# Uniform Calderón-Zygmund estimates
in multiscale elliptic homogenization

Weisheng Niu  W. Niu: School of Mathematical Science, Anhui University, Hefei,
China<EMAIL_ADDRESS>and  Jinping Zhuge  J. Zhuge: Morningside Center of
Mathematics, Academy of Mathematics and systems science, Chinese Academy of
Sciences, Beijing, China<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with the elliptic equation $-\text{\rm
div}(A_{\varepsilon}\nabla u_{\varepsilon})=\text{\rm div}f$ in a bounded
$C^{1}$ domain, where $A_{\varepsilon}$ takes a form of
$A_{\varepsilon}(x)=A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})$,
with $A(y_{1},y_{2},\cdots,y_{n})$ being 1-periodic in each $y_{i}$. We prove
the uniform Calderón-Zygmund estimate, namely, the uniform $L^{p}$ boundedness
of the linear map $f\mapsto\nabla u_{\varepsilon}$ for any $p\in(1,\infty)$
with a constant independent of small parameters
$(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n})\in(0,1]^{n}$. Our
result includes the uniform Calderón-Zygmund estimate in quasiperiodic
elliptic homogenization (even without the Diophantine condition), which was
previously unknown. The proof novelly combines the Dirichlet’s theorem on the
simultaneous Diophantine approximation from number theory, a technique of
reperiodization, reiterated periodic homogenization and a large-scale real-
variable argument. Using the idea of reperiodization, we also obtain some
large-scale or mesoscopic-scale Lipschitz estimates.

Keywords: Multiscale homogenization, Diophantine approximation, Calderón-
Zygmund estimate.

###### 2020 Mathematics Subject Classification:

35B27

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Local periodic homogenization
    2. 2.2 Reiterated correctors and effective matrix
  3. 3 Quantitative scale separation and scale reduction
    1. 3.1 Scale separation
    2. 3.2 Scale reduction and uniform approximation
  4. 4 The uniform Calderón-Zygmund estimates
  5. 5 Large-scale Lipschitz estimates

##  1\. Introduction

In this paper, we study the uniform regularity for a family of linear elliptic
equations or systems in divergence form in a bounded domain
$\Omega\subset\mathbb{R}^{d}$,

$\displaystyle-\text{\rm div}(A_{\varepsilon}(x)\nabla
u_{\varepsilon})=\text{\rm div}f,$ (1.1)

where the coefficient matrix takes a form of
$A_{\varepsilon}(x)=A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})$,
and $(\varepsilon_{i})_{1\leq i\leq n}\in(0,1]^{n}$ are small parameters
describing $n$ different oscillating scales of the coefficients. We assume
that $A(y_{1},y_{2},\cdots,y_{n}):\mathbb{R}^{d\times
n}\mapsto\mathbb{R}^{d\times d}$ is 1-periodic in each
$y_{i}\in\mathbb{R}^{d}$ with $1\leq i\leq n$. Due to the symmetry of
$y_{i}$’s, it is harmless to assume
$\varepsilon_{1}\geq\varepsilon_{2}\geq\cdots\geq\varepsilon_{n}>0$. The full
assumptions on $A$ will be given shortly.

The homogenization theorem (or quantitative convergence rates) and uniform
regularity are the fundamental questions in homogenization theory. These two
questions have been well answered for the equation (1.1) in the case of one
oscillating scale (i.e., $n=1$); see recent monographs [She18, AK22] and
references therein. For the case of multiple oscillating scales, we refer to
our recent work [NZ23] for a brief survey. It is important to point out that
before the work [NZ23], the homogenization theorem and uniform regularity were
considered closely interconnected and always appear together, in the sense
that the uniform regularity can be derived as a consequence of convergence
rates either by a compactness method [AL87, KLS13] or a quantitative method
(excess decay iteration [GNO20, AS16, She17] or a real-variable argument
[CP98]). For the equation (1.1) with $n\geq 2$, we have the qualitative
homogenization theorem or a quantitative convergence rate to a deterministic
homogenized equation only if the scales
$(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n})$ are separated or
well-separated [AB96], respectively. Recall that the scales
$(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n})\in(0,1]^{n}$ are
well-separated if $\varepsilon_{i+1}\lesssim\varepsilon_{i}^{\alpha_{i}}$ for
some $\alpha_{i}>1$. Consequently, under this scale-separation condition, the
uniform regularity estimates for (1.1) were established in [NSX20] by a
quantitative method. Yet, as pointed out by Avellaneda in [Ave96] (see also
[GH03]) such an assumption is not adequate for treating the most general
problems of transport and diffusion in self-similar (particularly random)
media. The main contribution of [NZ23] is that, without any separation
condition on $(\varepsilon_{i})_{1\leq i\leq n}$, we obtained the uniform
$C^{\alpha}$ estimate for any $\alpha\in(0,1)$ by the compactness method
combined with a novel scale-reduction process. This result was unexpected
since it shows for the first time that the uniform regularity holds stably
even without homogenization, and in some sense the averaging effect always
takes places across all the mesoscopic scales.

This paper is a continuation of our previous work [NZ23]. It was conjectured
in [NZ23] that with $f=0$ in (1.1) and without any scale-separation condition,
the solution admits uniform Lipschitz estimate. In this paper, we partially
solve the conjecture by proving a slightly weaker result, namely, the gradient
$\nabla u_{\varepsilon}$ is uniformly bounded in $L^{p}$ spaces for any
$p<\infty$. In particular, this recovers the uniform $C^{\alpha}$ estimate in
[NZ23] in view of the Sobolev embedding theorem. Moreover, our proof is
quantitative (in contrast to the compactness method in [NZ23]) in the sense
that the constant can be computed explicitly.

Before stating the main result, we list the assumptions on the coefficient
matrix $A$:

  * •

Strong ellipticity condition: there exists $\Lambda\in(0,1]$ so that

$\Lambda|\xi|^{2}\leq\xi\cdot
A(y_{1},\cdots,y_{n})\xi\leq\Lambda^{-1}|\xi|^{2}$ (1.2)

for every $\xi\in\mathbb{R}^{d}$ and for any
$(y_{1},\cdots,y_{n})\in\mathbb{R}^{d\times n}.$

  * •

Periodicity: for any $(y_{1},\cdots,y_{n})\in\mathbb{R}^{d\times n}$ and
$(z_{1},\cdots,z_{n})\in\mathbb{Z}^{d\times n}$,

$A(y_{1}+z_{1},\cdots,y_{n}+z_{n})=A(y_{1},\cdots,y_{n}).$ (1.3)

  * •

Smoothness: there exists $\tau\in(0,1]$ and $L>0$ such that

$|A(y_{1},\cdots,y_{n})-A(y_{1}^{\prime},\cdots,y_{n}^{\prime})|\leq
L\big{\\{}|y_{1}^{\prime}-y_{1}|+\cdots+|y_{n}^{\prime}-y_{n}|\big{\\}}^{\tau}$
(1.4)

for any
$(y_{1},\cdots,y_{n}),(y_{1}^{\prime},\cdots,y_{n}^{\prime})\in\mathbb{R}^{d\times
n}$.

We now state our main result.

######  Theorem 1.1.

Let $\Omega$ be a bounded $C^{1}$ domain. Assume $A$ satisfies (1.2), (1.3)
and (1.4) and $(\varepsilon_{i})_{1\leq i\leq n}\in(0,1]^{n}$. Assume
$1<p<\infty$. Let $u_{\varepsilon}$ be the weak solution to the Dirichlet
problem

$\displaystyle-\text{\rm div}\big{(}A_{\varepsilon}\nabla
u_{\varepsilon}\big{)}=\text{\rm div}f\,\text{ in }\Omega,\quad
u_{\varepsilon}=0\,\text{ on }\partial\Omega,$ (1.5)

where
$A_{\varepsilon}=A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})$
and $f\in L^{p}(\Omega)^{d}$. Then $\nabla u_{\varepsilon}\in
L^{p}(\Omega)^{d}$ and

$\displaystyle\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\|f\|_{L^{p}(\Omega)},$ (1.6)

where $C$ depends only on $d,n,\Lambda,p$, $(\tau,L)$ in (1.4), and $\Omega$.

The same uniform estimate holds if the Dirichlet boundary condition in (1.5)
is replaced by the compatible Neumann boundary condition

$\nu\cdot A_{\varepsilon}\nabla u_{\varepsilon}=-\nu\cdot f\,\text{ on
}\partial\Omega,$ (1.7)

where $\nu$ is the unit outer normal vector of $\partial\Omega$.

The new ingredient in the proof of Theorem 1.1 is the Dirichlet’s theorem (see
Theorem 3.1) on simultaneous Diophantine approximation from number theory,
which enters into a new technique of _reperiodization_ and allow us to
quantitatively separate one scale from the rest $n-1$ scales; see Section 3.1
for details. Precisely, for a given sequence
$(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n})$ in nonincreasing
order and any number $Q>1$, we can find a new 1-periodic matrix $A^{\sharp}$,
depending on $Q$ and the ratios $\varepsilon_{n}/\varepsilon_{i}$, such that
$A_{\varepsilon}(x)$ can be rewritten as

$A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})=A^{\sharp}(x/\varepsilon^{\prime}_{1},x/\varepsilon^{\prime}_{2},\cdots,x/\varepsilon^{\prime}_{n}),$
(1.8)

with some
$(\varepsilon_{1}^{\prime},\varepsilon_{2}^{\prime},\cdots,\varepsilon_{n}^{\prime})\in(0,\infty)^{n}$
and the last scale $\varepsilon_{n}^{\prime}$ is _$Q$ -separated_ from the
rest $\varepsilon_{i}^{\prime}$’s (i.e., $\varepsilon^{\prime}_{i}\geq
Q\varepsilon_{n}^{\prime}$ for all $1\leq i\leq n-1$). Theoretically, $Q$ can
be chosen as large as $(\varepsilon_{n}/r)^{-\sigma}$ with some $\sigma>0$,
yielding a well-separation condition. However, as a payoff the regularity of
$A^{\sharp}(y_{1},y_{2},\cdots,y_{n})$ in $y_{n}$ will become worse as $Q$
increases, which leads to a small-scale estimate depending on $Q$ in a blow up
argument. Fortunately, a sufficiently large $Q$ independent of
$(\varepsilon_{i})_{1\leq i\leq n}$ will be just enough for our proof. With
the $Q$-separation condition, we can perform a quantitative argument of the
reiterated homogenization and approximate the solution of the original
equation with $n$ oscillating scales by a solution of a new equation with at
most $n-1$ oscillating scales. Hence, an inductive argument on the number of
scales combined with a careful real-variable argument involving a _double-
averaging estimate_ will complete the proof.

It is crucial to point out that the proofs for $C^{\alpha}$ estimate (a type
of Schauder estimates) and $L^{p}$ gradient estimate (Calderón-Zygmund
estimate related to singular integrals) are essentially different, for the
latter is stronger and more informative. As in [NZ23], the $C^{\alpha}$
estimate can be derived with a qualitative $H$-convergence theorem and a
compactness method (the correctors are not needed; also see [She15, Theorem
3.1] or [SZ18, Theorem 6.1] for similar situations). However, the estimate of
gradient must involve a quantitative convergence rate or the fine properties
of correctors. While our proof of Theorem 1.1 also uses a scale-reduction
theorem and an inductive argument on the number scales, it is exactly the
Dirichlet’s theorem that helps us quantify the error in the scale-reduction
process and provides much stronger implications and more information about the
behavior of the solutions. In particular, it possibly explains the reason
behind the occurrence of the mesoscopic averaging effect in arbitrary
multiscale media.

As a direct consequence, Theorem 1.1 implies the uniform Calderón-Zygmund
estimate for the operator $-\text{\rm div}(A(x/\varepsilon)\nabla)$ with
arbitrary quasiperiodic coefficient $A$. We recall that $A(y)$ is said to be
quasiperiodic if $A(y)=B(My)$, where $B(w)$ is $1$-periodic in
$w\in\mathbb{R}^{N}$ with $1\leq N\in\mathbb{N}$ and $M$ is an $N\times d$
constant real matrix. Note that we do not have any restriction on the entries
of $M$ and therefore $A$ is allowed to be periodic with a very degenerate
periodic cell.

######  Corollary 1.2.

Let $\Omega$ be a bounded $C^{1}$ domain. Assume that $A(y)=B(My)$ is quasi-
periodic as above and $B$ satisfies (1.2), (1.3) and (1.4) (with $n=1$). Let
$1<p<\infty$, $f\in L^{p}(\Omega)^{d}$, and $u_{\varepsilon}$ be the weak
solution to

$\displaystyle-\text{\rm div}\big{(}A(x/\varepsilon)\nabla
u_{\varepsilon}\big{)}=\text{\rm div}f\,\text{ in }\Omega,\quad
u_{\varepsilon}=0\,\text{ on }\partial\Omega.$

Then $\nabla u_{\varepsilon}\in L^{p}(\Omega)^{d}$ and

$\displaystyle\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\|f\|_{L^{p}(\Omega)},$

where $C$ depends only on $d,\Lambda,p,\Omega,N$, and $(\tau,L)$ in (1.4).

In the setting of quasiperiodic homogenization, the convergence rate was first
obtained by Kozlov in [Koz78] under a Diophantine condition (a quantitative
ergodicity condition): for each row $M_{i}$ of $M$, $|M_{i}\cdot z|\geq
c_{0}|z|^{-\beta}$ for all $z\in\mathbb{Z}^{d}\setminus\\{0\\}$ and for some
$c_{0},\beta>0$. It was observed in [AL91] that the uniform regularity is
valid under Kozlov’s Diophantine condition. This condition was also required
in recent work [She15, AGK16, BG19] applied to the case of quasiperiodic
coefficients. In our Corollary 1.2, we do not need the Diophantine condition
and the constant in the estimate is independent of the entries of the matrix
$M$ (except for the dimension of $M$). This includes some typical degenerate
cases, such as periodic coefficients with a degenerate periodic cell (e.g., a
thin rectangle), quasiperiodic coefficients oscillating at two almost resonant
frequencies (e.g., the ratio of the frequencies is a Liouville number),
multiscale quasiperiodic coefficients, etc.

The proof of Corollary 1.2 based on Theorem 1.1 is simple and we include it
here. In fact, it suffices to write $A(x/\varepsilon)$ into a periodic
coefficient matrix with multiple oscillating scales and apply Theorem 1.1. To
this end, let $M=(M_{ij})_{1\leq i\leq N,1\leq j\leq d}$. Let
$y_{ij}\in\mathbb{R}^{d}$ with $1\leq i\leq N$ and $1\leq j\leq d$ be $Nd$
independent variables. Define a matrix with variables $y_{ij}$

$A^{+}(y_{11},\cdots,y_{ij},\cdots,y_{Nd})=B\big{(}\sum_{j=1}^{d}y_{1j}\cdot
e_{j},\cdots,\sum_{j=1}^{d}y_{Nj}\cdot e_{j}\big{)}.$

Then $A^{+}$ is 1-periodic in each $y_{ij}$ since $B(w)$ is 1-periodic.
Moreover, it is straightforward to verify

$A(x/\varepsilon)=B(Mx/\varepsilon)=A^{+}(M_{11}x/\varepsilon,\cdots,M_{ij}x/\varepsilon,\cdots,M_{Nd}x/\varepsilon).$
(1.9)

This reduces the quasiperiodic coefficient matrix with one oscillating scale
into a periodic coefficient matrix with $Nd$ oscillating scales. Clearly, in
this case, all the scales are possibly not well-separated. Moreover, if $B$ is
uniformly elliptic and Hölder continuous, so is $A^{+}$. As a result,
Corollary 1.2 follows readily from Theorem 1.1.

