\text {and for each}\; y\in \partial \Omega ~\text {there is a ball} \; B_\rho (z)\subset \Omega ~\text {that is tangent to}~ \partial \Omega ~ \text {at}\; y.\nonumber \\ \end{aligned}$$

(2.1)

Let $\phi : \overline{\Omega }\rightarrow \mathbb{R }, \phi \in C^{0,1}(\overline{\Omega }) \cap C^2(\Omega )$ be a convex function satisfying

$$\begin{aligned} \det D^2 \phi =g, \quad \quad 0 <\lambda \le g \le \Lambda \quad \text {in}\, \Omega . \end{aligned}$$

(2.2)

Assume further that on $\partial \Omega ,\phi$ separates quadratically from its tangent planes, namely

$$\begin{aligned} \rho \left| x-x_{0}\right| ^2 \le \phi (x)- \phi (x_{0})-\nabla \phi (x_{0}) \cdot (x- x_{0}) \le \rho ^{-1}\left| x-x_{0}\right| ^2, ~\forall x, x_{0}\in \partial \Omega .\nonumber \\ \end{aligned}$$

(2.3)

The section of $\phi$ centered at $x\in \overline{\Omega }$ with height $h$ is defined by

$$\begin{aligned} S_{\phi } (x, h) :=\Big \{y\in \overline{\Omega }: \quad \phi (y) < \phi (x) + \nabla \phi (x) \cdot (y- x) +h\Big \}. \end{aligned}$$

For $x\in \Omega$, we denote by $\bar{h}(x)$ the maximal height of all sections of $\phi$ centered at $x$ and contained in $\Omega$, that is,

$$\begin{aligned} \bar{h}(x): =\sup {\big \{h\ge 0| \quad S_{\phi }(x, h)\subset \Omega \big \}}. \end{aligned}$$

In this case, $S_{\phi }(x, \bar{h}(x))$ is called the maximal interior section of $\phi$ with center $x\in \Omega$.

Remark 2.1

In this paper, we denote by $c, \bar{c}, C, C_{1}, C_{2}, \theta _{0}, \theta _{*}, \ldots$, positive constants depending only on $\rho , \lambda , \Lambda , n$, and their values may change from line to line whenever there is no possibility of confusion. We refer to such constants as universal constants. Small universal constants decrease when $\lambda$ decreases and/or $\Lambda$ increases. Large universal constants increase when $\lambda$ decreases and/or $\Lambda$ increases, etc. Therefore, when $1-\epsilon \le \det D^2\phi \le 1+\epsilon$ with $0<\epsilon <1/2$, we can suppress the dependence of universal constants on $\epsilon$.

2.1 The localization theorem

In this subsection, we recall the main tool to study geometric properties of boundary sections of solutions to the Monge–Ampère equation: the Localization theorem at the boundary for solution to the Monge–Ampère equation (Theorem 2.2). Throughout this subsection, we assume that the convex domain $\Omega$ and the convex function $\phi$ satisfy (2.1)–(2.3). We now focus on sections centered at a point on the boundary $\partial \Omega$ and describe their geometry. Assume this boundary point to be $0$ and by (2.1), we can also assume that

$$\begin{aligned} B_\rho (\rho e_n) \subset \, \Omega \, \subset \{x_n \ge 0\} \cap B_{\frac{1}{\rho }}, \end{aligned}$$

(2.4)

where $\rho >0$ is the constant given by condition (2.1). After subtracting a linear function, we can assume further that

$$\begin{aligned} \phi (0)=0\quad \text{ and } \quad \nabla \phi (0)=0. \end{aligned}$$

(2.5)

If the boundary data has quadratic growth near $\{x_n=0\}$ then, as $h \rightarrow 0,\; S_{\phi }(0, h)$ is equivalent to a half-ellipsoid centered at 0. This is the content of the Localization Theorem proved by Savin [26, 27]. Precisely, this theorem reads as follows.

Theorem 2.2

(Localization Theorem [26, 27]) Assume that $\Omega$ satisfies (2.4) and $\phi$ satisfies (2.2), (2.5), and

$$\begin{aligned} \rho |x|^2 \le \phi (x) \le \rho ^{-1} |x|^2 \quad \text {on} \, \partial \Omega \cap \{x_n \le \rho \}. \end{aligned}$$

Then there exists a constant $k=k(\rho , \lambda , \lambda , n)>0$ such that for each $h \le k$ there is an ellipsoid $E_h$ of volume $\omega _{n}h^{n/2}$ satisfying

$$\begin{aligned} kE_h \cap \overline{\Omega }\, \subset \, S_{\phi }(0, h) \, \subset \, k^{-1}E_h \cap \overline{\Omega }. \end{aligned}$$

Moreover, the ellipsoid $E_h$ is obtained from the ball of radius $h^{1/2}$ by a linear transformation $A_h^{-1}$ (sliding along the $x_n=0$ plane)

$$\begin{aligned} A_hE_h&= h^{1/2}B_1,\, \det A_{h} =1,\, A_h(x) = x - \tau _h x_n,\\ \tau _h&= (\tau _1, \tau _2, \ldots , \tau _{n-1}, 0) \,\quad \text{ and } \quad |\tau _{h}| \le k^{-1} |\log h|. \end{aligned}$$

From Theorem 2.2 we also control the shape of sections that are tangent to $\partial \Omega$ at the origin.

Proposition 2.3

Let $\phi$ and $\Omega$ satisfy the hypotheses of the Localization Theorem 2.2 at the origin. Assume that for some $y \in \Omega$ the section $S_{\phi }(y, h) \subset \Omega$ is tangent to $\partial \Omega$ at $0$, i.e., $\partial S_{\phi }(y, h)\cap \partial \Omega =\{0\}$, for some $h \le c$ with $c$ universal. Then there exists a small positive constant $k_0<k$ depending on $\lambda$, $\Lambda$, $\rho$ and $n$ such that

$$\begin{aligned} \nabla \phi (y)&= a e_n \quad \text{ for } \text{ some } \quad a \in [k_0 h^{1/2}, k_0^{-1} h^{1/2}], \\&k_0 E_h \subset S_{\phi }(y, h) -y\subset k_0^{-1} E_h, \quad \quad k_0 h^{1/2} \le \text {dist}(y,\partial \Omega ) \le k_0^{-1} h^{1/2}, \quad \end{aligned}$$

with $E_h$ and $k$ the ellipsoid and constant defined in Theorem 2.2.

Proposition 2.3 is a consequence of Theorem 2.2 and was proved in [28].

The quadratic separation from tangent planes on the boundary for $\phi$ is a crucial assumption in the Localization Theorem (Theorem 2.2). This is the case for solutions to the Monge–Ampère equation with the right hand side bounded away from $0$ and $\infty$ on uniformly convex domains and smooth boundary data as proved in [27, Proposition 3.2].

Proposition 2.4

Let $\Omega \subset \mathbb{R }^{n}$ be a uniformly convex domain satisfying (2.1) and $\Vert \partial \Omega \Vert _{C^3}\le 1/\rho$. Let $\phi : \overline{\Omega }\rightarrow \mathbb{R }$, $\phi \in C^{0,1}(\overline{\Omega })\cap C^{2}(\Omega )$ be a convex function satisfying $\phi \mid _{\partial \Omega }\in C^{3}$ and

$$\begin{aligned} 0<\lambda \le \det D^{2} \phi \le \Lambda <\infty \quad \text{ in }\,\Omega . \end{aligned}$$

Then, on $\partial \Omega$, $\phi$ separates quadratically from its tangent planes, that is,

$$\begin{aligned} \rho _{0}\left| x-x_{0}\right| ^2 \le \phi (x)- \phi (x_{0})-\nabla \phi (x_{0}) \cdot (x- x_{0}) \le \rho _{0}^{-1}\left| x-x_{0}\right| ^2,~\forall x_{0}, x\in \partial \Omega \end{aligned}$$

for some constant $\rho _{0}>0$ depending only on $n, \rho ,\lambda , \Lambda ,\; \Vert \phi \Vert _{C^{3}(\partial \Omega )}$ and the uniform convexity of $\Omega$.

2.2 Properties of the rescaled functions and boundary regularity estimates

In this subsection, we discuss properties of solutions to the Monge–Ampère equation and its linearization under suitable rescalings and then use these properties to establish a boundary $C^{2, \alpha }$ estimates for solutions to the standard Monge–Ampère equation $\det D^2 w=1$ in our rescaled setting.

Let $\Omega$ and $\phi$ satisfy the hypotheses of the Localization Theorem at the origin. We know that for all $h \le k$, $S_{\phi }(0, h)$ satisfies

$$\begin{aligned} k E_h \cap \overline{\Omega }\subset S_{\phi }(0, h) \subset k^{-1} E_h\cap \overline{\Omega }, \end{aligned}$$

(2.6)

with $A_h$ being a linear transformation and

$$\begin{aligned} \det A_{h}&= 1,\quad E_h=A^{-1}_hB_{h^{1/2}}, \quad A_hx=x-\tau _hx_n,\quad \tau _h \cdot e_n=0, \\&\quad \Vert A_h^{-1}\Vert , \,\Vert A_h\Vert \le k^{-1} |\log h|. \end{aligned}$$

This gives for all $h\le k$

$$\begin{aligned} \overline{\Omega } \cap B^{+}_{h^{2/3}}\subset \overline{\Omega }\cap B^{+}_{ch^{1/2}/\left| \log h\right| }\subset S_{\phi }(0, h) \subset \overline{\Omega } \cap B^{+}_{C h^{1/2} \left| \log h\right| }\subset B^{+}_{h^{1/3} }. \end{aligned}$$

(2.7)

We denote the rescaled function of $\phi$ and the rescaled domain of $\Omega$ by

$$\begin{aligned} \phi _h(x):=\frac{\phi \left( h^{1/2}A^{-1}_hx\right) }{h}\quad \text{ and }\quad \Omega _h:= h^{-1/2}A_h \Omega . \end{aligned}$$

(2.8)

The function $\phi _h$, defined in $\overline{\Omega }_h$, is continuous and solves the Monge–Ampère equation

$$\begin{aligned} \det D^2 \phi _h=g_h(x), \quad \quad \lambda \le g_h(x):=g\left( h^{1/2}A_h^{-1}x\right) \le \Lambda . \end{aligned}$$

By (2.6), the section of $\phi _h$ centered at the origin and with height 1 satisfies

$$\begin{aligned} B_k^{+} \cap \overline{\Omega }_h \subset S_{\phi _{h}}(0, 1)=h^{-1/2}A_hS_{\phi } (0, h)\subset B_{k^{-1}}^+ \cap \overline{\Omega _{h}}. \end{aligned}$$

(2.9)

In what follows, we denote

$$\begin{aligned} U_{h}= S_{\phi _{h}}(0,1). \end{aligned}$$

(2.10)

Now, we discuss two natural rescalings for the linearized Monge–Ampère equation

$$\begin{aligned} \mathcal L _{\phi }u:= \Phi ^{ij} u_{ij}= f \quad \text{ in }\, \Omega . \end{aligned}$$

We focus on the boundary section $S_{\phi }(0, h)$ in the present setting of Theorem 2.2. $L^{\infty }$ -norm preserving rescaling. Consider the following rescaling of functions:

$$\begin{aligned} u_{h}(x):= u\left( h^{1/2} A^{-1}_{h}x\right) \, \quad \text{ and } \,\quad f_{h}(x):= h f\left( h^{1/2} A^{-1}_{h}x\right) ,\quad \text{ for } \, x\in \Omega _h. \end{aligned}$$

Simple computation gives

$$\begin{aligned} D^2 \phi _h= \left( A_{h}^{-1}\right) ^{t} D^2 \phi A_{h}^{-1},\quad D^2 u_h= h\left( A_{h}^{-1}\right) ^{t} D^2 u A_{h}^{-1}, \end{aligned}$$

and

$$\begin{aligned} \Phi _h: = \left( \det D^2 \phi _h\right) (D^2\phi _h)^{-1} = (\det D^2\phi ) A_{h} (D^2\phi )^{-1} (A_{h})^{t} = A_h \Phi (A_h)^{t}. \end{aligned}$$

Therefore, we find that

$$\begin{aligned} \mathcal L _{\phi _{h}} u_{h}= trace (\Phi _h D^2 u_h)=f_{h}~\text {in }~\Omega _{h},\quad \text{ and }\quad \Vert u_h\Vert _{L^{\infty }(\Omega _h)} = \Vert u\Vert _{L^{\infty }(\Omega )}. \end{aligned}$$

Thus this rescaling preserves the $L^{\infty }$-norm of $u$. Since $\Vert f_h\Vert _{L^{n}(\Omega _h)}= h^{1/2}\Vert f\Vert _{L^{n}(\Omega )}$ is small if $f\in L^{n}(\Omega )$ and $h$ small, we can expect that $u_h$ has some nice second derivative estimates, say their boundedness. Given this and as

$$\begin{aligned} D^{2} u (h^{1/2} A^{-1}_{h} x) = h^{-1} (A_{h})^{t} \, D^{2} u_{h}(x) \, A_{h}, \end{aligned}$$

it is again quite natural to expect that $|D^2u |$ behaves like $\frac{1}{h}$ in some part of the section $S_{\phi }(0, h).$ This is what we will prove in Lemma 5.2.

Almost $W^{2,\infty }$ -norm preserving rescaling. The next rescaling almost preserves the $L^{\infty }$-norm of $D^2u$. Under the following rescaling of functions

$$\begin{aligned} \tilde{u}_{h}(x):= h^{-1}u\left( h^{1/2} A^{-1}_{h}x\right) \quad \text{ and } \quad \tilde{f}_{h}(x):= f\left( h^{1/2} A^{-1}_{h}x\right) \quad \text{ for } \,x\in \Omega _h, \end{aligned}$$

we have $\mathcal L _{\phi _{h}} \tilde{u}_{h}= \tilde{f}_{h}~\text {in }~\Omega _{h}$ with

by changing variables and recalling that $\det A_h =1.$ As

$$\begin{aligned} D^{2} \tilde{u}_{h}(x) = \left( A_{h}^{-1}\right) ^{t}D^{2} u\left( h^{1/2} A^{-1}_{h} x\right) A_{h}^{-1}, \end{aligned}$$

the present rescaling almost preserves the $L^{\infty }$-norm of $D^2u$ since

$$\begin{aligned} \Vert D^2\tilde{u}_h\Vert _{L^{\infty }(\Omega _h)}\le k^{-2}\left| \log h\right| ^2 \Vert D^2 u\Vert _{L^{\infty }(\Omega )}. \end{aligned}$$

In principle, the $L^{\infty }$-norm preserving rescaling allows us to find some good points with controlled second derivatives for $u$. Having found them, we would like to propagate them by finding more similar points near by, maybe at the cost of a slightly larger bound on the second derivatives. This is the key technical point of the paper and almost $W^{2,\infty }$-norm preserving rescaling is the means for this; see Lemmas 5.2 and 5.4.

A variant of the $L^{\infty }$ -norm preserving rescaling is the following which applies to sections tangent to the boundary.

$L^{\infty }$ -norm preserving rescaling in a section tangent to the boundary. Consider a prototype section $S_{\phi }(y, h)$ with $h:= \bar{h}(y)\le c.$ By applying Proposition 2.3 to $S_{\phi }(y, h)$, we see that it is equivalent to an ellipsoid $E_h$, i.e.,

$$\begin{aligned} k_0 E_h \subset S_{\phi }(y, h)-y \subset k_{0}^{-1} E_h, \end{aligned}$$

where

$$\begin{aligned} E_h := h^{1/2}A_{h}^{-1}B_1 \quad \text{ with } \quad \det A_{h}=1,\quad \Vert A_{h}\Vert , \, \Vert A_h^{-1} \Vert \le C |\log h|. \end{aligned}$$

We use the following rescalings:

$$\begin{aligned} \tilde{\Omega }_{h} := h^{-1/2} A_{h} (\Omega -y), \end{aligned}$$

and for $x\in \tilde{\Omega }_{h}$

$$\begin{aligned}&\tilde{u}_{h}(x) := u \left( y + h^{1/2} A^{-1}_{h} x\right) , \\&\quad \tilde{\phi }_{h}(x) := h^{-1} \left[ \phi \left( y + h^{1/2} A^{-1}_{h} x\right) -\phi (y)-\nabla \phi (y)\cdot \left( h^{1/2}A^{-1}_h x\right) - h\right] . \end{aligned}$$

Then

$$\begin{aligned} B_{k_0}\subset \tilde{U}_h\equiv S_{\tilde{\phi }_{h}} (0, 1)\equiv h^{-1/2} A_{h} \big (S_{\phi }(y, h) -y\big )\subset B_{k_0^{-1}}. \end{aligned}$$

We have

$$\begin{aligned} \det D^{2} \tilde{\phi }_{h} (x)= \tilde{g}_{h}(x):= g \left( y + h^{1/2} A^{-1}_{h} x\right) , \quad \tilde{\phi }_{h} =0 \, \text{ on } \, \partial S_{\tilde{\phi }_{h}} (0, 1) \end{aligned}$$

and

$$\begin{aligned} \min _{ S_{\tilde{\phi }_{h}} (0, 1)}{\tilde{\phi }_{h}}=-1 =\tilde{\phi }_{h}(0). \end{aligned}$$

Also

$$\begin{aligned} \tilde{\Phi }_{h}^{ij}(\tilde{u}_{h})_{ij} = \tilde{f}_{h}(x) : = hf \left( y + h^{1/2} A^{-1}_{h} x\right) . \end{aligned}$$

Some properties of the rescaled function $\phi _h$ was established in [27] and [23, Lemma 4.2, Lemma 5.4]. For later use, we record them here.

Lemma 2.5

There exists a small constant $c=c(n,\rho ,\lambda ,\Lambda )>0$ such that if $h\le c$, then

• (a) for any $x,x_{0}\in \partial \Omega _{h}\cap B_{2/k}$ we have

  $$\begin{aligned} \frac{\rho }{4}\left| x-x_{0}\right| ^2 \le \phi _h(x) - \phi _h(x_{0}) \!-\!\nabla \phi _h(x_{0}) \cdot (x- x_{0})\le 4\rho ^{-1} \left| x- x_{0}\right| ^2.\quad \quad \quad \quad \end{aligned}$$

  (2.11)

• (b) if $r \le c$ small, we have

  $$\begin{aligned} |\nabla \phi _h| \le C r |\log r|^2 \quad \text{ in } \quad \overline{\Omega }_h \cap B_r. \end{aligned}$$

• (c) $\partial \Omega _h \cap B_{2/k}$ is a graph in the $e_n$ direction whose $C^{1,1}$ norm is bounded by $C h^{1/2}.$

• (d) $\phi _{h}$ satisfies in $U_{h}\equiv S_{\phi _{h}}(0,1)$ the hypotheses of Theorem 2.2 at all points on $\partial U_{h}\cap B_{c}$.

• (e) If $y\in U_{h}\cap B_{c^2}$ then the maximal interior section $S_{\phi _h}(y, \bar{h}(y))$ of $\phi _h$ in $U_{h}$ satisfies:

  $$\begin{aligned} c\ge \bar{h}(y)\ge k_0^2 \, \text {dist}^2 (y, \partial U_h)\quad \text{ and }\quad S_{\phi _h}(y, \bar{h}(y))\subset U_h\cap B_c. \end{aligned}$$

Proof

[23, Lemma 4.2] contains (a)–(c) while its proof implies (d). The statement (e) can be proved as in [23, Lemma 5.4] and we give a complete proof here. Let $y\in U_{h}\cap B_{c^2}$. Then it follows from property (d) and (2.7) that $y\in S_{\phi _h}(0, c^3)$. Hence, $\phi _h(y)\le c^3.$ By [22, Lemma 4.1] we obtain $S_{\phi _h}(0, c^3) \subset S_{\phi _h}(y, \theta _0 c^3)$ and consequently

$$\begin{aligned} {\bar{h}}(y)\le \theta _0 c^3. \end{aligned}$$

(2.12)

Thus, $\bar{h}(y)\le c$ if $c$ is small. Since $S_{\phi _h}(y, \bar{h}(y))$ is balanced around $y$, we can use Theorem 3.3.8 in [15] to conclude that

$$\begin{aligned} S_{\phi _h}(y, {\bar{h}}(y)) \subset B(y, K\,{\bar{h}}(y)^b) \end{aligned}$$

(2.13)

for some universal constants $K, b>0$.

From (2.12) and (2.13) we see that for $c$ small the section $S_{\phi _h}(y, \bar{h}(y))$ is tangent to $\partial \Omega _h$. Let $x_0 \in \partial S_{\phi _h}(y, \bar{h}(y)) \cap \partial \Omega _h$. Applying (2.11) to $x_0$ and $0$, and using property (b) and (2.13), we have

$$\begin{aligned} \frac{\rho }{4} |x_0|^2&\le \phi _h(x_0)= \phi _h(y)+\nabla \phi _h(y)\cdot (x_0 -y) +{\bar{h}}(y)\\&\le c^3 + CK\left| y\right| {\bar{h}}(y)^b\big |\log {|y|}\big |^2+{\bar{h}}(y). \end{aligned}$$

This together with the assumption $|y|< c^2$ and (2.12) implies that $|x_0|<c$. Now, thanks to (d) we can apply Proposition 2.3 at $x_0$ and obtain

$$\begin{aligned} k_0^2 \text {dist}^2 (y, \partial U_h)\le \bar{h}(y)\le k_{0}^{-2} \text {dist}^2 (y, \partial U_h). \end{aligned}$$

Since $S_{\phi _h}(y, \bar{h}(y))-y \subset k_{0}^{-1} E_h$, we find from the definition of $E_h$ and ${\bar{h}}(y)\le \theta _0 c^3$ that

$$\begin{aligned} S_{\phi _h}(y, \bar{h}(y))\subset y + k_{0}^{-1} E_h\subset B_{c^2 + k_0^{-1}k^{-1} \left| \bar{h}(y)\right| ^{1/2}\left| \log \bar{h}(y)\right| }\subset B_c \end{aligned}$$

if $c$ is universally small. $\square$

Remark 2.6

From now on, we fix a universally small constant $c\le k/2$, $c\ll 1$ depending only on $n, \rho , \lambda , \Lambda$ as in the Lemma 2.5.

The rest of this subsection is devoted to establishing boundary $C^{2,\alpha }$ estimates for the convex solution $w$ to the standard Monge–Ampère equation

$$\begin{aligned} \left\{ \begin{array}{ll} \det D^2 w =1 &{}\quad \text{ in }\ U_h :=S_{\phi _{h}}(0, 1),\\ w =\phi _h&{}\quad \text{ on }\ \partial U_h. \end{array}\right. \end{aligned}$$

(2.14)

For this, we first show in the next lemma that $w$ separates quadratically from its tangent planes on the boundary of $U_h$.

Lemma 2.7

Let $\Omega _h, \phi _h$ and $U_h$ be as in (2.8) and (2.10) with $h\le c$. Let $w\in C(\overline{U_h})$ be the convex solution to (2.14). Then there exist universal constants $\delta , \theta >0$ depending only on $n, \rho , \lambda , \Lambda$ such that for any $x_0 \in \partial U_h\cap B_{c}$,

$$\begin{aligned} x_{n+1}= \phi _h(x_0) + \langle \nabla \phi _h(x_0) -2\delta ^{1-n} k^{-1} \nu _{x_0}, x-x_0\rangle =: \bar{l}_{x_0}(x) \end{aligned}$$

is a supporting hyperplane in $\overline{U_h}$ to $w$ at $x_0$, and

$$\begin{aligned} \theta \, |x - x_0|^2 \le w(x) - \bar{l}_{x_0}(x) \le \theta ^{-1} \, | x- x_0|^2\quad \text{ for } \text{ all }\quad x\in \partial U_h. \end{aligned}$$

(2.15)

Here $\nu _{x_0}$ denotes the unit inner normal to $\partial \Omega _h$ at $x_0$.

Proof

For $x_0 \in \partial U_h\cap B_{c}$, let $l_{x_0}(x) := \phi _h(x_0) +\nabla \phi _h(x_0) \cdot (x- x_0)$. Then by Lemma 2.5(a),

$$\begin{aligned} \frac{\rho }{4} |x-x_0|^2 \le \phi _h(x) - l_{x_0}(x)\le \frac{4}{\rho } |x-x_0|^2 \quad \forall x\in \partial U_h \cap \partial \Omega _h. \end{aligned}$$

(2.16)

By Lemma 2.5(d) and a consequence of the Localization Theorem 2.2 (see (2.7)), there is $r_0>0$ universally small depending only on $n, \rho , \lambda , \Lambda$ such that

$$\begin{aligned} S_{\phi _h}(x_0, r_0)\subset B\left( x_0, \frac{k}{2}\right) \cap \overline{U_h}\subset B_{k}\cap \overline{U_h}. \end{aligned}$$

This gives $\phi _h(x) \ge l_{x_0}(x) + r_0$ for all $x\in \partial U_h {\setminus } \partial \Omega _h$, and consequently, by (2.9)

$$\begin{aligned} \phi _h(x) \ge l_{x_0}(x) + \frac{k^2 r_0}{4} |x- x_0|^2\quad \forall x\in \partial U_h {\setminus } \partial \Omega _h. \end{aligned}$$

(2.17)

Define

$$\begin{aligned} w^-(x)&:= l_{x_0}(x) + \delta \, \Big [|x - x_0|^2 - |(x - x_0) \cdot \nu _{x_0}|^2\Big ]\\&+ \delta ^{1-n} \, \Big [ |(x - x_0) \cdot \nu _{x_0}|^2 -2 k^{-1} (x - x_0) \cdot \nu _{x_0}\Big ]\quad \forall x\in \overline{U_h}, \end{aligned}$$

where

$$\begin{aligned} \delta :=\min {\left\{ \frac{\rho }{4}, \frac{k^2 r_0}{4}\right\} }. \end{aligned}$$

Then $w^-$ is a convex function in $\overline{U_h}$ satisfying $D^2 w^- = 2\delta \, [ I + (\delta ^{-n} -1) \, \nu _{x_0} \otimes \nu _{x_0}]$. Therefore,

$$\begin{aligned} \det D^2 w^- = (2\delta )^n \delta ^{-n} = 2^n> 1 =\det D^2 w \quad \text{ in }\quad U_h. \end{aligned}$$

(2.18)

For $x\in \partial U_h\cap \partial \Omega _h$, we obtain from $0\le (x - x_0) \cdot \nu _{x_0} \le 2 k^{-1}$ and the first inequality in (2.16) that

$$\begin{aligned} w^-(x) \le l_{x_0}(x) {+} \delta |x - x_0|^2 \le \phi _h(x)-\frac{\rho }{4} |x-x_0|^2 {+}\delta |x - x_0|^2\le \phi _h(x)= w(x). \end{aligned}$$

On the other hand, for $x\in \partial U_h{\setminus } \partial \Omega _h$ by using (2.17) we have

$$\begin{aligned} w^-(x)\le l_{x_0}(x) + \delta |x - x_0|^2 \le l_{x_0}(x) + \frac{k^2 r_0}{4} |x- x_0|^2 \le \phi _h(x)=w(x). \end{aligned}$$

Therefore, $w\ge w^-$ on $\partial U_h$. It follows from this, (2.18) and the comparison principle that $w(x)\ge w^-(x)$ in $\overline{U_h}$. Hence,

$$\begin{aligned} w(x)&\ge \bar{l}_{x_0}(x) + \delta \, \Big [|x - x_0|^2 - |(x - x_0) \cdot \nu _{x_0}|^2\Big ] + \delta ^{1-n} \, |(x - x_0) \cdot \nu _{x_0}|^2 \nonumber \\&\ge \bar{l}_{x_0}(x) + \delta |x - x_0|^2 \quad \text{ in }\quad \overline{U_h}. \end{aligned}$$

(2.19)

In particular, $w(x)\ge \bar{l}_{x_0}(x)$ for all $x\in \overline{U_h}$. Since $\bar{l}_{x_0}(x_0) = \phi _h(x_0) = w(x_0)$, we then conclude that $x_{n+1}=\bar{l}_{x_0}(x)$ is a supporting hyperplane in $\overline{U_h}$ to $w$ at $x_0$.