Finally, using a similar idea of scale separation, we can show the uniform
large-scale or mesoscopic-scale Lipschitz estimates. In particular, in the
case of two scales, i.e., $n=2$, for any $\alpha\in(0,1)$, we have for all
$\varepsilon_{2}^{1-\alpha}\leq r\leq 1$,

$\bigg{(}\fint_{B_{r}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{1/2}\leq
C_{\alpha}\bigg{\\{}\bigg{(}\fint_{B_{1}}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+\bigg{(}\fint_{B_{1}}|F|^{p}\bigg{)}^{1/p}\bigg{\\}},$
(1.10)

where $C_{\alpha}$ depends only on $d,\Lambda,p,\tau,L$ and $\alpha$; see
Theorem 5.1. For general cases $n\geq 3$, we have some mesoscopic-scale
Lipschitz estimates near any given scale; see Theorem 5.2.

Organization. The rest of the paper is organized as follows. In Section 2, we
recall some knowledge and known results from locally periodic homogenization
and reiterated homogenization. In Section 3, we use the Dirichlet’s theorem
and reperiodization technique to separate scales and reduce one scale by a
uniform approximation. The main theorem is proved in Section 4 by a real-
variable argument. In Section 5, we exploit the idea of scale separation to
obtain the uniform large-scale or mesoscopic-scale Lipschitz estimates.

Acknowledgements. W. Niu is supported by NNSF of China (12371106, 11971031).
J. Zhuge is partially supported by grants for Excellent Youth from the NSFC.

##  2\. Preliminaries

###  2.1. Local periodic homogenization

In this subsection, we recall some results concerning the homogenization of
the locally periodic operator $-\text{\rm div}(A(x,x/\varepsilon)\nabla)$ with
a scalar $\varepsilon\in(0,1)$, which will be used in our quantitative scale
reduction process in Section 3.

Suppose $A(x,y)$ satisfies the ellipticity condition (1.2) and is 1-periodic
in $y\in\mathbb{R}^{d}$. The effective (homogenized) matrix is given by

$\displaystyle\widehat{A}(x)=\fint_{\mathbb{T}^{d}}\Big{(}A(x,y)+A(x,y)\nabla_{y}\chi(x,y)\Big{)}dy,$
(2.1)

where $\chi(x,y)=(\chi_{1}(x,y),\dots,\chi_{d}(x,y))$ is the corrector given
by the cell problem

$\left\\{\begin{aligned} &-\text{\rm
div}_{y}\big{(}A(x,y)\nabla_{y}\chi_{j})=\text{\rm
div}_{y}\big{(}A(x,y)\nabla_{y}y^{j}\big{)}\quad\text{ in
}\mathbb{T}^{d}:=\mathbb{R}^{d}/\mathbb{Z}^{d},\\\
&\chi_{j}=\chi_{j}(x,y)\text{ is 1-periodic in }y,\\\
&\int_{\mathbb{T}^{d}}\chi_{j}(x,y)\,dy=0,\end{aligned}\right.$ (2.2)

for $1\leq j\leq d$. Here $y^{j}$ denotes the $j$th component of
$y\in\mathbb{R}^{d}$. Assume that

$\|\nabla_{x}A\|_{\infty}=\|\nabla_{x}A\|_{L^{\infty}(\mathbb{R}_{x}^{d}\times\mathbb{R}^{d}_{y})}<\infty.$
(2.3)

The standard energy estimates for (2.2) imply that

$\|\nabla_{y}\chi\|_{L^{2}(Y)}\leq
C,\quad\|\nabla_{x}\nabla_{y}\chi\|_{L^{\infty}(\mathbb{R}_{x}^{d};L^{2}(Y))}\leq
C\|\nabla_{x}A\|_{\infty}$ (2.4)

with $C$ depending only on $\Lambda,d.$ This combined with the definition of
$\widehat{A}$ yields

$\|\nabla_{x}\widehat{A}\|_{\infty}\leq C\|\nabla_{x}A\|_{\infty}.$

Likewise, if

$|A(x,y)-A(x^{\prime},y)|\leq L|x-x^{\prime}|^{\tau}$ (2.5)

for some $0<\tau\leq 1$ for any $x,x^{\prime},y\in\mathbb{R}^{d}.$ Then we
have

$\displaystyle|\widehat{A}(x)-\widehat{A}(x^{\prime})|\leq
CL|x-x^{\prime}|^{\tau}.$ (2.6)

Let $\varphi\in C_{0}^{\infty}(\mathbb{R}^{d})$ be supported in $B_{1/2}(0)$
such that $\varphi\geq 0$ and $\int_{\mathbb{R}^{d}}\varphi dx=1$. For
functions of the form $g^{\varepsilon}(x)=g(x,x/\varepsilon)$, we define a
partial-smoothing operator

$\displaystyle
S_{\varepsilon}(g^{\varepsilon})(x)=\varepsilon^{-d}\int_{\mathbb{R}^{d}}g(z,x/\varepsilon)\varphi((x-z)/\varepsilon)dz.$
(2.7)

The following theorem provides the convergence rate in $H^{1}(\Omega)$ for the
locally periodic operator. The proof may be found in [NSX20].

######  Theorem 2.1.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$. Let
$u_{\varepsilon}$ be the solution to $-\text{\rm div}(A(x,x/\varepsilon)\nabla
u_{\varepsilon})=F$ in $\Omega$ with $u_{\varepsilon}=g$ on $\partial\Omega,$
and $u_{0}$ the solution to $-\text{\rm div}(\widehat{A}(x)\nabla u_{0})=F$ in
$\Omega$ with $u_{0}=g$ on $\partial\Omega.$ Then we have

$\displaystyle\begin{split}&\|\nabla u_{\varepsilon}-\nabla
u_{0}-S_{\varepsilon}((\nabla_{y}\chi)^{\varepsilon}\nabla
u_{0})\|_{L^{2}(\Omega)}\\\ &\leq
C\varepsilon\big{\\{}(1+\|\nabla_{x}A\|_{\infty})\|\nabla
u_{0}\|_{L^{2}(\Omega)}+\|\nabla^{2}u_{0}\|_{L^{2}(\Omega\setminus\Omega_{3\varepsilon})}\big{\\}}+C\|\nabla
u_{0}\|_{L^{2}(\Omega_{4\varepsilon})},\end{split}$ (2.8)

where we have extended $u_{0}$ to the whole space $\mathbb{R}^{d},$
$\Omega_{t}=\big{\\{}x\in\Omega:\text{\rm dist}(x,\partial\Omega)<t\big{\\}}$,
and $C$ depends only on $d$, $\Lambda$ and $\Omega$.

We emphasize that in the above theorem, $A$ has no smoothness assumption in
$y$.

###  2.2. Reiterated correctors and effective matrix

Consider the operator with $n$ scales $-\text{\rm
div}(A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})\nabla)$,
where $A$ satisfies the ellipticity and periodicity conditions (1.2) and
(1.3). Assume that $\varepsilon_{n}$ is separated from $\varepsilon_{i}$ for
$1\leq i\leq n-1.$ By using the idea of reiterated homogenization, one can
view $(\varepsilon_{i})_{1\leq i\leq n-1}$ as parameters and homogenize the
finest scale $\varepsilon_{n}$ to obtain an operator of the same type but with
only $n-1$ scales.

Indeed rewrite

$\mathcal{A}_{\varepsilon}(x,y):=A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n-1},y).$

It is obvious that $\mathcal{A}_{\varepsilon}(x,y)$ satisfies (1.2), and is
locally 1-periodic in $y$. Let $\chi_{\varepsilon}(x,y)$ be the corrector
given by (2.2) with $A(x,y)$ replaced by $\mathcal{A}_{\varepsilon}(x,y)$, and
$\widehat{\mathcal{A}}_{\varepsilon}(x)$ the effective matrix of
$\mathcal{A}_{\varepsilon}$ given by (2.1). In view of the structure of
$\mathcal{A}_{\varepsilon}(x,y)$, the corresponding corrector
$\chi_{\varepsilon}(x,y)$ takes a form of

$\chi_{\varepsilon}(x,y)=\mathcal{X}(x/\varepsilon_{1},\cdots,x/\varepsilon_{n-1},y),$
(2.9)

where the reiterated corrector $\mathcal{X}=(\mathcal{X}_{j})$ is defined by
the cell problem

$\left\\{\begin{aligned} &-\text{\rm
div}_{y}\big{(}A(y_{1},\cdots,y_{n-1},y)\nabla_{y}\mathcal{X}_{j})=\text{\rm
div}_{y}\big{(}A(y_{1},\cdots,y_{n-1},y)\nabla_{y}y^{j}\big{)}\quad\text{ in
}\mathbb{T}^{d},\\\
&\mathcal{X}_{j}=\mathcal{X}_{j}(y_{1},\cdots,y_{n-1},y)\text{ is 1-periodic
in }y,\\\
&\int_{\mathbb{T}^{d}}\mathcal{X}_{j}(y_{1},\cdots,y_{n-1},y)\,dy=0,\end{aligned}\right.$
(2.10)

for $1\leq j\leq d$, which is also 1-periodic in $y_{1},\cdots,y_{n-1}$, due
to the periodicity of $A$. Moreover, $\widehat{\mathcal{A}}_{\varepsilon}(x)$
takes a form of

$\widehat{\mathcal{A}}_{\varepsilon}(x)=\widehat{\mathcal{A}}(x/\varepsilon_{1},\cdots,x/\varepsilon_{n-1})$
(2.11)

with

$\widehat{\mathcal{A}}(y_{1},y_{2},\cdots,y_{n-1})=\fint_{\mathbb{T}^{d}}A(y_{1},\cdots,y_{n-1},y)\Big{(}I+\nabla_{y}\mathcal{X}(y_{1},\cdots,y_{n-1},y)\Big{)}dy.$
(2.12)

It is obvious that $\widehat{\mathcal{A}}(y_{1},y_{2},\cdots,y_{n-1})$ is
1-periodic in each $y_{i}$ with $1\leq i\leq n-1$, and it is defined
independent of $(\varepsilon_{1},\cdots,\varepsilon_{n})$. Similar to the case
$n=1$, $\widehat{\mathcal{A}}$ satisfies the ellipticity condition (1.2).
Furthermore, similar to (2.6), if $A$ is Hölder continuous in $y_{i}$ for
$1\leq i\leq n-1$, so is $\widehat{\mathcal{A}}$.

##  3\. Quantitative scale separation and scale reduction

###  3.1. Scale separation

The key ingredient of this paper is the Dirichlet’s theorem on the
simultaneous Diophantine approximation. The connection between the Dirichlet’s
theorem and the regularity theory in homogenization is previously unknown. The
only loosely related notion is the Diophantine condition imposed in the
setting of quasiperiodic homogenization, as mentioned earlier. The Dirichlet’s
theorem appears to be a powerful tool (instead of a condition) that eventually
allows us to derive the uniform Calderón-Zygmund estimates in multiscale or
quasiperiodic homogenization without any additional conditions.

Let us first recall the Dirichlet’s theorem; see [Sch91].

######  Theorem 3.1 (Dirichlet (1842)).

Suppose that $\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$ are $m$ real numbers
and $Q>1$. There exist integers $q,p_{1},p_{2},\cdots,p_{m}$ such that $1\leq
q<Q^{m}$ and

$\sup_{1\leq i\leq m}\Big{|}\alpha_{i}-\frac{p_{i}}{q}\Big{|}<\frac{1}{qQ}.$
(3.1)

Roughly speaking, the Dirichlet’s theorem states that any real numbers can be
approximated by good rational numbers with a quantitative small error. In the
rest of this subsection, we apply the Dirichlet’s theorem to separate at least
one scale in the coefficient matrix oscillating at multiple (possibly
unseparated) scales.

Without loss of generality, hereafter we always assume

$1\geq\varepsilon_{1}\geq\varepsilon_{2}\geq\cdots\geq\varepsilon_{n}>0.$
(3.2)

Let $Q\gg 1$ be a large number. We emphasize that the fixed number $Q$ will be
chosen later independent of
$\\{\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n}\\}$. We say $a$ is
$Q$-separated from $b$ if $b\geq Qa$. This scale separation condition is
crucial in quantitative reiterated homogenization.

Given $Q>1$ as above, by the Dirichlet’s theorem with
$\alpha_{i}=\varepsilon_{n}/\varepsilon_{i}\in(0,1]$ for $i=1,2,\cdots,n-1$,
we can find integers $q,p_{1},p_{2},\cdots,p_{n-1}$ such that $1\leq
q<Q^{n-1}$ and

$\sup_{1\leq i\leq
n-1}\Big{|}\frac{\varepsilon_{n}}{\varepsilon_{i}}-\frac{p_{i}}{q}\Big{|}<\frac{1}{qQ}.$
(3.3)

Moreover, because of (3.2), we have $0\leq p_{i}\leq p_{i+1}\leq q<Q^{n-1}$.
Note that it is possible that $p_{i}=0$ if $\varepsilon_{i}$ is large enough
that $\varepsilon_{n}/\varepsilon_{i}<1/Q$ (i.e., $\varepsilon_{n}$ is already
$Q$-separated from $\varepsilon_{i}$).

Define, for $1\leq i\leq n-1$,

$\gamma_{i}:=\Big{|}\frac{\varepsilon_{n}}{\varepsilon_{i}}-\frac{p_{i}}{q}\Big{|}\quad\text{and}\quad
s_{i}=\text{sgn}\Big{(}\frac{\varepsilon_{n}}{\varepsilon_{i}}-\frac{p_{i}}{q}\Big{)}.$
(3.4)

Then, we can write

$\frac{1}{\varepsilon_{i}}=\frac{\gamma_{i}s_{i}}{\varepsilon_{n}}+\frac{p_{i}}{q}\frac{1}{\varepsilon_{n}}.$
(3.5)

Now, we define

$A^{\sharp}(y_{1},y_{2},\cdots,y_{n})=A(s_{1}y_{1}+p_{1}y_{n},s_{2}y_{2}+p_{2}y_{n},\cdots,s_{n-1}y_{n-1}+p_{n-1}y_{n},qy_{n}).$
(3.6)

The key insight here is that, since $s_{i}=\pm 1$ and
$p_{1},p_{2},\cdots,p_{n}$ are all integers, $A^{\sharp}$ is 1-periodic in
each $y_{i}$. We thereby will call the transformation from $A$ to $A^{\sharp}$
a technique of reperiodization. As a result of (3.5) and (3.6), we have

$A\big{(}\frac{x}{\varepsilon_{1}},\cdots,\frac{x}{\varepsilon_{n}}\big{)}=A^{\sharp}\big{(}\frac{\gamma_{1}x}{\varepsilon_{n}},\frac{\gamma_{2}x}{\varepsilon_{n}},\cdots,\frac{\gamma_{n-1}x}{\varepsilon_{n}},\frac{x}{q\varepsilon_{n}}\big{)}.$
(3.7)

In other words, the reperiodization allows us to rewrite the original periodic
matrix $A$ oscillating at scales
$\\{\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n}\\}$ into a new
periodic matrix $A^{\sharp}$ oscillating at different scales
$\\{\varepsilon_{n}/\gamma_{1},\varepsilon_{n}/\gamma_{2},\cdots,\varepsilon_{n}/\gamma_{n-1},q\varepsilon_{n}\\}$.
Note that if $p_{i}=0$ for some $i$, then the corresponding scale for that
(slower) variable does not change, i.e.,
$\varepsilon_{n}/\gamma_{i}=\varepsilon_{i}$. If $\gamma_{i}=0$ for some $i$,
then we just do not have the scale $\varepsilon_{n}/\gamma_{i}$ and we have
less oscillating scales after reperiodization (which is even better). Without
loss of generality, we may assume this does not happen.