We now show the second inequality in (2.15). For this, we first recall that $0\le \phi _h\le 1$ in $U_{h}$ and by Lemma 2.5(b), we find that for $M:= 1 + 2k^{-1} C \,c\left| \log c\right| ^2,$

$$\begin{aligned} \phi _h(x)\le \phi _h(x_0) +\nabla \phi _h(x_0) \cdot (x- x_0) + M\equiv l_{x_0}(x) + M\quad \forall x\in \overline{U_h}. \end{aligned}$$

(2.20)

We now compare $w$ with $w^{+}$ defined by

$$\begin{aligned} w^+(x) := l_{x_0}(x)+ 2\Theta k^{-1}\, (x - x_0)\cdot \nu _{x_0} + \Theta \, \Big [|x - x_0|^2 - | (x- x_0)\cdot \nu _{x_0}|^2\Big ] \quad \forall x\in \overline{U_h}, \end{aligned}$$

where

$$\begin{aligned} \Theta := \max {\left\{ \frac{4}{\rho }, \, \frac{4 M}{k^2}\right\} }. \end{aligned}$$

Clearly, $w^+$ is a convex function in $\overline{U_h}$ satisfying

$$\begin{aligned} \det D^2 w^+ =0< 1 =\det D^2 w \quad \text{ in }\, U_h. \end{aligned}$$

(2.21)

For $x\in \partial U_h\cap \partial \Omega _h$, we obtain from the second inequality in (2.16) and $\Theta \ge \frac{4}{\rho }$ that

$$\begin{aligned} w^+(x)&= l_{x_0}(x) + \Theta |x - x_0|^2 + \Theta \Big [2 k^{-1} \, (x - x_0) \cdot \nu _{x_0} - |(x - x_0) \cdot \nu _{x_0}|^2\Big ]\\&\ge \phi _h(x)-\frac{4}{\rho } |x-x_0|^2 +\Theta |x - x_0|^2\ge \phi _h(x)= w(x). \end{aligned}$$

For $x\in \partial U_h{\setminus } \partial \Omega _h$, we have $\left| x-x_0\right| \ge k/2$ and thus, by using (2.20) we obtain

$$\begin{aligned} w^+(x) \ge l_{x_0}(x)+ \Theta |x - x_0|^2 \ge \phi _h(x) - M + \frac{k^2 \Theta }{4} \ge \phi _h(x)=w(x). \end{aligned}$$

Therefore, $w\le w^+$ on $\partial U_h$. It follows from this, (2.21) and the comparison principle that $w\le w^+$ in $\overline{U_h}$. In particular,

$$\begin{aligned} w(x)&\le l_{x_0}(x)+ 2\Theta k^{-1}\, (x - x_0)\cdot \nu _{x_0} + \Theta \, |x - x_0|^2\\&= \bar{l}_{x_0}(x)+ 2 k^{-1} \big (\delta ^{1-n} +\Theta \big )\, (x - x_0)\cdot \nu _{x_0} + \Theta \, |x - x_0|^2 \quad \forall x\in \overline{U_h}. \end{aligned}$$

We then use Lemma 2.5(c) for $x\in \partial U_h \cap \partial \Omega _h$ and the fact that $k/2\le |x - x_0| \le 2/k$ for $x\in \partial U_h{\setminus } \partial \Omega _h,$ to conclude that

$$\begin{aligned} w(x) \le \bar{l}_{x_0}(x) + C|x - x_0|^2 \quad \forall x\in \partial U_h. \end{aligned}$$

This together with (2.19) gives the quadratic separation in (2.15). $\square$

Thanks to the quadratic separation property of $w$ in Lemma 2.7, we can now apply Savin’s boundary $C^{2,\alpha }$ estimates for solutions to the Monge–Ampère equations [27] to get boundary $C^{2,\alpha }$ estimates for $w$ when $\partial \Omega \cap B_{\rho }$ and $\phi \mid _{\partial \Omega \cap B_{\rho }}$ are $C^{2,\alpha }$ and $h$ is small.

Proposition 2.8

Let $\Omega$ and $\phi$ satisfy the hypotheses of the Localization Theorem 2.2 at the origin. Assume in addition that $\partial \Omega \cap B_{\rho }$ is $C^{2,\alpha }$ and $\phi \in C^{2,\alpha }(\partial \Omega \cap B_{\rho })$ for some $\alpha \in (0,1)$. Let $\Omega _h$, $\phi _h$, $U_h$ and $w$ be as in Lemma 2.7. Then there exists $h_0>0$ depending on $n, \lambda , \Lambda , \rho , \alpha$, $\Vert \partial \Omega \cap B_{\rho }\Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega \cap B_{\rho })}$ such that for any $h\le h_0$, we have

$$\begin{aligned}&\Vert w\Vert _{C^{2,\alpha } (\overline{B_{c}\cap U_h})} \le c_0^{-1} \quad \text{ and }\quad c_{0}I_{n}\le D^2 w\le c_{0}^{-1} I_{n}~\text {in}~ B_{c}\cap U_h \end{aligned}$$

(2.22)

for some $c_{0}>0$ depending only on $n, \lambda , \Lambda , \alpha$ and $\rho$.

Now, let us assume in addition that $\partial \Omega$ and $\phi |_{\partial \Omega }$ are $C^{2,\alpha }$ at the origin for some $\alpha \in (0,1)$, that is, we assume that for $x=(x', x_n)\in \partial \Omega \cap B_{\rho }$, we have

$$\begin{aligned} \left| x_{n}- q(x')\right| \le M\left| x'\right| ^{2+\alpha } \quad \text{ and }\quad \left| \phi - p(x')\right| \le M\left| x'\right| ^{2+\alpha }, \end{aligned}$$

where $p(x')$ and $q(x')$ are homogeneous quadratic polynomials.

If $h$ is sufficiently small, then the corresponding rescaling $\phi _{h}$ satisfies the hypotheses of $\phi$ in which the constant $M$ is replaced by an arbitrary small constant $\sigma$.

Lemma 2.9

([27, Lemma 7.4]) Given any $\sigma >0$, there exists a small positive constant $h = h_{0}(M,\sigma , \alpha , n,\lambda ,\Lambda ,\rho )$ such that on $\partial \Omega _{h}\cap B_{k^{-1}}$, we have

$$\begin{aligned} \left| x_{n}- q_{h}(x')\right| \le \sigma \left| x'\right| ^{2+\alpha },\quad \left| q_{h}(x')\right| \le \sigma \quad \text {and}\quad \left| \phi _{h}- p(x')\right| \le \sigma \left| x'\right| ^{2+\alpha }, \end{aligned}$$

where $q_{h}(x') := h^{1/2} q(x')$ is a homogeneous quadratic polynomial.

Remark 2.10

By inspecting the proof of Lemma 7.4 in [27], we see that the following more precise statement holds true: There exists $C= C(M, n, \lambda ,\Lambda , \rho )>0$ such that for any $h\le c$, on $\partial \Omega _{h}\cap B_{k^{-1}}$ we have

$$\begin{aligned} \left| x_{n}- q_{h}(x')\right| \le C h^{\frac{1+\alpha }{2}}\left| x'\right| ^{2+\alpha },\quad \left| q_{h}(x')\right| \le C h^{\frac{1}{2}}~ \quad \text {and}\quad ~ \left| \phi _{h}- p(x')\right| \le C h^{\frac{\alpha }{2}}\left| x'\right| ^{2+\alpha }. \end{aligned}$$

Proof of Proposition 2.8.

Let $M :=max{\big \{ \Vert \partial \Omega \cap B_{\rho }\Vert _{C^{2,\alpha }}, \Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega \cap B_{\rho })}\big \}}$ and let $h_0$ be the small constant in Lemma 2.9 corresponding to $M$ and $\sigma =1$. Then by our assumptions, Lemma 2.9, Remark 2.10 and Lemma 2.7, we can apply [27, Corollary 7.2] to conclude that there exist $C, \delta >0$ depending on $n,\lambda , \Lambda , \alpha$ and $\rho$ such that

$$\begin{aligned} \Vert w\Vert _{C^{2,\alpha }(\mathcal C _0 \cap B_\delta (0))}\le C, \end{aligned}$$

where $\mathcal C _0 := \{x\in \mathbb{R }^n_+ :\, |x'|\le x_n\}$ is the cone at the origin with opening $\theta = \pi /4$.

By varying the point under consideration, we then conclude in the similar fashion that

$$\begin{aligned} \Vert w\Vert _{C^{2,\alpha }(\mathcal C _{x_0} \cap B_\delta (x_0))}\le C \quad \forall x_0\in \partial \Omega _h \cap \overline{B_{c}}. \end{aligned}$$

(2.23)

Here $\mathcal C _{x_0} := \{x\in \mathbb{R }^n_+ :\, |x - x_0|^2\le 2 |(x- x_0)\cdot \nu _{x_0}|^2\}$ is the cone at $x_0$ with opening $\theta = \pi /4$ and in the direction of $\nu _{x_0}$, the unit inner normal to $\partial \Omega _h$ at $x_0$. As a consequence of (2.23) and Caffarelli’s interior $C^{2,\alpha }$ estimates [2], we obtain the first estimate in (2.22) from which the second estimate in (2.22) follows. $\square$

2.3 The classes $\mathcal P _{\lambda ,\Lambda , \rho , \kappa ,\alpha }$ and $\mathcal P _{\lambda ,\Lambda , \rho , \kappa , *}$

Fix $n, \rho , \lambda , \Lambda , \kappa$ and $\alpha$. We define the classes $\mathcal P _{\lambda ,\Lambda , \rho , \kappa , \alpha }$ and $\mathcal P _{\lambda ,\Lambda , \rho , \kappa , *}$ consisting of the triples $(\Omega , \phi , U)$ satisfying the following sets of conditions $(i)$–$(vii)$ and $(i)$–$(vi)$, respectively:

• (i) $0 \in \partial \Omega , U\subset \Omega \subset \mathbb{R }^{n}$ are bounded convex domains such that

  $$\begin{aligned} B_{k}^{+}\cap \overline{\Omega }\subset \overline{U}\subset B_{k^{-1}}^{+}\cap \overline{\Omega }.\end{aligned}$$

• (ii) $\phi :\overline{\Omega }\rightarrow \mathbb{R }^{+}$ is convex satisfying $\phi =1$ on $\partial U\cap \Omega$ and

  $$\begin{aligned}&\phi (0)=0, \quad \nabla \phi (0)=0, \quad \lambda \le \det D^{2} \phi \le \Lambda \, \text{ in } \, \Omega ,\\&\partial \Omega \cap \{\phi <1\}= \partial U\cap \{\phi <1\}. \end{aligned}$$

• (iii) (quadratic separation)

  $$\begin{aligned} \frac{\rho }{4}\left| x-x_{0}\right| ^2&\le \phi (x)- \phi (x_{0})-\nabla \phi (x_{0}) \cdot (x- x_{0})\\&\le \frac{4}{\rho }\left| x-x_{0}\right| ^2 \quad \quad \forall x, x_{0}\in \partial \Omega \cap B_{\frac{2}{k}}. \end{aligned}$$

• (iv) (flatness)

  $$\begin{aligned} \partial \Omega \cap \{\phi <1\} \subset G \subset \{x_n \le \kappa \} \end{aligned}$$

  where $G\subset B_{2/k}$ is a graph in the $e_n$ direction and its $C^{1,1}$ norm is bounded by $\kappa$.

• (v) (localization and gradient estimates) $\phi$ satisfies in $U$ the hypotheses of the Localization Theorem 2.2 at all points on $\partial U\cap B_{c}$ and

  $$\begin{aligned} \left| \nabla \phi \right| \le C_{0}~\quad \text {in}~U\cap B_{c}. \end{aligned}$$

• (vi) (Maximal sections around the origin) If $y\in U\cap B_{c^2}$ then the maximal interior section of $\phi$ in $U$ satisfies:

  $$\begin{aligned} c\ge \bar{h}(y)\ge k_0^2\, \text {dist}^2 (y, \partial U) \quad \text{ and }\quad S_{\phi _h}(y, \bar{h}(y))\subset U\cap B_c. \end{aligned}$$

• (vii) (Pogorelov estimates)

  $$\begin{aligned} \Vert \partial U\cap B_c\Vert _{C^{2,\alpha }}\le c_{0}^{-1} \end{aligned}$$

  and if $w$ is the convex solution to

  $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \det D^2 w =1 &{} \text{ in }\, U\\ w =\phi &{} \text{ on }\, \partial U, \end{array}\right. \end{aligned}$$

  (2.24)

  then

  $$\begin{aligned} \Vert w\Vert _{C^{2,\alpha }(\overline{B_{c}\cap U})}\le c_0^{-1} \quad \text{ and }\quad c_{0}I_{n}\le D^2 w\le c_{0}^{-1} I_{n}~\text {in }~ B_{c}\cap U. \end{aligned}$$

The constants $k, k_0, c, C_0$ above depend only on $n, \rho , \lambda , \Lambda$ and $c_0$ depends also on $\alpha$.

Remark 2.11

If $(\Omega , \phi , U)\in \mathcal P _{\lambda , \Lambda , \rho , \kappa , *}$ then the Pogorelov estimates in $(vii)$ might not hold. However, $\phi$ satisfies in $U$ the hypotheses of the Localization Theorem 2.2 at all points on $\partial U\cap B_{c}$. Thus, if $w$ is the convex solution to (2.24), then by inspecting the proof of Lemma 2.7, we see that $w$ separates quadratically from its tangent planes at any point $x_0\in \partial U\cap B_c$, that is,

$$\begin{aligned} \,\theta \, |x - x_0|^2 \le w(x) \!-\! w(x_0)\!-\!\nabla w(x_0)\cdot (x-x_0) \le \theta ^{-1} \, | x- x_0|^2\quad \text{ for } \text{ all }\, x\in \partial U. \end{aligned}$$

We summarize the discussion at the end of Sect. 2.2, Lemmas 2.5,  2.9 and Proposition 2.8 in the following proposition.

Proposition 2.12

Let $\Omega$ and $\phi$ satisfy the hypotheses of Theorem 2.2 at the origin. Assume in addition that $\partial \Omega \cap B_{\rho }$ is $C^{2,\alpha }$ and $\phi \in C^{2,\alpha }(\partial \Omega \cap B_{\rho })$ for some $\alpha \in (0,1)$. Then there exists $h_0>0$ depending only on $n, \lambda , \Lambda , \rho , \alpha ,\; \Vert \partial \Omega \cap B_{\rho }\Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega \cap B_{\rho })}$ such that for $h\le h_0$ we have

$$\begin{aligned} \big (\Omega _{h}, \phi _{h}, S_{\phi _{h}}(0, 1)\big )\in \mathcal P _{\lambda ,\Lambda , \rho , C h^{1/2}, \alpha }\quad \text {and}\quad \Vert \partial \Omega _{h}\cap B_{1/k}\Vert _{C^{2,\alpha }}\le C'\, h^{1/2}. \end{aligned}$$

Here $C$ depends only on $n, \lambda ,\Lambda$ and $\rho$; $C'$ depends only on $n, \lambda ,\Lambda , \rho ,$ $\Vert \partial \Omega \cap B_{\rho }\Vert _{C^{2,\alpha }},$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega \cap B_{\rho })}$.

2.4 Geometric properties of boundary sections of solutions to Monge–Ampère equation

In this subsection, we recall some important properties of boundary sections of solutions to the Monge–Ampère equations established in [22]: the engulfing and dichotomic properties, volume estimates, a covering theorem and strong type $p-p$ estimates for the maximal functions corresponding to small sections including boundary ones.

The engulfing property and volume estimates are summarized in the following theorem.

Theorem 2.13

Assume that $\Omega$ and $\phi$ satisfy (2.1)–(2.3). Then,

• a. (Engulfing property) There exists $\theta _{*}>0$ depending only on $\rho , \lambda , \Lambda$ and $n$ such that if $y\in S_{\phi }(x, t)$ with $x\in \overline{\Omega }$ and $t>0$, then $S_{\phi }(x, t)\subset S_{\phi }(y, \theta _{*}t).$

• b. (Volume estimates) There exist constants $c_{*}, C_{1}, C_{2}$ depending only on $\rho , \lambda , \Lambda$ and $n$ such that for any section $S_{\phi }(x, t)$ with $x\in \overline{\Omega }$ and $t\le c_{*}$, we have

  $$\begin{aligned} C_{1} t^{n/2}\le |S_{\phi }(x, t)|\le C_{2} t^{n/2}. \end{aligned}$$

Our next property is a dichotomy for sections of solutions to the Monge–Ampère equations: any section is either an interior section or included in a boundary section with a comparable height.

Proposition 2.14

$($Dichotomy$)$ Assume that $\Omega$ and $\phi$ satisfy (2.1)–(2.3). Let $S_{\phi }(x, t)$ be a section of $\phi$ with $x\in \overline{\Omega }$ and $t>0$. Then one of the following is true:

• (i) $S_{\phi }(x, 2t)$ is an interior section, that is, $S_{\phi }(x, 2t)\subset \Omega$;

• (ii) $S_{\phi }(x, 2 t)$ is included in a boundary section with comparable height, that is, there exists $z\in \partial \Omega$ such that $S_{\phi }(x, 2t)\subset S_{\phi }(z, \bar{c}t)$ for some constant $\bar{c}=\bar{c}(\rho ,\lambda , \Lambda , n)>0.$

Our covering theorem states as follows.

Theorem 2.15

(Covering theorem) Assume $\Omega$ and $\phi$ satisfy (2.1)–(2.3). Let $\mathcal{O }\subset \overline{\Omega }$ be a Lebesgue measurable set and $\epsilon >0$ small. Suppose that for each $x\in \mathcal{O }$ a section $S_{\phi }(x,t_x)$ is given with

$$\begin{aligned} \frac{|S_{\phi }(x,t_x) \cap \mathcal{O }|}{|S_{\phi }(x,t_x)|} = \epsilon . \end{aligned}$$

Then if $\sup \{t_{x}: x\in \mathcal O \}\,{<}\,\infty$, there exists a countable subfamily of sections $\{S_{\phi }(x_k,t_k)\}_{k=1}^\infty$ satisfying

$$\begin{aligned} \mathcal{O }\subset \bigcup _{k=1}^\infty {S_{\phi }(x_k, t_k)}\quad \text {and}\quad |\mathcal{O }|\le \sqrt{\epsilon } \, \big | \bigcup _{k=1}^\infty {S_{\phi }(x_k, t_k)}\big |. \end{aligned}$$

Finally, we have the following global strong-type $p-p$ estimates for the maximal function corresponding to small sections.

Theorem 2.16

(Strong-type p-p estimates) Assume that $\Omega$ and $\phi$ satisfy (2.1)–(2.3). For $f\in L^1(\Omega )$, define

$$\begin{aligned} \mathcal M (f)(x)=\sup _{t\le c} \dfrac{1}{|S_\phi (x,t)|}\int \limits _{S_\phi (x,t)}|f(y)|\, dy\quad \forall x\in \Omega . \end{aligned}$$

Then, for any $1<p<\infty$, there exists $C_p>0$ depending on $p,\; \rho ,\; \lambda ,\; \Lambda$ and $n$ such that

$$\begin{aligned} \Vert \mathcal M (f)\Vert _{L^{p}(\Omega )}\le C_p\, \Vert f\Vert _{L^{p}(\Omega )}. \end{aligned}$$

3 Global power decay and $W^{2,\delta }$ estimates

In this section, we establish preliminary power decay estimates for the distribution function of the second derivatives of solutions to the linearized Monge–Ampère equations and also their global $W^{2,\delta }$ estimates. We also show under suitable geometric conditions, the cofactor matrices of the Hessian matrices of two convex functions defined on the same domain are close if their Monge–Ampère measures and boundary values are close in the $L^{\infty }$ norm.

We begin this section by recalling the definitions, introduced in [17], of the quasi distance $d(x,x_0)$ generated by a convex function $\phi$ and the set $G_M(u, \Omega )$ where the function $u$ is touched from above and below by “quasi paraboloids” generated by this quasi distance.

Definition 3.1

Let $\Omega$ be a bounded convex set in $\mathbb{R }^n$ and let $\phi \in C^1(\Omega )$ be a convex function. For any $x\in \Omega$ and $x_0\in \Omega$, we define the quasi distance $d(x, x_0)$ by

$$\begin{aligned} d(x,x_0)^2 := \phi (x) - \phi (x_0) -\nabla \phi (x_0)\cdot (x-x_0). \end{aligned}$$

Definition 3.2

Let $\Omega$ and $\phi$ be as in Definition 3.1. For $u\in C(\Omega )$ and $M>0$, we define

$$\begin{aligned} G_M(u,\Omega )&= \left\{ {\bar{x}}\in \Omega : u \text { is differentiable at } \bar{x} \hbox { and } |u(x)-u(\bar{x})- \nabla u(\bar{x})\cdot (x-\bar{x})|\right. \\&\quad \left. \le \frac{M}{2} d(x,\bar{x})^2\,\forall x\in \Omega \right\} . \end{aligned}$$

We call $\frac{M}{2}d(x,\bar{x})^2$ and $-\frac{M}{2}d(x,\bar{x})^2$ quasi paraboloids of opening $M$ generated by $\phi$. When we would like to emphasize the dependence of $d(x, x_0)$ on $\phi$, we write $d_{\phi }(x, x_0).$ Likewise, we write $G_{M}(u,\Omega , \phi )$ to indicate the dependence on $\phi$ of the set $G_{M}(u,\Omega )$. Notice that for $\phi (x) =\left| x\right| ^2$, we have $d(x, \bar{x})=\left| x-\bar{x}\right|$ is the Euclidean distance.

In the next lemma, we show that if the quasi distance $d(x, x_0)$ is bounded from below by the Euclidean distance $\left| x-x_0\right|$ around $x_0$ then it is also bounded from above by a multiple of this Euclidean distance around $x_0$. This lemma is a slight modification of [15, Lemma 6.2.1].

Lemma 3.3

Assume $\Omega$ satisfies (2.1) and let $\phi \in C(\overline{\Omega })$ be a convex function satisfying $\lambda \le \det D^2\phi \le \Lambda$ in $\Omega$ and $\phi =0$ on $\partial \Omega$. There exists $c=c(n,\lambda ,\Lambda ,\rho )>0$ such that if $x_0\in \Omega$ and

$$\begin{aligned} d(x, x_0)^2 \ge \sigma \, |x-x_0|^2~ \text { in } ~ B_r(x_0)\subset \Omega \quad \text {for some}\, r>0, \end{aligned}$$

then for all $x$ in a small neighborhood of $x_0$, we have

$$\begin{aligned} d(x,x_0)^2 \le \frac{1}{c^2 \sigma ^{n-1}}\, |x-x_0|^2. \end{aligned}$$

Proof

Let $\varphi (x) := \phi (x)- \phi (x_0)- \nabla \phi (x_0)\cdot (x-x_0)$. Then the strict convexity of $\phi$ implies that there exists $\delta >0$ such that $S_\phi (x_{0}, \delta ) := \{x\in \Omega : \varphi (x) <\delta \}\subset B_r(x_0)$. Therefore by the proof of Lemma 6.2.1 in [15], we have $\varphi (x)\le C(n,\lambda ,\Lambda ,\rho )\sigma ^{-n+1}\, |x-x_0|^2$ for all $x\in \Omega$ satisfying $\varphi (x)\le \delta$, which gives the conclusion of the lemma. $\square$

The following lemma allows us to estimate the distribution function of $D^2u$. It is the starting point for our proofs of Theorems 1.1 and 1.2 and the global version of [17, Lemma 2.7].

Lemma 3.4

Let $\Omega , \phi$ and $c$ be as in Lemma 3.3, and $u\in C^2(\Omega )$. Define

$$\begin{aligned} A^{loc}_\sigma := \left\{ x_0\in \Omega : d(x, x_0)^2\ge \sigma \, |x-x_0|^2, \text {for all}\; x\; \text {in some neighborhood of}\; x_0 \right\} \!. \end{aligned}$$

Then for any $m>1$ and $\beta >0$, we have

$$\begin{aligned} \left\{ x\in \Omega : |D_{i j} u(x)| > \beta ^m\right\} \subset \left( \Omega \setminus A^{\text{ loc }}_{\left( c\beta ^{\frac{m -1}{2}}\right) ^{\frac{-2}{n-1}}}\right) \cup \left( \Omega \setminus G_\beta (u,\Omega )\right) .\quad \end{aligned}$$

(3.1)

Proof

Let $\gamma := \beta ^{\frac{m -1}{2}}$. If $\bar{x}\in A^{\text{ loc }}_{(c\gamma )^{\frac{-2}{n-1}}} \cap G_\beta (u,\Omega )$, then

$$\begin{aligned} -\frac{\beta }{2} \,d(x,\bar{x})^2 \le u(x)-u(\bar{x}) -\nabla u(\bar{x})\cdot (x-\bar{x}) \le \frac{\beta }{2} \,d(x,\bar{x})^2 \end{aligned}$$

for each $x\in \Omega$. Since $\bar{x}\in A^{\text{ loc }}_{(c\gamma )^{\frac{-2}{n-1}}}$, these together with Lemma 3.3 yield

$$\begin{aligned} -\frac{\beta \gamma ^2}{2} |x-\bar{x}|^2 \le u(x)-u(\bar{x}) -\nabla u(\bar{x})\cdot (x-\bar{x}) \le \frac{\beta \gamma ^2}{2} |x-\bar{x}|^2 \end{aligned}$$

for all $x$ in a small neighborhood of $\bar{x}$, and so $|D_{ij}u(\bar{x})|\le \beta \gamma ^2 = \beta ^{m}$. Thus we have proved that

$$\begin{aligned} A^{\text{ loc }}_{(c\gamma )^{\frac{-2}{n-1}}} \cap G_\beta (u,\Omega ) \subset \big \{x\in \Omega : |D_{ij}u(x)|\le \beta ^m, \;\hbox {for}\; i,j=1,\dots ,n\big \} \end{aligned}$$

and the lemma follows by taking complements. $\square$

3.1 Power decay estimates

In order to derive global $W^{2,p}$ estimates for solutions $u$ to the linearized Monge–Ampère equation, we will need to estimate the distribution function

$$\begin{aligned} F(\beta ) :=\big | \{x\in \Omega : |D_{ij}u(x)| > \beta ^m\} \big | \end{aligned}$$

for some suitable choice of $m>1$. It follows from Lemma 3.4 that this can be done if one can get appropriate decay estimates for

$$\begin{aligned} F_1(\beta ) := |\Omega \setminus A^{\text{ loc }}_{(c\beta ^{\frac{m -1}{2}})^{\frac{-2}{n-1}}}|\quad \text {and}\quad F_2(\beta ) := |\Omega \setminus G_\beta (u,\Omega )|. \end{aligned}$$

Notice that the function $F_1(\beta )$ involves only the solution $\phi$ of the Monge–Ampère equation and its power decay is given in the next theorem.

Theorem 3.5

Assume $\Omega$ satisfies (2.1) and $\partial \Omega \in C^{1,1}$. Let $\phi \in C(\overline{\Omega })$ be a convex function such that $1-\epsilon \le \det D^2\phi \le 1+\epsilon$ in $\Omega$ and (2.3) holds, where $0<\epsilon < 1/2$. Then there exists a positive constant $M$ depending only on $n$ and $\rho$ such that

$$\begin{aligned} \big |\Omega \setminus A^{\text{ loc }}_{s^{-2}}\big | \le C'(\epsilon ,n,\rho , \Vert \partial \Omega \Vert _{C^{1,1}}) \quad s^{\dfrac{\ln \sqrt{C \epsilon }}{\ln M}}\quad \text{ for } \text{ all }\quad s>0. \end{aligned}$$

(3.2)

In particular, for $s = (c\beta ^{\frac{m-1}{2}})^{\frac{1}{n-1}}$, we get

$$\begin{aligned} F_{1}(\beta )\le C'(\epsilon ,n,\rho , \Vert \partial \Omega \Vert _{C^{1,1}}) \, \beta ^{-\frac{m-1}{2(n-1) \ln M }\ln \frac{1}{\sqrt{C\epsilon }}}\quad \forall \beta >0. \end{aligned}$$

The small power decay estimates for $F_2(\beta )$ are given in the following proposition. It is the boundary version of Proposition 3.4 in [17].

Proposition 3.6

Assume that $\Omega$ and $\phi$ satisfy the assumptions (2.1)–(2.3). Assume in addition that $\partial \Omega \in C^{1,1}$. Suppose $u\in C^1(\Omega )\cap W^{2,n}_{loc}(\Omega )$, $|u|\le 1$ in $\Omega$ and $\mathcal{L }_\phi u=f$ in $\Omega$ with $\Vert f\Vert _{L^n(\Omega )}\le 1$. Then there exist $\tau =\tau (n,\lambda ,\Lambda ,\rho )\in (0, 1/2)$ and $C= C(\rho , \lambda , \Lambda , n, \Vert \partial \Omega \Vert _{C^{1,1}})>0$ such that

$$\begin{aligned} F_2(\beta )=\big |\Omega \setminus G_{\beta }(u,\Omega )\big | \le \dfrac{C}{\beta ^\tau }\quad \text{ for } \text{ all }\quad \beta >0. \end{aligned}$$

The next result is a variant of Proposition 3.6 which will be important for the density and improved power decay estimates in Sect. 5.1.

Proposition 3.7

Let $(\Omega , \phi , U)$ be in the class $\mathcal P _{\lambda ,\Lambda , \rho , \kappa ,*}$. Suppose $u\in C(\Omega )\cap C^1(U)\cap W^{2,n}_{loc}(U)$, $|u|\le 1$ in $\Omega$ and $\mathcal{L }_\phi u=f$ in $U$ with $\Vert f\Vert _{L^n(U\cap B_c)}\le 1$. Then there exist $\tau =\tau (n,\lambda ,\Lambda ,\rho )\in (0, 1/2)$ and $C= C(n,\lambda ,\Lambda ,\rho , \kappa )>0$ such that

$$\begin{aligned} \big |(U\cap B_{c^2})\setminus G_{\beta }(u,\Omega )\big | \le \dfrac{C}{\beta ^\tau } \, |U\cap B_{c^2}|\quad \text{ for } \text{ all }\quad \beta >0. \end{aligned}$$

The above inequality also holds if $U\cap B_{c^2}$ is replaced by $S_{\phi }(0, r)$ for any universal constant $r$ satisfying $r\le c^6$.