The new scales and the new 1-periodic matrix $A^{\sharp}$ have the following
crucial properties:

  * •

The smallest scale $q\varepsilon_{n}$ is $Q$-separated from
$\varepsilon_{n}/\gamma_{i}$ for each $1\leq i\leq n-1$. In fact, (3.3)
implies

$\frac{\varepsilon_{n}/\gamma_{i}}{q\varepsilon_{n}}=\frac{1}{q\gamma_{i}}\geq
Q.$ (3.8)

This is the key property for our application.

  * •

If $A$ is Hölder continuous in $y_{i}$ for $1\leq i\leq n-1$, i.e.,

$\displaystyle|A(y_{1},\cdots,y_{n-1},y_{n})-A(y_{1}^{\prime},\cdots,y_{n-1}^{\prime},y_{n})|$
(3.9) $\displaystyle\leq
L\big{\\{}|y_{1}^{\prime}-y_{1}|+\cdots+|y_{n-1}^{\prime}-y_{n-1}|\big{\\}}^{\tau}$

with $0<L,0<\tau\leq 1$ for any
$(y_{1},\cdots,y_{n}),(y_{1}^{\prime},\cdots,y_{n-1}^{\prime},y_{n})\in\mathbb{R}^{d\times
n}$, then $A^{\sharp}$ is also Hölder continuous in $y_{i}$ for $1\leq i\leq
n-1$ with the same constants $(\tau,L)$. However, $A^{\sharp}$ may lose good
regularity in $y_{n}$ since many periods have been compressed into a single
1-periodic cell.

  * •

The construction of $A^{\sharp}$ relies on the choice of $Q$ as well as the
ratios $\varepsilon_{n}/\varepsilon_{i}$ with $1\leq i\leq n-1$. In other
words, $A^{\sharp}$ is scale-invariant. Also note that $Q$ is a dimensionless
parameter.

It is important to note that the above scale separation process can only
separate the smallest scale from the rest $n-1$ larger scales, while the
relationships among the rest $n-1$ larger scales cannot be determined. This
allows us to perform a one-scale reduction by using the idea of reiterated
homogenization.

###  3.2. Scale reduction and uniform approximation

In this subsection we use the scale separation technique in Section 3.1 to
reduce at least one scale. Precisely, we shall prove that the solution of
(1.1) can be uniformly approximated by a solution to the equation of the same
type but with at most $n-1$ oscillating scales. The following is the main
theorem.

######  Theorem 3.2.

Suppose that $A$ satisfies the assumptions (1.2), (1.3) and (3.9). Let
$u_{\varepsilon}$ be a weak solution to

$\displaystyle-\text{\rm
div}\big{(}A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon}\big{)}=F\quad\text{in }B_{2r}:=B(0,2r).$ (3.10)

Then for any $Q>1$, there exists a coefficient matrix
$A^{\flat}=A^{\flat}(y_{1},\cdots,y_{n-1})$, scales
$\varepsilon_{k}^{\prime}\in(0,\infty)$ with $k=1,\cdots,n-1$, and a weak
solution $u^{\flat}_{\varepsilon^{\prime}}$ to

$-\text{\rm
div}\big{(}A^{\flat}(x/\varepsilon^{\prime}_{1},x/\varepsilon^{\prime}_{2},\cdots,x/\varepsilon^{\prime}_{n-1})\nabla
u^{\flat}_{\varepsilon^{\prime}}\big{)}=F\quad\text{in }B_{r},$

such that

$\|\nabla u^{\flat}_{\varepsilon^{\prime}}\|_{L^{2}(B_{r})}\leq
C\big{\\{}\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2r})}+r\|F\|_{L^{2}(B_{2r})}\big{\\}},$ (3.11)

and

$\|\nabla u_{\varepsilon}-\nabla
u^{\flat}_{\varepsilon^{\prime}}-U_{\varepsilon^{\prime}}\|_{L^{2}(B_{r})}\leq
C\bigg{\\{}\bigg{(}\frac{Q^{n-1}\varepsilon_{n}}{r}\bigg{)}^{\sigma}+Q^{-\tau}\bigg{\\}}\big{\\{}\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2r})}+r\|F\|_{L^{2}(B_{2r})}\big{\\}}$ (3.12)

for some $\sigma>0,$ where $U_{\varepsilon^{\prime}}$ is given by

$U_{\varepsilon^{\prime}}(x)=(\varepsilon_{n}^{\prime})^{-d}\int_{\mathbb{R}^{d}}\varphi\bigg{(}\frac{x-z}{\varepsilon_{n}^{\prime}}\bigg{)}(\nabla_{y}\mathcal{X}^{\sharp})\bigg{(}\frac{z}{\varepsilon_{1}^{\prime}},\cdots,\frac{z}{\varepsilon^{\prime}_{n-1}},\frac{x}{\varepsilon_{n}^{\prime}}\bigg{)}\nabla
u^{\flat}_{\varepsilon^{\prime}}(z)dz,$ (3.13)

$\mathcal{X}^{\sharp}$ is the corrector given by (2.10) with $A$ replaced by
$A^{\sharp}$, and $\varepsilon^{\prime}_{n}$ is given by (3.18). Moreover, the
constant $C$ depends only on $d,n,\Lambda$ and $(\tau,L)$ in (3.9), and the
assumptions (1.2), (1.3) and (3.9) are preserved for $A^{\flat}$.

The proof of Theorem 3.2 relies on the scale separation technique in Section
3.1 and the following approximation theorem that pertains to the quantitative
reiterated homogenization.

######  Theorem 3.3.

Suppose $A$ satisfies the assumption (1.2), (1.3), and (3.9). Assume
$\varepsilon_{n}$ is $Q$-separated from $\varepsilon_{k},k=1,\cdots,n-1$. Let
$u_{\varepsilon}$ be a weak solution to

$-\text{\rm
div}\big{(}A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon}\big{)}=F\quad\text{ in }B_{2}.$ (3.14)

Then there exist a 1-periodic matrix
$\widehat{\mathcal{A}}=\widehat{\mathcal{A}}(y_{1},\cdots,y_{n-1})$ with $n-1$
scales and a weak solution $\widehat{u}_{\varepsilon}$ to

$-\text{\rm
div}\big{(}\widehat{\mathcal{A}}(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n-1})\nabla\widehat{u}_{\varepsilon}\big{)}=F\quad\text{
in }B_{1},$

such that

$\|\nabla\widehat{u}_{\varepsilon}\|_{L^{2}(B_{1})}\leq C\big{\\{}\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2})}+\|F\|_{L^{2}(B_{2})}\big{\\}},$ (3.15)

and

$\|\nabla
u_{\varepsilon}-\nabla\widehat{u}_{\varepsilon}-S_{\varepsilon_{n}}((\nabla_{y}\mathcal{X})^{\varepsilon}\nabla\widehat{u}_{\varepsilon})\|_{L^{2}(B_{1})}\leq
C(Q^{-\tau}+\varepsilon_{n}^{\sigma})\big{\\{}\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2})}+\|F\|_{L^{2}(B_{2})}\big{\\}},$ (3.16)

for some $\sigma>0$, where $\mathcal{X}$ and $\widehat{\mathcal{A}}$ are
defined by (2.10) and (2.12), respectively, and

$S_{\varepsilon_{n}}((\nabla_{y}\mathcal{X})^{\varepsilon}\nabla\widehat{u}_{\varepsilon})(x)=(\varepsilon_{n})^{-d}\int_{\mathbb{R}^{d}}\varphi\bigg{(}\frac{x-z}{\varepsilon_{n}}\bigg{)}(\nabla_{y}\mathcal{X})\bigg{(}\frac{z}{\varepsilon_{1}},\cdots,\frac{z}{\varepsilon_{n-1}},\frac{x}{\varepsilon_{n}}\bigg{)}\nabla\widehat{u}_{\varepsilon}(z)dz.$
(3.17)

Moreover, the constant $C$ depends only on $d,n,\Lambda$ and $(\tau,L)$ in
(3.9).

We first prove Theorem 3.2, assuming Theorem 3.3.

###### Proof of Theorem 3.2.

We first consider the case $r=1$. Given
$1\geq\varepsilon_{1}\geq\varepsilon_{2}\geq\cdots\geq\varepsilon_{n}>0$ and
$Q>1$, by the Dirichlet’s theorem, we can find integers
$q,p_{1},p_{2},\cdots,p_{n-1}$ such that $1\leq q<Q^{n-1}$, and (3.3)-(3.8)
hold. Set

$\displaystyle\varepsilon_{i}^{\prime}$
$\displaystyle=\varepsilon_{i}/\gamma_{i},\quad\text{for }i=1,2,\cdots,n-1,$
(3.18) $\displaystyle\varepsilon_{n}^{\prime}$
$\displaystyle=q\varepsilon_{n}.$

By (3.7) and (3.10), we know that $u_{\varepsilon}$ satisfies

$\displaystyle-\text{\rm
div}\big{(}A^{\sharp}(x/\varepsilon^{\prime}_{1},\cdots,x/\varepsilon^{\prime}_{n})\nabla
u_{\varepsilon}\big{)}=F\,\text{ in }B_{2}.$ (3.19)

By (3.8), $\varepsilon_{n}^{\prime}$ is $Q$-separated from
$\varepsilon_{i}^{\prime}$ for all $i=1,2,\cdots,n-1$. Thus, by Theorem 3.3,
there exist
$\widehat{\mathcal{A}}^{\sharp}=\widehat{\mathcal{A}}^{\sharp}(y_{1},\cdots,y_{n-1})$
given by (2.12) with $A$ replaced by $A^{\sharp}$, and a weak solution
$\widehat{u}_{\varepsilon^{\prime}}$ to

$-\text{\rm
div}\big{(}\widehat{\mathcal{A}}^{\sharp}(x/\varepsilon^{\prime}_{1},\cdots,x/\varepsilon^{\prime}_{n-1})\nabla\widehat{u}_{\varepsilon^{\prime}}\big{)}=F\quad\text{in
}B_{1},$ (3.20)

such that

$\|\nabla\widehat{u}_{\varepsilon^{\prime}}\|_{L^{2}(B_{1})}\leq C\\{\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2})}+\|F\|_{L^{2}(B_{2})}\\},$ (3.21)

and

$\displaystyle\|\nabla
u_{\varepsilon}-\nabla\widehat{u}_{\varepsilon^{\prime}}-S_{\varepsilon_{n}^{\prime}}((\nabla_{y}\mathcal{X}^{\sharp})^{\varepsilon^{\prime}}\nabla\widehat{u}_{\varepsilon^{\prime}})\|_{L^{2}(B_{1})}$
(3.22) $\displaystyle\leq
C(Q^{-\tau}+(\varepsilon^{\prime}_{n})^{\sigma})\\{\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2})}+\|F\|_{L^{2}(B_{2})}\\}$ $\displaystyle\leq
C(Q^{-\tau}+(Q^{n-1}\varepsilon_{n})^{\sigma})\\{\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2})}+\|F\|_{L^{2}(B_{2})}\\},$

for some $\sigma>0$, where $\mathcal{X}^{\sharp}$ is the corrector defined in
(2.10) with $A$ replaced by $A^{\sharp}$, and

$\displaystyle\begin{split}S_{\varepsilon_{n}^{\prime}}((\nabla_{y}\mathcal{X}^{\sharp})^{\varepsilon^{\prime}}\nabla\widehat{u}_{\varepsilon^{\prime}})&=(\varepsilon_{n}^{\prime})^{-d}\int_{\mathbb{R}^{d}}\varphi\bigg{(}\frac{x-z}{\varepsilon_{n}^{\prime}}\bigg{)}(\nabla_{y}\mathcal{X}^{\sharp})\left(\frac{z}{\varepsilon_{1}^{\prime}},\cdots,\frac{z}{\varepsilon^{\prime}_{n-1}},\frac{x}{\varepsilon_{n}^{\prime}}\right)\nabla\widehat{u}_{\varepsilon^{\prime}}(z)dz.\end{split}$

We rename
$A^{\flat}=\widehat{\mathcal{A}}^{\sharp},u^{\flat}_{\varepsilon^{\prime}}=\widehat{u}_{\varepsilon^{\prime}}$,
$U_{\varepsilon^{\prime}}=S_{\varepsilon_{n}^{\prime}}((\nabla_{y}\mathcal{X}^{\sharp})^{\varepsilon^{\prime}}\nabla\widehat{u}_{\varepsilon^{\prime}})$,
and derive the desired estimate in the case $r=1$. Finally, note that
$A^{\sharp}$ and $A^{\flat}$ are scale-invariant. The estimate for general
$r>0$ follows immediately by rescaling. The proof is complete.

∎

We now provide the proof of Theorem 3.3, following the idea of [NSX20].

###### Proof of Theorem 3.3.

We consider the matrix

$\mathcal{A}_{\varepsilon}(x,y):=A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n-1},y).$
(3.23)

Recall that $\mathcal{A}_{\varepsilon}(x,y)$ is strongly elliptic and locally
1-periodic in $y$. In view of (1.4), $\mathcal{A}_{\varepsilon}(x,y)$ is
Hölder continuous in $x$, and

$\displaystyle|\mathcal{A}_{\varepsilon}(x^{\prime},y)-\mathcal{A}_{\varepsilon}(x,y)|\leq
L\sum_{k=1}^{n-1}\varepsilon_{k}^{-\tau}|x-x^{\prime}|^{\tau}.$ (3.24)

In order to apply Theorem 2.1, we need to find an approximate matrix
$\mathcal{A}^{\rm app}_{\varepsilon}=\mathcal{A}^{\rm app}_{\varepsilon}(x,y)$
which is Lipschitz in the $x$ variable. In fact, we define

$\mathcal{A}^{\rm
app}_{\varepsilon}(x,y)=\varepsilon_{n}^{-d}\int_{\mathbb{R}^{d}}\varphi\Big{(}\frac{x-z}{\varepsilon_{n}}\Big{)}\mathcal{A}_{\varepsilon}(z,y)dz,$
(3.25)

with $\varphi$ given as in Section 2.1. Then $\mathcal{A}^{\rm
app}_{\varepsilon}$ satisfies the ellipticity condition (1.2), is 1-periodic
in $y$, and

$\|\mathcal{A}_{\varepsilon}-\mathcal{A}_{\varepsilon}^{\rm
app}\|_{\infty}\leq
CL\sum_{k=1}^{n-1}\varepsilon_{k}^{-\tau}\varepsilon_{n}^{\tau}\quad\text{ and
}\quad\|\nabla_{x}\mathcal{A}_{\varepsilon}^{\rm app}\|_{\infty}\leq
CL\sum_{k=1}^{n-1}\varepsilon_{k}^{-\tau}\varepsilon_{n}^{\tau-1},$ (3.26)

where $C$ depends only on $d$ and $\tau$.