As a consequence of the power decay estimates for $F_1(\beta )$ and $F_2(\beta )$ in Theorem 3.5 and Proposition 3.6, we find that the decay for $F(\beta )$ when $0<\epsilon < 1/2$ is given by

$$\begin{aligned} F(\beta )\le C(\epsilon ,n,\rho , \Vert \partial \Omega \Vert _{C^{1,1}}) \, \beta ^{-\frac{m-1}{2(n-1)\ln M }\ln \frac{1}{\sqrt{C\epsilon }}} + C\beta ^{-\tau }. \end{aligned}$$

Since $\frac{m-1}{2(n-1)\ln M }\ln \frac{1}{\sqrt{C\epsilon }} \rightarrow \infty$ as $\epsilon \rightarrow 0$, we obtain global $W^{2,\delta }$ estimates for all $\delta <\tau /m<1/2$ for solutions to the linearized Monge–Ampère equation $\mathcal L _{\phi } u = f$ provided that $f\in L^{n}(\Omega )$ and $\epsilon$ is small, that is, $\det D^2\phi$ is close to a constant. However, in the next subsection, we offer a more direct proof of global $W^{2,\delta }$ estimates based on interior estimates without resorting to decay estimates of the distribution function of the second derivatives. Another advantage of this proof is that it works for all Monge–Ampère measures $\det D^2\phi$ bounded away from $0$ and $\infty$.

Remark 3.8

It is now clear that the obstruction to higher integrability of $|D^2 u|$ is the small exponent $\tau$ in the decay estimates for $\big |\Omega \setminus G_{\beta }(u,\Omega )\big |$ given by Proposition 3.6. Most of the paper is devoted to developing tools to improve the decay estimates for $\big |\Omega {\setminus } G_{\beta }(u,\Omega )\big |$. In particular, the global stability of cofactor matrices and an approximation lemma in the next two sections will be employed for this purpose.

3.2 Global $W^{2,\delta }$ estimates

In this subsection, we obtain global $W^{2,\delta }(\Omega )$ estimates ($\delta >0$ small) for solutions to the linearized Monge–Ampère equation $\mathcal{L }_{\phi }u=f$ when $\det D^{2}\phi$ is only bounded away from $0$ and $\infty$ and under natural assumptions on the domain $\Omega$ and the boundary data of $\phi$.

Our main theorem in this subsection is the following.

Theorem 3.9

Assume $\Omega$ and $\phi$ satisfy the assumptions (2.1)–(2.3). Assume in addition that $\partial \Omega \in C^{1,1}$. Let $u \in C(\overline{\Omega })\cap C^1(\Omega )\cap W^{2, n}_{loc}(\Omega )$ be a solution of

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal{L }_\phi u =f &{}\quad \text{ in }\, \Omega ,\\ u=0&{}\quad \text{ on }\, \partial \Omega . \end{array}\right. \end{aligned}$$

Then there exist $p=p(\rho ,\lambda ,\Lambda ,n)>0$ and $C= C(\rho , \lambda , \Lambda , n, \Vert \partial \Omega \Vert _{C^{1,1}})>0$ such that

$$\begin{aligned} \Vert D^2 u\Vert _{L^{p}(\Omega )}\le C\Vert f\Vert _{L^{n}(\Omega )}. \end{aligned}$$

The rest of this subsection is devoted to proving this theorem. The idea is to cover $\Omega$ by maximal interior sections whose shapes are under control by Proposition 2.3 and then apply the interior $W^{2,\delta }$ estimates of Gutiérrez and Tournier [18] in these sections. Furthermore, since we can control the number of these sections within certain height due to the $C^{1,1}$ regularity of the boundary $\partial \Omega$, the global estimates follow by adding interior ones.

For reader’s convenience, we recall Gutiérrez-Tournier’s $W^{2,\delta }$ estimates.

Theorem 3.10

([18, Theorem 6.3]) Let $\Omega$ be a convex domain such that $B_{k_0}\subset \Omega \subset B_{k_{0}^{-1}}$. Let $\phi \in C^{2}(\Omega )$ be a convex function satisfying $\lambda \le \det D^{2}\phi \le \Lambda ~\text { in }~\Omega$ and $\phi =0~\text { on }~\partial \Omega .$ Let $u\in C^{1}(\Omega )\cap W^{2,n}_{loc}(\Omega )$ be a solution of $\mathcal L _{\phi } u=f$ in $\Omega$. Then, given $\alpha _{0}\in (0,1)$, there exist positive constants $\delta$ and $C$ depending only on $\alpha _{0}, k_0, \lambda , \Lambda$ and $n$ such that

$$\begin{aligned} \Vert D^2 u\Vert _{L^{\delta }\big (S_{\phi }(x_0, -\alpha _0\phi (x_0))\big )}\le C \Big (\Vert u\Vert _{L^{\infty }(\Omega )} + \Vert f\Vert _{L^{n}(\Omega )}\Big ), \end{aligned}$$

where $x_0\in \Omega$ is such that $\min _{\Omega }\phi =\phi (x_0)$.

Let $0<p< \min \{\delta , \frac{1}{2}\}$ where $\delta =\delta (\rho ,\lambda , \Lambda , n)>0$ is a small number appearing in Theorem 3.10 corresponding to $\alpha _{0} =1/2$ and $k_0=k_0(\rho , n, \lambda , \Lambda )$ given by Proposition 2.3.

We will show that the conclusion of Theorem 3.9 holds for the above choice of $p$. To achieve this, we first estimate the $L^{p}$ norm of $D^{2} u$ in the interior of each maximal interior section.

Lemma 3.11

Assume $\Omega$ and $\phi$ satisfy the assumptions (2.1)–(2.3). Let $u \in C(\overline{\Omega })\cap C^1(\Omega )\cap W^{2, n}_{loc}(\Omega )$ be a solution of

$$\begin{aligned} \mathcal{L }_\phi u =f ~ \text{ in } ~\Omega ,\quad \text{ and }\quad u =0 ~ \text{ on } ~ \partial \Omega . \end{aligned}$$

Then, there exists a constant $C>0$ depending only on $p, \rho , \lambda , \Lambda$ and $n$ such that

$$\begin{aligned}&\Vert D^{2} u\Vert _{L^{p}\big (S_{\phi }\left( y, \frac{\bar{h}(y)}{2}\right) \big )}\\&\quad \le C {\bar{h}}(y)^{\frac{n}{2p}-1}|\log \bar{h}(y)|^{2} \, \Big (\Vert u\Vert _{L^{\infty }\big (S_{\phi }(y, \bar{h}(y))\big )} + \bar{h}(y)^{1/2}\Vert f\Vert _{L^{n}\big (S_{\phi }(y, \bar{h}(y))\big )} \Big ) \end{aligned}$$

for all $y\in \Omega$ satisfying $\bar{h}(y)\le c$.

Proof

Let $h:= \bar{h}(y)$ with $\bar{h}(y)\le c.$ We now define the rescaled domain $\tilde{\Omega }_{h}$ and rescaled functions $\tilde{\phi }_{h}$, $\tilde{u}_{h}$ and $\tilde{f}_h$ as in Sect. 2.2 that preserve the $L^{\infty }$ -norm in a section tangent to the boundary. For simplicity, let us denote $\tilde{S}_{t}(0) := S_{\tilde{\phi }_{h}} (0, t)$ for $t>0$. Then by Theorem 3.10, we have

$$\begin{aligned} \Vert D^{2}\tilde{u}_{h}\Vert _{L^{p}\left( \tilde{S}_{\frac{1}{2}}(0)\right) }\le C(p,\rho ,\lambda , \Lambda , n) \, \Big (\Vert \tilde{u}_{h}\Vert _{L^{\infty }(\tilde{S}_{1}(0))} + \Vert \tilde{f}_{h}\Vert _{L^{n}(\tilde{S}_{1}(0))}\Big ). \quad \quad \end{aligned}$$

(3.3)

Using the fact

$$\begin{aligned} D^{2} u\left( y + h^{1/2} A^{-1}_{h} x\right) = h^{-1} (A_{h})^{t} \, D^{2} \tilde{u}_{h}(x) \, A_{h}, \end{aligned}$$

we obtain

$$\begin{aligned} \int \limits _{S_{\phi }\left( y,\frac{h}{2}\right) }|D^{2} u(z)|^{p} \, dz&= h^{\frac{n}{2}-p} \int \limits _{\tilde{S}_{\frac{1}{2}}(0)} |A_{h}^{t} \, D^{2} \tilde{u}_{h}(x) \, A_{h}|^{p}\, dx\\&\le C \,h^{\frac{n}{2}-p} \, |\log h|^{2p} \int \limits _{\tilde{S}_{\frac{1}{2}}(0)} |D^{2} \tilde{u}_{h}(x)|^{p}\, dx. \end{aligned}$$

It follows that

$$\begin{aligned} \Vert D^{2} u\Vert _{L^{p}\big (S_{\phi }\left( y, \frac{h}{2}\right) \big )}\le C h^{\frac{n}{2p} -1}|\log h|^{2} \, \Vert D^{2}\tilde{u}_{h}\Vert _{L^{p}\big (\tilde{S}_{\frac{1}{2}}(0)\big )}. \end{aligned}$$

(3.4)

Moreover, we have

$$\begin{aligned} \Vert \tilde{f}_{h}\Vert _{L^{n}(\tilde{S}_{1}(0))} = h^{\frac{1}{2}}\Vert f\Vert _{L^{n}(S_{\phi }(y, h))}\quad \text{ and }\quad \Vert \tilde{u}_{h}\Vert _{L^{\infty }(\tilde{S}_{1}(0))} = \, \Vert u\Vert _{L^{\infty }(S_{\phi }(y, h))}. \quad \quad \quad \end{aligned}$$

(3.5)

Combining (3.3)–(3.5), we obtain the desired estimate stated in our lemma. $\square$

Finally, we will use the following Vitali covering lemma proved by Savin in [28]; see also [22, Lemma 2.5] for a more general covering result.

Lemma 3.12

([28, Lemma 2.3]) Assume $\Omega$ and $\phi$ satisfy the assumptions (2.1)–(2.3). Then there exists a sequence of disjoint sections $S_{\phi }(y_{i}, \delta _{0}\bar{h}(y_i))$ with $\delta _0= \delta _0(\lambda , \Lambda , n)>0$ such that

$$\begin{aligned} \Omega \subset \bigcup _{i=1}^{\infty } S_{\phi }\left( y_{i}, \frac{{\bar{h}}(y_i)}{2}\right) . \end{aligned}$$

Proof of Theorem 3.9.

It follows from Proposition 2.3 (see also [28, Lemma 2.2]) that if $y\in \Omega$ with $\bar{h}(y)\le c$ then

$$\begin{aligned}&S_{\phi }(y, \bar{h}(y))\subset y + k_{0}^{-1}E_h\subset D_{C\bar{h}(y)^{1/2}}\\&\quad : =\left\{ x\in \overline{\Omega }: \, \text {dist}(x, \partial \Omega )\le C\bar{h}(y)^{1/2}\right\} \!,\quad C:= 2k_{0}^{-2}. \end{aligned}$$

By Lemma 3.12, we have

$$\begin{aligned} \int \limits _{\Omega } |D^{2} u|^{p} dx \le \sum _{i=1}^\infty \int \limits _{S_{\phi }\left( y_{i}, \frac{{\bar{h}}(y_i)}{2}\right) } |D^2 u|^{p} dx. \end{aligned}$$

There is a finite number of sections $S_{\phi }(y_i, \bar{h}(y_i))$ with $\bar{h}(y_i)\ge c$ and, by Theorem 3.10, we have in each such section

$$\begin{aligned} \int \limits _{S_{\phi }\left( y_{i}, \frac{{\bar{h}}(y_i)}{2}\right) } |D^{2} u|^{p} \le C \, \big (\Vert u\Vert _{L^{\infty }(\Omega )} + \Vert f\Vert _{L^{n}(\Omega )}\big )^{p}. \end{aligned}$$

Now, for $d\le c$ we consider the family $\mathcal F _{d}$ of sections $S_{\phi }(y_{i}, \bar{h}(y_i)/2)$ such that $d/2<\bar{h}(y_i)\le d$. Let $M_{d}$ be the number of sections in $\mathcal F _{d}$. We claim that

$$\begin{aligned} M_{d}\le C_{b} d^{\frac{1}{2}-\frac{n}{2}} \end{aligned}$$

(3.6)

for some constant $C_{b}$ depending only on $\rho , n, \lambda , \Lambda$ and $\Vert \partial \Omega \Vert _{C^{1,1}}.$ Indeed, we first note that, by [15, Corollary 3.2.4] (see also Theorem 2.13(b)), there exists a constant $C= C(n,\lambda ,\Lambda , \rho )>0$ such that

$$\begin{aligned} |S_{\phi }(y_{i}, \delta _{0}\bar{h}(y_i))| \ge C \bar{h}(y_i)^{n/2} \ge C d ^{n/2}. \end{aligned}$$

Since $S_{\phi }(y_{i}, \delta _{0}\bar{h}(y_i))\subset D_{Cd^{1/2}}$ are disjoint, we find that

$$\begin{aligned} M_d Cd^{n/2}\le \sum _{i\in \mathcal F _{d}} |S_{\phi }(y_{i},\delta _{0} \bar{h}(y_i))|\le |D_{Cd^{1/2}}|\le C_{*}d^{1/2} \end{aligned}$$

for some constant $C_{*}$ depending only on $n$ and $\Vert \partial \Omega \Vert _{C^{1,1}}$. Thus (3.6) holds.

It follows from Lemma 3.11 and (3.6) that

$$\begin{aligned} \sum _{i\in \mathcal F _{d}} \int \limits _{S_{\phi }\left( y_{i}, \frac{{\bar{h}}(y_i)}{2}\right) } |D^{2} u|^{p}&\le C M_{d}d^{\frac{n}{2} -p}|\log d|^{2p} \, \Big (\Vert u\Vert _{L^{\infty }(\Omega )} + \Vert f\Vert _{L^{n}(\Omega )}\Big )^{p}\\&\le C d^{\frac{1}{2} -p}|\log d|^{2p}\, \Big (\Vert u\Vert _{L^{\infty }(\Omega )} + \Vert f\Vert _{L^{n}(\Omega )}\Big )^{p}. \end{aligned}$$

Adding these inequalities for the sequence $d = c2^{-k}, k= 0, 1, 2, \ldots ,$ and noting that

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\Omega )} \le C(n, \rho , \lambda , \Lambda ) \Vert f\Vert _{L^{n}(\Omega )}, \end{aligned}$$

by the ABP estimate, we obtain the desired global $L^p$ estimate for $D^2 u$. $\square$

3.3 Proofs of the power decay estimates

Proof of Theorem 3.5.

Let $\{S_{\phi }(y_i, \bar{h}(y_i)/2)\}$ be the sequence of sections covering $\Omega$ given by Lemma 3.12. In what follows we will use the notations as in the proof of Lemma 3.11. We then have

$$\begin{aligned} \big |\Omega \setminus A^{\text{ loc }}_{s^{-2}}\big |&\le \sum _{i=1}^\infty {\big |S_{\phi }(y_i, \bar{h}(y_i)/2)\setminus A^{\text{ loc }}_{s^{-2}}\big | }\nonumber \\&\le \sum _{k=0}^\infty \sum _{i\in \mathcal{F }_{c2^{-k}}}{\big |S_{\phi }(y_i, \bar{h}(y_i)/2)\setminus A^{\text{ loc }}_{s^{-2}}\big | }\nonumber \\&+ \sum _{i: \, {\bar{h}}({y_i})> c}{\big |S_{\phi }(y_i, \bar{h}(y_i)/2)\setminus A^{\text{ loc }}_{s^{-2}}\big | }=: I + II. \end{aligned}$$

(3.7)

Let us first estimate the summation $I$ corresponding to sections with ${\bar{h}}(y_i)\le c$. Consider a prototype section $S_{\phi }(y, h)$ with $h:=\bar{h}(y)\le c$. Proposition 2.3 tells us that $S_{\phi }(y, h)$ is equivalent to an ellipsoid $E_h$, i.e.,

$$\begin{aligned} k_0 E_h \subset S_{\phi }(y, h)-y \subset k_{0}^{-1} E_h, \end{aligned}$$

where

$$\begin{aligned} E_h := h^{1/2}A_{h}^{-1}B_1, \quad \text{ with } \quad \det A_{h}=1,\quad \Vert A_{h}\Vert , \, \Vert A_{h}^{-1} \Vert \le k^{-1} |\log h|. \end{aligned}$$

Here $k, k_{0}$ depend only on $n$ and $\rho$. Let $T(x) := h^{-1/2} A_h(x-y)$. Define $\tilde{U}_h:=T(S_{\phi }(y, h))$ and

$$\begin{aligned} \tilde{\phi }_h(z) := h^{-1} \Big [\phi (T^{-1}z) -\phi (y) -\nabla \phi (y)\cdot (T^{-1}z -y)-h\Big ] \quad \text{ for }\, z\in \tilde{U}_h. \end{aligned}$$

Then $B_{k_0} \subset \tilde{U}_h\equiv S_{\tilde{\phi }_h}(0,1)\subset B_{k_0^{-1}}$, $1-\epsilon \le \det D^2\tilde{\phi }_h \le 1+\epsilon$ in $\tilde{U}_h$ and $\tilde{\phi }_h =0$ on $\partial \tilde{U}_h$. By [15, Theorem 3.3.10], there exists $\eta _{0}= \eta _{0}(n,\rho )>0$ such that

$$\begin{aligned} S_{\tilde{\phi }_h}(x, t)\Subset \tilde{U}_h\quad \text { for all}~ x\in S_{\tilde{\phi }_h}(0, 1/2)\quad \text {and}\quad t\le \eta _{0}. \end{aligned}$$

Now, let

$$\begin{aligned} \tilde{D}_{s}^{\frac{1}{2}} :=\left\{ x\in S_{\tilde{\phi }_h}(0, 1/2): S_{\tilde{\phi }_h}(x, t)\subset B(x, s\sqrt{t}), ~\forall t\le \eta _{0}\right\} \!. \end{aligned}$$

Then, by [17, Theorem 2.8], we obtain

$$\begin{aligned} |S_{\tilde{\phi }_h}(0, 1/2)\setminus \tilde{D}^{\frac{1}{2}}_{s}| \le \frac{|\tilde{U}_h|}{(C\epsilon )^2}\, s^{-p_\epsilon }, \end{aligned}$$

where $p_\epsilon := -\dfrac{\ln \sqrt{C \epsilon }}{\ln M}$ with $C, M>0$ is a constant depending only on $n$ and $\rho$. Let

$$\begin{aligned} \tilde{A}_\sigma := \left\{ \bar{z}\in \tilde{U}_h: \tilde{\phi }_h(z)\ge \tilde{\phi }_h(\bar{z}) + \nabla \tilde{\phi }_h(\bar{z})\cdot (z-\bar{z}) + \sigma \, |z-\bar{z}|^2, \quad \forall z\in \tilde{U}_h\right\} . \end{aligned}$$

Since $\tilde{D}^{\frac{1}{2}}_{s} = S_{\tilde{\phi }_h}(0, 1/2) \cap \tilde{A}_{s^{-2}}$ by [15, Theorem 6.2.2], we can rewrite the above inequality as

$$\begin{aligned} |T\big (S_{\phi }(y, h/2)\big )\setminus \tilde{A}_{s^{-2}}| \le C(\epsilon ,n,\rho ) \, s^{-p_\epsilon }. \end{aligned}$$

(3.8)

Let us relate $\tilde{A}_{s^{-2}}$ to $A^{\text{ loc }}_\sigma$. Since $|x-\bar{x}|\le \Vert A_h^{-1}\Vert \, | A_h (x-\bar{x})|\le k^{-1} h^{1/2} |\log h| |Tx - T\bar{x}|$, we have

$$\begin{aligned} \tilde{A}_{s^{-2}}&= T \left\{ \bar{x}\in S_\phi (y, h): \tilde{\phi }_h(Tx)\ge \tilde{\phi }_h(T\bar{x}) + \nabla \tilde{\phi }_h(T\bar{x})\cdot (T x-T\bar{x})\right. \\&\qquad \left. + s^{-2}\, |T x-T\bar{x}|^2, \, \forall x\in S_\phi (y, h)\right\} \\&\subset T \left\{ \bar{x}\in S_\phi (y, h): \phi (x)\ge \phi (\bar{x}) + \nabla \phi (\bar{x})\cdot ( x-\bar{x})\right. \\&\qquad \left. + (k^{-1} s |\log h|)^{-2}\, | x-\bar{x}|^2, \, \forall x\in S_\phi (y, h)\right\} \\&\subset T\left( S_\phi (y, h) \cap A^{\text{ loc }}_{(k^{-1} s |\log h|)^{-2}}\right) . \end{aligned}$$

We infer from this and (3.8) that

$$\begin{aligned} |S_\phi (y, h/2)\setminus A^{\text{ loc }}_{(k^{-1} s |\log h|)^{-2}}|&\le C(\epsilon ,n,\rho ) |\det T|^{-1} \, s^{-p_\epsilon }\\&= C(\epsilon ,n,\rho ) \, h^{n/2} \, s^{-p_\epsilon }\quad \forall s>0, \end{aligned}$$

or equivalently,

$$\begin{aligned} |S_\phi (y, h/2)\setminus A^{\text{ loc }}_{s^{-2}}| \le C(\epsilon ,n,\rho ) \, h^{n/2} |\log h|^{p_\epsilon } \, s^{-p_\epsilon }\quad \forall s>0. \end{aligned}$$

Thus the summation $I$ in (3.7) can be estimated as follows

$$\begin{aligned} I&\le C(\epsilon , n, \rho ) s^{-p_\epsilon } \sum _{k=0}^\infty \sum _{i\in \mathcal{F }_{c 2^{-k}}}{\bar{h}(y_i)^{n/2} |\log \bar{h}(y_i)|^{p_\epsilon } }\nonumber \\&\le C(\epsilon ,n,\rho ) s^{-p_\epsilon } \sum _{k=0}^\infty {(c 2^{-k})^{n/2} |\log (c 2^{-k-1})|^{p_\epsilon } M_{c 2^{-k}}}\nonumber \\&\le C \, s^{-p_\epsilon } \sum _{k=0}^\infty {(c 2^{-k})^{1/2} |\log (c 2^{-k-1})|^{p_\epsilon } } \le C \, s^{-p_\epsilon }. \end{aligned}$$

(3.9)

Note that $C$ depends on $\epsilon , n, \rho$ and $\Vert \partial \Omega \Vert _{C^{1,1}}$, and we have used the bound (3.6) for $M_d$ to obtain the third inequality.

Next let us estimate the summation $II$ corresponding to sections $S_{\phi }(y_i, \bar{h}(y_i)/2)$ with $\bar{h} (y_i) >c$. Since the family $\{S_\phi (y_i, \delta _0 {\bar{h}}(y_i))\}$ is disjoint, we infer from the lower bound on volume of sections and $\Omega \subset B_{1/\rho }$ that

$$\begin{aligned} \#\{i: {\bar{h}}(y_i)> c\} \le C(n,\rho ). \end{aligned}$$

Also, by using the standard normalization for interior sections and [17, Theorem 2.8] we get

$$\begin{aligned} |S_\phi (y_i, \bar{h}(y_i)/2)\setminus A^{\text{ loc }}_{s^{-2}}| \le C(\epsilon ,n,\rho ) \, s^{-p_\epsilon } \quad \text{ for } \text{ all }\; i\; \hbox {with}\quad {\bar{h}}(y_i)> c. \end{aligned}$$

Therefore,

$$\begin{aligned} II \le \#\{i: {\bar{h}}(y_i)> c\} \, \big [ C(\epsilon ,n,\rho ) \, s^{-p_\epsilon } \big ]\le C(\epsilon ,n,\rho ) \, s^{-p_\epsilon } \quad \forall s>0. \end{aligned}$$

(3.10)

By combining (3.7), (3.9) and (3.10) we obtain

$$\begin{aligned} \big |\Omega \setminus A^{\text{ loc }}_{s^{-2}}\big | \le I + II&\le C(\epsilon ,n,\rho , \Vert \partial \Omega \Vert _{C^{1,1}}) \, s^{-p_\epsilon }\\&= C(\epsilon ,n,\rho , \Vert \partial \Omega \Vert _{C^{1,1}}) \quad s^{\dfrac{\ln \sqrt{C \epsilon }}{\ln M}}. \end{aligned}$$

$\square$

The proof of Theorem 3.5 can also be employed to give the proof of Proposition 3.6.

Proof of Theorem 3.6.

Let $\{S_{\phi }(y_{i},\bar{h}(y_i)/2)\}$ be the sequence of sections covering $\Omega$ given by Lemma 3.12. Then we have

$$\begin{aligned} \big |\Omega \setminus G_{\beta }(u,\Omega )\big |&\le \sum _{i:\, {\bar{h}}(y_i) >c}\big |S_{\phi }\left( y_{i},\frac{{\bar{h}}(y_i)}{2}\right) \setminus G_{\beta }(u,\Omega )\big |\nonumber \\&\quad + \sum _{k=0}^\infty \sum _{i\in \mathcal{F }_{c 2^{-k}}}{\big |S_{\phi }\left( y_{i},\frac{{\bar{h}}(y_i)}{2}\right) \setminus G_{\beta }(u,\Omega )\big | }. \end{aligned}$$

(3.11)

By using [17, Proposition 3.4] and arguing as in estimating the term $II$ in the proof of Theorem 3.5, we see that there exist constants $C,\tau >0$ depending only on $n, \lambda , \Lambda$ and $\rho$ with $\tau <1/2$ such that

$$\begin{aligned} \sum _{i:\, {\bar{h}}(y_i) >c}\big |S_{\phi }\left( y_{i},\frac{{\bar{h}}(y_i)}{2}\right) {\setminus } G_{\beta }(u,\Omega )\big | \!\le \! \sum _{i:\, {\bar{h}}(y_i) >c}{\frac{C}{\beta ^\tau }} = \frac{C}{\beta ^\tau } \, \#\{i: {\bar{h}}(y_i)> c\} \le \frac{C}{\beta ^\tau }.\nonumber \\ \end{aligned}$$

(3.12)

To estimate the last expression in (3.11), let us consider a prototype section $S_{\phi }(y, h)$ with $h:=\bar{h}(y)\le c$. We now define the rescaled domains $\tilde{\Omega }_{h}, \tilde{U}_h$ and rescaled functions $\tilde{\phi }_{h}$, $\tilde{u}_{h}$ and $\tilde{f}_h$ as in Sect. 2.2 that preserve the $L^{\infty }$ -norm in a section tangent to the boundary. Then

$$\begin{aligned} \Vert \tilde{f}_h\Vert _{L^n(\tilde{U}_h)} =h^{1/2} \Vert f\Vert _{L^n(S_{\phi }(y, h))}\le h^{1/2} \Vert f\Vert _{L^n(\Omega )} \le 1. \end{aligned}$$

(3.13)

Therefore, we can apply [17, Proposition 3.4] to obtain for $T(x): = h^{-1/2} A_h(x-y)$

$$\begin{aligned} \big |T(S_{\phi }(y, h/2)) \setminus G_{\beta }(\tilde{u}_h,\tilde{\Omega }_h, \tilde{\phi }_h)\big |&= \big | S_{\tilde{\phi }_h}(0,1/2) \setminus G_{\beta }(\tilde{u}_h,\tilde{\Omega }_h, \tilde{\phi }_h)\big |\\&\le \dfrac{C}{\beta ^\tau }\quad \text{ for } \text{ all }\quad \beta >0. \end{aligned}$$

But as $\tilde{u}_h \in C^1(\tilde{U}_h)$ and $d_{\tilde{\phi }_h}(Tx, T \bar{x})^2 =h^{-1} d(x, \bar{x})^2$ for all $x,\bar{x}\in \Omega$, we get

$$\begin{aligned} T(S_{\phi }(y, h/2)) \cap G_{\beta }(\tilde{u}_h,\tilde{\Omega }_h, \tilde{\phi }_h)= T\Big ( S_{\phi }(y, h/2)\cap G_{\beta h^{-1}}( u, \Omega ) \Big ). \end{aligned}$$

Thus we infer from the above inequality that

$$\begin{aligned} \big |S_{\phi }(y, h/2) \setminus G_{\beta h^{-1}}( u, \Omega ) \big | \le \dfrac{C}{\beta ^\tau } |\det T|^{-1} = \dfrac{C}{\beta ^\tau } h^{\frac{n}{2}}, \end{aligned}$$

or equivalently,

$$\begin{aligned} \big |S_{\phi }(y,h/2) \setminus G_{\beta }( u, \Omega ) \big | \le \dfrac{C}{\beta ^\tau } h^{\frac{n}{2} -\tau } \quad \text{ for } \text{ all }\quad \beta >0. \end{aligned}$$

This together with the estimate (3.6) for $M_d$ yields

$$\begin{aligned}&\sum _{k=0}^\infty \sum _{i\in \mathcal{F }_{c 2^{-k}}}{\big |S_{\phi }(y_{i}, {\bar{h}}(y_i)/2)\setminus G_{\beta }(u,\Omega )\big | }\nonumber \\&\quad \le \frac{C}{\beta ^\tau } \sum _{k=0}^\infty \sum _{i\in \mathcal{F }_{c2^{-k}}}{{\bar{h}}(y_i)^{\frac{n}{2} -\tau } } \le \frac{C}{\beta ^\tau } \sum _{k=0}^\infty (c2^{-k})^{\frac{n}{2} -\tau } M_{c2^{-k}}\le \frac{C'}{\beta ^\tau } \sum _{k=0}^\infty (c2^{-k})^{\frac{1}{2} -\tau } \le \frac{C'}{\beta ^\tau }\nonumber \\ \end{aligned}$$

(3.14)

provided that $\tau <1/2$. Here $C'$ also depends on $\Vert \partial \Omega \Vert _{C^{1,1}}$. The desired estimate is now obtained by combining (3.11), (3.12) and (3.14). $\square$

To prove Proposition 3.7, we use the following localized version at the boundary of Lemma 3.12.