Let $u_{\varepsilon}$ be the weak solution to (3.14). Then by (3.23), it
satisfies

$-\text{\rm div}\big{(}\mathcal{A}_{\varepsilon}(x,x/\varepsilon_{n})\nabla
u_{\varepsilon}\big{)}=F\quad\text{ in }B_{2}.$ (3.27)

Let $u_{\varepsilon}^{\rm app}$ be the weak solution to

$-\text{\rm div}\big{(}\mathcal{A}^{\rm
app}_{\varepsilon}(x,x/\varepsilon_{n})\nabla u_{\varepsilon}^{\rm
app}\big{)}=F\quad\text{in }B_{3/2},\quad\text{and}\quad u_{\varepsilon}^{\rm
app}=u_{\varepsilon}\quad\text{on }\partial B_{3/2}.$ (3.28)

Combining (3.27), (3.28) and the first inequality in (3.26), we can apply the
energy estimate to the equation of $u_{\varepsilon}-u_{\varepsilon}^{\rm app}$

$\left\\{\begin{aligned} &-\text{\rm div}\big{(}\mathcal{A}^{\rm
app}_{\varepsilon}(x,x/\varepsilon_{n})\nabla(u_{\varepsilon}^{\rm
app}-u_{\varepsilon})\big{)}=\text{\rm div}\big{(}(\mathcal{A}^{\rm
app}_{\varepsilon}(x,x/\varepsilon_{n})-\mathcal{A}_{\varepsilon}(x,x/\varepsilon_{n}))\nabla
u_{\varepsilon}\big{)}\quad\text{in }B_{3/2},\\\ &u_{\varepsilon}^{\rm
app}-u_{\varepsilon}=0\quad\text{on }\partial B_{3/2},\end{aligned}\right.$
(3.29)

to obtain

$\displaystyle\int_{B_{3/2}}|\nabla(u_{\varepsilon}-u_{\varepsilon}^{\rm
app})|^{2}$ $\displaystyle\leq
CL^{2}\sum_{k=1}^{n-1}\varepsilon_{k}^{-2\tau}\varepsilon_{n}^{2\tau}\int_{B_{3/2}}|\nabla
u_{\varepsilon}|^{2}.$ (3.30)

Next, we apply Theorem 2.1 to the equation (3.28). Let $\chi^{\rm
app}_{\varepsilon}(x,y)$ be the corrector given by (2.2) with $A(x,y)$
replaced by $\mathcal{A}^{\rm app}_{\varepsilon}(x,y)$, and
$\widehat{\mathcal{A}}^{\rm app}_{\varepsilon}(x)$ the corresponding effective
coefficient matrix given by (2.1). Let $\widehat{u}_{\varepsilon}^{\rm app}$
be the solution to

$\displaystyle-\text{\rm div}\big{(}\widehat{\mathcal{A}}^{\rm
app}_{\varepsilon}(x)\nabla\widehat{u}_{\varepsilon}^{\rm
app}\big{)}=F\quad\text{in
}B_{5/4},\quad\text{and}\quad\widehat{u}_{\varepsilon}^{\rm
app}=u_{\varepsilon}^{\rm app}\quad\text{on }\partial B_{5/4}.$ (3.31)

Thanks to Theorem 2.1,

$\displaystyle\int_{B_{5/4}}|\nabla u_{\varepsilon}^{\rm
app}-\nabla\widehat{u}_{\varepsilon}^{\rm
app}-S_{\varepsilon_{n}}((\nabla_{y}\chi^{\rm
app}_{\varepsilon})^{\varepsilon_{n}}\nabla u_{\varepsilon}^{\rm app})|^{2}$
(3.32) $\displaystyle\leq
C\varepsilon_{n}^{2}(1+\|\nabla_{x}\mathcal{A}_{\varepsilon}^{\rm
app}\|_{\infty})^{2}\int_{B_{5/4}}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{2}$
$\displaystyle\qquad+C\varepsilon_{n}^{2}\int_{B_{(5/4)-3\varepsilon_{n}}}|\nabla^{2}\widehat{u}_{\varepsilon}^{\rm
app}|^{2}+C\int_{B_{5/4}\setminus
B_{(5/4)-4\varepsilon_{n}}}|\nabla\widehat{u}_{\varepsilon}^{\rm app}|^{2}.$

By the Hölder’s inequality, the last term on the right-hand side of (3.32) is
bounded by

$\displaystyle\int_{B_{5/4}\setminus
B_{(5/4)-4\varepsilon_{n}}}|\nabla\widehat{u}_{\varepsilon}^{\rm app}|^{2}\leq
C\varepsilon_{n}^{1-\frac{2}{q}}\bigg{(}\int_{B_{5/4}}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{q}\bigg{)}^{2/q}\quad\text{ for }q>2.$ (3.33)

On the other hand, by the interior $H^{2}$ estimate for the equation (3.28),
for any $B(z,2\rho)\subset B_{3/2}$,

$\int_{B(z,\rho)}|\nabla^{2}\widehat{u}_{\varepsilon}^{\rm app}|^{2}\leq
C\int_{B(z,2\rho)}|F|^{2}+C\big{(}\|\nabla_{x}\mathcal{A}^{\rm
app}_{\varepsilon}\|_{\infty}^{2}+\rho^{-2}\big{)}\int_{B(z,2\rho)}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{2}.$

Then it follows by a covering argument that

$\displaystyle\int_{B_{(5/4)-3\varepsilon_{n}}}|\nabla^{2}\widehat{u}_{\varepsilon}^{\rm
app}|^{2}\,dx$ (3.34) $\displaystyle\leq
C\int_{B_{5/4}}|F|^{2}+C\|\nabla_{x}\mathcal{A}^{\rm
app}_{\varepsilon}\|^{2}_{\infty}\int_{B_{5/4}}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{2}+\varepsilon_{n}^{-\frac{2}{q}-1}\bigg{(}\int_{B_{5/4}}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{q}\bigg{)}^{2/q},$

where the Hölder’s inequality has been used for the last integral. By Meyers’
estimate for the elliptic equations (3.27), (3.28) and (3.31), we have for
some $q>2$,

$\displaystyle\begin{split}&\bigg{(}\int_{B_{3/2}}|\nabla
u_{\varepsilon}|^{q}\bigg{)}^{2/q}\leq
C\int_{B_{2}}|F|^{2}+C\int_{B_{2}}|\nabla u_{\varepsilon}|^{2},\\\
&\bigg{(}\int_{B_{3/2}}|\nabla u_{\varepsilon}^{\rm
app}|^{q}\bigg{)}^{2/q}\leq
C\int_{B_{3/2}}|F|^{2}+C\bigg{(}\int_{B_{3/2}}|\nabla
u_{\varepsilon}|^{q}\bigg{)}^{2/q},\\\
&\bigg{(}\int_{B_{5/4}}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{q}\bigg{)}^{2/q}\leq
C\int_{B_{5/4}}|F|^{2}+C\bigg{(}\int_{B_{5/4}}|\nabla u_{\varepsilon}^{\rm
app}|^{q}\bigg{)}^{2/q},\end{split}$ (3.35)

which implies that

$\displaystyle\bigg{(}\int_{B_{5/4}}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{q}\bigg{)}^{2/q}\leq C\int_{B_{2}}|F|^{2}+C\int_{B_{2}}|\nabla
u_{\varepsilon}|^{2}.$ (3.36)

By taking (3.33) and (3.34) into (3.32), and using (3.36) we obtain that

$\displaystyle\int_{B_{5/4}}|\nabla u_{\varepsilon}^{\rm
app}-\nabla\widehat{u}_{\varepsilon}^{\rm
app}-S_{\varepsilon_{n}}((\nabla_{y}\chi_{\varepsilon}^{\rm
app})^{\varepsilon_{n}}\nabla\widehat{u}_{\varepsilon}^{\rm app})|^{2}$ (3.37)
$\displaystyle\leq
C\big{\\{}\varepsilon_{n}^{1-\frac{2}{q}}+L^{2}\sum_{k=1}^{n-1}\varepsilon_{k}^{-2\tau}\varepsilon_{n}^{2\tau}\big{\\}}\bigg{\\{}\int_{B_{2}}|\nabla
u_{\varepsilon}|^{2}+\int_{B_{2}}|F|^{2}\bigg{\\}}.$

Now, let $\chi_{\varepsilon}(x,y)$ be the corrector given by (2.2) with
$A(x,y)$ replaced by $\mathcal{A}_{\varepsilon}(x,y)$, and
$\widehat{\mathcal{A}}_{\varepsilon}(x)$ the effective matrix of
$\mathcal{A}_{\varepsilon}$ given by (2.1) (recall that
$\chi_{\varepsilon}(x,y)$ and $\widehat{\mathcal{A}}_{\varepsilon}(x)$ take
the forms of (2.9) and (2.11)). Note that
$\chi_{\varepsilon}-\chi_{\varepsilon}^{\rm
app}=(\chi_{\varepsilon,j}-\chi_{\varepsilon,j}^{\rm app})$ satisfies

$\displaystyle-\text{\rm
div}_{y}\big{(}\mathcal{A}_{\varepsilon}(x,y)(\nabla_{y}\chi_{\varepsilon,j}-\nabla_{y}\chi_{\varepsilon,j}^{\rm
app})\big{)}$ $\displaystyle=\text{\rm
div}_{y}\big{(}(\mathcal{A}_{\varepsilon}(x,y)-\mathcal{A}_{\varepsilon}^{\rm
app}(x,y))\nabla_{y}\chi^{app}_{\varepsilon,j}\big{)}+\text{\rm
div}_{y}\big{(}(\mathcal{A}_{\varepsilon}(x,y)-\mathcal{A}_{\varepsilon}^{\rm
app}(x,y))e_{j}\big{)}\quad\text{ in }\mathbb{T}^{d}.$

Thus, the standard energy estimate and the first inequality in (3.26) imply
that

$\displaystyle\sup_{x\in\mathbb{R}^{d}}\|\nabla_{y}\chi_{\varepsilon,j}(x,\cdot)-\nabla_{y}\chi^{\rm
app}_{\varepsilon,j}(x,\cdot)\|_{L^{2}(\mathbb{T}^{d})}\leq
CL\sum_{k=1}^{n-1}\varepsilon_{k}^{-\tau}\varepsilon_{n}^{\tau},$ (3.38)

where $C$ depends only on $d,\Lambda$ and $\tau$. This together with the
definitions of $\widehat{\mathcal{A}}_{\varepsilon}^{\rm app}$ and
$\widehat{\mathcal{A}}_{\varepsilon}$ gives

$\|\widehat{\mathcal{A}}_{\varepsilon}^{\rm
app}-\widehat{\mathcal{A}}_{\varepsilon}\|_{\infty}\leq
CL\sum_{k=1}^{n-1}\varepsilon_{k}^{-\tau}\varepsilon_{n}^{\tau}.$ (3.39)

Let $\widehat{u}_{\varepsilon}$ be the weak solution to

$\displaystyle-\text{\rm
div}\big{(}\widehat{\mathcal{A}}_{\varepsilon}(x)\nabla\widehat{u}_{\varepsilon}\big{)}=F\quad\text{
in }B_{5/4},\quad\text{ and }\quad\widehat{u}_{\varepsilon}=\widehat{u}^{\rm
app}_{\varepsilon}\quad\text{ on }\partial B_{5/4}.$ (3.40)

Similar to (3.30), we apply the standard energy estimate to the equation for
$\widehat{u}_{\varepsilon}^{\rm app}-\widehat{u}_{\varepsilon}$,

$\left\\{\begin{aligned} &-\text{\rm
div}\big{(}\widehat{\mathcal{A}}_{\varepsilon}(x,x/\varepsilon_{n})\nabla(\widehat{u}_{\varepsilon}-\widehat{u}_{\varepsilon}^{\rm
app})\big{)}=\text{\rm
div}\big{(}(\widehat{\mathcal{A}}_{\varepsilon}(x,x/\varepsilon_{n})-\widehat{\mathcal{A}}^{\rm
app}_{\varepsilon}(x,x/\varepsilon_{n}))\nabla\widehat{u}_{\varepsilon}^{\rm
app}\big{)}\quad\text{in }B_{5/4},\\\
&\widehat{u}_{\varepsilon}-\widehat{u}_{\varepsilon}^{\rm app}=0\quad\text{on
}\partial B_{5/4},\end{aligned}\right.$ (3.41)

and use (3.39) to obtain

$\displaystyle\begin{split}\int_{B_{5/4}}|\nabla(\widehat{u}_{\varepsilon}-\widehat{u}_{\varepsilon}^{\rm
app})|^{2}&\leq
CL^{2}\sum_{k=1}^{n-1}\varepsilon_{k}^{-2\tau}\varepsilon_{n}^{2\tau}\int_{B_{5/4}}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{2}\\\ &\leq
CL^{2}\sum_{k=1}^{n-1}\varepsilon_{k}^{-2\tau}\varepsilon_{n}^{2\tau}\bigg{\\{}\int_{B_{2}}|\nabla
u_{\varepsilon}|^{2}+\int_{B_{2}}|F|^{2}\bigg{\\}},\end{split}$ (3.42)

where the energy estimate for $\widehat{u}_{\varepsilon}^{\rm app}$ (or
(3.36)) is also used in the last inequality.

Finally, combining (3.30), (3.38) and (3.42), we may replace the approximate
solutions $u_{\varepsilon}^{\rm app},\widehat{u}_{\varepsilon}^{\rm
app},\chi_{\varepsilon}^{\rm app}$ on the left-hand side of (3.37) by
$u_{\varepsilon},\widehat{u}_{\varepsilon},\chi_{\varepsilon}$, respectively.
In particular, by the property of the smoothing operator $S_{\varepsilon_{n}}$
(see e.g., [She18, NSX20]), we have

$\displaystyle\begin{split}&\int_{B_{1}}|S_{\varepsilon_{n}}\big{(}(\nabla_{y}\chi_{\varepsilon})^{\varepsilon_{n}}\nabla\widehat{u}_{\varepsilon}\big{)}-S_{\varepsilon_{n}}\big{(}(\nabla_{y}\chi_{\varepsilon}^{\rm
app})^{\varepsilon_{n}}\nabla\widehat{u}_{\varepsilon}^{\rm
app}\big{)}|^{2}\\\ &\leq
2\int_{B_{1}}\big{\\{}|S_{\varepsilon_{n}}\big{(}(\nabla_{y}\chi_{\varepsilon})^{\varepsilon_{n}}(\nabla\widehat{u}_{\varepsilon}-\nabla\widehat{u}_{\varepsilon}^{\rm
app})\big{)}|^{2}+|S_{\varepsilon_{n}}\big{(}((\nabla_{y}\chi_{\varepsilon}^{\rm
app})^{\varepsilon_{n}}-(\nabla_{y}\chi_{\varepsilon})^{\varepsilon_{n}})\nabla\widehat{u}_{\varepsilon}^{\rm
app}\big{)}|^{2}\big{\\}}\\\ &\leq
C\int_{B_{1+\varepsilon_{n}}}|\nabla\widehat{u}_{\varepsilon}-\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{2}+CL^{2}\sum_{k=1}^{n-1}\varepsilon_{k}^{-2\tau}\varepsilon_{n}^{2\tau}\int_{B_{1+\varepsilon_{n}}}|\nabla\widehat{u}_{\varepsilon}^{\rm
app}|^{2}\\\ &\leq
CL^{2}\sum_{k=1}^{n-1}\varepsilon_{k}^{-2\tau}\varepsilon_{n}^{2\tau}\bigg{\\{}\int_{B_{2}}|\nabla
u_{\varepsilon}|^{2}+\int_{B_{2}}|F|^{2}\bigg{\\}}.\end{split}$ (3.43)

It follows that

$\displaystyle\int_{B_{5/4}}|\nabla
u_{\varepsilon}-\nabla\widehat{u}_{\varepsilon}-S_{\varepsilon_{n}}((\nabla_{y}\chi)^{\varepsilon}\nabla\widehat{u}_{\varepsilon})|^{2}$
(3.44) $\displaystyle\leq
C\big{\\{}\varepsilon_{n}^{1-\frac{2}{q}}+L^{2}\sum_{k=1}^{n-1}\varepsilon_{k}^{-2\tau}\varepsilon_{n}^{2\tau}\big{\\}}\bigg{\\{}\int_{B_{2}}|\nabla
u_{\varepsilon}|^{2}+\int_{B_{2}}|F|^{2}\bigg{\\}}.$

Recall that $\varepsilon_{n}$ is $Q$-separated from
$\varepsilon_{k},k=1,\cdots,n-1$, i.e., $\varepsilon_{k}/\varepsilon_{n}\geq
Q$ for $k=1,\cdots,n-1.$ In view of (2.9) and (2.11), the desired estimate
(3.16) follows immediately from (3.44), while the estimate (3.15) follows
directly from (3.42). The proof is complete. ∎

By similar arguments as above, it is not difficult to prove the boundary
version of the uniform approximation theorem.