Lemma 3.13

Assume $(\Omega , \phi , U)\in \mathcal P _{\lambda , \Lambda , \rho , \kappa , *}$ and let $w$ be the solution to (2.24). Let $\psi$ denote one of the functions $\phi$ and $w$. Then there exists a sequence of disjoint sections $\{S_{\psi }(y_{i}, \delta _{0}\bar{h}(y_i))\}_{i=1}^\infty$, where $\delta _0=\delta _0(n,\lambda ,\Lambda )$, $y_i\in U\cap B_{c^2}$ and $S_{\psi }(y_{i},\bar{h}(y_i))$ is the maximal interior section of $\psi$ in $U$, such that

$$\begin{aligned} U\cap B_{c^2}\subset \bigcup _{i=1}^{\infty } S_{\psi }\left( y_{i}, \frac{{\bar{h}}(y_i)}{2}\right) . \end{aligned}$$

(3.15)

Moreover, we have

$$\begin{aligned} S_{\psi }(y_{i},\bar{h}(y_i))\subset U\cap B_c,\quad \bar{h}(y_i)\le c. \end{aligned}$$

(3.16)

If we let $M^{loc}_{d}$ denote the number of sections $S_{\psi }(y_{i}, \bar{h}(y_i)/2)$ such that $d/2<\bar{h}(y_i)\le d\le c$, then

$$\begin{aligned} M^{loc}_{d}\le C_{b} d^{\frac{1}{2}-\frac{n}{2}} \end{aligned}$$

(3.17)

for some constant $C_{b}$ depending only on $\rho , n, \lambda , \Lambda$ and $\Vert \partial \Omega \cap B_{\rho }\Vert _{C^{1,1}}.$

Proof

By Remark 2.11, we can use Proposition 2.3 to get the same conclusion as in Lemma 2.5(e) for sections of $\psi$ with centers in $U\cap B_{c^2}$. All these sections thus satisfy (3.16) and are equivalent to ellipsoids. In particular, $\psi$ is strictly convex in $U\cap B_c.$ Furthermore,

$$\begin{aligned} S_{\psi }(y_{i},\bar{h}(y_i)) \subset \Big \{x\in B_c\cap U: \text {dist}(x, \partial \Omega \cap \partial U)\le 2k_0^{-1} \bar{h}(y_i)^{1/2}\Big \}. \end{aligned}$$

With this in mind and assuming that the sequence $\{S_{\psi }(y_{i}, \delta _{0}\bar{h}(y_i))\}_{i=1}^\infty$ is disjoint and satisfies (3.15), we argue similarly as in deriving the estimate (3.6) for $M_d$ to obtain (3.17).

It remains to establish the covering (3.15). The crucial point in the proof of Lemma 3.12 is the engulfing property of interior sections which hold for strictly convex solution to the Monge–Ampère equation with bounded right hand side. By our discussion above, $\psi$ is strictly convex in $U\cap B_c$ and thus we obtain (3.15). For completeness, we include the proof here, taken almost verbatim from [28]. By the engulfing property of interior sections of strictly convex solution to the Monge–Ampère equation with bounded right hand side, we can choose $\delta _0$ depending only on $n, \lambda , \Lambda$ with the following property. If $y, z\in B_{c^2}\cap U$ with

$$\begin{aligned} S_{\psi }(y, \delta _0\bar{h}(y))\cap S_{\psi }(z, \delta _0\bar{h}(z))\ne \emptyset \quad \text {and}\quad 2\bar{h}(y) \ge \bar{h}(z) \end{aligned}$$

then

$$\begin{aligned} S_{\psi }(z, \delta _0\bar{h}(z)) \subset S_{\psi }(y, \bar{h}(y)/2). \end{aligned}$$

We choose $S_{\psi }(y_1, \delta _0\bar{h}(y_1))$ from all sections $S_{\psi }(y, \delta _0\bar{h}(y))$, $y\in U\cap B_{c^2}$ such that

$$\begin{aligned} \bar{h}(y_1)\ge \frac{1}{2}\sup _{y}\bar{h}(y) \end{aligned}$$

then choose $S_{\psi }(y_2, \delta _0\bar{h}(y_2))$ as above but only from the remaining sections $S_{\psi }(y, \delta _0\bar{h}(y))$ that are disjoint from $S_{\psi }(y_1, \delta _0\bar{h}(y_1))$, then $S_{\psi }(y_3, \delta _0\bar{h}(y_3))$, etc. Consequently, we easily obtain

$$\begin{aligned} U\cap B_{c^2}\subset \bigcup _{y\in U\cap B_{c^2}}S_{\psi }(y, \delta _0\bar{h}(y))\subset \bigcup _{i=1}^{\infty }S_{\psi }(y_i, \delta _0\bar{h}(y_i)). \end{aligned}$$

$\square$

Proof of Theorem 3.7.

Our proof is similar to that of Proposition 3.6 using Lemma 3.13. In the proof of Proposition  3.6, we replace $\Omega {\setminus }G_{\beta }(u, \Omega )$ by $(U\cap B_{c^2}){\setminus } G_{\beta }(u, \Omega )$, the covering of $\Omega$ using Lemma 3.12 by the covering of $U\cap B_{c^2}$ using Lemma 3.13. By (3.16), the first term of the right hand side of (3.11) disappears. For the second term of the right hand side of (3.11), we estimate as in the rest of the proof of Proposition 3.6. Note that, since all sections in the covering for $U\cap B_{c^2}$ satisfy $S_{\phi }(y_i, \bar{h}(y_i))\subset B_c\cap U$, instead of (3.13), we now have

$$\begin{aligned} \Vert \tilde{f}\Vert _{L^n(T(S_{\phi }(y, h)))} =h^{1/2} \Vert f\Vert _{L^n(S_{\phi }(y, h))}\le h^{1/2} \Vert f\Vert _{L^n(U\cap B_c)} \le 1. \end{aligned}$$

In (3.14), we replace $M_d$ by $M^{loc}_d$ and use (3.17) to estimate it. The conclusion of Proposition 3.7 follows. Note that by (2.7), we have $S_{\phi }(0, r)\subset U\cap B_{c^2}$ if $r\le c^6$ and the last remark of the proposition follows. $\square$

3.4 Global stability of cofactor matrices

In this subsection, we prove that, under suitable geometric conditions, the cofactor matrices of the Hessian matrices of two convex functions defined on the same domain are close if their Monge–Ampère measures and boundary values are close in the $L^{\infty }$ norm.

We first start with a stability result at the boundary for the second derivatives and the cofactor matrices of functions in the class $\mathcal P$.

Proposition 3.14

Assume $(\Omega , \phi , U)\in \mathcal P _{1-\epsilon , 1+ \epsilon , \rho , \kappa , *}$. Let $w\in C(\overline{U})$ be the convex solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \det D^2 w =1&{} \quad \text{ in }\, U\\ w =\phi &{} \quad \text{ on }\, \partial U. \end{array}\right. \end{aligned}$$

Then the following statements hold.

• (i) For any $p>1$, there exist $\epsilon _0=\epsilon _0(p, n,\rho )>0$ and $C =C(p,n,\rho ,\kappa )>0$ such that

  $$\begin{aligned} \Vert D^{2}\phi -D^{2} w\Vert _{L^{p}(B_{c^2}\cap U)}\le C \epsilon ^{\frac{\delta }{n (2p - \delta )}}\quad \text{ for } \text{ all }\, \epsilon \le \epsilon _0. \end{aligned}$$

• (ii) Assume in addition that $(\Omega , \phi , U)\in \mathcal P _{1-\epsilon , 1+ \epsilon , \rho , \kappa , \alpha }$. Then for any $q\ge 1$, there exist $\epsilon _0=\epsilon _0(q, n,\rho )>0$ and $C =C(q,n,\rho ,\kappa ,\alpha )>0$ such that

  $$\begin{aligned} \Vert \Phi -\mathrm{W }\Vert _{L^{q}(B_{c^2}\cap U)}\le C \epsilon ^{\frac{(n-1)\delta }{n (2nq - \delta )}}\quad \text{ for } \text{ all }\, \epsilon \le \epsilon _0.\end{aligned}$$

Here $\delta =\delta (n, \rho )>0$, and $\Phi , \mathrm{W }$ are the matrices of cofactors of $D^2\phi$ and $D^2 w$, respectively.

Proof

• (i) Our conclusion follows from the following claims.

Claim 1. There exist $\epsilon _0=\epsilon _0(p, n,\rho )>0$ small and $C_0 =C_0(p,n,\rho ,\kappa )>0$ such that

$$\begin{aligned} \Vert D^{2}\phi \Vert _{L^{2p}(B_{c^2}\cap U)} + \Vert D^{2} w\Vert _{L^{2p}\left( B_{c^2}\cap U\right) }\le C_{0}\quad \text{ whenever }\quad \epsilon \le \epsilon _0. \end{aligned}$$

Claim 2. There exist $\delta =\delta (n,\rho )\in (0,1/2)$ and $C=C(n,\rho ,\kappa )>0$ such that

$$\begin{aligned} \Vert D^{2}\phi -D^{2} w\Vert _{L^{\delta }(B_{c^2}\cap U)}\le C \epsilon ^{1/n}\quad \text{ for } \text{ all }\quad \epsilon <\frac{1}{2}. \end{aligned}$$

(3.18)

Indeed, let $\theta \in (0, 1)$ be such that

$$\begin{aligned} \frac{1}{p} =\frac{\theta }{2p} + \frac{1-\theta }{\delta }. \end{aligned}$$

Then $1-\theta = \delta / (2p -\delta )$ and by the interpolation inequality we get

$$\begin{aligned} \Vert D^{2}\phi -D^{2} w\Vert _{L^{p}(B_{c^2}\cap U)}&\le \Vert D^{2}\phi -D^{2} w\Vert ^{\theta }_{L^{2p} (B_{c^2}\cap U)} \Vert D^{2}\phi -D^{2} w\Vert ^{1-\theta }_{L^{\delta }(B_{c^2}\cap U)}\\&\le C \epsilon ^{\frac{1-\theta }{n}} = C \epsilon ^{\frac{\delta }{n (2p - \delta )}}. \end{aligned}$$

We now turn to the proofs of the claims.

Claim 1 is essentially Savin’s global $W^{2, p}$ estimates for the Monge–Ampère equations [28]. For the proof in our setting, we use Lemma 3.13 and follow his arguments. For completeness, we include the proof here. Let $\psi$ denote one of the functions $\phi$ and $w$. Then by Lemma 3.13, there exists a sequence of disjoint sections $\{S_{\psi }(y_{i}, \delta _{0}\bar{h}(y_i))\}_{i=1}^\infty$, where $y_i\in U\cap B_{c^2}$ and $S_{\psi }(y_{i},\bar{h}(y_i))$ is the maximal interior section of $\psi$ in $U$, such that

$$\begin{aligned} U\cap B_{c^2}\subset \bigcup _{i=1}^{\infty } S_{\psi }\left( y_{i}, \frac{{\bar{h}}(y_i)}{2}\right) . \end{aligned}$$

Moreover, we have

$$\begin{aligned} S_{\psi }(y_{i},\bar{h}(y_i))\subset U\cap B_c,\quad \bar{h}(y_i)\le c. \end{aligned}$$

We will prove that: There exist $\epsilon _0=\epsilon _{0}(p, \rho , n)>0$ small and $C= C(p, \rho , n)>0$ such that for $\epsilon \le \epsilon _{0}$, we have

$$\begin{aligned} \int \limits _{S_{\psi }\left( y, \frac{\bar{h}(y)}{2}\right) } \left| D^2 \psi \right| ^{2p}\le C \,{\bar{h}}(y)^{\frac{n}{2}}|\log \bar{h}(y)|^{4p} \quad \forall y\in U\cap B_{c^2}. \end{aligned}$$

(3.19)

Given this, we can complete the proof of Claim 1 as follows. We have

$$\begin{aligned} \int \limits _{U\cap B_{c^2}}\left| D^2\psi \right| ^{p}\le \sum _{i=1}^\infty \int \limits _{S_{\psi }\left( y_{i}, \frac{\bar{h}(y_{i})}{2}\right) } \left| D^2 \psi \right| ^{2p} =\sum _{k=0}^\infty \sum _{i\in \mathcal{F }_{c2^{-k}}}\int \limits _{S_{\psi }\left( y_{i}, \frac{\bar{h}(y_{i})}{2}\right) } \left| D^2 \psi \right| ^{2p}, \nonumber \\ \end{aligned}$$

(3.20)

where $\mathcal F _{d}$ is the family of sections $S_{\psi }(y_{i}, \frac{\bar{h}(y_{i})}{2})$ such that $d/2<\bar{h}(y_i)\le d\le c.$ By (3.19), we have for each $S_{\psi }(y_{i}, \frac{\bar{h}(y_{i})}{2})\in \mathcal F _{d}$,

$$\begin{aligned} \int \limits _{S_{\psi }\left( y, \frac{\bar{h}(y)}{2}\right) } \left| D^2 \psi \right| ^{2p}\le C \left| \log d\right| ^{4p} \left| S_{\psi } (y_i, \delta _0 \bar{h}(y_i))\right| \end{aligned}$$

and since

$$\begin{aligned} S_{\psi } (y_i, \delta _0 \bar{h}(y_i))\subset \{x\in B_c\cap U: \text {dist}(x, \partial \Omega \cap \partial U)\le 2k_0^{-1} d^{1/2}\} \end{aligned}$$

are disjoint, we find

$$\begin{aligned} \sum _{i\in \mathcal{F }_{d}}\int \limits _{S_{\psi }\left( y_{i}, \frac{\bar{h}(y_{i})}{2}\right) } \left| D^2 \psi \right| ^{2p}\le C_1\left| \log d\right| ^{4p} d^{1/2} \end{aligned}$$

where $C_1$ now depends also on $\kappa$ which is the upper bound for $\Vert \partial \Omega \cap B_{c}\Vert _{C^{1,1}}$. Therefore, Claim 1 easily follows from (3.20) by adding these inequalities for $d= c2^{-k},\; k=0, 1, \dots$.

It remains to prove (3.19). Let $h:= \bar{h}(y)$. Then $h\le c.$ By applying Proposition 2.3 to $S_{\psi }(y, h)$, we find that it is equivalent to an ellipsoid $E_h$, i.e.,

$$\begin{aligned} k_0 E_h \subset S_{\psi }(y, h)-y \subset k_{0}^{-1} E_h, \end{aligned}$$

where $E_h := h^{1/2}A_{h}^{-1}B_1$ with $\det A_{h}=1$ and $\Vert A_{h}\Vert , \, \Vert A_h^{-1} \Vert \le C |\log h|.$ We use the following rescalings similar to those in Sect. 2.2:

$$\begin{aligned} \tilde{\Omega }_{h} := h^{-1/2} A_{h} (\Omega -y), \end{aligned}$$

and for $x\in \tilde{\Omega }_{h}$

$$\begin{aligned} \tilde{\psi }_{h}(x) := h^{-1} \left[ \psi \left( y + h^{1/2} A^{-1}_{h} x\right) -\psi (y)-\nabla \psi (y)\cdot \left( h^{1/2}A^{-1}_h x\right) -h\right] . \end{aligned}$$

Then

$$\begin{aligned} B_{k_0}\subset S_{\tilde{\psi }_{h}} (0, 1)\equiv h^{-1/2} A_{h} \big (S_{\phi }(y, h) -y\big )\subset B_{k_0^{-1}}. \end{aligned}$$

We have

$$\begin{aligned} \det D^{2} \tilde{\psi }_{h} (x)= \det D^2 \psi \left( y + h^{1/2} A^{-1}_{h} x\right) \quad \text{ and }\quad \tilde{\psi }_{h}= 0 \, \text{ on } \, \partial S_{\tilde{\psi }_{h}} (0, 1). \end{aligned}$$

For simplicity, we denote $\tilde{S}_{t}(0) := S_{\tilde{\psi }_{h}} (0, t)$ for $t>0.$ If $\psi =\phi$ then by Caffarelli’s interior $W^{2,p}$ estimates for the Monge–Ampère equation [2], we have

$$\begin{aligned} \int \limits _{\tilde{S}_{\frac{1}{2}}(0)}\left| D^2 \tilde{\psi }_{h}\right| ^{2p}\le C \end{aligned}$$

if $\epsilon \le \epsilon _{0}$ small depending only on $p, \rho$ and $n$. If $\psi =w$ then as $\det D^2 w=1$, the above inequality obviously holds. Using the fact

$$\begin{aligned} D^{2} \psi \left( y + h^{1/2} A^{-1}_{h} x\right) = A_{h}^{t} \, D^{2} \tilde{\psi }_{h}(x) \, A_{h}, \end{aligned}$$

we obtain (3.19) from

$$\begin{aligned} \int \limits _{S_{\phi }\left( y,\frac{h}{2}\right) }|D^{2} \psi (z)|^{2p} \, dz&= h^{\frac{n}{2}} \int \limits _{\tilde{S}_{\frac{1}{2}}(0)} |A_{h}^{t} \, D^{2} \tilde{\psi }_{h}(x) \, A_{h}|^{2p}\, dx\\&\le C \,h^{\frac{n}{2}} \, |\log h|^{4p} \int \limits _{\tilde{S}_{\frac{1}{2}}(0)} |D^{2} \tilde{\psi }_{h}(x)|^{p}\ \le C \,h^{\frac{n}{2}} \, |\log h|^{4p}. \end{aligned}$$

Finally, we verify Claim 2 by proving (3.18). As in [16, Lemma 3.4], we note that the difference $v:= \phi -w$ is a subsolution (supersolution) of linearized Monge–Ampère equations with bounded right hand side, corresponding to the potentials $w$ and $\phi$ respectively. We cover $U\cap B_{c^2}$ by sections of $w$ and $\phi$ using Lemma 3.13. In each of these sections, we can use the one-sided $W^{2,\delta }$ estimates of Gutiérrez-Tournier [18]. Then, adding these estimates as in the proof of Theorem 3.9, we get (3.18). The details are as follows.

Consider the operator $\mathcal M u:= (\det D^{2}u)^{1/n}$ and its linearized operator

$$\begin{aligned} \hat{\mathcal{L }}_{u} v:= \frac{1}{n} (\det D^{2}u)^{1/n} \text {trace}\big ((D^{2} u)^{-1} D^{2} v\big ). \end{aligned}$$

Notice that $\hat{\mathcal{L }}_{u} v$ and the operator $\mathcal L _{u} v$ defined in (1.1) are related by

$$\begin{aligned} \mathcal L _{u} v= n (\det D^{2}u)^{\frac{n-1}{n}} \hat{\mathcal{L }}_{u}v. \end{aligned}$$

Let $v:= \phi -w$ and $g:= \det D^2\phi .$ Since $\mathcal M$ is concave, we obtain

$$\begin{aligned} g^{1/n}-1 = \mathcal M \phi -\mathcal M w \le \hat{\mathcal{L }}_{w} v\end{aligned}$$

and hence

$$\begin{aligned} \mathcal L _{w} v = n (\det D^{2}w)^{\frac{n-1}{n}} \hat{\mathcal{L }}_{w}v\ge -n |g^{1/n}-1|. \end{aligned}$$

(3.21)

We also have $\hat{\mathcal{L }}_{\phi } v \le \mathcal M \phi - \mathcal M w \le |g^{1/n}-1|$ and thus

$$\begin{aligned} \mathcal L _{\phi } v = n (\det D^{2}\phi )^{\frac{n-1}{n}} \hat{\mathcal{L }}_{\phi }v\le n (1+\epsilon )^{\frac{n-1}{n}} |g^{1/n}-1|\le 2n |g^{1/n}-1|.\quad \quad \end{aligned}$$

(3.22)

On the other hand, it follows from the maximum principle ([19, Lemma 3.1]) that

$$\begin{aligned} \Vert v\Vert _{L^{\infty }(U)}\le C_{n} \text{ diam }\,(U) \Vert g^{1/n}-1\Vert _{L^{n}(U)}. \end{aligned}$$

(3.23)

We cover $U\cap B_{c^2}$ by sections of $w$ using Lemma 3.13. From (3.21) and (3.23), we can use Gutiérrez-Tournier’s one-sided $W^{2,\delta }$ estimates [18] instead of Theorem 3.10 in each of these sections to estimate the $L^{\delta }$ norm of $(D^2 v)^{+}$. After that, we argue as in the proof of Theorem 3.9, and taking into account Lemma 3.13 again to obtain $\delta _{1}=\delta _1(n,\rho )\in (0,1/2)$ and $C_{1}=C_1(n,\rho ,\kappa )>0$ such that

$$\begin{aligned} \Vert (D^2 v)^{+}\Vert _{L^{\delta _{1}}(U\cap B_{c^2})}&\le C_{1}\Big ( \Vert v\Vert _{L^{\infty }(U\cap B_c)} + \Vert (\mathcal L _{w}v)^{-}\Vert _{L^{n}(U\cap B_c)}\Big )\nonumber \\&\le C_{1} \Vert g^{1/n}-1\Vert _{L^{n}(U)}. \end{aligned}$$

(3.24)

Similarly, from (3.22), (3.23) and by covering $U\cap B_{c^2}$ by sections of $\phi$, we obtain

$$\begin{aligned} \Vert (D^2 v)^{-}\Vert _{L^{\delta _{2}}(U\cap B_{c^2})}&\le C_{2}\Big (\Vert v\Vert _{L^{\infty }(U\cap B_c)} + \Vert (\mathcal L _{\phi }v)^{+}\Vert _{L^{n}(U\cap B_c)}\Big )\nonumber \\&\le C_{2} \Vert g^{1/n}-1\Vert _{L^{n}( U)}. \end{aligned}$$

(3.25)

Let $\delta :=\min \{\delta _{1}, \delta _{2}\}.$ Then, from (3.24) and (3.25), we obtain (3.18) as desired since

$$\begin{aligned} \Vert D^2 v\Vert _{L^{\delta }(U\cap B_{c^2})} \le C \Vert g^{1/n}-1\Vert _{L^{n}(U)} \le C \epsilon ^{1/n}. \end{aligned}$$

(ii) The key to the proof is the following estimate

$$\begin{aligned}&\Vert \Phi -\mathrm{W }\Vert _{L^{q}(U\cap B_{c^2})}\nonumber \\&\quad \le C_{n}\left( \epsilon + \Vert D^2 w\Vert ^{n-1}_{L^{\infty }(U\cap B_{c^2})} \Vert D^2\phi -D^2 w\Vert ^{n-1}_{L^{qn}(U\cap B_{c^2})}\right) \Vert D^2 \phi \Vert ^{n-1}_{L^{qn}(U\cap B_{c^2})}\nonumber \\ \end{aligned}$$

(3.26)

which can be deduced from the proof of Lemma 3.5 in [16].

As in Claim 1 in the proof of part (i), we have

$$\begin{aligned} \Vert D^2 \phi \Vert _{L^{q n}(U\cap B_{c^2})} \le C_{0}\quad \text{ for } \text{ all }\, \epsilon \le \epsilon _0, \end{aligned}$$

(3.27)

where $\epsilon _{0}=\epsilon _0(q,n,\rho )>0$ small and $C_0 =C_0(q,n,\rho ,\kappa )>0$.

On the other hand, by (vii) in the definition of the class $\mathcal P$, we have

$$\begin{aligned} \Vert D^2 w\Vert _{L^{\infty }(U\cap B_{c^2})} \le C_{1}(n,\alpha ,\rho ). \end{aligned}$$

(3.28)

Putting (3.26)–(3.28) together, we obtain for $\epsilon \le \epsilon _0$

$$\begin{aligned} \Vert \Phi -\mathrm{W }\Vert _{L^{q}(U\cap B_{c^2})} \le C_{n} \left( \epsilon + C_{1}^{n-1} \Vert D^2\phi -D^2 w\Vert ^{n-1}_{L^{qn}\left( U\cap B_{c^2}\right) }\right) C_{0}^{n-1}. \end{aligned}$$

By applying part (i) of this proposition to $p= qn,$ we then get the desired conclusion. $\square$

We also obtain the following global stability of matrices of cofactors.

Lemma 3.15

(Global stability of cofactor matrices) Let $\Omega \subset \mathbb{R }^{n}$ be a uniformly convex domain satisfying (2.1) and $\Vert \partial \Omega \Vert _{C^3}\le 1/\rho$. For any $q\ge 1$, there exist $C, \epsilon _0>0$ depending only on $q$, $n$ and $\rho$ with the following property. If $\phi ,\,w\in C(\overline{\Omega })$ are convex functions satisfying

$$\begin{aligned} \left\{ \begin{array}{ll} 1-\epsilon \le \det D^2 \phi \le 1+\epsilon &{}\quad \mathrm{in}\, \Omega \\ \phi =0 &{}\quad \mathrm{on}\,\partial \Omega \end{array}\right. \qquad \mathrm{and}\qquad \left\{ \begin{array}{ll} \det D^2 w = 1 &{}\quad \mathrm{in}\, \Omega \\ w=0 &{}\quad \mathrm{on}\,\partial \Omega , \end{array}\right. \end{aligned}$$

then for some small constant $\delta >0$ depending only on $n$ and $\rho$, we have

$$\begin{aligned} \Vert \Phi -\mathrm{W }\Vert _{L^q(\Omega )}\le C \epsilon ^{\frac{(n-1)\delta }{n (2nq - \delta )}}\quad \text{ for } \text{ all }\quad \epsilon \le \epsilon _0. \end{aligned}$$

Proof

The proof follows the lines of the proof of Proposition 3.14 using Proposition 2.4. Here we choose $U=\Omega$, replace $U\cap B_{c^2}$ by $\Omega$ and use the covering Lemma 3.12. The estimate (3.28) is now a classical result of Caffarelli–Nirenberg–Spruck [6]. $\square$

3.5 Global $W^{2, 1 +\epsilon }$ estimates for convex solutions

In this subsection, we establish the global $W^{2, 1 +\epsilon }$ estimates for convex solutions to the linearized Monge–Ampère equations. These estimates are simple consequence of the global stability of cofactor matrices in Sect. 3.4.

Theorem 3.16

Let $\Omega$ be a uniformly convex domain satisfying (2.1) with $\partial \Omega \in C^{3}$. Let $\phi \in C(\overline{\Omega })\cap C^{2}(\Omega )$ be a convex function satisfying

$$\begin{aligned} 0<\lambda \le \det D^{2}\phi \le \Lambda ~\text { in }~\Omega \quad \text{ and }\quad \phi =0 ~\text { on }~\partial \Omega . \end{aligned}$$

Let $v$ be the convex solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \Phi ^{ij} v_{ij} =f &{}\quad \mathrm{in}\, \Omega ,\\ v =0&{}\quad \mathrm{on}\, \partial \Omega , \end{array}\right. \end{aligned}$$

where $f\in L^\infty (\Omega )$. Then, there exist $\gamma >1$ and $C>0$ depending only on $\lambda , \Lambda , n$ and $\Omega$ such that

$$\begin{aligned} \Vert D^2 v\Vert _{L^{\gamma }(\Omega )}\le C\Vert f\Vert _{L^{\infty }(\Omega )}. \end{aligned}$$

(3.29)

Remark 3.17

• (i) De Phillipis–Figalli–Savin [9] and Schmidt [29] discovered the interior $W^{2, 1+\epsilon }$ estimates for convex solution $\phi$ to the Monge–Ampère equation

  $$\begin{aligned} \det D^2 \phi =g ~ \text{ in }\, \Omega ,\quad \text {and}\quad \phi =0~ \text{ on }\, \partial \Omega , \end{aligned}$$

  with $0<\lambda \le g\le \Lambda$. In these works, the convexity of $\phi$ plays a crucial role, especially in giving a bound for $|D^2 \phi |$ by $\Delta \phi .$ Since $\Phi ^{ij}\phi _{ij} = n \det D^2\phi = ng,$ our theorem is a natural extension of De Phillipis–Figalli–Savin’s and Schmidt’s estimates.

• (ii) The convexity of $v$ and standard arithmetic-geometric inequality give

  $$\begin{aligned} f = \Phi ^{ij}v_{ij} =\text {trace}(\Phi \, D^2v)\ge n (\det \Phi )^{1/n}(\det D^2 v)^{1/n}\ge 0. \end{aligned}$$

• (iii) It would be interesting to remove the convexity of $v$ in the statement of Theorem 3.16.

Now, we proceed with the proof of Theorem 3.16. To do this, we first establish the following Sobolev stability result.