######  Theorem 3.4.

Let $\Omega$ be a bounded $C^{1}$ domain. Suppose that $A$ satisfies the
assumptions (1.2), (1.3) and (3.9). Let
$x_{0}\in\partial\Omega,B_{r}=B(x_{0},r)$ with $0<r<r_{0}=r_{0}(\Omega)$ and
$u_{\varepsilon}$ be a weak solution to

$\displaystyle-\text{\rm
div}\big{(}A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon}\big{)}=F\,\text{ in }B_{2r}\cap\Omega,\quad
u_{\varepsilon}=0\,\text{ on }B_{2r}\cap\partial\Omega.$ (3.45)

Then for any $Q>1$, there exist a coefficient matrix
$A^{\flat}=A^{\flat}(y_{1},\cdots,y_{n-1})$, scales
$\varepsilon_{k}^{\prime}\in(0,\infty)$ with $k=1,\cdots,n-1$, and a weak
solution $u^{\flat}_{\varepsilon^{\prime}}$ to

$-\text{\rm
div}\big{(}A^{\flat}(x/\varepsilon^{\prime}_{1},x/\varepsilon^{\prime}_{2},\cdots,x/\varepsilon^{\prime}_{n-1})\nabla
u^{\flat}_{\varepsilon^{\prime}}\big{)}=F\quad\text{in }B_{r},\quad
u^{\flat}_{\varepsilon^{\prime}}=0\text{ on }B_{r}\cap\partial\Omega,$

such that

$\|\nabla u^{\flat}_{\varepsilon^{\prime}}\|_{L^{2}(B_{r}\cap\Omega)}\leq
C\big{\\{}\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2r}\cap\Omega)}+r\|F\|_{L^{2}(B_{2r}\cap\Omega)}\big{\\}},$
(3.46)

and

$\|\nabla u_{\varepsilon}-\nabla
u^{\flat}_{\varepsilon^{\prime}}-U_{\varepsilon^{\prime}}\|_{L^{2}(B_{r}\cap\Omega)}\leq
C\bigg{\\{}\bigg{(}\frac{Q^{n-1}\varepsilon_{n}}{r}\bigg{)}^{\sigma}+Q^{-\tau}\bigg{\\}}\big{\\{}\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{2r}\cap\Omega)}+r\|F\|_{L^{2}(B_{2r}\cap\Omega)}\big{\\}}$
(3.47)

for some $\sigma>0,$ where $U_{\varepsilon^{\prime}}$ is given by

$U_{\varepsilon^{\prime}}(x)=(\varepsilon_{n}^{\prime})^{-d}\int_{\mathbb{R}^{d}}\varphi\bigg{(}\frac{x-z}{\varepsilon_{n}^{\prime}}\bigg{)}(\nabla_{y}\mathcal{X}^{\sharp})\bigg{(}\frac{z}{\varepsilon_{1}^{\prime}},\cdots,\frac{z}{\varepsilon^{\prime}_{n-1}},\frac{x}{\varepsilon_{n}^{\prime}}\bigg{)}\nabla
u^{\flat}_{\varepsilon^{\prime}}(z)dz,$ (3.48)

and $\mathcal{X}^{\sharp}$ is the corrector given by (2.2) with $A$ replaced
by $A^{\sharp}$ (we have extended $u^{\flat}_{\varepsilon^{\prime}}(z)$ by
zero across the boundary). Moreover, the constant $C$ depends only on
$d,n,\Lambda$ and $(\tau,L)$ in (3.9), and the assumptions (1.2), (1.3) and
(1.4) are preserved for $A^{\flat}$.

##  4\. The uniform Calderón-Zygmund estimates

In this section we prove the uniform Calderón-Zygmund estimate for the
equation (1.1). Let us first consider the interior estimate.

######  Theorem 4.1.

Assume $A$ satisfies the assumptions (1.2), (1.3) and (1.4). Let
$u_{\varepsilon}$ be a weak solution to

$\displaystyle-\text{\rm
div}\big{(}A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon}\big{)}=\text{\rm div}f\quad\text{in }5B_{0},$ (4.1)

with $f\in L^{p}(5B_{0})^{d}$ for some $2<p<\infty$, where
$B_{0}=B(x_{0},r_{0})$ for some $x_{0}\in\mathbb{R}^{d},r_{0}>0$. Then $\nabla
u_{\varepsilon}\in L^{p}(B_{0})^{d}$ and

$\displaystyle\bigg{(}\fint_{B_{0}}|\nabla
u_{\varepsilon}|^{p}\bigg{)}^{1/p}\leq
C\bigg{\\{}\bigg{(}\fint_{5B_{0}}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+\bigg{(}\fint_{5B_{0}}|f|^{p}\bigg{)}^{1/p}\bigg{\\}},$
(4.2)

where $C$ depends only on $d,\Lambda,p,n,$ and $(\tau,L)$ in (1.4).

The proof of Theorem 4.1 relies on a real-variable argument involving a
double-averaging estimate. We define the averaging operator as

$M_{t}[f](x):=\bigg{(}\fint_{B_{t}(x)}|f|^{2}\bigg{)}^{1/2}.$ (4.3)

We need the averaging operator in the real-variable argument for two reasons.
The first reason is that the approximation in (3.12) is meaningful only if
$r\gg Q^{n-1}\varepsilon_{n}$ for sufficiently large $Q\gg 1$. This forces us
to take the large-scale averaging estimate above the scale
$t=CQ^{n-1}\varepsilon_{n}$ for sufficiently large $C$ and ignore the possible
irregularity below that scale. The second reason to introduce another
averaging is due to the special structure of the corrector
$\mathcal{X}^{\sharp}$ in $U_{\varepsilon^{\prime}}$ given by (3.13). Recall
that $\mathcal{X}^{\sharp}=\mathcal{X}^{\sharp}(y_{1},y_{2},\cdots,y_{n})$ is
1-periodic in each $y_{i}$. But its gradient in $y_{n}$ does not have an
$L^{p}$ estimate uniformly in $Q$ before we prove it in case of $n$
oscillating scales (otherwise we will run into a circular reasoning), which
causes a big problem in the real-variable argument. To resolve this issue, we
take the averaging for another time at scale
$t=\varepsilon^{\prime}_{n}=q\varepsilon_{n}$, a scale comparable to the size
of the periodic cell (depending on $Q$ as $1<q\leq Q^{n-1}$). This allows us
to use only the periodic structure of $\mathcal{X}^{\sharp}$ and the $L^{2}$
energy estimate (independent of $Q$). Overall, the principle is that we have
to make sure all the constants, except for $\eta$, in the real-variable
argument (Theorem 4.2 below) are independent of $Q$.

######  Theorem 4.2.

[She23, Theorem 2.1 and Remark 2.4] Let $p_{0}>2$, $\mathcal{F}\in
L^{2}(4B_{0})$ and $G\in L^{p}(4B_{0})$ for some $2<p<p_{0}$, where $B_{0}$ is
a ball in $\mathbb{R}^{d}$. Let $0\leq t<c_{0}\text{diam}(B_{0})$ be a fixed
number. Suppose that for each ball $B\subset 2B_{0}$ with the property that
$t<\text{diam}(B)\leq c_{0}\text{diam}(B_{0})$, there exists two measurable
functions $\mathcal{F}_{B}$ and $\mathcal{R}_{B}$ on $2B$ such that
$|\mathcal{F}|\leq|\mathcal{F}_{B}|+|\mathcal{R}_{B}|$ on $2B$, and

$\displaystyle\bigg{(}\fint_{2B}|\mathcal{R}_{B}|^{p_{0}}\bigg{)}^{1/p_{0}}\leq
C_{1}\bigg{\\{}\bigg{(}\fint_{4B}|\mathcal{F}|^{2}\bigg{)}^{1/2}+\sup_{B\subset
B^{\prime}\subset
4B_{0}}\bigg{(}\fint_{B^{\prime}}|G|^{2}\bigg{)}^{1/2}\bigg{\\}},$ (4.4)
$\displaystyle\bigg{(}\fint_{2B}|\mathcal{F}_{B}|^{2}\bigg{)}^{1/2}\leq
C_{2}\sup_{B\subset B^{\prime}\subset
4B_{0}}\bigg{(}\fint_{B^{\prime}}|G|^{2}\bigg{)}^{1/2}+\eta\bigg{(}\fint_{4B}|\mathcal{F}|^{2}\bigg{)}^{1/2},$
(4.5)

where $C_{1},C_{2}>0,0<c_{0}<1$, and $\eta\geq 0$. Then there exists
$\eta_{0}>0$, depending only on $C_{1},C_{2},c_{0},p,p_{0},$ with the property
that if $0\leq\eta<\eta_{0}$, then

$\displaystyle\bigg{(}\fint_{B_{0}}M_{t}[\mathcal{F}]^{p}\bigg{)}^{1/p}\leq
C\bigg{\\{}\bigg{(}\fint_{4B_{0}}|\mathcal{F}|^{2}\bigg{)}^{1/2}+\bigg{(}\fint_{4B_{0}}|G|^{p}\bigg{)}^{1/p}\bigg{\\}},$
(4.6)

where $C$ depends only on $d,C_{1},C_{2},c_{0},p$ and $p_{0}$. If $t=0$, then
(4.6) is replaced by

$\displaystyle\bigg{(}\fint_{B_{0}}|\mathcal{F}|^{p}\bigg{)}^{1/p}\leq
C\bigg{\\{}\bigg{(}\fint_{4B_{0}}|\mathcal{F}|^{2}\bigg{)}^{1/2}+\bigg{(}\fint_{4B_{0}}|G|^{p}\bigg{)}^{1/p}\bigg{\\}}.$
(4.7)

###### Proof of Theorem 4.1.

Let us first consider the case $f=0$. We use an inductive argument on the
number of scales to prove the theorem. Note that the result is well-known for
$n=1$ (see e.g., [She18]). Assume the theorem is true if the number of scales
is strictly less than $n$, which means that for any
$\infty>\varepsilon_{1}\geq\varepsilon_{2}\geq\cdots\geq\varepsilon_{n-1}>0$,
the solution $v_{\varepsilon}$ to

$-\text{\rm div}\big{(}A(x/\varepsilon_{1},\cdots,x/\varepsilon_{n-1})\nabla
v_{\varepsilon}\big{)}=0\quad\text{in }B_{2r}$

satisfies the uniform interior $W^{1,p}$ estimate

$\displaystyle\bigg{(}\fint_{B_{3r/2}}|\nabla
v_{\varepsilon}|^{p}\bigg{)}^{1/p}\leq C_{p}\bigg{(}\fint_{B_{2r}}|\nabla
v_{\varepsilon}|^{2}\bigg{)}^{1/2}$ (4.8)

for any $0<r<\infty$ and any $2<p<\infty$.

We next prove the theorem for $n$ scales by using Theorem 4.2 with
$\mathcal{F}:=M_{\varepsilon_{n}^{\prime}}[\nabla u_{\varepsilon}]$ with
$\varepsilon_{n}^{\prime}=q\varepsilon_{n}$ given in (3.18). Fix any
$p\in(2,\infty)$ and $p_{1}\in(2,p)$.

Let $Q>1$ to be determined later, and
$\varepsilon_{n}^{\prime}=q\varepsilon_{n}\leq
Q^{n-1}\varepsilon_{n}<r<c_{0}r_{0}$ for some fixed $c_{0}\in(0,1)$. For each
ball $B_{r}=B_{r}(x)$ such that $B_{r}\subset 2B_{0}=B(x_{0},2r_{0})$, we need
to construct $\mathcal{F}_{B}$ and $\mathcal{R}_{B}$, which satisfy (4.4) and
(4.5), respectively. Since $u_{\varepsilon}$ is a solution to $-\text{\rm
div}\big{(}A(x/\varepsilon_{1},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon}\big{)}=0$ in $B_{4r}\subset 4B_{0}$, by Theorem 3.2, there
exist a 1-periodic matrix $A^{\flat}=A^{\flat}(y_{1},y_{2},\cdots,y_{n-1})$,
scales
$\varepsilon^{\prime}=(\varepsilon_{1}^{\prime},\cdots,\varepsilon_{n-1}^{\prime})$,
and a solution $u_{\varepsilon^{\prime}}^{\flat}(x)$ to

$\displaystyle-\text{\rm
div}\big{(}A^{\flat}(x/\varepsilon^{\prime}_{1},\cdots,x/\varepsilon^{\prime}_{n-1})\nabla
u_{\varepsilon^{\prime}}^{\flat}\big{)}=0\quad\text{in }B_{2r},$ (4.9)

such that

$\displaystyle\|\nabla u_{\varepsilon^{\prime}}^{\flat}\|_{L^{2}(B_{2r})}\leq
C\|\nabla u_{\varepsilon}\|_{L^{2}(B_{4r})},$ (4.10)

and

$\displaystyle\begin{split}&\|\nabla u_{\varepsilon}-\nabla
u_{\varepsilon^{\prime}}^{\flat}-U_{\varepsilon^{\prime}}\|_{L^{2}(B_{2r})}\leq
C\bigg{\\{}Q^{-\tau}+\bigg{(}\frac{Q^{n-1}\varepsilon_{n}}{r}\bigg{)}^{\sigma}\bigg{\\}}\|\nabla
u_{\varepsilon}\|_{L^{2}(B_{4r})},\end{split}$ (4.11)

where $U_{\varepsilon^{\prime}}$ is given by (3.13). Let

$\displaystyle\mathcal{F}_{B}(x)=M_{\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon}-\nabla
u_{\varepsilon^{\prime}}^{\flat}-U_{\varepsilon^{\prime}}](x)\,\,\text{ and
}\,\,\mathcal{R}_{B}(x)=M_{\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon^{\prime}}^{\flat}-U_{\varepsilon^{\prime}}](x).$

By the triangle inequality, we have
$|\mathcal{F}|\leq|\mathcal{F}_{B}|+|\mathcal{R}_{B}|.$

We recall a basic property of the averaging operator $M_{t}$. If $F\in
L^{2}_{\rm loc}(\mathbb{R}^{d})$, then for any $r\geq t>0$, we have

$\fint_{B_{r}}M_{t}[F]^{2}\leq C\fint_{B_{r+t}}|F|^{2}\leq
C\fint_{B_{2r}}|F|^{2}\leq C\fint_{B_{2r}}M_{t}[F]^{2}.$ (4.12)

This can be shown easily by the Fubini’s theorem.

Consequently, by the assumption $r>\varepsilon_{n}^{\prime}$, we have

$\fint_{B_{r}}|\mathcal{F}_{B}|^{2}\leq C\fint_{B_{2r}}|\nabla
u_{\varepsilon}-\nabla
u_{\varepsilon^{\prime}}^{\flat}-U_{\varepsilon^{\prime}}|^{2},$ (4.13)

and

$\fint_{B_{4r}}|\nabla u_{\varepsilon}|^{2}\leq
C\fint_{B_{4r}}|\mathcal{F}|^{2}.$ (4.14)

Combining the last two inequalities with (4.11), we have

$\displaystyle\fint_{B_{r}}|\mathcal{F}_{B}|^{2}\leq
C\bigg{\\{}Q^{-\tau}+\bigg{(}\frac{Q^{n-1}\varepsilon_{n}}{r}\bigg{)}^{\sigma}\bigg{\\}}^{2}\fint_{B_{4r}}|\mathcal{F}|^{2}.$
(4.15)

We now choose $Q=Q_{0}>1$ large enough such that $CQ_{0}^{-\tau}<\eta_{0}/2$,
and then take $r>k_{0}Q_{0}^{n-1}\varepsilon_{n}$ for some $k_{0}$ large
enough such that
$CQ_{0}^{(n-1)\sigma}(\varepsilon_{n}/r)^{\sigma}<\eta_{0}/2,$ where
$\eta_{0}$ is given by Theorem 4.2 depending only on
$C_{1},C_{2},c_{0},p,p_{1}.$ It follows that $\mathcal{F}_{B}$ satisfies
condition (4.5) for $r>k_{0}Q_{0}^{n-1}\varepsilon_{n}$ with $C_{2}=0$. In the
following, we will verify that $\mathcal{R}_{B}$ satisfies (4.4) and the
constant $C_{1}$ is independent of $Q$.

Note that

$\displaystyle\begin{split}\fint_{B_{r}}|\mathcal{R}_{B}|^{p}&=\fint_{B_{r}}M_{\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon^{\prime}}^{\flat}-U_{\varepsilon^{\prime}}]^{p}\\\ &\leq
2^{p-1}\fint_{B_{r}}|M_{\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon^{\prime}}^{\flat}](z)|^{p}dz+2^{p-1}\fint_{B_{r}}|M_{\varepsilon_{n}^{\prime}}[U_{\varepsilon^{\prime}}](z)|^{p}dz.\end{split}$
(4.16)

Since $u_{\varepsilon^{\prime}}^{\flat}$ satisfies (4.9), and the coefficient
matrix $A^{\flat}$ involves at most $n-1$ scales and is periodic and Hölder
continuous $y_{i}=x/\varepsilon_{i}^{\prime}$ for $i=1,\cdots,n-1$. By the
inductive assumption, we know that $u_{\varepsilon^{\prime}}^{\flat}$
satisfies (4.8), which combined with (4.10) implies

$\displaystyle\bigg{(}\fint_{B_{3r/2}}|\nabla
u_{\varepsilon^{\prime}}^{\flat}|^{p}\bigg{)}^{1/p}\leq
C\bigg{(}\fint_{B_{4r}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{1/2},$

for any $k_{0}\varepsilon_{n}^{\prime}\leq k_{0}Q_{0}^{n-1}\varepsilon_{n}\leq
r\leq c_{0}r_{0}$. By the Hölder’s inequality and Fubini’s theorem,

$\displaystyle\begin{split}\fint_{B_{r}}|M_{\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon^{\prime}}^{\flat}](z)|^{p}dz&\leq\fint_{B_{r}}\fint_{B_{\varepsilon_{n}^{\prime}}(z)}|\nabla
u_{\varepsilon^{\prime}}^{\flat}(y)|^{p}dydz\\\ &\leq C\fint_{B_{3r/2}}|\nabla
u_{\varepsilon^{\prime}}^{\flat}(y)|^{p}dy\\\ &\leq
C\bigg{(}\fint_{B_{4r}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{p/2}\\\ &\leq
C\bigg{(}\fint_{B_{4r}}|\mathcal{F}|^{2}\bigg{)}^{p/2},\end{split}$ (4.17)

where we have used (4.12) in the last step.

To bound the second term on the right hand side of (4.16), we have to take
advantage of the periodic structure of $\mathcal{X}^{\sharp}$ and it is right
here that the averaging operator $M_{\varepsilon_{n}^{\prime}}$ plays a key
role. We observe that, for each $y\in B_{r}$,

$\displaystyle\begin{split}|M_{\varepsilon_{n}^{\prime}}[U_{\varepsilon^{\prime}}](y)|^{2}&=\fint_{B_{\varepsilon_{n}^{\prime}}(y)}\bigg{|}(\varepsilon_{n}^{\prime})^{-d}\int_{\mathbb{R}^{d}}\varphi\bigg{(}\frac{x-z}{\varepsilon_{n}^{\prime}}\bigg{)}(\nabla_{y}\mathcal{X}^{\sharp})\bigg{(}\frac{z}{\varepsilon_{1}^{\prime}},\cdots,\frac{z}{\varepsilon^{\prime}_{n-1}},\frac{x}{\varepsilon_{n}^{\prime}}\bigg{)}\nabla
u_{\varepsilon^{\prime}}^{\flat}(z)dz\bigg{|}^{2}dx\\\
&\leq\fint_{B_{\varepsilon_{n}^{\prime}}(y)}(\varepsilon_{n}^{\prime})^{-d}\int_{\mathbb{R}^{d}}\varphi\bigg{(}\frac{x-z}{\varepsilon_{n}^{\prime}}\bigg{)}\bigg{|}(\nabla_{y}\mathcal{X}^{\sharp})\bigg{(}\frac{z}{\varepsilon_{1}^{\prime}},\cdots,\frac{z}{\varepsilon^{\prime}_{n-1}},\frac{x}{\varepsilon_{n}^{\prime}}\bigg{)}\nabla
u_{\varepsilon^{\prime}}^{\flat}(z)\bigg{|}^{2}dzdx\\\
&\leq\sup_{z\in\mathbb{R}^{d}}\fint_{B_{1}}\bigg{|}(\nabla_{y}\mathcal{X}^{\sharp})\bigg{(}\frac{z}{\varepsilon_{1}^{\prime}},\cdots,\frac{z}{\varepsilon^{\prime}_{n-1}},y\bigg{)}\bigg{|}^{2}dy\fint_{B_{2\varepsilon_{n}^{\prime}}(y)}|\nabla
u_{\varepsilon^{\prime}}^{\flat}(z)|^{2}dz\\\ &\leq
C\fint_{B_{2\varepsilon_{n}^{\prime}}(y)}|\nabla
u_{\varepsilon^{\prime}}^{\flat}(z)|^{2}dz=C|M_{2\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon^{\prime}}^{\flat}](y)|^{2},\end{split}$

where we have used the $L^{2}$ energy estimate for $\mathcal{X}^{\sharp}$ (see
the equation (2.10)) and $C$ depends only on $d$ and $\Lambda$. This combined
with (4.17) implies that

$\displaystyle\begin{aligned}
\fint_{B_{r}}|M_{\varepsilon_{n}^{\prime}}[U_{\varepsilon^{\prime}}](z)|^{p}&\leq
C\fint_{B_{r}}|M_{2\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon^{\prime}}^{\flat}](y)|^{p}dy\\\ &\leq
C\fint_{B_{4r/3}}|M_{\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon^{\prime}}^{\flat}](y)|^{p}dy\leq
C\bigg{(}\fint_{B_{4r}}|\mathcal{F}|^{2}\bigg{)}^{p/2},\end{aligned}$ (4.18)

for any $k_{0}Q_{0}^{n-1}\varepsilon_{n}\leq r<c_{0}r_{0}$. We point out that
we cannot simply take the $L^{p}$ norm of $U_{\varepsilon^{\prime}}$ directly
since $\|\nabla\mathcal{X}^{\sharp}\|_{L^{p}(B_{1},dy_{n})}$ depends on $q$
and hence on $Q$, due to the loss of regularity of
$A^{\sharp}(y_{1},y_{2},\cdots,y_{n})$ in $y_{n}$ caused by reperiodization.

By taking (4.17) and (4.18) into (4.16), we find that $\mathcal{R}_{B}$
satisfies (4.4) for $k_{0}Q_{0}^{n-1}\varepsilon_{n}<r<c_{0}r_{0}$ (without
the $G$ part) with $C_{1}$ independent of $Q$. Thus the conditions of Theorem
4.2 are all satisfied. Thanks to (4.6), we obtain that

$\displaystyle\bigg{(}\fint_{B_{0}}M_{\varepsilon_{n}^{\prime\prime}}[M_{\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon}]]^{p_{1}}\bigg{)}^{1/{p_{1}}}\leq
C\bigg{(}\fint_{4B_{0}}|M_{\varepsilon_{n}^{\prime}}[\nabla
u_{\varepsilon}](x)|^{2}\bigg{)}^{1/2},$ (4.19)

where $\varepsilon_{n}^{\prime}=q\varepsilon_{n}$ and
$\varepsilon_{n}^{\prime\prime}=k_{0}Q_{0}^{n-1}\varepsilon_{n}$ and
$p_{1}\in(2,p)$. This is a double-averaging estimate since we have taken
average twice in the $L^{p}$ norm on the left-hand side. At this stage (after
using Theorem 4.2), it is safe to remove one averaging in (4.19) by (4.12). In
fact, since $\varepsilon_{n}^{\prime\prime}>\varepsilon_{n}^{\prime}$, (4.19)
implies

$\bigg{(}\fint_{B_{0}}M_{2\varepsilon_{n}^{\prime\prime}}[\nabla
u_{\varepsilon}]^{p_{1}}\bigg{)}^{1/{p_{1}}}\leq
C\bigg{(}\fint_{5B_{0}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{1/2}.$ (4.20)

Next, we claim that for each $y\in B_{0}$, we have

$|\nabla u_{\varepsilon}(y)|\leq CM_{2\varepsilon_{n}^{\prime\prime}}[\nabla
u_{\varepsilon}](y)=C\bigg{(}\fint_{B_{2\varepsilon_{n}^{\prime\prime}}(y)}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}.$ (4.21)

This pointwise Lipschitz estimate follows from a well-known blow-up argument
at small scales and the fact
$\varepsilon_{n}^{\prime\prime}=k_{0}Q_{0}^{n-1}\varepsilon_{n}$. Inserting
this into (4.20), we arrive at

$\bigg{(}\fint_{B_{0}}|\nabla u_{\varepsilon}|^{p_{1}}\bigg{)}^{1/p_{1}}\leq
C\bigg{(}\fint_{5B_{0}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{1/2}.$ (4.22)

Since $p_{1}\in(2,p)$ and $p\in(2,\infty)$ are arbitrary with $C$ depending on
$p,p_{1}$, we actually have shown the uniform interior $W^{1,p}$ estimate for
any $p\in(0,\infty)$ in the case of $f=0$.

Finally, we consider general case with $f\in L^{p}(5B_{0})^{d}$. Let
$u_{\varepsilon}$ be a solution to (4.1). For each ball $B=B_{r}$ such that
$4B\subset 2B_{0}$, let $u_{\varepsilon,1}\in H^{1}_{0}(4B)$ be the solution
to

$\displaystyle-\text{\rm
div}\big{(}A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon,1}\big{)}=\text{div }f\,\text{ in }4B,$ (4.23)

and $u_{\varepsilon,2}$ the solution to

$\displaystyle-\text{\rm
div}\big{(}A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon,2}\big{)}=0\,\text{ in }4B,\,\text{ and
}\,u_{\varepsilon,2}=u_{\varepsilon}\,\text{ on }\partial(4B).$ (4.24)

Thus, $u_{\varepsilon}=u_{\varepsilon,1}+u_{\varepsilon,2}$ in $4B$. The
energy estimate for (4.23) implies that

$\bigg{(}\fint_{4B}|\nabla u_{\varepsilon,1}|^{2}\bigg{)}^{1/2}\leq
C\bigg{(}\fint_{4B}|f|^{2}\bigg{)}^{1/2},$

while the uniform $W^{1,p}$ estimates for $u_{\varepsilon,2}$ proved above
implies that

$\displaystyle\bigg{(}\fint_{2B}|\nabla
u_{\varepsilon,2}|^{p_{0}}\bigg{)}^{1/p_{0}}$ $\displaystyle\leq
C\bigg{(}\fint_{4B}|\nabla u_{\varepsilon,2}|^{2}\bigg{)}^{1/2}$
$\displaystyle\leq C\bigg{(}\fint_{4B}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+C\bigg{(}\fint_{4B}|f|^{2}\bigg{)}^{1/2}$

for some $p_{0}\in(p,\infty)$. Let $\mathcal{F}=|\nabla u_{\varepsilon}|$,
$\mathcal{F}_{B}=|\nabla u_{\varepsilon,1}|$, and $\mathcal{R}_{B}=|\nabla
u_{\varepsilon,2}|$. We obtain (4.2) from Theorem 4.2 immediately (with
$t=0$). ∎

Theorem 4.1 provides the uniform interior $W^{1,p}$ estimate for the equation
(1.1). Based on Theorem 3.4 and a boundary version of Theorem 4.2 (See Theorem
4.1 and Remark 4.2 in [She23].), we can follow the same arguments to prove the
following uniform boundary $W^{1,p}$ estimate.

######  Theorem 4.3.

Let $\Omega$ be a bounded $C^{1}$ domain. There exists $r_{0}=r_{0}(\Omega)$
such that the following statement is true. Let $B=B(x,r)$ with
$x\in\partial\Omega$, $0<r<r_{0}$. Assume $A$ satisfies the assumption (1.2),
(1.3) and (1.4). Let $u_{\varepsilon}$ be a weak solution to

$\displaystyle-\text{\rm
div}\big{(}A(x/\varepsilon_{1},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon}\big{)}=\text{\rm div}f\,\text{ in }B_{2r}\cap\Omega,\quad
u_{\varepsilon}=0\,\text{ on }B_{2r}\cap\partial\Omega,$ (4.25)

with $f\in L^{p}(B_{2r}\cap\Omega)^{d}$ for some $2<p<\infty$. Then $\nabla
u_{\varepsilon}\in L^{p}(B_{r}\cap\Omega)^{d}$ and

$\displaystyle\bigg{(}\fint_{B_{r}\cap\Omega}|\nabla
u_{\varepsilon}|^{p}\bigg{)}^{1/p}\leq
C\bigg{\\{}\bigg{(}\fint_{B_{2r}\cap\Omega}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+\bigg{(}\fint_{B_{2r}\cap\Omega}|f|^{p}\bigg{)}^{1/p}\bigg{\\}},$
(4.26)

where $C$ depends only on $d,\Lambda,p,n,(\tau,L)$ in (1.4), and the $C^{1}$
character of $\Omega$.

We finally provide the proof of our main results.