Proposition 3.18

(Sobolev stability estimates) Let $\Omega$ be a uniformly convex domain with $\partial \Omega \!\in \! C^{3}$. Let $\phi _{k}\!\in \! C(\overline{\Omega })\!\cap \! C^{2}(\Omega )$ ($k\!=\!1,2$) be convex Aleksandrov solutions of

$$\begin{aligned} \det D^2 \phi _{k} =g_{k}~ \text{ in }\, \Omega ,\quad \text{ and }\quad \phi _{k} =0~ \text{ on }\, \partial \Omega , \end{aligned}$$

with $0<\lambda \le g_{k}\le \Lambda$ in $\Omega$. Then there exist $\gamma >1,\; \alpha \in (0,1)$ and $C>0$ depending only on $n, \lambda , \Lambda$ and $\Omega$ such that

$$\begin{aligned} \Vert D^2\phi _{1}-D^2\phi _2\Vert _{L^{\gamma }(\Omega )}\le C \Vert g_{1}- g_{2}\Vert ^{\frac{\alpha }{n}}_{L^{1}(\Omega )}. \end{aligned}$$

(3.30)

Proof

The interior counterpart of our proposition was established by De Phillipis–Figalli [8]. Here, we will prove the boundary version with a different method. Our proof relies on the $W^{2,\delta }$ estimates of Gutiérrez-Tournier [18] for solutions to the linearized Monge–Ampère equation.

First, using Proposition 2.4, [19, Lemma 3.1] and arguing as in the proof of (3.18) in Proposition 3.14, we find a small $\delta >0$ and $C_{1}>0$ depending only on $n, \lambda , \Lambda$ and $\Omega$ such that

$$\begin{aligned} \Vert D^2\phi _{1}-D^2\phi _2\Vert _{L^{\delta }(\Omega )}\le C_{1} \Vert g^{\frac{1}{n}}_{1}- g^{\frac{1}{n}}_{2}\Vert _{L^{n}(\Omega )} \le C_{1} \Vert g_{1}- g_{2}\Vert ^{\frac{1}{n}}_{L^{1}(\Omega )}. \end{aligned}$$

(3.31)

Second, using De Phillipis–Figalli–Savin’s and Schmidt’s interior $W^{2, 1+\epsilon }$ estimates for solutions to the Monge–Ampère equation [9, 29] and arguing as in [28], we obtain the following global $W^{2, 1+\epsilon }$ estimates

$$\begin{aligned} \Vert D^2\phi _{1}\Vert _{L^{\gamma _{1}}(\Omega )} + \Vert D^2\phi _{2}\Vert _{L^{\gamma _{1}}(\Omega )} \le C_{2}, \end{aligned}$$

(3.32)

where $\gamma _{1}>1$ and $C_{2}>0$ depend only on $n, \lambda , \Lambda ,$ and $\Omega$.

We now choose $\alpha \in (0,1)$ sufficiently close to $0$ so that

$$\begin{aligned} \frac{1}{\gamma }:= \frac{\alpha }{\delta } + \frac{1-\alpha }{\gamma _{1}}<1, \end{aligned}$$

i.e., $\gamma >1.$ Then by the interpolation inequality, we obtain

$$\begin{aligned} \Vert D^2\phi _{1}-D^2\phi _2\Vert _{L^{\gamma }(\Omega )}\le \Vert D^2\phi _{1}-D^2\phi _2\Vert ^{\alpha }_{L^{\delta }(\Omega )} \Vert D^2\phi _{1}-D^2\phi _2\Vert ^{1-\alpha }_{L^{\gamma _{1}}(\Omega )} \end{aligned}$$

which together with (3.31) and (3.32) yields the estimate (3.30). $\square$

Proof of Theorem 3.16.

For any $t\in (0, \Vert f\Vert ^{-1}_{L^{\infty }(\Omega )}]$, we have $\phi =\phi + tv~\text {on}~\partial \Omega$ and, by the convexity of $v$, $\phi \ge \phi + tv~\text {in}~\Omega .$ Thus

$$\begin{aligned} \lambda \le \det D^2 \phi \le \det D^{2} (\phi + tv). \end{aligned}$$

Moreover by the concavity of the map $\phi \mapsto \log \det D^2 \phi ,$ we obtain

$$\begin{aligned} \log \det D^2 (\phi + tv)\le log \det D^2 \phi + t \phi ^{ij}v_{ij} = log \det D^2\phi + t\frac{f}{det D^2 \phi }. \end{aligned}$$

Therefore,

$$\begin{aligned} 0\le \det D^2 (\phi + tv) - \det D^2 \phi \le (\det D^2 \phi ) \, \left( e^{\frac{tf}{\det D^2\phi }}-1\right) \le \Lambda \big (e^{\frac{1}{\lambda }}-1\big ) \quad \text{ in }\,\Omega . \end{aligned}$$

Applying the stability result in Proposition 3.18, we can find $\alpha , C>0, \gamma >1$ depending only on $n, \lambda , \Lambda$ and $\Omega$ such that

$$\begin{aligned} \Vert t D^2 v\Vert _{L^{\gamma }(\Omega )}\le C \Vert \det D^2 (\phi + tv) - \det D^2 \phi \Vert ^{\frac{\alpha }{n}}_{L^{1}(\Omega )} \le C \Vert \Lambda (e^{\frac{1}{\lambda }}-1)\Vert ^{\frac{\alpha }{n}}_{L^{1}(\Omega )}. \end{aligned}$$

The estimate (3.29) follows by taking $t= \Vert f\Vert ^{-1}_{L^{\infty }(\Omega )}$. $\square$

4 Global Hölder estimates and approximation lemma

In this section, we establish global Hölder continuity estimates for solutions to the linearized Monge–Ampère equation under natural assumptions on the domain, Monge–Ampère measure and Hölder continuous boundary data. We then use these Hölder estimates to prove approximation lemmas allowing us to approximate the solution $u$ to $\mathcal L _{\phi } u =f$ by smooth solutions of linearized Monge–Ampère equations associated with convex functions $w$ whose Monge–Ampère measures are close to that of $\phi$.

4.1 Global Hölder estimates

In this subsection, we derive global Hölder estimates for solutions to the linearized Monge–Ampère equation in convex domains when the right hand side is assumed to be in $L^{n}$ and the boundary data is Hölder continuous. These estimates extend the global Hölder estimates in [21] where the domains are assumed to be uniformly convex.

Our first main theorem is concerned with Hölder estimates in a neighborhood of a boundary point. Its precise statement is as follows.

Theorem 4.1

Assume $\Omega$ and $\phi$ satisfy (2.1), (2.2), (2.4) and if $x_0\in \partial \Omega \cap B_\rho$ then

$$\begin{aligned} \rho \left| x-x_{0}\right| ^2 \le \phi (x)- \phi (x_{0})-\nabla \phi (x_{0}) \cdot (x- x_{0}) \le \rho ^{-1}\left| x-x_{0}\right| ^2, ~\forall x\in \partial \Omega .\nonumber \\ \end{aligned}$$

(4.1)

Let $u \in C\big (B_{\rho }\cap \overline{\Omega }\big ) \cap W^{2,n}_{loc}(B_{\rho }\cap \Omega )$ be a solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \Phi ^{ij}u_{ij} = f &{}\quad \text {in} \, B_{\rho }\cap \Omega , \\ u = \varphi &{}\quad \text {on}\, \partial \Omega \cap B_{\rho }, \end{array} \right. \end{aligned}$$

where $\varphi \in C^{\alpha }(\partial \Omega \cap B_{\rho })$ for some $\alpha \in (0,1)$. Then, there exist constants $\beta , C >0$ depending only on $\lambda , \Lambda , n, \alpha$ and $\rho$ such that

$$\begin{aligned} |u(x)-u(y)|&\le C|x-y|^{\beta }\Big ( \Vert u\Vert _{L^{\infty }(\Omega \cap B_{\rho })} + \Vert \varphi \Vert _{C^\alpha (\partial \Omega \cap B_{\rho })} + \Vert f\Vert _{L^{n}(\Omega \cap B_{\rho })} \Big ),\\&\quad \forall x, y\in \Omega \cap B_{\rho /2}. \end{aligned}$$

As an immediate consequence of Theorem 4.1, we obtain the following estimates which are the global counterparts of Caffarelli–Gutiérrez’s interior Hölder estimates for solutions to the linearized Monge–Ampère equation [4].

Theorem 4.2

Assume $\Omega$ and $\phi$ satisfy (2.1)–(2.3). Let $u \in C(\overline{\Omega })\cap W^{2, n}_{loc}(\Omega )$ be a solution to

$$\begin{aligned} \Phi ^{ij}u_{ij}= f ~\text { in}\, \Omega ,\quad \text{ and }\quad u = \varphi ~\text { on}\,\partial \Omega , \end{aligned}$$

where $\varphi \in C^{\alpha }(\partial \Omega )$ for some $\alpha \in (0,1)$. Then, there exist constants $\beta , C >0$ depending only on $\lambda , \Lambda , n, \alpha$ and $\rho$ such that

$$\begin{aligned} |u(x)-u(y)|\le C|x-y|^{\beta }\Big ( \Vert u\Vert _{L^{\infty }(\Omega )} + \Vert \varphi \Vert _{C^\alpha (\partial \Omega )} + \Vert f\Vert _{L^{n}(\Omega )} \Big ),~\forall x, y\in \Omega . \end{aligned}$$

The key to the proof of Theorem 4.1 is the following boundary Hölder estimates.

Proposition 4.3

Let $\phi$ and $u$ be as in Theorem 4.1. Then, there exist $\delta , C$ depending only on $\lambda , \Lambda , n, \alpha , \rho$ such that, for any $x_{0}\in \partial \Omega \cap B_{\rho /2}$, we have

$$\begin{aligned}&|u(x)-u(x_{0})|\le C|x-x_{0}|^{\frac{\alpha }{\alpha +3n}} \Big ( \Vert u\Vert _{L^{\infty }(\Omega \cap B_{\rho })} + \Vert \varphi \Vert _{C^\alpha (\partial \Omega \cap B_{\rho })} + \Vert f\Vert _{L^{n}(\Omega \cap B_{\rho })} \Big ),\\&\quad ~\forall x\in \Omega \cap B_{\delta }(x_{0}). \end{aligned}$$

Proof of Theorem 4.1.

The boundary Hölder estimates in Proposition 4.3 combined with the interior Hölder continuity estimates of Caffarelli–Gutiérrez [4] and Savin’s Localization Theorem [26–28] gives the global Hölder estimates in Theorem 4.1. The precise arguments are almost the same as the proof of [21, Theorem 1.4]. Since [21, Theorem 1.4] is a global result and our theorem is local, we indicate some differences in the arguments. It suffices to prove the theorem for $x, y\in B_{c^2}\cap \Omega$. We use the quadratic separation (4.1) and Proposition 2.3 to show that if $y\in \Omega \cap B_{c^2}$ then the maximal interior section $S_{\phi }(y, \bar{h}(y))$ is contained in $\Omega \cap B_c$ and so tangent to $\partial \Omega$ at $x_{0}\in \partial \Omega \cap B_{c}$ (see Lemma 2.5(e)). Using this fact, Caffarelli–Gutiérrez’s interior Hölder estimates [4] and Proposition 4.3, we obtain as in [21]

$$\begin{aligned}&\left| u(z_1)-u(z_2)\right| \le \left| z_1-z_2\right| ^{\beta }\Big ( \Vert u\Vert _{L^{\infty }(\Omega \cap B_{\rho })} + \Vert \varphi \Vert _{C^\alpha (\partial \Omega \cap B_{\rho })} + \Vert f\Vert _{L^{n}(\Omega \cap B_{\rho })} \Big )\\&\quad \forall z_1, z_2\in S_{\phi }\left( y, \frac{\bar{h}(y)}{2}\right) . \end{aligned}$$

The rest of the argument is the same as in [21]. $\square$

The proof of Proposition 4.3 is based on a construction of suitable barriers. Assume $\phi$ and $\Omega$ satisfy the assumptions in the proposition. We also assume for simplicity that $\phi (0)=0\, \text{ and } \, \nabla \phi (0)=0.$ We now construct a supersolution as in [23, Lemma 6.2].

Lemma 4.4

(Supersolution) Given $\delta$ universally small ($\delta \le \rho$), define

$$\begin{aligned} \tilde{\delta }:= \frac{\delta ^3}{2}\quad \text{ and } \quad M_{\delta }:= \frac{2^{n-1} \Lambda ^n}{\lambda ^{n-1}}\frac{1}{\delta ^{3n-3}} \equiv \frac{\Lambda ^{n}}{(\lambda \tilde{\delta })^{n-1}}.\nonumber \\ \end{aligned}$$

Then the function

$$\begin{aligned} w_{\delta }(x', x_n):= M_{\delta } x_{n} + \phi - \tilde{\delta }|x'|^2 - \frac{\Lambda ^n}{( \lambda \tilde{\delta })^{n-1}} x_{n}^2\quad \text{ for }\quad (x', x_n)\in \overline{\Omega } \end{aligned}$$

satisfies

$$\begin{aligned} \mathcal L _{\phi } (w_{\delta }):= \Phi ^{ij}(w_{\delta })_{ij} \le - n\Lambda \quad \text{ in }\,\Omega , \end{aligned}$$

and

$$\begin{aligned} w_{\delta } \ge 0 ~\text { on }~ \partial (\Omega \cap B_\delta ), \quad w_{\delta } \ge \frac{\delta ^3}{2} ~\text { on }~ \Omega \cap \partial B_{\delta }. \end{aligned}$$

Proof

We recall from (2.7) that

$$\begin{aligned} \overline{\Omega }\cap B^{+}_{ch^{1/2}/\left| \log h\right| }\subset S_{\phi }(0, h) \subset \overline{\Omega } \cap B^{+}_{C h^{1/2} \left| \log h\right| }. \end{aligned}$$

The first inclusion gives $\phi \le h~\text { in }~ \overline{\Omega }\cap B^{+}_{ch^{1/2}/\left| \log h\right| }$ and hence for $x$ close to the origin,

$$\begin{aligned} \phi (x)\le C\left| x\right| ^2\left| \log \left| x\right| \right| ^2. \end{aligned}$$

Similarly, the second inclusion gives

$$\begin{aligned} \phi (x)\ge c\left| x\right| ^2\left| \log \left| x\right| \right| ^{-2}\ge \left| x\right| ^3 \end{aligned}$$

for $x$ close to the origin. In conclusion, we have

$$\begin{aligned} \left| x\right| ^3 \le \phi (x) \le C\left| x\right| ^2 \left| \log {|x|}\right| ^2 \end{aligned}$$

(4.2)

if $\left| x\right| \le \delta$ for $\delta$ universally small. Therefore, the choice of $\tilde{\delta }$ gives

$$\begin{aligned} \phi (x)- \tilde{\delta }|x'|^2 \ge \left| x\right| ^3-\tilde{\delta }\left| x\right| ^2\ge \frac{1}{2}\left| x\right| ^3 = \tilde{\delta }\quad \text{ on } \, \Omega \cap \partial B_{\delta }. \end{aligned}$$

On the other hand, the choice of $M_{\delta }$ implies that

$$\begin{aligned} M_\delta x_n - \frac{\Lambda ^n}{( \lambda \tilde{\delta })^{n-1}} x_{n}^2 \ge 0 \quad \quad \text {on}\,\overline{ \Omega \cap B_{\delta }}. \end{aligned}$$

Hence, $w_\delta \ge \tilde{\delta }$ on $\Omega \cap \partial B_{\delta }$ while on $\partial \Omega \cap B_{\delta }$, the quadratic separation (4.1) and $\delta \le \rho$ give

$$\begin{aligned} w_{\delta } \ge \phi - \tilde{\delta }|x'|^2 \ge 0. \end{aligned}$$

As a consequence, we obtain the desired inequalities for $w_{\delta }$ on $\partial (\Omega \cap B_{\delta })$.

It remains to prove that $\mathcal L _{\phi } (w_{\delta })\le - n\Lambda$ in $\Omega$. If we denote

$$\begin{aligned} q(x):=\frac{1}{2} \left( \tilde{\delta }|x'|^2 + \frac{\Lambda ^n}{( \lambda \tilde{\delta })^{n-1}} x_{n}^2 \right) , \end{aligned}$$

then

$$\begin{aligned} \det D^2 q=\frac{\Lambda ^n}{\lambda ^{n-1}}, \quad D^2q \ge \tilde{\delta }I. \end{aligned}$$

Using the matrix inequality

$$\begin{aligned} \text {trace}(AB) \ge n (\det A \det B)^{1/n}~\text { for }~ A, B~\text { symmetric }~\ge 0, \end{aligned}$$

we get

$$\begin{aligned} \mathcal L _\phi q=\text {trace}(\Phi \, D^2 q) \ge n (\det (\Phi ) \det D^{2} q)^{1/n} = n \left( (\det D^{2}\phi )^{n-1}\frac{\Lambda ^n}{\lambda ^{n-1}}\right) ^{1/n}\ge n\Lambda . \end{aligned}$$

Since $\mathcal L _\phi x_n=0,$ we find

$$\begin{aligned} \mathcal L _\phi w_{\delta }&= L_\phi (M_{\delta }x_n + \phi - 2 q) = \Phi ^{ij}\phi _{ij}- 2 \mathcal L _\phi q\\&= n\det D^2\phi - 2\mathcal L _\phi q\le -n\Lambda \quad \text{ in }\,\Omega . \end{aligned}$$

$\square$

Proof of Theorem 4.3.

Our proof follows closely the proof of Proposition 2.1 in [21]. We can suppose that $K:= \Vert u\Vert _{L^{\infty }(\Omega \cap B_{\rho })} + \Vert \varphi \Vert _{C^\alpha (\partial \Omega \cap B_{\rho })} + \Vert f\Vert _{L^{n}(\Omega \cap B_{\rho })}$ is finite. By working with the function $v:= u/K$ instead of $u$, we can assume in addition that

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\Omega \cap B_{\rho })} + \Vert \varphi \Vert _{C^\alpha (\partial \Omega \cap B_{\rho })} + \Vert f\Vert _{L^{n}(\Omega \cap B_{\rho })}\le 1 \end{aligned}$$

and need to show that the inequality

$$\begin{aligned} |u(x)-u(x_{0})|\le C|x-x_{0}|^{\frac{\alpha }{\alpha +3n}}\quad \forall x\in \Omega \cap B_{\delta }(x_{0}) \end{aligned}$$

(4.3)

holds for all $x_0\in \Omega \cap B_{\rho /2}$, where $\delta$ and $C$ depends only on $\lambda , \Lambda , n, \alpha$ and $\rho$.

We prove (4.3) for $x_{0} =0$. However, our arguments apply to all points $x_0\in \Omega \,\cap \, B_{\rho /2}$ with obvious modifications. For any $\varepsilon \in (0,1)$, we consider the functions

$$\begin{aligned} h_{\pm }(x) := u(x)- u(0)\pm \epsilon \pm \frac{6}{\delta _2^3} w_{\delta _2} \end{aligned}$$

in the region

$$\begin{aligned} A:= \Omega \cap B_{\delta _{2}}(0), \end{aligned}$$

where $\delta _{2}$ is small to be chosen later and the function $w_{\delta _2}$ is as in Lemma 4.4. We remark that $w_{\delta _2}\ge 0$ in $A$ by the maximum principle. Observe that if $x\in \partial \Omega$ with $|x|\le \delta _{1}(\epsilon ):= \epsilon ^{1/\alpha }$ then,

$$\begin{aligned} |u(x)-u(0)| =|\varphi (x)-\varphi (0)| \le |x|^{\alpha } \le \epsilon . \end{aligned}$$

(4.4)

On the other hand, if $x\in \Omega \cap \partial B_{\delta _2}$ then from Lemma 4.4, we obtain $\frac{6}{\delta _2^3} w_{\delta _2}(x)\ge 3.$ It follows that, if we choose $\delta _{2}\le \delta _{1}$ then from (4.4) and $|u(x)-u(0)\pm \epsilon |\le 3$, we get

$$\begin{aligned} h_{-}\le 0, \, h_{+}\ge 0~\text { on}\,\partial A. \end{aligned}$$

Also from Lemma 4.4, we have

$$\begin{aligned} \mathcal L _{\phi } h_{+}\le f, \, \mathcal L _{\phi } h_{-}\ge f~\text { in}\,A. \end{aligned}$$

Hence the ABP estimate applied in $A$ gives

$$\begin{aligned} h_{-}\le C_{1}(n, \lambda )diam (A) \Vert f\Vert _{L^{n}(A)}\le C_{1}(n, \lambda )\delta _{2}~\text { in }~ A \end{aligned}$$

(4.5)

and

$$\begin{aligned} h_{+}\ge - C_{1}(n, \lambda )diam (A) \Vert f\Vert _{L^{n}(A)}\ge -C_{1}(n,\lambda )\delta _{2}~\text { in }~ A. \end{aligned}$$

(4.6)

By restricting $\epsilon \le C_{1}(n, \lambda )^{\frac{-\alpha }{1-\alpha }}$, we can assume that

$$\begin{aligned} \delta _{1} = \epsilon ^{1/\alpha }\le \frac{\epsilon }{C_{1}(n,\lambda )}. \end{aligned}$$

Then, for $\delta _{2}\le \delta _{1}$, we have $C_{1}(n,\lambda )\delta _{2}\le \epsilon$ and thus, for all $x\in A$, we obtain from (4.5) and (4.6) that

$$\begin{aligned} |u(x)-u(0)|\le 2\epsilon + \frac{6}{\delta _2^3} w_{\delta _2}(x). \end{aligned}$$

Note that, by construction and the estimate (4.2) for the function $\phi$, we have in $A$

$$\begin{aligned} w_{\delta _2}(x)\le M_{\delta _2} x_n + \phi (x)\le M_{\delta _2}\left| x\right| + C\left| x\right| ^2\left| \log \left| x\right| \right| ^2\le 2M_{\delta _2}\left| x\right| . \end{aligned}$$

Therefore, choosing $\delta _{2}= \delta _{1}$ and recalling the choice of $M_{\delta _2}$, we get

$$\begin{aligned} |u(x)-u(0)|\le 2\epsilon + \frac{12 M_{\delta _2}}{\delta _2^3}\left| x\right| = 2\epsilon + \frac{C_{2}(n,\lambda , \Lambda )}{\delta _{2}^{3n}}|x| = 2\epsilon + C_{2} \epsilon ^{-3n/\alpha }|x|\nonumber \\ \end{aligned}$$

(4.7)

for all $x,\epsilon$ satisfying the following conditions

$$\begin{aligned} |x|\le \delta _{1}(\epsilon ):= \epsilon ^{1/\alpha },\quad \epsilon \le C_1(n,\lambda )^{\frac{-\alpha }{1-\alpha }} =: c_{1}. \end{aligned}$$

(4.8)

Finally, let us choose $\epsilon = |x|^{\frac{\alpha }{\alpha + 3n}}.$ It satisfies the conditions in (4.8) if

$$\begin{aligned} |x|\le \min \left\{ c_{1}^{\frac{\alpha +3n}{\alpha }}, 1\right\} =: \delta . \end{aligned}$$

Then, by (4.7), we have $|u(x)-u(0)| \le (2 + C_2) |x|^{\frac{\alpha }{\alpha + 3n}}\,$ for all $\, x\in \Omega \cap B_{\delta }(0)$. $\square$

4.2 Global approximation lemma

In this subsection, we prove an approximation lemma that allows us to compare the solution $u$ to the linearized Monge–Ampère equation $\mathcal L _{\phi } u =f$ to smooth solutions $h$ of linearized Monge–Ampère equations $\mathcal L _{w}h =0$ associated with convex functions $w$ satisfying $\det D^2 w =1$. We will estimate the difference $u-h$ in terms of the $L^{n}$-norms of $f$ and $\Phi -\mathrm{W }$ where $\Phi =(\Phi ^{ij})$ and $\mathrm{W }= (\mathrm{W }^{ij})$ are the matrices of cofactors of $D^2\phi$ and $D^2w$, respectively. Therefore, in light of the global stability of cofactor matrices in Sect. 3.4, $u$ is well-approximated by $h$ provided that $\det D^2\phi$ is close to $1$. This approximation lemma will play a key role in Sect. 5 where we use it to get power decay estimates for the distribution function of the second derivatives of $u$ that are more refined than those provided by Proposition 3.7.

Our approximation lemma, relevant for data of the type $\big (\Omega _{h}, \phi _{h}, S_{\phi _{h}}(0, 1)\big )$, states as follows.

Lemma 4.5

Assume $(\Omega , \phi , U)\in \mathcal P _{\frac{1}{2}, \frac{3}{2}, \rho , \kappa , \alpha }.$ Let $r := c^2/4$ where $c$ is as in Remark 2.6. Suppose that $u\in C(\overline{U})\cap W^{2,n}_{loc}(U)$ is a solution of $\,\Phi ^{i j} u_{i j} = f\,$ in $U\cap B_{4r}$ with

$$\begin{aligned}\Vert u\Vert _{L^{\infty }(U\cap B_{4r})} +\Vert u\Vert _{C^{2,\alpha }(\partial U\cap B_{4r})} \le 1. \end{aligned}$$

Let $w$ be defined as in $(vii)$ of the definition of the class $\mathcal P$. Assume $h$ is a solution of

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\mathrm{W }^{i j} h_{i j} = 0 \quad \text{ in } \, B_{2r}\cap U\\ &{}h =u\qquad \qquad \text{ on } \,\partial (B_{2r}\cap U). \end{array}\right. \end{aligned}$$

Then, there exist $C>0$ and $\gamma \in (0,1)$ depending only on $n, \rho$ and $\alpha$ such that

$$\begin{aligned} \Vert h\Vert _{C^{1,1}(\overline{ B_{r} \cap U})} \le C, \end{aligned}$$

(4.9)

and if $\Vert \Phi - \mathrm{W }\Vert _{L^n( B_{2r}\cap U)}\le r^4$ then

$$\begin{aligned}&\Vert u-h\Vert _{ L^\infty (B_{r}\cap U)} +\Vert f-\text {trace}([\Phi -\mathrm{W }]D^2h)\Vert _{ L^n( B_{r}\cap U)}\\&\quad \le C \left\{ \big (1+ \Vert u\Vert _{C^{1/2}(\partial U\cap B_{4r})} \big )\, \Vert \Phi - \mathrm{W }\Vert _{L^n( B_{2r}\cap U)}^\gamma + \Vert f\Vert _{L^n(U\cap B_{4r})} \right\} . \end{aligned}$$

Proof

Observe first that by $(vii)$ in the definition of the class $\mathcal P$, the following $C^{2,\alpha }$ and Pogorelov estimates hold

$$\begin{aligned} \Vert \partial U\!\cap \! B_{4r}\Vert _{C^{2,\alpha }} \le c_{0}^{-1},~ \Vert w\Vert _{C^{2,\alpha }(\overline{U\cap B_{4r}})} \le c_{0}^{-1},~c_{0}I_{n}\le D^2 w\le c_{0}^{-1} I_{n}~\text { in }~ B_{4r}\!\cap U.\nonumber \\ \end{aligned}$$

(4.10)

Therefore, $W^{ij} \partial _{ij}$ is a uniformly elliptic differential operator with $C^{\alpha }$ coefficients. Hence, we can employ the standard boundary $C^{2,\alpha }$-estimates for linear uniformly elliptic equation and obtain (4.9) since

$$\begin{aligned} \Vert h\Vert _{C^{1,1}(\overline{ B_{r} \cap U})}&\le \Vert h\Vert _{C^{2,\alpha }(\overline{ B_{r} \cap U})} \le C(n, \rho , \alpha ) \big ( \Vert u\Vert _{L^\infty ( B_{2r} \cap U)} + \Vert u\Vert _{C^{2,\alpha }(\partial U\cap B_{2r})}\big )\\&\le C(n, \rho ,\alpha ). \end{aligned}$$

Next, since $(\Omega , \phi , U)\in \mathcal P _{\frac{1}{2}, \frac{3}{2}, \rho , \kappa , \alpha }$, by Remark 2.11, the domain $U$ and function $\phi$ satisfy (2.1), (2.2) and (2.4) and (4.1). Therefore, it follows from Theorem 4.1 with $C^{1/2}$ boundary data that there exist constants $C>0$ and $\beta \in (0,1)$ depending only on $n$ and $\rho$ such that

$$\begin{aligned} \Vert u\Vert _{C^\beta (\overline{ B_{2r} \cap U})} \le C\Big (\Vert u\Vert _{L^\infty ( B_{4r} \cap U)} + \Vert u\Vert _{C^{1/2}(\partial U \cap B_{4r})} + \Vert f\Vert _{L^n(B_{4r} \cap U)}\Big ) \le C \Theta ,\nonumber \\ \end{aligned}$$

(4.11)

where

$$\begin{aligned} \Theta :=1 + \Vert u\Vert _{C^{1/2}(\partial U \cap B_{4r})} + \Vert f\Vert _{L^n(B_{4r} \cap U)}. \end{aligned}$$

In view of (4.10), (4.11), and the standard global Hölder estimates for linear uniformly elliptic equations (see [14, Corollary 9.29], [3, Proposition 4.13] and [35, Theorem 1.10]), we can find constants $C>0$ and $\beta ' \in (0, \beta )$ depending only on $n, \rho$ and $\alpha$ such that

$$\begin{aligned} \Vert h\Vert _{C^{\beta '}(\overline{ B_{2r} \cap U})} \le C\Big (\Vert h\Vert _{L^\infty (B_{2r} \cap U)} + \Vert u\Vert _{C^{\beta }(\partial (U\cap B_{2r}))} \Big ) \le C \Theta . \end{aligned}$$

(4.12)

Now let $0<\delta <r$. Then we claim that

$$\begin{aligned} \Vert u-h\Vert _{L^\infty (\partial (B_{2r-\delta }\cap U)) } \le C\delta ^{\beta '} \Theta , \end{aligned}$$

(4.13)

and

$$\begin{aligned} \Vert D^2 h\Vert _{L^\infty (B_{2r-\delta }\cap U) } \le C \delta ^{\beta ' -2-\alpha } \Theta . \end{aligned}$$

(4.14)

To prove (4.13), we verify that $|(u-h)(x)|\le C\delta ^{\beta '} \Theta$ for all $x\in \partial (B_{2r-\delta }\cap U)$. Indeed, if $x\in \partial (B_{2r-\delta }\cap U)$ then we can find $y\in \partial (B_{2r}\cap U)$ such that $|x-y| \le \delta$. Since $u-h =0$ on $\partial (B_{2r}\cap U)$, we get from (4.11) and (4.12) that

$$\begin{aligned} |(u-h)(x)| = |(u-h)(x) - (u-h)(y)| \le |u(x)-u(y)| + |h(x)-h(y)| \le C \delta ^{\beta '} \Theta . \end{aligned}$$

To prove (4.14), let $x_0\in B_{2r-\delta }\cap U$. If $B_{\delta /2}(x_0)\subset B_{2r}\cap U$, then we can apply interior $C^{1,1}$-estimates to $h-h(x_0)$ in $B_{\delta /2}(x_0)$ and use (4.12) to get

$$\begin{aligned} \Vert D^2 h(x_0)\Vert \le C \delta ^{-2} \Vert h-h(x_0)\Vert _{L^\infty \left( B_{\frac{\delta }{2}}(x_0)\right) } \le C \delta ^{\beta '-2} \Theta . \end{aligned}$$

In case , then there exists $z_0\in B_{2r-\delta }\cap \partial U \subset \partial \Omega$ such that $x_0\in B_{\delta /2}(z_0)$. Hence since $B_\delta (z_0)\cap U \subset B_{2r}\cap U$ and by applying boundary $C^{2,\alpha }$-estimates to $h-h(x_0)$ in $B_{\delta /2}(z_0)\cap U$ we obtain

$$\begin{aligned}&\Vert D^2(h-h(x_0)) \Vert _{C^{\alpha }(B_{\frac{\delta }{2}}(z_0)\cap U)} \\&\quad \!\le \! C \delta ^{-(2+\alpha )} \Big ( \Vert h-h(x_0)\Vert _{L^\infty (B_{\delta }(z_0)\cap U)} \!+\!\sum _{k=1}^{2} \delta ^{k+\alpha }\Vert D^k( u-h(x_0))\Vert _{C^{\alpha }(\partial U\cap B_{\delta }(z_0))}\Big )\\&\quad \le C \delta ^{-(2+\alpha )} \Big (\delta ^{\beta '} \Theta +\delta ^{1+\alpha }\Big )\le C \delta ^{\beta ' -2-\alpha } \Theta . \end{aligned}$$

It follows that $\Vert D^2 h(x_0)\Vert \le C \delta ^{\beta '-2-\alpha } \Theta$, and thus (4.14) is proved.

Having (4.13) and (4.14), we now complete the proof of the lemma. Observe that $u-h\in W^{2,n}_{loc}( B_{2r}\cap U)$ is a solution of

$$\begin{aligned} \Phi ^{ij} (u-h)_{ij}=f - \Phi ^{ij} h_{i j} = f- [ \Phi ^{i j} - \mathrm{W }^{ij}] h_{i j} =: F \quad \text{ in }\, B_{2r}\cap U. \end{aligned}$$

The ABP estimate together with (4.13) and (4.14) gives

$$\begin{aligned}&\Vert u-h\Vert _{L^\infty ( B_{2r-\delta }\cap U) } +\Vert F\Vert _{L^n(B_{2r-\delta }\cap U)}\\&\quad \le \Vert u-h\Vert _{L^\infty (\partial (B_{2r-\delta }\cap U)) } + C_n \Vert F\Vert _{L^n(B_{2r-\delta }\cap U)} \\&\quad \le \Vert u-h\Vert _{L^\infty (\partial (B_{2r-\delta }\cap U)) } +C_n \Vert D^2 h\Vert _{L^\infty (B_{2r-\delta }\cap U) } \Vert \Phi -\, \mathrm{W }\Vert _{L^n(B_{2r}\cap U)} \\&\qquad +\, C_n \Vert f\Vert _{L^n(U\cap B_{2r})} \\&\quad \le C \Big ( \delta ^{\beta '} + \delta ^{\beta '-2-\alpha }\Vert \Phi - \mathrm{W }\Vert _{L^n(B_{2r}\cap U)}\Big ) \, \Theta + C_n \Vert f\Vert _{L^n(U\cap B_{2r})}. \end{aligned}$$

If $\Vert \Phi - \mathrm{W }\Vert _{L^n(B_{2r}\cap U)}\le r^4$ then by taking $\delta = \Vert \Phi - \mathrm{W }\Vert _{L^n(B_{2r}\cap U)}^{\frac{1}{2 +\alpha }},$ we obtain the desired inequality with $\gamma =\beta '/(2+\alpha )$ since

$$\begin{aligned}&\Vert u-h\Vert _{L^\infty (B_{r}\cap U) } +\Vert F\Vert _{L^n(B_{r}\cap U)} \le C \Vert \Phi - \mathrm{W }\Vert _{L^n(B_{2r}\cap U)}^{\frac{\beta '}{2+\alpha }} \Theta + C_n \Vert f\Vert _{L^n(U\cap B_{2r})}\\&\quad \le C \Big \{ \big ( 1 + \Vert u\Vert _{C^{1/2}(\partial U\cap B_{4r})}\big )\, \Vert \Phi - \mathrm{W }\Vert _{L^n(B_{2r}\cap U)}^{\frac{\beta '}{2+\alpha }} + \Vert f\Vert _{L^n( U\cap B_{4r})}\Big \}. \end{aligned}$$

$\square$

We end this subsection with a result allowing us to estimate the measure of the set where the quasi distance generated by $\phi$ is bounded from below by certain multiple of the Euclidean distance. This set, when restricted to sections of $\phi$, has almost full measure if the Monge–Ampère measure $\det D^2\phi$ is close to a constant. Its precise statement is as follows.

Lemma 4.6

Assume that $(\Omega , \phi , U)\in \mathcal P _{1-\epsilon , 1+ \epsilon , \rho , \kappa , \alpha }$ where $0<\epsilon <1/2$. Define

$$\begin{aligned} A_{\sigma }\! :=\! \left\{ \tilde{x}\in U: \phi (x)\ge \phi (\tilde{x}) + \nabla \phi (\tilde{x})\cdot (x-\tilde{x}) + \frac{\sigma }{2}\, |x-\tilde{x}|^2, \quad \forall x\in B_{c^2} \cap U\right\} .\nonumber \\ \end{aligned}$$

(4.15)

Then there exist $\sigma =\sigma (n,\rho ,\alpha )>0$ and $C= C(n, \rho , \alpha , \kappa )>0$ such that

$$\begin{aligned} \big | S_{\phi }(0, c^9)\setminus A_\sigma \big | \le C \epsilon ^{1/3n}\, | S_{\phi }(0, c^9)|. \end{aligned}$$

Proof

We first note that (2.7) implies

$$\begin{aligned} | S_{\phi }(0, c^9)| \ge |B_{c^6}\cap U|\ge C. \end{aligned}$$

(4.16)

Let $w$ be defined as in $(vii)$ in the definition of the class $\mathcal P$. Then the following boundary Pogorelov estimates hold

$$\begin{aligned} c_{0}I_{n}\le D^2 w\le c_{0}^{-1}I_{n}~\text { in }\, B_{c^2}\cap U. \end{aligned}$$

(4.17)

Let $\Gamma$ be the convex envelope of $\phi -\frac{w}{2}$ in $U\cap B_{c^2}$. We claim that there exists $C>0$ depending only on $n$, $\rho , \alpha$ and $\kappa$ such that

$$\begin{aligned} \left| \left\{ \Gamma = \phi -\frac{w}{2}\right\} \cap S_{\phi }\big (0, c^9\big ) \big | \ge \big (1-C\epsilon ^{1/3n}\big )\big | S_{\phi }\big (0, c^9\big )\right| . \end{aligned}$$

(4.18)

Assume this claim for a moment. Then by using (4.17) and arguing as in the proof of [15, Theorem 6.1.1], we obtain the desired conclusion. For completeness, we include the proof.

Let the contact set be

$$\begin{aligned} \mathcal C :=\left\{ x\in U\cap B_{c^2}: \Gamma (x)= \phi (x)-\frac{w(x)}{2}\right\} . \end{aligned}$$

We assert that for $\sigma := c_0/2$, we have

$$\begin{aligned} \mathcal C \cap S_{\phi }(0, c^9)\subset A_{\sigma }\cap S_{\phi }(0, c^9). \end{aligned}$$

It then follows from (4.18) that

$$\begin{aligned} |S_{\phi }(0, c^9)\setminus A_{\sigma }|\le |S_{\phi }(0, c^9)\setminus \mathcal C |\le C \epsilon ^{1/3n}|S_{\phi }(0, c^9)|. \end{aligned}$$

We now proceed with the proof of the claim. Let $x_0\in \mathcal C \cap S_{\phi }(0, c^9)$, and let $l_{x_0}$ be a supporting hyperplane to $\Gamma$ at $x_0$. Since $x_0\in \mathcal C$, we have $l_{x_0}(x_0) =\phi (x_0) - \frac{1}{2}w(x_0)$ and

$$\begin{aligned} \phi (x) \ge l_{x_0}(x) +\frac{w(x)}{2}\quad \text{ for } \text{ all }\, x\in U\cap B_{c^2}. \end{aligned}$$

(4.19)

On the other hand, if $x\in U\cap B_{c^2}$ then the Taylor formula and the first inequality in the Pogorelov estimates (4.17) give

$$\begin{aligned}&w(x)-w(x_0)-\nabla w(x_0)\cdot (x-x_0)\nonumber \\&\quad =\int \limits _{0}^{1} t \int \limits _{0}^{1} \langle D^2 w(x_0 +\theta t (x-x_0))\cdot (x-x_0), x- x_0\rangle d\theta dt\\&\quad \ge \int \limits _{0}^{1} t c_0\left| x-x_0\right| ^2 dt = \frac{c_0}{2}\left| x-x_0\right| ^2. \end{aligned}$$

Combining this with (4.19), we deduce that

$$\begin{aligned} \phi (x)\ge l(x) + \frac{c_0}{4}\left| x-x_0\right| ^2 \quad \forall x\in U\cap B_{c^2}, \end{aligned}$$

where $l(x)$ is the supporting hyperplane to $\phi$ at $x_0$ in $U\cap B_{c^2}$ given by

$$\begin{aligned} l(x) := l_{x_0}(x) + \frac{1}{2} w(x_0) + \frac{1}{2} \nabla w(x_0) \cdot (x-x_0). \end{aligned}$$

Therefore $x_0\in A_{\sigma }$ with $\sigma = c_0/2$, proving the assertion.

It remains to prove (4.18). The idea is to compare the image of the gradient mappings of convex functions which are close in $L^{\infty }$-norms. This idea goes back to Caffarelli (see [2, Lemma 2] and also [24, Lemma 6.2]). Since our setting near the boundary is a bit different, we sketch the proof.

By (4.16), it suffices to consider the case $\epsilon \ll 1$. By the maximum principle ([19, Lemma 3.1]), we have

$$\begin{aligned} \Vert \phi -w\Vert _{L^{\infty }(U)}\le C_{n} \text{ diam }\,(U) \Vert (\det D^2\phi )^{1/n}-1\Vert _{L^{n}(U)} \le C\, \epsilon ^{1/n}\equiv \bar{\epsilon }. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{1}{2}w-\bar{\epsilon }\le \phi -\frac{w}{2}\le \frac{1}{2} w +\bar{\epsilon }\quad \text {in}\, U\cap B_{c^2} \end{aligned}$$

and since $w$ is convex, we have

$$\begin{aligned} \frac{1}{2}w-\bar{\epsilon }\le \Gamma \le \frac{1}{2} w +\bar{\epsilon }\quad \text {in}\, U\cap B_{c^2}. \end{aligned}$$

Let

$$\begin{aligned} V_{1}=\Big \{x\in U\cap B_{c^2}: \, \text {dist}\big (x, \partial (U\cap B_{c^2})\big )>\delta \Big \}. \end{aligned}$$

Then, using (4.17), we will show that for $1\gg \delta > \bar{\epsilon }$ to be chosen later, we have

$$\begin{aligned} \left| \left\{ \Gamma = \phi -\frac{w}{2}\right\} \cap V_1 \right| \ge (1-C\delta )|V_1| \end{aligned}$$

(4.20)

for some $C$ depends on $n$, $\rho , \alpha$ and $\kappa$. Indeed, let

$$\begin{aligned} V_2=\Big \{x\in U\cap B_{c^2}: \, \text {dist}\big (x, \partial (U\cap B_{c^2})\big )>2\delta \Big \}. \end{aligned}$$

For $x_0\in V_2$, consider

$$\begin{aligned} v^*(x) :=\frac{1}{2} w(x) -\bar{\epsilon } + \delta \left( r^2- \left| x-x_0\right| ^2\right) \end{aligned}$$

where

$$\begin{aligned} \delta = 2\root 3 \of {\bar{\epsilon }}\sim \epsilon ^{1/3n},~\delta >r>\sqrt{\frac{2\bar{\epsilon }}{\delta }}. \end{aligned}$$

Then $B(x_0, r)\subset V_1$ and

$$\begin{aligned} v^*\le \Gamma ~\text { on }~ \partial B(x_0, r) \quad \text {and}\quad v^* \ge \Gamma ~\text { in }~ B\left( x_0, r- \frac{2\bar{\epsilon }}{\delta r}\right) . \end{aligned}$$

It follows that $\nabla v^*\big (B\big (x_0, r- \frac{2\bar{\epsilon }}{\delta r}\big ) \subset \nabla \Gamma (B(x_0, r)\big ).$ Hence

$$\begin{aligned} \nabla v^*(V_2)\subset \nabla \Gamma (V_1). \end{aligned}$$

(4.21)

From the $C^2$ bound on $w$ in (4.17), we have

$$\begin{aligned} D^2 v^*= \frac{1}{2}D^2 w -2\delta I_n&= \frac{1}{2} \big (1 - 4 c_0^{-1}\delta \big ) D^2 w + 2\delta \big (c_0^{-1} D^2 w- I_n\big )\\&\ge \frac{1}{2} \big (1 - 4 c_0^{-1}\delta \big ) D^2 w. \end{aligned}$$

Therefore, using $\det D^2 w=1$, we obtain

$$\begin{aligned} |\nabla v^* (V_2)|=\int \limits _{V_2}\det D^2 v^* \ge \left( \frac{1}{2^{n}}- C_{1}\delta \right) |V_2|. \end{aligned}$$

(4.22)

Next, as $\Gamma$ is convex with $\Gamma \in C^{1,1}(U\cap B_{c^2})$ and $\det D^2\Gamma =0$ a.e. outside $\mathcal C$, we have

$$\begin{aligned} |\nabla \Gamma (V_1)|= |\nabla \Gamma (V_1\cap \mathcal C )| = \int \limits _{V_1\cap \mathcal C }\det D^2\Gamma . \end{aligned}$$

(4.23)

We now estimate $\det D^2 \Gamma$ from above. For this, observe that for any $x\in \mathcal C$, the function $\phi -\frac{1}{2}w-\Gamma$ attains its local minimum value $0$ at $x$. Hence,

$$\begin{aligned} D^2\Gamma (x) \le D^2 \left( \phi -\frac{1}{2}w\right) (x) \end{aligned}$$

at any twice differentiable point of $\Gamma$ and $\phi$. Therefore, this inequality holds for a.e $x\in \mathcal C$ by Aleksandrov theorem. Note that for symmetric, nonnegative matrices $A$ and $B$, we have

$$\begin{aligned} (\det (A+B))^{1/n}\ge (\det A)^{1/n} + (\det B)^{1/n}. \end{aligned}$$

Thus, for a.e $x\in \mathcal C$, we have

$$\begin{aligned} (\det D^2\Gamma (x))^{1/n}&\le \left( \det D^2 \left( \phi -\frac{1}{2}w\right) (x)\right) ^{1/n}\\&\le (\det D^2\phi )^{1/n}-\left( \det D^2 \left( \frac{1}{2}w\right) (x)\right) ^{1/n}\\&\le (1+\epsilon )^{1/n}-\frac{1}{2}\le \frac{1}{2} +\frac{\epsilon }{n}. \end{aligned}$$

Combining with (4.23) gives

$$\begin{aligned} |\nabla \Gamma (V_1)|\le \left( \frac{1}{2^n} + C_2\epsilon \right) |V_1\cap \mathcal C |. \end{aligned}$$

We infer from this, (4.21) and (4.22) that

$$\begin{aligned} |V_1\cap \mathcal C |\ge \frac{1-2^n C_1\delta }{1 + 2^{n}C_2\epsilon }\left| V_2\right| \ge (1-C_3 \delta )\left| V_2\right| \end{aligned}$$

(4.24)

for $\epsilon \le \epsilon _0$ with $\epsilon _0$ is a small universal constant.

By $(iv)$ in the definition of the class $\mathcal P$, we have $\Vert B_{c^2} \cap \partial U\Vert _{C^{1,1}}\le \kappa .$ Consequently,

$$\begin{aligned} \big |(U\cap B_{c^2}) \backslash V_2 \big | \le C_4\delta \quad \text {and}\quad \left| V_2\right| \ge \left| V_1\right| - C_4\delta \end{aligned}$$

for some $C_4>0$ depending only on $n, \rho$ and $\kappa$. Combining the above inequalities with (4.24), we easily obtain (4.20).

It follows from (4.20), the inclusion $\{\Gamma <\phi -\frac{w}{2}\}\subset U\cap B_{c^2}$ and (4.16) that

$$\begin{aligned}&\left| \left\{ \Gamma <\phi -\frac{w}{2}\right\} \cap S_{\phi }(0, c^9)\right| \\&\quad \le \left| \left\{ \Gamma <\phi -\frac{w}{2}\right\} \cap V_1\right| + |(U\cap B_{c^2}) \backslash V_1| \le C\delta |V_1| + C_4\delta \\&\quad \le C_5\delta \le C\delta | S_{\phi }(0, c^9)| \le C \epsilon ^{1/3n} | S_{\phi }(0, c^9)|. \nonumber \end{aligned}$$

This gives the claim (4.18) and the proof is complete. $\square$

5 Density and global $W^{2,p}$ estimates

In this section we will prove global $W^{2,p}$ estimates for solutions to the linearized Monge–Ampère equations as stated in the introduction. The key tools are density estimates and a covering lemma.

5.1 Density estimates

In this subsection, by using the approximation lemma in Sect. 4.2 together with the stability of cofactor matrices established in Sect. 3.4, we improve density estimates obtained in Sect. 3 when the Monge–Ampère measure $\det D^{2}\phi$ is close to $1$.

Our first lemma improves the power decay estimates in Proposition 3.7 which say that for $(\Omega , \phi , U)\in \mathcal P _{\lambda , \Lambda , \rho , \kappa , *}$, the quantity $|S_{\phi }(0, r)\setminus G_{N}(u, \Omega ) |$ decays like $CN^{-\tau }$. Here, we improve $C$ by roughly a factor of when $*$ is replaced by $\alpha$, $\lambda$ and $\Lambda$ are close to $1$, and $W$ is the matrix of cofactors of $D^2 w$ of the solution to the Monge–Ampère equation $\det D^2 w=1$ with the same boundary values as $\phi$. The precise statement is as follows.

Lemma 5.1

Assume $(\Omega , \phi , U)\in \mathcal P _{1-\epsilon , 1+ \epsilon , \rho , \kappa , \alpha }$ where $0<\epsilon <1/2$. Let $r= c^2/4$. Suppose $u\in C(\Omega )\cap C^1(U)\cap W^{2, n}_{loc}(U)$ is a solution of $\mathcal{L }_\phi u=f$ in $U$ that satisfies

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(U)} + \Vert u\Vert _{C^{2,\alpha }(\partial U\cap B_{4r})} \le 1, \end{aligned}$$

and has at most quadratic growth in the sense that

$$\begin{aligned} |u(x)|\le C^*\, \big [ 1+ d(x, x_0)^2\big ] ~\text { in }~ \Omega {\setminus } U~ \text { for some }~ x_0\in B_{r/2}\cap U. \end{aligned}$$

(5.1)

Then there exist $\tau =\tau (n,\rho )>0$ and $N_0 = N_0(C^*, n,\rho , \alpha )>0$ such that for $N\ge N_0$ we have

$$\begin{aligned} |G_{N}(u,\Omega ) \cap S_{\phi }(0, c^9)|\ge \left\{ 1- C\big ( N^{-\tau } \delta _0^\tau +\epsilon ^{1/3n} \big )\right\} \, | S_{\phi }(0, c^9)| \end{aligned}$$

provided that $\Vert \Phi - \mathrm{W }\Vert _{L^n(B_{2r} \cap U)} \le r^4$. Here $C= C(n, \rho , \alpha ,\kappa )>0$, $\mathrm{W },\, \gamma$ are from Lemma 4.5, and

Proof

Let $h$ be the solution of

$$\begin{aligned} \mathrm{W }^{i j} h_{i j} = 0~ \text{ in } \quad B_{2r} \cap U,\quad \text{ and }\quad h =u~ \text{ on } \,\partial ( B_{2r} \cap U). \end{aligned}$$

By Lemma 4.5 and since $U\subset B_{k^{-1}}$, there exists $C_0$ depending only on $n, \rho$ and $\alpha$ such that

$$\begin{aligned}&\Vert h\Vert _{C^{1,1}(\overline{ B_{r} \cap U})} \le C_0,\end{aligned}$$

(5.2)

$$\begin{aligned}&\Vert u-h\Vert _{L^\infty (\overline{ B_{r} \cap U})} + \Vert f-\text {trace}([\Phi -\mathrm{W }] D^2h)\Vert _{L^n(\overline{ B_{r} \cap U})} \le C_{0} \,\delta _0 =:\delta _0'.\quad \quad \quad \end{aligned}$$

(5.3)

We now consider $h|_{B_{r} \cap U}$ and then extend $h$ outside $B_{r} \cap U$ continuously such that

$$\begin{aligned} \left\{ \begin{array}{l} h(x) = u(x) \quad \forall x\in \Omega \setminus (B_{2r} \cap U),\\ \Vert u-h\Vert _{L^\infty (\Omega )} =\Vert u-h\Vert _{L^\infty (B_{r} \cap U)}. \end{array} \right. \end{aligned}$$

The maximum principle gives $\Vert h\Vert _{L^\infty (B_{r} \cap U)} \le \Vert u\Vert _{L^\infty (U)}\le 1,$ and thus

$$\begin{aligned} u(x) -2 \le h(x) \le u(x) +2 \quad \text{ for } \text{ all }\, x\in \Omega . \end{aligned}$$

(5.4)

We claim that if $N\ge N_0$, then

$$\begin{aligned} \left( B_{\frac{r}{2}} \cap U\right) \cap A_{\sigma }\subset G_N(h,\Omega ) \end{aligned}$$

(5.5)

where $\sigma =\sigma (n,\rho ,\alpha )>0$ is the constant given by Lemma 4.6 and the set $A_{\sigma }$ is defined by (4.15).

Indeed, let $\bar{x}\in (B_{\frac{r}{2}} \cap U)\cap A_{\sigma }$. By (5.2) we have

$$\begin{aligned} \left| h(x) -[ h (\bar{x}) + \nabla h (\bar{x})\cdot ( x- \bar{x})]\right| \le C_0 |x- \bar{x}|^2\quad \text {for all } x\in \overline{B_{r} \cap U}, \end{aligned}$$

and since $\bar{x}\in A_{\sigma }$

$$\begin{aligned} d(x, \bar{x})^2 = \phi (x) -[\phi (\bar{x}) +\nabla \phi (\bar{x})\cdot ( x-\bar{x})] \ge \frac{\sigma }{2} |x-\bar{x}|^2\quad \forall x\in B_{4r} \cap U.\quad \quad \quad \end{aligned}$$

(5.6)

Therefore,

$$\begin{aligned} \big | h(x) -[ h (\bar{x}) +\nabla h (\bar{x})\cdot ( x- \bar{x})] \big |\le \frac{2 C_0}{\sigma } \, d(x, \bar{x})^2\quad \forall x \in \overline{B_{r} \cap U}. \end{aligned}$$

(5.7)

We next show that by increasing the constant on the right hand side of (5.7), that the resulting inequality holds for all $x$ in $\Omega$.

To see this, we first observe that by the maximum principle $\max _{U}\phi = \max _{\partial U}\phi =1$ and by the gradient estimates $(v)$ in the definition of the class $\mathcal P$ and $x_0\in U\cap B_{r/2}$, we have

$$\begin{aligned} d(x, x_0)^2&= d(x, \bar{x})^2 + [\phi (\bar{x}) -\phi (x_0) -\langle \nabla \phi (x_0), \bar{x} -x_0\rangle ]\nonumber \\&+\, \langle \nabla \phi (\bar{x}) -\nabla \phi (x_0), x-\bar{x}\rangle \nonumber \\&\le d(x, \bar{x})^2 + C_1 (1 + |x -\bar{x}| ) \qquad \text{ for } \text{ all } \, x\in \Omega \end{aligned}$$

(5.8)

for some universal $C_1$ depending only on $n$ and $\rho$.

Next, we observe that if $c_{1}= \sigma r/4$ then

$$\begin{aligned} d(x, \bar{x})^2 \ge c_1 |x-\bar{x}| \quad \forall x\in \overline{\Omega }\setminus \overline{ B_{r} \cap U}. \end{aligned}$$

(5.9)

Indeed, by (5.6) and the fact that $\bar{x}\in B_{\frac{r}{2}} \cap U$, the above inequality holds for all $x\in U\cap \partial B_{r}$. Now for $x\in \overline{\Omega }\setminus \overline{B_{r} \cap U}$ we can choose $\hat{x} \in U \cap \partial B_{r}$ and $\lambda \in (0,1)$ satisfying $\hat{x} = \lambda x + (1-\lambda ) \bar{x}$. Then since $d(\hat{x}, \bar{x})^2 \ge c_1 |\hat{x} -\bar{x}|$ and the function $z\mapsto d(z,\bar{x})^2$ is convex, we obtain

$$\begin{aligned} \lambda d(x, \bar{x})^2 + (1-\lambda ) d(\bar{x}, \bar{x})^2 \ge c_1 |\lambda x + (1-\lambda ) \bar{x} -\bar{x}| = c_1\lambda |x-\bar{x}| \end{aligned}$$

which gives $d(x, \bar{x})^2 \ge c_1 |x-\bar{x}|$ and hence (5.9) is proved.

We are ready to show that (5.7) holds for all $x\in \Omega$ but with a bigger constant on the right hand side. Let $x\in \Omega \setminus \overline{B_{r} \cap U}$. Then, recalling $\bar{x}\in B_{\frac{r}{2}} \cap U$ and by (5.9), we have

$$\begin{aligned} d(x, \bar{x})^2 \ge c_{1}r/2 =: c_2. \end{aligned}$$

We can estimate using (5.2) and (5.4),

$$\begin{aligned} \left| h(x) -[ h (\bar{x}) + \nabla h (\bar{x})\cdot ( x- \bar{x})]\right|&\le |h(x) - h(\bar{x})| + C_0 |x-\bar{x}|\nonumber \\&\le |u(x)| + C_0 ( |x-\bar{x}|+ 1). \quad \quad \end{aligned}$$

(5.10)

Consider the following cases:

Case 1: $x\in \overline{U}\setminus \overline{B_{r} \cap U}$. Using (5.10) and the above lower bound for $d(x, \bar{x})^2$, we obtain

$$\begin{aligned} \left| h(x) -[ h (\bar{x}) + \nabla h (\bar{x})\cdot ( x- \bar{x}) ]\right|&\le 1 + C_0 ( |x-\bar{x}|+ 1)\le 1 + C_0(2k^{-1}+ 1)\\&\le C_{2} \, d(x,\bar{x})^2. \end{aligned}$$

Case 2: $x\in \Omega \setminus \overline{U}$. Using (5.10), (5.1), (5.8), (5.9) and the bound $d(x, \bar{x})^2\ge c_2$, we find that

$$\begin{aligned} \left| h(x) -[ h (\bar{x}) + \nabla h (\bar{x})\cdot ( x- \bar{x})]\right|&\le C^*\,[ 1+ d(x, x_0)^2] +C_0 ( |x-\bar{x}|+ 1) \\&\le C^*\,d(x, \bar{x})^2 +C_3 ( |x-\bar{x}|+ 1)\le C_4 \, d(x, \bar{x})^2. \end{aligned}$$

Therefore if we choose

$$\begin{aligned} N_0 := \max {\left\{ \frac{ 4 C_0}{\sigma }, 2 C_2, 2 C_4 \right\} }, \end{aligned}$$

then it follows from the above considerations and (5.7) that

$$\begin{aligned} \left| h(x) -[ h (\bar{x}) + \nabla h (\bar{x})\cdot ( x- \bar{x})]\right| \le \frac{N_0}{2} \, d(x, \bar{x})^2 \quad \text{ for } \text{ all }\, x \in \Omega . \end{aligned}$$

This means $\bar{x}\in G_{N_0}(h,\Omega )\subset G_{N}(h,\Omega )$ for all $N\ge N_0$. Thus claim (5.5) is proved.

Next let

$$\begin{aligned} u'(x) := \frac{(u-h)(x)}{\delta _0'}, \quad \text{ for }\, x\in \Omega . \end{aligned}$$

We infer from (5.3) and the way $h$ was initially defined and extended that

$$\begin{aligned}&\Vert u'\Vert _{L^\infty (\Omega )}=\frac{1}{\delta _0'} \Vert u-h\Vert _{L^\infty (B_{r} \cap U)}\le 1,\\&\mathcal{L }_\phi u' =\frac{1}{\delta _0'} [\mathcal{L }_\phi u -\mathcal{L }_\phi h ] =\frac{1}{\delta _0'} \Big [f-\text {trace}([\Phi -\mathrm{W }] D^2 h) \Big ] =: f'(x) \quad \text{ in }\, B_{r} \cap U. \end{aligned}$$

Notice that $\Vert f'\Vert _{L^n(B_{r} \cap U)}\le 1$ by (5.3). Thus we can apply Proposition 3.7 to get

$$\begin{aligned} \big | S_{\phi }(0, c^9)\setminus G_{\frac{N}{\delta _0'}}(u',\Omega )\big | \le C \left( \frac{\delta _0'}{N}\right) ^\tau \, | S_{\phi }(0, c^9)|. \end{aligned}$$

As $G_{\frac{N}{\delta _0'}}(u',\Omega ) = G_{N}(u-h,\Omega )$, we then conclude

$$\begin{aligned} | S_{\phi }(0, c^9)| - |G_{N}(u -h,\Omega ) \cap S_{\phi }(0, c^9)|\le C \left( \frac{\delta _0}{N} \right) ^{\tau } ~ | S_{\phi }(0, c^9)| \end{aligned}$$

yielding

$$\begin{aligned} \left\{ 1- C \left( \frac{\delta _0}{N} \right) ^{\tau } \right\} ~ | S_{\phi }(0, c^9)|&\le |G_{N}(u -h,\Omega ) \cap S_{\phi }(0, c^9)|\\&\le |G_{N}(u -h,\Omega ) \cap S_{\phi }(0, c^9)\cap A_{\sigma }| +\left| S_{\phi }(0, c^9)\setminus A_{\sigma }\right| \\&\le |G_{N}(u -h,\Omega ) \cap S_{\phi }(0, c^9)\cap A_{\sigma }| + C \epsilon ^{1/3n} \, | S_{\phi }(0, c^9)|, \end{aligned}$$

where the last inequality is by Lemma 4.6. Consequently,

$$\begin{aligned} |G_{N}(u -h,\Omega ) \cap S_{\phi }(0, c^9) \cap A_{\sigma }| \!\ge \! \left\{ 1\!-\! C \Big [ \left( \frac{\delta _0}{N} \right) ^{\tau } \!+ \epsilon ^{1/3n} \Big ] \right\} ~ | S_{\phi }(0, c^9)|.\quad \quad \quad \end{aligned}$$

(5.11)

Next observe that $G_{N}(u -h,\Omega ) \cap S_{\phi }(0, c^9)\cap A_{\sigma }\subset G_{N}(u -h,\Omega ) \cap G_N(h,\Omega )$ by (5.5). Therefore,

$$\begin{aligned} G_{N}(u -h,\Omega ) \cap S_{\phi }(0, c^9) \cap A_{\sigma } \subset G_{2 N}(u,\Omega ) \cap S_{\phi }(0, c^9) \end{aligned}$$

which together with (5.11) gives the conclusion of the lemma. $\square$

Having the improved decay estimates in Lemma 5.1, we can now proceed with density estimates when $\det D^2\phi$ is close to a constant. Our next lemma is concerned with second derivative estimates for solutions to $\mathcal L _{\phi } u=f$. It roughly says that in each section $S_{\phi }(x, t)$ with small height $t$, we can find a very large portion (as close to the full measure as we want) where $u$ has second derivatives bounded in a controllable manner. The bound on $D^2 u$ is made more precise by using the openings of the quasi paraboloids that touch $u$ from below and above. So far, we have no a priori information on the boundedness of $D^2u$. However, we can still hope for a bound of order $\frac{1}{t}$ for $|D^2u|$ in $S_{\phi }(x, t)$ as explained in Sect. 2.2 using an $L^{\infty }$-norm rescaling of our solution. This heuristic idea explains the factor $\frac{N}{t}$ in the estimate of Lemma 5.2 and the way the solution is rescaled in the proof.

Lemma 5.2

Assume $\Omega$ satisfies (2.1) and $\phi \in C^{0,1}(\overline{\Omega })$ is a convex function satisfying (2.3) and

$$\begin{aligned}1-\epsilon \le \det D^2\phi \le 1+\epsilon \quad \text{ in }\,\Omega . \end{aligned}$$

Assume in addition that $\partial \Omega \in C^{2,\alpha }$ and $\phi \in C^{2,\alpha }(\partial \Omega )$ for some $\alpha \in (0,1)$. Let $u\in C^1(\Omega ) \cap W^{2,n}_{loc}(\Omega )$ be a solution of $\mathcal{L }_\phi u=f$ in $\Omega$ with $u=0$ on $\partial \Omega$ and $\Vert u\Vert _{L^\infty (\Omega )}\le 1$. Let $0<\epsilon _0<1$. Then there exists $\epsilon >0$ depending only on $\epsilon _0, n, \rho$ and $\alpha$ such that for any $x\in \overline{\Omega }$ and $t\le c_1$ we have

$$\begin{aligned} \left| G_{\frac{N}{t}}(u,\Omega ) \cap S_\phi (x, t)\right| \ge \left\{ 1-\epsilon _0 -C \left( \frac{\sqrt{t}}{N}\right) ^\tau \Vert f\Vert _{L^n(\Omega )}^\tau \right\} \left| S_\phi (x, t)\right| \quad \forall N\ge N_1.\nonumber \\ \end{aligned}$$

(5.12)

Here $\tau =\tau (n,\rho );\; C$ and $N_1$ depend only on $n, \rho$ and $\alpha ;\; c_1>0$ is small depending only on $n, \rho , \alpha ,\; \Vert \partial \Omega \Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega )}$.

Proof

If $\epsilon$ is small then by the global $W^{2,p}$ estimates for solutions to the Monge–Ampère equations [28, Theorem 1.2], we have $\phi \in W^{2, 2n}(\Omega )$ and hence $\phi \in C^{1}(\overline{\Omega })$.

Let us first consider the case $x\in \partial \Omega$. We can assume that $x=0, \phi (0)=0$ and $\nabla \phi (0)=0.$ By the Localization Theorem 2.2, we have

$$\begin{aligned} k E_{t} \cap \overline{\Omega }\subset S_{\phi }(0, t) \subset k^{-1} E_{t}\cap \overline{\Omega }, \end{aligned}$$

where $E_{t} := A_{t}^{-1} B_{t^{1/2}}$ with $A_{t} x= x-\tau _{t} x_n$ and

$$\begin{aligned} \tau _{t}\cdot e_n =0, \quad \quad \Vert A_{t}^{-1}\Vert ,\, \Vert A_{t}\Vert \le k^{-1} |\log {t}|. \end{aligned}$$

We now define the rescaled domains $\Omega _{t}, U_{t}$ and rescaled functions $\phi _{t}$ and $u_{t}$ as in Sect. 2.2 that preserve the $L^{\infty }$ -norm of $u$. We have

$$\begin{aligned} \mathcal L _{\phi _{t}} u_{t}(y)= tf(T^{-1} y) =: f_{t}(y) \end{aligned}$$

where $T:=t^{-1/2} A_{t}$ and

$$\begin{aligned} \Vert u_{t} \Vert _{L^\infty (\Omega _{t})} =\Vert u \Vert _{L^\infty (\Omega )}\le 1, \quad u_{t} = 0 \quad \text{ on }\, \partial U_{t} \cap B_k. \end{aligned}$$

Moreover, we have from Proposition 2.12 that

$$\begin{aligned} (\Omega _{t}, \phi _{t}, U_{t})\in \mathcal P _{1-\epsilon , 1+\epsilon , \rho , Ct^{1/2}, \alpha } \subset \mathcal P _{1-\epsilon , 1+\epsilon , \rho , 1, \alpha } \end{aligned}$$

if $t\le \tilde{c}$, where $\tilde{c}>0$ is a small constant depending only on $n, \rho , \alpha$, $\Vert \partial \Omega \Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega )}$.

Now, applying Lemma 5.1 with $C^*=1$, we obtain

$$\begin{aligned} |G_{N}(u_{t}, \Omega _{t}, \phi _t) \cap S_{\phi _t}(0, c^9) |\ge \left\{ 1- C\big (N^{-\tau } \delta _0^{\tau } + \epsilon ^{1/3n} \big )\right\} \, | S_{\phi _t}(0, c^9)| \end{aligned}$$

for any $N\ge N_0=N_0(n,\rho , \alpha )$. Here

(5.13)

$\gamma$ is given by Lemma 4.5, $w_{t}$ is the function in $(vii)$ in the definition of the class $\mathcal P$ associated with the triple $(\Omega _{t}, \phi _{t}, U_{t})$ and $\mathrm{W }_{t}$ is the cofactor matrix of $D^2 w_{t}$. This together with the stability of cofactor matrices in Proposition 3.14 implies the existence of $\epsilon = \epsilon (\epsilon _0, n, \rho , \alpha )>0$ such that for $r:=c^9$, we have

where $\beta =\beta (n,\rho )<1$ is a universal constant to be chosen later.

As $S_{\phi _t}(0, r) =T(S_{\phi }(0, rt))$, it is easy to see that for $G_{N}(u,\Omega ,\phi ) = G_{N}(u,\Omega )$,

$$\begin{aligned}&G_{N}( u_{t}, \Omega _{t}, \phi _t) \cap S_{\phi _t}(0, r) = T \Big ( G_{\frac{N}{t}}(u, \Omega , \phi ) \cap S_{\phi }(0, rt)\Big ). \end{aligned}$$

Therefore we conclude that

This is equivalent to

(5.14)

giving (5.1) for any $N^{'}\ge N_1\equiv N_0 r$ and $t\le r\tilde{c}$.

Next we consider the situation that $x\in \Omega$. We then have the following possibilities:

Case 1: $t\le h/2$, where $h :={\bar{h}}(x)$.

If $h\ge c$ where $c$ is defined in Proposition 2.3 then the estimate (5.1) is an easy consequence of the interior density estimates [17, Lemma 4.3] which we now recall.

Lemma 5.3

([17, Lemma 4.2]) Let $0<\alpha _0<1$ and $\Omega$ be a convex domain in $\mathbb{R }^{n}$ satisfying $B_{k_0}\subset \Omega \subset B_{k_0^{-1}}$ and $u\in C^1(\Omega )\cap W^{2, n}_{loc}(\Omega )$ be a solution of $\Phi ^{ij} u_{ij} = f$ in $\Omega$ with $\Vert u\Vert _{L^{\infty }(\Omega )}\le 1,$ where $\phi \in C(\overline{\Omega })$ is a convex function satisfying $\phi =0$ on $\partial \Omega$. Let $0<\epsilon _{0}<1$. There exists $\epsilon >0$ depending only on $\epsilon _0, \alpha _0, k_0$ and $n$ such that if

$$\begin{aligned} 1-\epsilon \le \det D^2 \phi \le 1+ \epsilon \quad \text{ in }\,\Omega , \end{aligned}$$

then for any section $S_{\phi }(x_0, \frac{t_0}{\alpha _0})\subset \Omega _{\frac{\alpha _0 +1}{2}}:=\{x\in \Omega : \phi (x) < (1-\frac{\alpha _0 + 1}{2})\min _{\Omega }\phi \}$, we have

for every $N\ge N_0$. Here $C, \tau , N_0$ are positive constants depending only on $\alpha _0, n$ and $k_0.$

Now we consider the remaining situation in Case 1 when $h\le c.$ We define the rescaled domain $\tilde{\Omega }_{h}$ and rescaled functions $\tilde{\phi }_{h}$, $\tilde{u}_{h}$ and $\tilde{f}_h$ as in Sect. 2.2 that preserve the $L^{\infty }$ -norm in a section tangent to the boundary. Now, we apply Lemma 5.3 to the domain $S_{\tilde{\phi }_{h}} (0, 1)$ with $\alpha _0 =3/4$, $x_0=0$ and $t_0 = t/h\le 1/2$, noting that $(S_{\tilde{\phi }_h}(0,1))_{\alpha }= S_{\tilde{\phi }_h}(0,\alpha )$ for all $\alpha >0$. Thus,

(5.15)

Let $Ty := h^{-1/2}A_{h}(y-x)$. Then

$$\begin{aligned} G_{\frac{N h}{t}}\left( \tilde{u}_h, S_{\tilde{\phi }_{h}} (0, 1), \tilde{\phi }_h\right) \cap S_{\tilde{\phi }_{h}} \left( 0, \frac{t}{h}\right) = T\big (G_{\frac{N}{t}}(u,\Omega ) \cap S_{\phi }(x, t)\big ). \end{aligned}$$

Changing variables in (5.15) gives

and hence (5.12) holds.

Case 2: $h/2 <t\le r\tilde{c}/\bar{c}\equiv c_{1}$ where $\bar{c}>1$ is the constant in Proposition 2.14. Then by Proposition 2.14, we know that $S_{\phi }(x, 2 t) \subset S_{\phi }(z, \bar{c} t)$ for some $z\in \partial \Omega$, and by Theorem 2.13(b),

$$\begin{aligned} C_{1} t^{n/2}\le |S_{\phi }(x, t)|\le C_{2} t^{n/2}\quad \forall t\le c_0. \end{aligned}$$

Using these inequalities and the estimate (5.14) in the case of boundary section, we get

This implies (5.1) as desired by choosing $\beta = C_1 C_2^{-1} \bar{c}^{\frac{-n}{2}}$ and $c_1= r\tilde{c}/\bar{c}= c^9\tilde{c}/\bar{c}$. $\square$

The next lemma is a key technical ingredient in our global $W^{2, p}$ estimates. It propagates a point in a given section where the solution $u$ of $\mathcal L _{\phi } u=f$ has bounded second derivative to almost all points in that section. More precisely, it says that if in a small section $S_{\phi }(x, t)$ we can find a point where $u$ is touched from above and below by quasi paraboloids of opening $\gamma$ generated by $\phi$ then on a set of nearly full measure of $S_{\phi }(x, t)$, $u$ is touched from above and below by quasi paraboloids of opening $N\gamma$ for some controllable constant $N$, provided that $\det D^2\phi$ is close to a constant.

Lemma 5.4

Assume $\Omega$ is uniformly convex satisfying (2.1) and $\phi \in C^{0,1}(\overline{\Omega })$ is a convex function satisfying (2.3) and

$$\begin{aligned} 1-\epsilon \le \det D^2\phi \le 1+\epsilon \quad \text{ in }\,\Omega . \end{aligned}$$

Assume in addition that $\partial \Omega \in C^{2,\alpha }$ and $\phi \in C^{2,\alpha }(\partial \Omega )$ for some $\alpha \in (0,1)$. Let $u\in C^1(\Omega ) \cap W^{2,n}_{loc}(\Omega )$ be a solution of $\mathcal{L }_\phi u=f$ in $\Omega$ and $u=0$ on $\partial \Omega$. Let $0<\epsilon _0<1$. Then there exists $\epsilon >0$ depending only on $\epsilon _0, n, \rho$ and $\alpha$ such that for any $x\in \overline{\Omega }$, $t\le c_2$ and $S_\phi (x,t)\cap G_\gamma (u,\Omega )\ne \emptyset$ we have

(5.16)

for all $\tilde{x}\in S_\phi (x,t)$ and $N\ge N_2$. Here $\tau$ and $\Theta$ depend only on $n$ and $\rho$; $C$, $c_2$ and $N_2$ depend only on $n, \rho , \alpha$, the uniform convexity of $\Omega$, $\Vert \partial \Omega \Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega )}$.

Proof

As explained in the proof of Lemma 5.2, we have $\phi \in C^{1}(\Omega )\cap W^{2, 2n}(\Omega )$ if $\epsilon$ is small.

Let us first consider the case $x\in \partial \Omega$. We can assume that $x=0$, $\phi (0)=0$ and $\nabla \phi (0)=0.$ Let $h=\theta t$ where $\theta =\theta (n,\rho )>1$ will be chosen later. Let $A_{h}$ be the affine transformation as in the Localization Theorem 2.2. We now define the rescaled domains $\Omega _{h}, U_{h}$ and rescaled functions $\phi _{h},$ $\tilde{u}_{h}$ and $\tilde{f}_h$ as in Sect. 2.2 that almost preserve the $L^{\infty }$ -norm of $D^2u$. Let $T= h^{-1/2}A_{h}$.

Let $\bar{x}\in S_\phi (0,t)\cap G_\gamma (u,\Omega )$ and $\bar{y} :=T\bar{x}$. Then

$$\begin{aligned} -\gamma \, d(x,\bar{x})^2 \le u(x)-u(\bar{x}) - \nabla u(\bar{x})\cdot (x-\bar{x}) \le \gamma \, d(x,\bar{x})^2,\quad \forall x\in \Omega . \end{aligned}$$

By changing variables and recalling that $\Omega _h= T (\Omega ), \tilde{u}_h(y)= h^{-1} u(T^{-1} y)$, we get

$$\begin{aligned} -\gamma \, \dfrac{d(T^{-1}y,T^{-1}\bar{y})^2}{ \theta t}&\le \tilde{u}_h(y)- \tilde{u}_h(\bar{y}) - \nabla \tilde{u}_h(\bar{y})\cdot (y-\bar{y})\nonumber \\&\le \gamma \, \dfrac{d(T^{-1}y,T^{-1}\bar{y})^2}{ \theta t},~ \forall y\in \Omega _h. \end{aligned}$$

(5.17)

Since $\bar{x}\in S_{\phi }(0, t) \subset S_{\phi }(0, \theta t),$ we have by the engulfing property of sections in Theorem 2.13(a) $S_{\phi }(0, \theta t)\subset S_\phi (\bar{x}, \theta ^2 t).$ It follows that $d(x,\bar{x})^2\le \theta ^2 t$ for $x\in S_{\phi }(0, \theta t)$ yielding $d(T^{-1}y,T^{-1}\bar{y})^2\le \theta ^2 t$ for all $y\in U_h:= T(S_{\phi }(0, h))$. Consequently, if we define

$$\begin{aligned} v(y) := \dfrac{1 }{\theta \gamma }\left[ \tilde{u}_h(y)- \tilde{u}_h(\bar{y}) - \nabla \tilde{u}_h(\bar{y})\cdot (y-\bar{y})\right] \!,\quad y\in \Omega _h, \end{aligned}$$

(5.18)

then $|v| \le 1$ in $U_h$. Thanks to Lemma 5.5 below we get for $t\le c_{\alpha }$

$$\begin{aligned} \Vert v\Vert _{C^{2, \alpha }(\partial {U_h} \cap B_{k})}\le C_{\alpha }, \end{aligned}$$

(5.19)

where $c_{\alpha }, C_{\alpha }$ depend only on $n, \rho ,\alpha$, the uniform convexity of $\Omega$, $\Vert \partial \Omega \Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega )}$. By (5.17) we have

$$\begin{aligned} |v(y)|\le \frac{1}{\theta ^2 t} d(T^{-1} y, T^{-1} \bar{y})^2 \le \frac{1}{\theta } d_{ \phi _h}( y, \bar{y})^2\quad \forall y\in T(\Omega ), \end{aligned}$$

(5.20)

where we recall $\theta t= h$ and

$$\begin{aligned} d_{\phi _h}(y,\bar{y})^2 := \phi _h(y) -\phi _h(\bar{y}) - \nabla \phi _h (\bar{y})\cdot ( y - \bar{y}) =h^{-1}\, d(T^{-1}y,T^{-1}\bar{y})^2. \end{aligned}$$

Moreover

$$\begin{aligned} \mathcal L _{\phi _h} v= (\theta \gamma )^{-1} \mathcal L _{\phi _h}\tilde{u_h}=(\theta \gamma )^{-1}\tilde{f}_h\equiv (\theta \gamma )^{-1} f(T^{-1} y)=: \tilde{f}(y). \end{aligned}$$

Because $\bar{x}\in S_{\phi }(0, t)$, we have $\bar{y} = T \bar{x}\in S_{\tilde{\phi }}(0, \frac{1}{\theta }).$ Hence, we can choose $\theta >1$ depending on $n, \rho , k$ such that $\bar{y}\in B_{\frac{c^2}{8}}\cap \tilde{U}$. With this choice of $\theta$, we have by Proposition 2.12

$$\begin{aligned} (\Omega _h, \phi _h, U_h)\in \mathcal P _{1-\epsilon , 1+\epsilon , \rho , Ch^{1/2}, \alpha } \subset \mathcal P _{1-\epsilon , 1+\epsilon , \rho , 1, \alpha } \end{aligned}$$

if $t\le \tilde{c}$, where $\tilde{c}>0$ is a small constant depending only on $n, \rho , \alpha$, $\Vert \partial \Omega \Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega )}$. Here we can choose $\tilde{c}\le c_{\alpha }$, and hence it also depends on the uniform convexity of $\Omega$.

Thus, using (5.19) and (5.20), we can apply Lemma 5.1 to $\bar{v}:= v/C_{\alpha }$ to obtain

$$\begin{aligned} |G_{N}(\bar{v}, \Omega _h, \phi _h) \cap S_{\tilde{\phi }}(0, c^9)|\ge \left\{ 1- C\big ( N^{-\tau }\delta _0^{\tau } +\epsilon ^{1/3n}\big )\right\} \, | S_{\phi _h}(0, c^9)| \end{aligned}$$

for any $N\ge N_0$, where $\delta _0$ is as in (5.13). This together with the stability of cofactor matrices in Proposition 3.14 implies the existence of $\epsilon = \epsilon (\epsilon _0, n,\rho ,\alpha )>0$ such that

where for simplicity we have denoted

$$\begin{aligned} r:= c^9 \end{aligned}$$

and $\beta =\beta (n,\rho )<1$ is a universal constant to be chosen later. It follows that

As $S_{\phi _h}(0, r) =T(S_{\phi }(0, \theta rt))$ and $\bar{v}(y) =\frac{1}{C_{\alpha }\theta ^2 \gamma t} \big [u(T^{-1}y) - u(\bar{x}) - \nabla u(\bar{x}) \cdot (T^{-1}y -\bar{x} ) \big ]$, it is easy to see that

$$\begin{aligned}&G_{N}(\bar{v},\Omega _h, \phi _h) \cap S_{\phi _h}(0, r) =T \Big ( G_{C_{\alpha }N\theta \gamma }(u,\Omega ) \cap S_{\phi }(0, r\theta t)\Big ). \end{aligned}$$

Therefore, by the volume estimates in Theorem 2.13(b), we conclude that

By setting $N'= C_{\alpha }N\theta$, $\beta ' = C_1^{-1} C_2 \theta ^{n/2}\beta$, we can rewrite this as

(5.21)

for any $N'\ge N_2\equiv C_{\alpha }N_0\theta$ and $t\le r\tilde{c}$. From Theorem  2.13(a) we have $S_\phi (x,\frac{\theta }{r} t)\subset S_\phi (\tilde{x}, \frac{\theta _{*}\theta }{r} t)$ for any $\tilde{x}\in S_\phi (x, t)$. Therefore, by Theorem 2.13(b), we see that (5.21) yields (5.16).

Next we consider the situation that $x\in \Omega$. We then have the following possibilities:

Case 1: $t\le h/2$, where $h :={\bar{h}}(x)$. This case can be handled as Case 1 of Lemma 5.2, using now [17, Lemma 4.5] and affine transformations similar to the ones at the beginning of the proof of this lemma.

Case 2: $h/2 <t\le r\tilde{c}/\bar{c}\equiv c_2$, where $\bar{c}>1$ is the constant in Proposition 2.14. Then by Proposition 2.14, we know that $S_{\phi }(x, 2 t) \subset S_{\phi }(z, \bar{c} t)$ for some $z\in \partial \Omega$. Thus, by the estimate (5.21) in the case of boundary section, we get

(5.22)

For any $\tilde{x}\in S_\phi (x, t) \subset S_\phi (z,\frac{ \theta \bar{c}}{r}t)$, we get $S_\phi (z, \frac{\theta \bar{c}}{r}t) \subset S_\phi (\tilde{x}, \frac{\theta _{*}\theta \bar{c}}{r}t)$ by the engulfing property in Theorem 2.13. Now, using (5.22) and the volume estimates in this theorem, we find that

This gives (5.16) with $\Theta := \theta _{*} \theta \bar{c}/r$ if we choose $\beta$ such that $\beta ' C_1^{-1} C_2 \bar{c}^{n/2} = \beta C_1^{-2} C_2^2 (\theta \bar{c})^{n/2} =1$. $\square$

In the next lemma we prove that the function $v$ defined as in the proof of Lemma 5.4 has uniform $C^{2, \alpha }$ bound on $\partial U_h\cap B_{k}$.

Lemma 5.5

Let $v$ be defined as in (5.18). There exist $C_{\alpha }, c_{\alpha }>0$ depending only on $n, \rho ,\alpha$, the uniform convexity of $\Omega$, $\Vert \partial \Omega \Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega )}$ such that for $t\le c_{\alpha }$, we have

$$\begin{aligned} \Vert v\Vert _{C^{2, \alpha } (\partial U_h\cap B_{k}^{+})} \le C_{\alpha }. \end{aligned}$$

(5.23)

Proof

Since $\partial \Omega$ is $C^{2,\alpha }$ at the origin and $\Omega$ is uniformly convex, we have

$$\begin{aligned}\left| x_n- q(x')\right| \le M\left| x'\right| ^{2+\alpha } \quad \text{ for }\, x=(x', x_n)\in \partial \Omega \cap B_{\rho },\end{aligned}$$

where $q(x')$ is a homogeneous quadratic polynomial with

$$\begin{aligned} D^2_{x'} q \ge C^{-1}I_{n-1}. \end{aligned}$$

(5.24)

Recall $h= \theta t.$ Then it follows from the definition of $U_h$ and Proposition 2.12 that

$$\begin{aligned} \left| x_{n}- h^{1/2}q(x')\right| \le C h^{\frac{1+\alpha }{2}}\left| x'\right| ^{2+\alpha } \quad \text{ on }\, \partial U_h \cap B_{k}^{+} \end{aligned}$$

(5.25)

if $h\le h_0$, where $h_0, C$ depend only on $n, \rho ,\alpha$ and the $C^{2, \alpha }$ norms of $\partial \Omega$ and $\phi |_{\partial \Omega }$ at the origin. Hence by combining with (5.24), we see that if $h\le h_0$ ($h_0$ now depends also on the uniform convexity of $\Omega$) then on $\partial U_h\cap B_{k}^{+}$,

$$\begin{aligned} \frac{1}{2} h^{1/2} q (x')\le x_n \le 2 h^{1/2} q(x'). \end{aligned}$$

(5.26)

Let

$$\begin{aligned} l(y) =\frac{ -1}{\theta \gamma } \big [\tilde{u}_h(\bar{y}) + \nabla \tilde{u}_h(\bar{y})\cdot (y-\bar{y})\big ]. \end{aligned}$$

Then $l(y) =v(y)$ for $y \in \partial U_h\cap B_{k}^{+}$. Since $\left| v\right| \le 1$ in $U_h$, we find that

$$\begin{aligned} \left| l(y)- l(z)\right| = \frac{1}{\theta \gamma }\left| \nabla \tilde{u}_h(\bar{y})\cdot (y-z)\right| \le 2\quad \forall y, z\in \partial U_h\cap B_{k}^{+}. \end{aligned}$$

(5.27)

All constants in this lemma, unless otherwise indicated, depend only on $n, \rho ,\alpha$, the uniform convexity of $\Omega$ and the $C^{2, \alpha }$ norms of $\partial \Omega$ and $\phi |_{\partial \Omega }$.

We now divide the proof into three steps.

Step 1. $l$ is uniformly Lipschitz at the origin: there exists $L>0$ such that

$$\begin{aligned} \left| l(z)-l(0)\right| \le L \left| z\right| \quad \forall z\in \partial U_h\cap B_{k^2}^{+}. \end{aligned}$$

Take $z\in \partial U_h\,\cap \, B_{k^2}^{+}\backslash \{0\}.$ Let $\mathcal C$ be the curve which is the intersection of $\partial U_h\,\cap \, B_{k}^{+}$ and the vertical plane $(P)$ passing through $z$ and the origin. Let $p$ and $q$ be the intersection of $\mathcal C$ with $\partial B_{k}^{+}.$ We now have a plane curve $\mathcal C$ in $(P)$ which can be assumed to be the usual $xy$-plane. It is easy to see from (5.24)–(5.26) that $\mathcal C$ is a graph in the $y$-direction $\mathcal C = \{(x, \varphi (x))\}$ with $C^{1, 1}$ norm comparable to $h^{1/2}$, that is

$$\begin{aligned} C^{-1} h^{1/2}\le \varphi ^{''}(x)\le Ch^{1/2}. \end{aligned}$$

Note that, this also follows from the proof of [23, Lemma 4.2] for the case of uniformly convex domains $\Omega$.

Since $\left| p\right| =\left| q\right| =k$, we find that

$$\begin{aligned} y_{p}\sim h^{1/2}, y_{q}\sim h^{1/2}, ~ \left| x_p\right| \sim k, \left| x_q\right| \sim k. \end{aligned}$$

Without loss of generality, we can assume that $y_{p}\le y_{q}$ and $x_p<0< x_q$, that is, $p$ is on the left half-plane while $q$ is on the right half-plane. The horizontal line through $p$ intersects $\mathcal C$ at another point $q^{'}.$ Since $\varphi ^{''}\le Ch^{1/2}$ and $y_{q^{'}}= y_{p}\sim h^{1/2}$, we must have $x_{q^{'}}\sim k.$ In particular, $z$ lies on the arc $p0q^{'}.$ We can assume that $z$ lies on the arc $0q^{'}.$ Now, take a ray emanating from $q^{'}$ and parallel to $0z.$ This ray is exactly $q^{'}0$ when $z\equiv q^{'}$ and it is $q^{'}p$ when $z\rightarrow 0.$ Thus, by continuity, there must be a point $m$ on the arc $0p$ such that $q^{'}m$ is parallel to $0z$. Clearly, $\left| q^{'}-m\right| \ge x_{q^{'}}\sim k.$ Using $z= \frac{\left| z\right| }{\left| q^{'}-m\right| } (q^{'}-m),$ we find from (5.27) that

$$\begin{aligned} \left| l(z)-l(0)\right|&= \frac{1}{\theta \gamma }\left| \nabla \tilde{u}_h(\bar{y})\cdot z\right| =\frac{\left| z\right| }{\left| q^{'}-m\right| } \frac{1}{\theta \gamma }\left| \nabla \tilde{u}_h(\bar{y})\cdot (q^{'}-m)\right| \\&\le \frac{\left| z\right| }{\left| q^{'}-m\right| } \le L\left| z\right| . \end{aligned}$$

Thus, $l$ is Lipschitz at $0$.

Step 2. Let $\frac{1}{\theta \gamma }\nabla \tilde{u}_h(\bar{y}) = (a', a_n).$ Then

$$\begin{aligned} \left| a'\right| \le 2L\quad \text{ and }\quad \left| a_n\right| h^{1/2}\le CL. \end{aligned}$$

First, we note that the projection of $\partial U_h\cap B_{k}$ on $\{x_n=0\}$ contains a ball of radius comparable to $k$. By rotating coordinates in $\{x_n=0\}$, we can assume that $a'= (A, 0,\ldots , 0).$ Take a curve $\mathcal C =\big \{(x, 0, \ldots , 0,\varphi (x))\mid -k^2 \le x\le k^2\big \}$ in $\partial U_h\cap B_{k}$ that lies in the $x_1x_n$ plane. Note that $\varphi (x)\sim h^{1/2} x^2.$ By the Lipschitz property of $l$ in Step 1, we have

$$\begin{aligned} \frac{1}{\theta \gamma }\big |\nabla \tilde{u}_h(\bar{y})\cdot (x, 0, \ldots , 0,\varphi (x))\big | = \left| A x + a_n \varphi (x)\right| \le L \sqrt{x^2 + (\varphi (x))^2} \le 2L\left| x\right| . \end{aligned}$$

Dividing the above inequalities by $x$ and then letting $x\rightarrow 0$, we get the desired bound

$$\begin{aligned} \left| a'\right| =\left| A\right| \le 2L. \end{aligned}$$

As a consequence, we have

$$\begin{aligned} \left| a_n \varphi (x)\right| \le \left| Ax\right| + 2L\left| x\right| \le 4L\left| x\right| . \end{aligned}$$

Using the lower bound on the growth of $\varphi$ and evaluating at $\left| x\right| \sim k^2$, we obtain

$$\begin{aligned} \left| a_n\right| h^{1/2} \le CL. \end{aligned}$$

Step 3. We have

$$\begin{aligned} \Vert v\Vert _{C^{2, \alpha } \left( \partial U_h\cap B_{k}^{+}\right) }= \Vert l\Vert _{C^{2, \alpha } \left( \partial U_h\cap B_{k}^{+}\right) } \le C. \end{aligned}$$

Recall from (5.25) that $\partial U_h\cap B_{k}$ is a graph in the $e_n$ direction, that is,

$$\begin{aligned} \partial U_h\cap B_{k} = \big \{(x', \psi (x')): \, |x'|\le C_k\big \}, \end{aligned}$$

with the following properties:

$$\begin{aligned} (a)\, \Vert \nabla \psi \Vert _{L^{\infty }} + \Vert D^2 \psi \Vert _{L^{\infty }}\le C h^{1/2},\qquad (b)\, \Vert D^2 \psi \Vert _{C^{\alpha }} \le Ch^{\frac{1+\alpha }{2}}. \end{aligned}$$

For $y\in \partial U_h\cap B_{k}$, we have $y =(x', \psi (x'))$ and

$$\begin{aligned} l(y) = l(0) -\frac{1}{\theta \gamma }\nabla \tilde{u}_h( \bar{y}) \cdot y = l(0) -a' \cdot x' -a_n \psi (x') \end{aligned}$$

where $l(0)$ is a constant bounded by $1$. Clearly, the $C^{2, \alpha }$ bound for $l$ on $\partial U_h\cap B_{k}$ now follows from $(a)-(b)$ and Step 2. $\square$

5.2 Global $W^{2,p}$ estimates

In this subsection we will use the density estimates established in Sect. 5.1 to derive global $W^{2,p}$-estimates for solution $u$ of the linearized equation $\mathcal{L }_\phi u=f$ when $f\in L^q(\Omega )$ for some $q>n$ as stated in Theorems 1.1 and 1.2.

Proof of Theorem 1.1.

The assumptions on $\Omega$ and $\phi$ in the statement of our theorem imply that $\Omega$ satisfy (2.1) for some $\rho >0$ and, by Proposition 2.4, $\phi$ satisfies (2.3). Thus, $\Omega$ and $\phi$ satisfy the conditions of Lemmas 5.2 and  5.4.

By the ABP estimate, it suffices to establish our $W^{2,p}$ estimates in the form

$$\begin{aligned} \Vert D^2u\Vert _{L^p(\Omega )}\le C\Big ( \Vert u\Vert _{L^\infty (\Omega )} +\Vert f\Vert _{L^{q}(\Omega )}\Big ). \end{aligned}$$

We first observe that by working with the function $v :=\dfrac{\epsilon u}{\epsilon \Vert u\Vert _{L^\infty (\Omega )} +\Vert f\Vert _{L^{q}(\Omega )}}$ instead of $u$, it is enough to show that there exist $\epsilon , C>0$ depending only on $p$, $q$, $n$ and $\Omega$ such that if $1-\epsilon \le \det D^2\phi \le 1+\epsilon$ in $\Omega$, $\phi =u=0$ on $\partial \Omega$, $\mathcal{L }_\phi u=f$ in $\Omega$, $\Vert u\Vert _{L^\infty (\Omega )}\le 1$ and $\Vert f\Vert _{L^{q}(\Omega )}\le \epsilon$, then

$$\begin{aligned} \Vert D^2u\Vert _{L^p(\Omega )}\le C. \end{aligned}$$

(5.28)

Notice that $u\in W^{2,s}_{loc}(\Omega )$ for any $n< s<q$ as a consequence of $W^{2,p}_{loc}$ estimates in [17].

Let $N_{*}= \max \{N_1, N_2\}$ where $N_1$ and $N_2$ are the large constants in Lemmas 5.2 and 5.4 and $\hat{c}=\min \{c_1, c_2\}$ where $c_1$ and $c_2$ are the small constants in the above lemmas. Fix $M\ge N_{*}$ so that $1/M< \hat{c}$. Next select $0<\epsilon _0<1/2$ such that

$$\begin{aligned} M^{q} \sqrt{2 \epsilon _0} =\frac{1}{2} \end{aligned}$$

and $\epsilon =\epsilon (\epsilon _0, n,\Omega )=\epsilon (p,q,n,\Omega )$ be the smallest of the constants in Lemmas 5.2 and 5.4. With this choice of $\epsilon$, we are going to show that (5.28) holds. Applying Lemma 5.2 to the function $u$ and using $\Vert f\Vert _{L^{q}(\Omega )}\le \epsilon$ we obtain

$$\begin{aligned} \big |S_\phi (x,t)\cap G_{\frac{M}{t}}(u,\Omega )\big | \ge \left( 1-\epsilon _0 - C \epsilon ^\tau \right) \, |S_\phi (x,t)| \end{aligned}$$

as long as $x\in \overline{\Omega }$ and $t\le \hat{c}$. By taking $\epsilon$ even smaller if necessary we can assume $C \epsilon ^\tau <\epsilon _0$. Then it follows from the above inequality that

$$\begin{aligned} \big |S_\phi (x,t)\setminus G_{\frac{M}{t}}(u,\Omega )\big | \le 2\epsilon _0 \, |S_\phi (x,t)| \quad \text{ for } \text{ any }\, x\in \overline{\Omega },\, t\le \hat{c}. \end{aligned}$$

(5.29)

Let $1/h \le \hat{c}$. For $x\in \Omega \setminus G_{h M}(u,\Omega )$, define

$$\begin{aligned} g(t) := \dfrac{\big |(\Omega \setminus G_{h M}(u,\Omega )) \cap S_\phi (x,t)\Big |}{|S_\phi (x,t)|}. \end{aligned}$$

We have $\lim _{t\rightarrow 0} g(t)=1.$ Also, if $1/h \le t \le \hat{c}$, then (5.29) gives

$$\begin{aligned} \big |(\Omega \setminus G_{h M}(u,\Omega )) \cap S_\phi (x,t)\big |&\le |S_\phi (x,t)\setminus G_{h M}(u,\Omega )|\\&\le |S_\phi (x,t)\setminus G_{M/t}(u,\Omega )| \le 2\epsilon _0 \, |S_\phi (x,t)|. \end{aligned}$$

Therefore $g(t)\le 2\epsilon _0$ for $t\in [1/h, \hat{c}].$ Then by continuity of $g$, there exists $t_{x}\le 1/h$ such that $g(t_{x})=2\epsilon _0$.

Thus for any $x\in \Omega \setminus G_{h M}(u,\Omega )$ there is $t_{x}\le 1/h \le \hat{c}$ satisfying

$$\begin{aligned} \big |\big (\Omega \setminus G_{h M}(u,\Omega )\big ) \cap S_\phi (x,t_{x})\big | = 2\epsilon _0 \, |S_\phi (x,t_{x})|. \end{aligned}$$

(5.30)

We now claim that (5.30) implies

$$\begin{aligned} S_\phi (x,t_{x}) \subset \big (\overline{\Omega }\setminus G_h (u,\Omega )\big ) \cup \big \{z\in \overline{\Omega }: \mathcal M (f^n)(z)> (c^* M h)^n\big \}, \end{aligned}$$

(5.31)

where $c^* :=(\frac{\epsilon _0}{C})^{1/\tau }$, and

$$\begin{aligned} \mathcal M (F)(z) :=\sup _{t\le \hat{c}} \frac{1}{|S_{\phi }(z, t)|}\int \limits _{S_\phi (z,t)}|F(y)|\, dy\quad \forall z\in \overline{\Omega }. \end{aligned}$$

Indeed, since otherwise there exists $\bar{x}\in S_\phi (x,t_{x})\cap G_h(u,\Omega )$ such that $\mathcal M (f^n)(\bar{x})\le (c^* M h)^n$. Note also that $t_{x} \le \hat{c}$. Then by Lemma 5.4 applied to $u$ we get

$$\begin{aligned} \big |S_\phi (x,t_{x})\cap G_{h M}(u,\Omega )\big | > (1-2 \epsilon _0) \, |S_\phi (x,t_{x})| \end{aligned}$$

yielding

$$\begin{aligned} \big |\big (\Omega \setminus G_{h M}(u,\Omega )\big )\cap S_\phi (x,t_{x})\big | \le \big |S_\phi (x,t_{x})\setminus G_{h M}(u,\Omega )\big | < 2\epsilon _0 \,|S_\phi (x,t_{x})|. \end{aligned}$$

This is a contradiction with (5.30) and so (5.31) is proved. We infer from (5.30), (5.31) and Theorem 2.15 that

$$\begin{aligned} |\Omega {\setminus } G_{h M}(u,\Omega )| \le \sqrt{2\epsilon _0}\left[ |\Omega {\setminus } G_{h}(u,\Omega )| + \big |\{x\in \Omega : \mathcal M (f^n)(x)> (c^* M h)^n\}\big |\right] ,\nonumber \\ \end{aligned}$$

(5.32)

as long as $1/h \le \hat{c}$.

For $k=0,1,\dots$, set

$$\begin{aligned} a_k:= |\Omega \setminus G_{M^k}(u,\Omega )| \quad \text {and}\quad b_k:=\big |\{x\in \Omega : \mathcal M (f^n)(x)> (c^* M M^k)^n\}\big |. \end{aligned}$$

Let $h=M$, then we get from (5.32) that $a_2\le \sqrt{2\epsilon _0}(a_1 + b_1)$. Next let $h=M^2$, then $a_3\le \sqrt{2\epsilon _0}(a_2 + b_2) \le 2\epsilon _0 a_1 + 2\epsilon _0 b_1 + \sqrt{2\epsilon _0}\, b_2$. Continuing in this way we conclude that

$$\begin{aligned} |\Omega {\setminus } G_{M^{k+1}}(u,\Omega )| =a_{k+1}\le \left( \sqrt{2\epsilon _0}\right) ^k a_1 \!+\! \sum _{i=1}^{k}{\left( \sqrt{2\epsilon _0}\right) ^{(k+1)-i} b_i}\!\quad \text{ for }\, k\!=\!1,2,\ldots \nonumber \\ \end{aligned}$$

(5.33)

We are now ready to prove (5.28). We have

$$\begin{aligned}&\int \limits _{\Omega }{|D_{i j} u|^p ~dx} = p\int \limits _0^{\infty }{t^{p-1} \big |\{x\in \Omega : \, |D_{i j} u(x)| >t \}\big | ~dt}\\&\quad = p \int \limits _0^{M^{\frac{q}{p}}}{t^{p-1} \big |\{x\in \Omega : |D_{i j} u(x)| >t \}\big | ~dt}\\&\quad \quad + p\sum _{k=1}^{\infty }\int \limits _{M^{\frac{qk}{p}}}^{M^{\frac{q(k+1)}{p}}}{t^{p-1} \big |\left\{ x\in \Omega : |D_{i j} u(x)| >t \right\} \big | ~dt}\nonumber \\&\quad \le |\Omega | M^{q} +\big (M^{q} -1 \big ) \sum _{k=1}^{\infty }{M^{qk} \left| \left\{ x\in \Omega : ~|D_{i j} u(x)| > M^{\frac{qk}{p}} \right\} \right| } \nonumber \\&\quad \le |\Omega | M^{q} +\big (M^{q} -1 \big ) \left[ \sum _{k=1}^{\infty }{M^{qk} \bigg |\Omega \setminus A^{\text{ loc }}_{\left( c M^{\frac{k(q-p)}{2p}}\right) ^{\frac{-2}{n-1}}}\bigg |} + \sum _{k=1}^{\infty }{M^{qk} \big |\Omega \setminus G_{M^k}(u,\Omega )\big |}\right] \nonumber \\&\quad \le |\Omega | M^{q} +\big (M^{q}\! -\!1 \big ) \left[ C(n,\epsilon ,\Omega ) \sum _{k=1}^{\infty }{ M^{k\left( q \!+\! \left( \frac{q}{p}\!-\!1\right) \frac{\ln {\sqrt{C \epsilon }}}{C}\right) }} \!+\! \sum _{k=1}^{\infty }{M^{qk} \big |\Omega \setminus G_{M^k}(u,\Omega )\big |}\right] ,\nonumber \end{aligned}$$

where we used (3.1) with $m =q/p>1$ and $\beta =M^k$ in the second inequality and used (3.2) in the last inequality. Since $\epsilon >0$ is small, the first summation in the last expression is finite and hence (5.28) will follow if we can show that $\sum _{k=1}^{\infty }{M^{kq} |\Omega \setminus G_{M^k}(u,\Omega )|}\le C$. For this, let us employ (5.33) to obtain

$$\begin{aligned}&\sum _{k=1}^{\infty }{M^{kq} |\Omega \setminus G_{M^k}(u,\Omega )|}\\&\quad \le a_1 \sum _{k=1}^{\infty } {M^{kq} \left( \sqrt{2\epsilon _0}\right) ^{k-1} } + \sum _{k=1}^{\infty } \sum _{i=0}^{k-1}{M^{kq} \left( \sqrt{2\epsilon _0}\right) ^{k-i} b_i}\\&\quad =\frac{ a_1}{\sqrt{2\epsilon _0}} \sum _{k=1}^{\infty } {\big (M^{q} \sqrt{2\epsilon _0}\big )^{k} } + \left[ \sum _{j=1}^{\infty }{\big (M^{q} \sqrt{2\epsilon _0}\big )^j}\right] \left[ \sum _{i=0}^{\infty } { M^{iq} b_i}\right] \\&\quad =\frac{ a_1}{\sqrt{2\epsilon _0}} \sum _{k=1}^{\infty } {2^{-k} } + \left[ \sum _{j=1}^{\infty }{2^{-j}}\right] \left[ \sum _{i=0}^{\infty } { M^{iq} b_i}\right] = \frac{ a_1}{\sqrt{2\epsilon _0}} + \sum _{i=0}^{\infty } { M^{iq} b_i}. \end{aligned}$$

But as $f^n \in L^{\frac{q}{n}}(\Omega )$ and $q>n$, by the strong-type estimate in Theorem 2.16 we have

$$\begin{aligned} \int \limits _{\Omega } \left| \mathcal M (f^n)(x)\right| ^{\frac{q}{n}} dx&\le C(n, q, \rho ) \int \limits _{\Omega }\left| f^{n}(x)\right| ^{\frac{q}{n}} dx \\&\le C(n, q, \rho )\Vert f\Vert ^{q}_{L^{q}(\Omega )}\le C(n, q, \rho ) \end{aligned}$$

implying $\sum _{i=0}^{\infty } { (M^n)^{i \frac{q}{n}} b_i}\le C$. Thus $\sum _{k=1}^{\infty }{M^{kq} |\Omega \setminus G_{M^k}(u,\Omega )|}\le C$ and (5.28) is proved. $\square$

Finally, we prove Theorem 1.2.

Proof of Theorem 1.2.

It suffices to prove the theorem for the case $\varphi =0$ since $\tilde{u}:= u-\varphi \in C(\overline{\Omega })\cap W^{2,n}_{loc}(\Omega )$ is the solution to the linearized Monge–Ampère equation

$$\begin{aligned} \mathcal{L }_\phi \tilde{u} =\tilde{f} \, \text{ in } \, \Omega , \quad \text{ and }\quad \tilde{u} = 0\, \text{ on } \, \partial \Omega , \end{aligned}$$

where $\tilde{f}:=f\,{-}\,\Phi ^{ij}\varphi _{ij}\in L^{q}(\Omega ).$ Indeed, since $g\in C(\overline{\Omega })$, we have $\phi \in W^{2, \frac{(n-1)q s}{s -q}}(\Omega )$ by Savin’s global $W^{2,p}$ estimates [28]. Thus $\Phi ^{ij}\in L^{\frac{q s }{s -q}}(\Omega )$ for all $i, j$ and hence $\tilde{f}:= f-\Phi ^{ij}\varphi _{ij}\in L^{q}(\Omega )$.

In view of Theorem 1.1 and the interior $W^{2,p}$ estimates obtained in [17], the theorem follows by localizing boundary sections of $\phi$ using Theorem 2.2. For completeness, we sketch the proof.

The assumptions on $\Omega$ and $\phi$ imply that $\Omega$ satisfies (2.1) for some $\rho >0$ and $\phi$ satisfies (2.3). Let $\epsilon$ be the small constant given by an analogous version of Theorem 1.1 which will be explained later. In particular, $\epsilon$ depends only on $n, p, q, \lambda , \Lambda , \rho$ and $\alpha$. Let $c$ be as in Remark 2.6.

Since $g\in C(\overline{\Omega })$, we can find $m\le c$ depending only on $\epsilon$, $\lambda$ and the modulus of continuity of $g$ such that

$$\begin{aligned} \left| g(x)-g(y)\right| \le \lambda \epsilon ~\text { for all }~ x, y\in \overline{\Omega }~\text { satisfying }~\left| x-y\right| \le m. \end{aligned}$$

Hence it follows from (2.7) that for $s\le m^3$ and any boundary point $y\in \partial \Omega$, we have

$$\begin{aligned} \left| g(x)-g(y)\right| \le \lambda \epsilon ~\text { for all } ~ x\in S_{\phi }(y, s). \end{aligned}$$

(5.34)

Let us consider a boundary point $y\in \partial \Omega$ and for simplicity we assume that $y=0$. We can assume further that $\Omega$ satisfies (2.5), $\phi (0)=0$ and $\nabla \phi (0) =0$. Then by the Localization Theorem, there is a linear map $T_s = s^{-1/2} A_s$ such that

$$\begin{aligned} \overline{\Omega } \cap B_{k}\subset T_s(S_\phi (0, s))\subset \overline{\Omega }\cap B_{k^{-1}}, \end{aligned}$$

(5.35)

where $\det A_s =1$ and $\Vert A_s\Vert , \, \Vert A_s^{-1}\Vert \le k^{-1} |\log {s}|$. By working with the function $g(0)^{\frac{-1}{n}}\phi (x)$ instead of $\phi (x)$ and using (5.34), we can also assume that $g(0) =1$ and

$$\begin{aligned} 1-\epsilon \le g\le 1 + \epsilon \quad \text{ in }\quad S_\phi (0, s). \end{aligned}$$

We now define the rescaled domains $U_s :=T_s(S_\phi (0,s))$, $\Omega _{s} :=T_s(\Omega )$ and the rescaled functions $\phi _s$, $u_s := u\circ T^{-1}_s,\; f_s$ as in Sect. 2.2 that preserve the $L^{\infty }$ -norm of $u$. We claim that

$$\begin{aligned} \Vert D^2 u_s\Vert _{L^p\big ( S_{\phi _s}(0, c^9)\big )} \le C \Big (\Vert u_s\Vert _{L^\infty (U_s)} +\Vert f_s\Vert _{L^{q}(U_s)} \Big ), \end{aligned}$$

(5.36)

where $C>0$ depends only on $p, q, n, \rho ,\lambda , \Lambda , \alpha$, the uniform convexity of $\partial \Omega$, $\Vert \partial \Omega \Vert _{C^{2,\alpha }}$ and $\Vert \phi \Vert _{C^{2,\alpha }(\partial \Omega )}$. Then by rescaling back as in the proof of Lemma 3.11 we obtain

$$\begin{aligned} \Vert D^2 u\Vert _{L^p\big ( S_{\phi }(y, c^9 s)\big )}&\le C s^{\frac{n}{2p}-1} |\log {s}|^2 \Vert u\Vert _{L^\infty (\Omega )} + C s^{\frac{n}{2}(\frac{1}{p}-\frac{1}{q})} |\log {s}|^2 \Vert f\Vert _{L^{q}(\Omega )}\nonumber \\&\le C(s) \, \left( \Vert u\Vert _{L^\infty (\Omega )} + \Vert f\Vert _{L^{q}(\Omega )}\right) \qquad \forall y\in \partial \Omega . \end{aligned}$$

(5.37)

Let $\delta := c^9 s$. By (2.7), we know that $S_{\phi }(y,\delta )\supset \overline{\Omega }\cap B(y, \delta ^{2/3}).$ Therefore if we let

$$\begin{aligned} \Omega _{\delta ^{2/3}} :=\{x\in \Omega : \text {dist}(x,\partial \Omega )>\delta ^{2/3}\}, \end{aligned}$$

then we can cover the $\delta ^{2/3}$ neighborhood of $\Omega$, that is $\Omega {\setminus } \Omega _{\delta ^{2/3}}$, by a finite number of boundary sections $\{S_{\phi }(y_j, \delta )\}_{j=1}^{N}$. Then by adding (5.37) over the family $\{S_{\phi }(y_j, \delta )\}_{j=1}^{N}$, we arrive at the $W^{2,p}$ estimate at the boundary

$$\begin{aligned} \Vert D^2 u\Vert _{L^{p}\left( \Omega \setminus \Omega _{\delta ^{2/3}}\right) }\le C\left( \Vert u\Vert _{L^{\infty }(\Omega )} + \Vert f\Vert _{L^{q}(\Omega )}\right) \!. \end{aligned}$$

On the other hand, by the interior estimate in [17, Theorem 1.1], we also have

$$\begin{aligned} \Vert D^2 u\Vert _{L^{p}( \Omega _{\delta ^{2/3}})}\le C(\Vert u\Vert _{L^{\infty }(\Omega )} + \Vert f\Vert _{L^{q}(\Omega )}). \end{aligned}$$

Our Theorem 1.2 follows from the above inequalities.

We now indicate how to obtain the claim (5.36). The proof consists of reviewing the proof of Theorem 1.1. By (2.7), we have

$$\begin{aligned} S_{\phi _s}(0, c^9)\subset U_{s}\cap B_{c^3}. \end{aligned}$$

We use Lemma 3.13 to cover $U_{s}\cap B_{c^2}$. We restrict our estimates on the distribution function for the second derivatives in Lemma 3.4 to $U_s\cap B_{c^2}$. Lemma 5.2 holds with obvious changes for the data $(\Omega _s, \phi _s, U_s)$. So does Lemma 5.4 provided that we have an analogous version of Lemma 5.5 for our data $(\Omega _s, \phi _s, U_s)$. Precisely, let $S_{\phi _s}(y_0, h)$ be a section of $\phi _s$ in $U_s$ such that $y_0\in \partial U_s\cap B_{c^3}$ and $S_{\phi _s}(y_0, h)\cap G_\gamma (u_s, U_s, \phi _s) \ne \emptyset$ for some $\gamma >0$ (say, $\bar{y}\in S_{\phi _s}(y_0, h)\cap G_\gamma (u_s, U_s, \phi _s)\,$). By Lemma 2.5 and the Localization Theorem 2.2, there exists an affine map $\tilde{T}_h$ such that

$$\begin{aligned} \tilde{T}_h (y_0) = y_0 \quad \text{ and }\quad \overline{U_s} \cap B_{k}(y_0)\subset \tilde{U}_h := \tilde{T}_h(S_{\phi _s}(y_0, \theta h) )\subset \overline{U_s}\cap B_{k^{-1}}(y_0). \end{aligned}$$

Here $\theta >1$ is the same constant at the beginning of the proof of Lemma 5.4. We need to show that the $C^{2,\alpha }$ norm on the boundary $\partial \tilde{U}_h \cap B_k(y_0)$ of the following function

$$\begin{aligned} \tilde{v}(z) := \dfrac{1 }{\theta \gamma h }\left[ u_s(\tilde{T}_h^{-1} z)- u_s(\bar{y}) - \nabla u_s(\bar{y})\cdot (\tilde{T}_h^{-1} z-\bar{y})\right] ,\quad z\in \tilde{T}_h(U_s) \end{aligned}$$

is bounded by a constant which is independent of the uniform convexity of $U_s$. The function $\tilde{v}$ is defined in a similar way to the definition of the function $v$ in (5.18). We note that the uniform convexity of the boundary $\partial \Omega$ plays a key role in the proof of Lemma 5.5. Thus we can not obtain the desired result by repeating the proof of Lemma 5.5 for our data $(\Omega _s, \phi _s, U_s)$ since the uniform convexity of $\partial \Omega _s$ deteriorates as $s\rightarrow 0$. However, we can get away from this as follows.

Let $T := \tilde{T}_h \circ T_s$. Then $T$ normalizes the section $S_\phi (T_s^{-1}y_0, \theta hs)$, and

$$\begin{aligned} \Vert T\Vert \le k^{-2} (\theta h s)^{-1/2} \, |\log {(\theta h)}| \, |\log {s}|, \quad \Vert T^{-1}\Vert \le k^{-2} (\theta h s)^{1/2} \, |\log {(\theta h)}| \, |\log {s}|. \end{aligned}$$

Moreover, $\bar{x} := T_s^{-1}(\bar{y})\in S_\phi (T_s^{-1}y_0, \theta hs) \cap G_{\gamma s^{-1} }(u, \Omega ,\phi )$ and

$$\begin{aligned} T\left( S_\phi (T_s^{-1}y_0, \theta hs)\right) = \tilde{T}_h (S_{\phi _s}(y_0, \theta h))=\tilde{U}_h. \end{aligned}$$

Therefore, by reviewing the proof of Lemma 5.5 we see that the function

$$\begin{aligned} v(y) := \frac{1}{\theta (\gamma s^{-1}) hs}\left[ u(T^{-1} y)- u(\bar{x}) - \nabla u(\bar{x})\cdot (T^{-1} y-\bar{x})\right] ,\quad y\in T(\Omega ) \end{aligned}$$

satisfies

$$\begin{aligned} \Vert v\Vert _{C^{2,\alpha }\big (\partial \tilde{U}_h \cap B_k(\tilde{T}_h(y_0))\big )}\le C_\alpha \end{aligned}$$

(5.38)

with $C_\alpha$ depending on the uniform convexity of $\Omega$. But since $\tilde{T}_h(y_0) = y_0$ and $\tilde{v} \equiv v$ on $\tilde{U}_h$ as $u_s(y) = u(T_s^{-1} y)$, we conclude that the $C^{2,\alpha }$ norm of $\tilde{v}$ on $\partial \tilde{U}_h \cap B_k(y_0)$ is bounded by the same constant $C_\alpha$ in (5.38). Hence the claim (5.36) follows as explained above. $\square$