###### Proof of Theorem 1.1.

Note that the case $p=2$ follows from standard energy estimates, while the
case $p>2$ is a direct consequence of Theorems 4.1 and 4.3 together with a
covering argument. Finally, the case $1<p<2$ follows from the case $p>2$ and a
standard duality argument. ∎

##  5\. Large-scale Lipschitz estimates

In this section, we investigate the large-scale Lipschitz estimates for (1.1).
We distinguish between the cases of two scales and more scales. We recall that
under the scale-separation condition: there exists a positive integer $N$ such
that

$\left(\frac{\varepsilon_{i+1}}{\varepsilon_{i}}\right)^{N}\leq\frac{\varepsilon_{i}}{\varepsilon_{i-1}}\quad\text{
for }1\leq i\leq n-1,$ (5.1)

the large-scale Lipschitz estimate for the multiscale elliptic operators has
been derived in [NSX20]. Indeed, let $u_{\varepsilon}$ be the weak solution to

$-\text{\rm div}(A(x,x/\varepsilon_{1},\cdots,x/\varepsilon_{n})\nabla
u_{\varepsilon})=F\quad\text{ in }B_{1}$

with $F\in L^{p}(B_{1})$ for some $p>d$. Suppose $A(x,y_{1},\cdots,y_{n})$ is
strongly elliptic, periodic in $y_{i}$ for $1\leq i\leq n$, and Hölder
continuous in $x,y_{1},\cdots,y_{n-1}$ (no smoothness is needed for $y_{n}$).
Then, under the condition (5.1), for any $0<\varepsilon_{n}\leq r<1$,

$\displaystyle\bigg{(}\fint_{B_{r}}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}\leq C\bigg{\\{}\bigg{(}\fint_{B_{1}}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+\bigg{(}\fint_{B_{1}}|F|^{p}\bigg{)}^{1/p}\bigg{\\}},$
(5.2)

where $C$ is independent of $\varepsilon$.

However, as we have stated before, without any scale separation condition the
full-scale uniform Lipschitz estimate for (1.1) seems difficult and still
remains open. The following theorems provide suboptimal results in this
direction using the idea of scale separation similar to Section 3.1.

######  Theorem 5.1.

Assume $A(y_{1},y_{2})$ satisfies (1.2), (1.3) and (3.9) with $n=2$. For
$B_{1}=B_{1}(0)$, let $u_{\varepsilon}$ be a weak solution of $-\text{\rm
div}(A_{\varepsilon}\nabla u_{\varepsilon})=F$ in $B_{1}$ with $F\in
L^{p}(B_{1})$ for some $p>d$. Then for any $\alpha>0$, and any
$\varepsilon_{2}^{1-\alpha}\leq r\leq 1$, we have

$\bigg{(}\fint_{B_{r}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{1/2}\leq
C_{\alpha}\bigg{\\{}\bigg{(}\fint_{B_{1}}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+\bigg{(}\fint_{B_{1}}|F|^{p}\bigg{)}^{1/p}\bigg{\\}},$
(5.3)

where $C_{\alpha}$ depends only on $d,\Lambda,p,\alpha,$ and $(\tau,L)$ in
(3.9).

###### Proof.

Applying the argument in Section 3.1 and by (3.7), for any $Q>1$ we can find a
1-periodic matrix $A^{\sharp}(y_{1},y_{2})$, Hölder continuous in $y_{1}$, and
write

$A\big{(}\frac{x}{\varepsilon_{1}},\frac{x}{\varepsilon_{2}}\big{)}=A^{\sharp}\big{(}\frac{\gamma_{1}x}{\varepsilon_{2}},\frac{x}{q\varepsilon_{2}}\big{)}=A^{\sharp}\big{(}\frac{x}{\varepsilon_{1}^{\prime}},\frac{x}{\varepsilon_{2}^{\prime}}\big{)},$
(5.4)

where $1\leq q\leq Q,\varepsilon_{1}^{\prime}=\varepsilon_{2}/\gamma_{1}$ and
$\varepsilon_{2}^{\prime}=q\varepsilon_{2}$. Let
$Q=\varepsilon_{2}^{-\alpha}$. Then
$\varepsilon_{2}\leq\varepsilon_{2}^{\prime}=q\varepsilon_{2}\leq\varepsilon_{2}^{1-\alpha}$
and $\varepsilon_{1}^{\prime}/\varepsilon_{2}^{\prime}\geq
Q=\varepsilon_{2}^{-\alpha}$. Thus $\varepsilon_{2}^{\prime}$ and
$\varepsilon_{1}^{\prime}$ are well-separated. Moreover, $u_{\varepsilon}$
satisfies the equation

$-\text{\rm
div}(A^{\sharp}(x/\varepsilon_{1}^{\prime},x/\varepsilon_{2}^{\prime})\nabla
u_{\varepsilon})=F\qquad\text{in }B_{1}.$ (5.5)

Thanks to (5.2), we have for $\varepsilon_{2}^{\prime}\leq r\leq 1$,

$\bigg{(}\fint_{B_{r}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{1/2}\leq
C_{\alpha}\bigg{\\{}\bigg{(}\fint_{B_{1}}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+\bigg{(}\fint_{B_{1}}|F|^{p}\bigg{)}^{1/p}\bigg{\\}}.$
(5.6)

This implies the desired estimate since
$\varepsilon_{2}^{\prime}\leq\varepsilon_{2}^{1-\alpha}$. ∎

######  Theorem 5.2.

Assume $A(y_{1},\cdots,y_{n})$ satisfies (1.2), (1.3) and (1.4) with $n\geq
3$. Let $u_{\varepsilon}$ be a weak solution of $-\text{\rm
div}(A_{\varepsilon}\nabla u_{\varepsilon})=F$ in $B_{1}$ with $F\in
L^{p}(B_{1})$ for some $p>d$. Then for any $\alpha\in(0,1/n]$ and
$\delta\in[\varepsilon_{n},1)$, there exists $1\leq q\leq\delta^{-n\alpha}$,
such that for any $q\delta\leq r\leq r_{1}:=\min\\{q\delta^{1-\alpha},1\\}$,
we have

$\bigg{(}\fint_{B_{r}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{1/2}\leq
C_{\alpha}\bigg{\\{}\bigg{(}\fint_{B_{r_{1}}}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+r_{1}^{1-d/p}\bigg{(}\fint_{B_{r_{1}}}|F|^{p}\bigg{)}^{1/p}\bigg{\\}},$
(5.7)

where $C_{\alpha}$ depends only on $d,n,\Lambda,p,\alpha$, and $(\tau,L)$ in
(1.4). Moreover, if $\delta=\varepsilon_{j}$ for some $1\leq j\leq n$, then
$\alpha$ can be taken from $(0,1/(n-1)]$ and $1\leq
q\leq\delta^{-(n-1)\alpha}$.

###### Proof.

Fix $\delta\in[\varepsilon_{n},1)$. Let $Q=\delta^{-\alpha}>1$ with
$\alpha\in(0,1/n]$. By the Dirichlet’s theorem, there exist
$q,p_{1},p_{2},\cdots,p_{n}$ such that $1<q\leq Q^{n}=\delta^{-n\alpha}$ and

$\big{|}\frac{\delta}{\varepsilon_{i}}-\frac{p_{i}}{q}\big{|}\leq\frac{1}{qQ},$
(5.8)

where $i=1,2,\cdots,n$. As in Section 3.1, we set

$\gamma_{i}:=\Big{|}\frac{\delta}{\varepsilon_{i}}-\frac{p_{i}}{q}\Big{|}\quad\text{and}\quad
s_{i}=\text{sgn}\Big{(}\frac{\delta}{\varepsilon_{i}}-\frac{p_{i}}{q}\Big{)},$

and write

$\frac{1}{\varepsilon_{i}}=\frac{\gamma_{i}s_{i}}{\delta}+\frac{p_{i}}{q}\frac{1}{\delta}.$

Define

$A_{1}^{\sharp}(y_{1},y_{2},\cdots,y_{n},y_{n+1})=A(s_{1}y_{1}+p_{1}y_{n+1},\cdots,s_{n}y_{n}+p_{n}y_{n+1}).$
(5.9)

As a consequence, we have

$A\big{(}\frac{x}{\varepsilon_{1}},\cdots,\frac{x}{\varepsilon_{n}}\big{)}=A_{1}^{\sharp}\big{(}\frac{\gamma_{1}x}{\delta},\cdots,\frac{\gamma_{n}x}{\delta},\frac{x}{q\delta}\big{)}.$
(5.10)

The last identity shows that we can rewrite original coefficient matrix $A$
with $n$ scales as a new 1-periodic matrix $A_{1}^{\sharp}$ with $n+1$ scales.
Moreover, the smallest scale $q\delta$ is at least $Q$-separated from the
remaining $n$ scales, i.e.,

$\frac{\delta/\gamma_{i}}{q\delta}\geq Q.$ (5.11)

Hence, by a blow-up argument and (5.2) we have

$\bigg{(}\fint_{B_{r}}|\nabla u_{\varepsilon}|^{2}\bigg{)}^{1/2}\leq
C_{\alpha}\bigg{\\{}\bigg{(}\fint_{B_{r_{1}}}|\nabla
u_{\varepsilon}|^{2}\bigg{)}^{1/2}+r_{1}^{1-d/p}\bigg{(}\fint_{B_{r_{1}}}|F|^{p}\bigg{)}^{1/p}\bigg{\\}}$
(5.12)

for any $r$ with $q\delta\leq r\leq r_{1}\leq\min\\{Qq\delta,1\\}$. This is
the desired estimate. To make sure $q\delta<1$, we require $q\delta\leq
Q^{n}\delta=\delta^{1-n\alpha}<1$ which gives $\alpha\leq 1/n$.

Finally, in the particular case $\delta=\varepsilon_{j}$ for some $1\leq j\leq
n$, $\delta/\varepsilon_{j}=1$ leads to a trivial approximation in (5.8). Thus
we will only need (5.8) for $i\neq j$ and $1\leq q\leq Q^{n-1}$. The rest of
the proof follows similarly. ∎

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The proof is a combination of proof of (1.) and (2.). Any fixed function $g$
can be written as $g_{+}-g_{-}$ where both $g_{+}$ and $g_{-}$ are non-
negative. In particular $g_{+}(x)=\max\\{g(x),0\\}$ and
$g_{-}(x)=\max\\{-g(x),0\\}$. As before, let
$\\{(l_{1},u_{1}),\dots,(l_{N_{1}},u_{N_{1}})\\}$ be a $\delta/2$ cover of
$\mathcal{F}$. Now for any $f\in\mathcal{F}$, there exists $1\leq j\leq N$
such that $l(x)\leq f(x)\leq u(x)$ for all $x$ almost surely. Therefore, we
have:

$l(x)g_{+}(x)\leq g_{+}(x)f(x)\leq u(x)g_{+}(x),\ \ \text{ and }\ \
l(x)g_{-}(x)\leq g_{-}(x)f(x)\leq u(x)g_{-}(x)\,,$

for all $x$. Therefore:

$l(x)g_{+}(x)-u(x)g_{-}(x)\leq g(x)f(x)\leq u(x)g_{+}(x)-l(x)g_{-}(x)\,.$

Furthermore:

$\displaystyle\left\|u(x)g_{+}(x)-l(x)g_{-}(x)-l(x)g_{+}(x)+u(x)g_{-}(x)\right\|$
$\displaystyle=\left\|(u(x)-l(x))g(x)\right\|$
$\displaystyle\leq\|g\|_{\infty}\frac{\delta}{\|g\|_{\infty}}\leq\delta\,.$

This completes the proof.

  4. 4.

Given $\delta>0$, let $[l_{1},u_{1}],\ldots,[l_{N},u_{N}]$ denote the
$\delta$-bracketing set of $\mathcal{F}$ where $N={\sf
exp}(\mathbb{H}_{\|\cdot\|}(\delta,\mathcal{F}))$. It suffices to show that
$[l_{1+},u_{1+}],\ldots,[l_{N+},u_{N+}]$ is a $\delta$-bracket of
$\mathcal{F}_{+}$. To see this, given an $f\in\mathcal{F}$, let $1\leq
i_{f}\leq N$ be such that $l_{i_{f}}\leq f\leq u_{i_{f}}$ and $\lVert
u_{i_{f}}-l_{i_{f}}\rVert_{\infty}\leq\delta$. Then $l_{i_{f}+}\leq f_{+}\leq
u_{i_{f}+}$ and $\lVert u_{i_{f}+}-l_{i_{f}+}\rVert_{\infty}\leq\lVert
u_{i_{f}}-l_{i_{f}}\rVert_{\infty}\leq\delta$. This completes the proof.

∎

###### Proof of Lemma 2.1.

Observe that the above function is well-defined as the function

$\beta_{q}-\frac{q}{n}\left(1+\sum_{k\geq 0\ :\
2^{-k}\sigma\geq\delta}\mathbb{H}_{\phi}(2^{-k}\sigma,\mathcal{F})\right)$

is $>0$ at $q=0$ and $\leq 0$ for $q=n$. This also implies
$\tilde{q}_{n,\beta}(\delta)\geq 1$. Further note that if
$\delta_{1}\leq\delta_{2}$, then
$\mathbb{H}_{\phi}({\delta_{1}},\mathcal{F})\geq\mathbb{H}_{\phi}({\delta_{2}},\mathcal{F})$.
Consequently,

$\displaystyle\beta(\tilde{q}_{n,\beta}(\delta_{2}))$
$\displaystyle\leq\frac{\tilde{q}_{n,\beta}(\delta_{2})}{n}\left(1+\sum_{k\geq
0\ :\
2^{-k}\sigma\geq\delta_{2}}\mathbb{H}_{\phi}(2^{-k}\sigma,\mathcal{F})\right)$
$\displaystyle\leq\frac{\tilde{q}_{n,\beta}(\delta_{1})}{n}\left(1+\sum_{k\geq
0\ :\
2^{-k}\sigma\geq\delta_{2}}\mathbb{H}_{\phi}(2^{-k}\sigma,\mathcal{F})\right).$

Therefore
$\tilde{q}_{n,\beta}(\delta_{1})\leq\tilde{q}_{n,\beta}(\delta_{2})$. As
$\Lambda_{\phi,\beta}(q)$ is non-decreasing in $q$, the conclusion follows. ∎

###### Proof of Lemma A.1.

Choose any arbitrary $t>\left\lVert h\right\rVert_{\phi,P}$ and sample $X\sim
P$. By using the inequality

$\phi^{c}(1)+\phi\left(\frac{h^{2}(x)}{t^{2}}\right)\geq\frac{h^{2}(x)}{t^{2}},$

we get:

$\displaystyle\mathbb{E}h^{2}(X)\leq t^{2}\left(\phi^{c}(1)+1\right).$

As $\phi\in\Phi$, we have $\phi^{c}(1)\in[0,\infty)$. Consequently, setting
$c_{\phi}:=\sqrt{1+\phi^{c}(1)}$ and taking infimum over $t\geq\left\lVert
h\right\rVert_{\phi,P}$ completes the proof. ∎

###### Proof.

We will use the covariance inequality from [124, Theorem 1.1] which implies
that

$|{\sf cov}(h(X_{i}),h(X_{j}))|\leq\int_{0}^{\beta(|i-j|)}Q_{h}^{2}(u)\,du.$

Note that $Q_{h}(U)$, $U\sim\mathrm{Unif}[0,1]$ has the same distribution as
$|h(X_{1})|$. Therefore, by Lemma A.1, the right hand side of the above
display is finite. Also note that by Lemma A.1, it follows that:

$\sum_{i=1}^{q}{\sf var}(h(X_{i}))\leq qc_{\phi}^{2}\left\lVert
h\right\rVert_{\phi,P}^{2}.$

Combining the two displays above, we get:

$\displaystyle{\sf var}\left(\sum_{i=1}^{q}h(X_{i})\right)$
$\displaystyle=\sum_{i}{\sf var}(h(X_{i}))+2\sum_{i<j}{\sf
cov}(h(X_{i}),h(X_{j}))$ $\displaystyle\leq qc_{\phi}^{2}\left\lVert
h\right\rVert_{\phi,P}^{2}+2\sum_{i<j}\int_{0}^{\beta(|i-j|)}Q_{h}^{2}(u)\,du$
$\displaystyle=qc_{\phi}^{2}\left\lVert
h\right\rVert_{\phi,P}^{2}+2\sum_{i<j}\int_{0}^{1}\mathds{1}\left(u\leq\beta(|i-j|)\right)Q_{h}^{2}(u)\,du$
$\displaystyle\leq qc_{\phi}^{2}\left\lVert
h\right\rVert_{\phi,P}^{2}+2q\sum_{k=1}^{q-1}\int_{0}^{1}\mathds{1}(u\leq\beta(k))Q_{h}^{2}(u)\,du.$
(D.1)

Note that by Markov’s inequality, given any $s>0$ and any $t>\left\lVert
h\right\rVert_{\phi,P}$, we have:

$\displaystyle P(|h(X)|\geq s)$
$\displaystyle\leq\frac{\mathbb{E}\phi\left(\frac{h^{2}(X)}{t^{2}}\right)}{\phi\left(\frac{s^{2}}{t^{2}}\right)}\leq\frac{1}{\phi\left(\frac{s^{2}}{t^{2}}\right)}.$

Next, given any $u>0$, we choose

$s=t\sqrt{\phi^{-1}\left(\frac{1}{u}\right)}$

in which case,

$P(|h(X)|\geq s)\leq u\implies Q_{h}(u)\leq
t\sqrt{\phi^{-1}\left(\frac{1}{u}\right)}.$

Now by taking infimum over $t>\left\lVert h\right\rVert_{\phi,P}$, we get:

$Q_{h}^{2}(u)\leq\left\lVert
h\right\rVert_{\phi,P}^{2}\phi^{-1}\left(\frac{1}{u}\right).$ (D.2)

The above observation implies

$\sum_{k=1}^{q-1}\int_{0}^{1}\mathds{1}(u\leq\beta(k))Q_{h}^{2}(u)\,du\leq\left\lVert
h\right\rVert_{\phi,P}^{2}\sum_{k=1}^{q-1}\int_{0}^{\beta(k)}\phi^{-1}\left(\frac{1}{u}\right)\,du\leq\left\lVert
h\right\rVert_{\phi,P}^{2}\Lambda_{\phi,\beta}(q).$

Plugging the above observation into (D) completes the proof. ∎

###### Proof of A.1.

In order to prove A.1, we will need two existing results from the literature
which we state here without proof. The first result is a variant of Berbee’s
coupling Lemma, which was proved independently in [8] and [59].

######  Proposition D.1.

Let $\\{X_{i}\\}_{i\geq 1}$ be a countable sequence of random elements taking
values in some Polish space $\mathcal{X}$. For any $j\geq 1$, let
$b_{j}:=\beta(X_{j},\ \\{X_{i}\\}_{i>j})$ (see Definition 2.1). Then there
exists a sequence $\\{X_{i}^{*}\\}_{i\geq 1}$ of independent random variables
such that for any $j\geq 1$, $X_{j}^{*}$ has the same distribution as $X_{j}$
and $P(X_{j}\neq X_{j}^{*})\leq b_{j}$.

The next result is the celebrated Bernstein’s inequality (see [9]).

######  Proposition D.2.

Suppose $Y_{1},\dots,Y_{n}$ are centered independent random variables with
$|Y_{i}|\leq b$ and ${\sf var}(Y_{i})\leq\sigma^{2}_{Y}$. Then we have:

$\mathbb{P}\left(\left|\sum_{i=1}^{n}Y_{i}\right|\geq t\right)\leq 2{\sf
exp}{\left(-\frac{t^{2}}{2\left(n\sigma^{2}_{Y}+\frac{1}{3}bt\right)}\right)}$

for all $t>0$.

We are now in position to prove A.1.

Fix $q>0$. By Berbee’s coupling D.1, we can construct another sequence
$\\{X_{i}^{0}\\}_{i\geq 1}$ of random elements in $\mathcal{X}$ such that:

  1. 1.

The sub-sequence
$\mathbf{X}^{0}_{k}:=\\{X^{0}_{qk+1},\dots,X^{0}_{q(k+1)}\\}\overset{\mathscr{L}}{=}\\{X_{qk+1},\dots,X_{q(k+1)}\\}=:\mathbf{X}_{k}$.

  2. 2.

The sequence $\\{\mathbf{X}^{0}_{2k}\\}_{k\geq 1}$ (respectively
$\\{\mathbf{X}^{0}_{2k-1}\\}_{k\geq 1}$) are independent.

  3. 3.

$\mathbb{P}\left(\mathbf{X}^{0}_{k}\neq\mathbf{X}_{k}\right)\leq\beta_{q}$.

For the rest of the calculation, assume without loss of generality $Pf=0$ for
all $f\in\mathcal{F}$. At the expense of potentially changing the universal
constant $K$, we can assume without loss of generality that $n$ is an even
multiple of $q$. For $k\leq(n/q)-1$, define:

  1. 1.

$Y_{k}(f):=\sum\limits_{i=q(k-1)+1}^{qk}f(X_{i})$

  2. 2.

$Y^{0}_{k}(f)=\sum\limits_{i=q(k-1)+1}^{qk}f(X^{0}_{i})$

A simple application of triangle inequality yields:

$\displaystyle\mathbb{E}\left[\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{i=1}^{n}f(X_{i})\right|\right]$
$\displaystyle\leq\mathbb{E}\left[\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{i=1}^{n}f(X^{0}_{i})\right|\right]+\mathbb{E}\left[\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(f(X_{i})-f(X^{0}_{i})\right)\right|\right]$
$\displaystyle\leq\mathbb{E}\left[\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{i=1}^{n}f(X^{0}_{i})\right|\right]+\frac{2b}{\sqrt{n}}\mathbb{E}\left[\sum_{i=1}^{n}\mathds{1}_{X_{i}\neq
X_{i}^{0}}\right]$
$\displaystyle\leq\mathbb{E}\left[\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{i=1}^{n}f(X^{0}_{i})\right|\right]+2b\sqrt{n}\beta_{q}\,.$
(D.3)

We next bound the first term of the above inequality.

From Lemma A.2 we have:

${\sf var}\left(\frac{Y_{k}^{0}}{\sqrt{q}}\right)={\sf
var}\left(\frac{Y_{k}}{\sqrt{q}}\right)\leq\sigma^{2}\left(c_{\phi}^{2}+2(1+\Lambda_{\phi,\beta}(q))\right)=\sigma^{2}\pi^{2}_{\phi,\beta}(q):=2\tau_{q}^{2}\,,$
(D.4)

where $\pi_{\phi,\beta}(q)$ is defined in (A.1). Further

$\max_{k\geq 1}\max_{f\in\mathcal{F}}|Y_{k}(f)|\vee|Y_{k}^{0}(f)|\leq qb\,.$

Therefore, for any $f\in\mathcal{F}$ we have by the Bernstein’s inequality
D.2:

$\displaystyle\mathbb{P}\left(\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{n/2q}Y^{0}_{2k}\right|>t\right)$
$\displaystyle\leq 2{\sf
exp}{\left(-\frac{1}{2}\frac{nt^{2}}{\frac{n}{2q}2q\tau_{q}^{2}+\frac{1}{3}qb\sqrt{n}t}\right)}$
$\displaystyle=2{\sf
exp}{\left(-\frac{1}{2}\frac{t^{2}}{\tau_{q}^{2}+\frac{1}{3}\frac{qb}{\sqrt{n}}t}\right)}$
$\displaystyle=2{\sf
exp}{\left(-\frac{t^{2}}{2\tau_{q}^{2}+\frac{2}{3}\frac{qb}{\sqrt{n}}t}\right)}$
$\displaystyle\leq 2{\sf
exp}{\left(-\frac{t^{2}}{2\max\left\\{2\tau_{q}^{2},\frac{2}{3}\frac{qb}{\sqrt{n}}t\right\\}}\right)}$
$\displaystyle=2{\sf
exp}{\left(-\min\left\\{\frac{t^{2}}{4\tau_{q}^{2}},\frac{3t\sqrt{n}}{4qb}\right\\}\right)}$
$\displaystyle\leq 2{\sf exp}{\left(-\frac{t^{2}}{4\tau_{q}^{2}}\right)}+2{\sf
exp}{\left(-\frac{3t\sqrt{n}}{4qb}\right)}$

Therefore, taking union bound, we have:

$\mathbb{P}\left(\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{n/2q}Y^{0}_{2k}\right|>t\right)\leq
2{\sf
exp}{\left(-\left[\frac{t^{2}}{4\tau_{q}^{2}}-\log{|\mathcal{F}|}\right]_{+}\right)}+2{\sf
exp}{\left(-\left[\frac{3t\sqrt{n}}{4qb}-\log{|\mathcal{F}|}\right]_{+}\right)}$

Same calculation will go through for the odd sequence $\\{Y^{0}_{2k-1}\\}$.
Therefore we have:

$\mathbb{P}\left(\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{n/q}Y^{0}_{k}\right|>t\right)\leq
4{\sf
exp}{\left(-\left[\frac{t^{2}}{16\tau_{q}^{2}}-\log{|\mathcal{F}|}\right]_{+}\right)}+4{\sf
exp}{\left(-\left[\frac{3t\sqrt{n}}{8qb}-\log{|\mathcal{F}|}\right]_{+}\right)}$

Integrating with respect to $t$, we have:

$\displaystyle\mathbb{E}\left[\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{n/q}Y^{0}_{k}\right|\right]$
$\displaystyle=\int_{0}^{\infty}\mathbb{P}\left(\max_{f\in\mathcal{F}}\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{n/q}Y^{0}_{k}\right|>t\right)\
dt$ $\displaystyle\leq 4\int_{0}^{\infty}{\sf
exp}{\left(-\left[\frac{t^{2}}{16\tau_{q}^{2}}-\log{|\mathcal{F}|}\right]_{+}\right)}\
dt+4\int_{0}^{\infty}{\sf
exp}{\left(-\left[\frac{3t\sqrt{n}}{8qb}-\log{|\mathcal{F}|}\right]_{+}\right)}\
dt$ $\displaystyle\leq 4\int_{0}^{4\tau_{q}\sqrt{\log{|\mathcal{F}|}}}{\sf
exp}{\left(-\left[\frac{t^{2}}{16\tau_{q}^{2}}-\log{|\mathcal{F}|}\right]_{+}\right)}\
dt+4\int_{4\tau_{q}\sqrt{\log{|\mathcal{F}|}}}^{\infty}{\sf
exp}{\left(-\left[\frac{t^{2}}{16\tau_{q}^{2}}-\log{|\mathcal{F}|}\right]_{+}\right)}\
dt$
$\displaystyle\qquad+4\int_{0}^{\frac{8qb\log{|\mathcal{F}|}}{3\sqrt{n}}}{\sf
exp}{\left(-\left[\frac{3t\sqrt{n}}{8qb}-\log{|\mathcal{F}|}\right]_{+}\right)}\
dt+4\int_{\frac{8qb\log{|\mathcal{F}|}}{3\sqrt{n}}}^{\infty}{\sf
exp}{\left(-\left[\frac{3t\sqrt{n}}{8qb}-\log{|\mathcal{F}|}\right]_{+}\right)}\
dt$ $\displaystyle\leq
16\tau_{q}\sqrt{\log{|\mathcal{F}|}}+4|\mathcal{F}|\int_{4\tau_{q}\sqrt{\log{|\mathcal{F}|}}}^{\infty}{\sf
exp}{\left(-\frac{t^{2}}{16\tau_{q}^{2}}\right)}\
dt+\frac{32qb\log{|\mathcal{F}|}}{3\sqrt{n}}$
$\displaystyle\qquad+4|\mathcal{F}|\int_{\frac{8qb\log{|\mathcal{F}|}}{3\sqrt{n}}}^{\infty}{\sf
exp}{\left(-\frac{3t\sqrt{n}}{8qb}\right)}\ dt$ $\displaystyle\leq
16\tau_{q}\sqrt{\log{|\mathcal{F}|}}+16\tau_{q}+\frac{32qb\log{|\mathcal{F}|}}{3\sqrt{n}}+\frac{32qb}{3\sqrt{n}}$
$\displaystyle=32\tau_{q}\sqrt{1+\log{|\mathcal{F}|}}+\frac{32qb}{3\sqrt{n}}\left(1+\log{|\mathcal{F}|}\right)\,.$

This along with (D.3) completes the proof. ∎

###### Proof of Lemma A.3.

We begin by noting that for $t\leq\varepsilon$, we have:

$\displaystyle Q_{h}(t)\leq\frac{\omega_{q,h}(\varepsilon)}{\sqrt{B_{q}(t)}}.$
(D.5)

As $Q_{h}^{-1}(Q_{h}(\varepsilon))\leq\varepsilon$, we consequently get,

$\displaystyle\lVert
h\mathds{1}(|h|>Q_{h}(\varepsilon))\rVert_{L_{1}(P)}=\int_{0}^{Q_{h}^{-1}(Q_{h}(\varepsilon))}Q_{h}(t)\,dt\leq\omega_{q,h}(\varepsilon)\int_{0}^{\varepsilon}\frac{1}{\sqrt{B_{q}(t)}}\,dt.$

The last inequality above follows from (D.5). Next we observe that $B_{q}$ is
an integral of a bounded decreasing function and hence, is concave, with
$B_{q}(0)=0$. Hence we have:

$B_{q}(t)=B_{q}\left(\varepsilon\frac{t}{\varepsilon}+\left(1-\frac{t}{\varepsilon}\right)0\right)\geq\frac{t}{\varepsilon}B_{q}(\varepsilon)\,.$

Using this we obtain:

$\lVert
h\mathds{1}(|h|>Q_{h}(\varepsilon))\rVert_{L_{1}(P)}\leq\frac{\omega_{q,h}(\varepsilon)\sqrt{\varepsilon}}{\sqrt{B_{q}(\varepsilon)}}\int_{0}^{\varepsilon}\frac{\,dt}{\sqrt{t}}=\frac{2\omega_{q,h}(\varepsilon)\varepsilon}{\sqrt{B_{q}(\varepsilon)}}.$

This proves (A.4). The proof of (A.6) follows from (A.4) and the following
observations. Note that as $Q_{h}$ is decreasing:

$\displaystyle\omega_{q,h}^{2}(\varepsilon)\leq\sup_{t\leq\varepsilon}\
Q_{h}^{2}(t)B_{q}(t)\leq\sup_{t\leq\varepsilon}\
\int_{0}^{t}Q_{h}^{2}(u)\left(\sum_{k=0}^{q}\mathds{1}(u\leq\beta(k))\right)\,du.$

By invoking (D.2), we then have for all $\varepsilon\in[0,1]$:

$\displaystyle\omega_{q,h}(\varepsilon)\leq\lVert
h\rVert_{\phi,P}\sqrt{\Lambda_{\phi,\beta}(q)}.$ (D.6)

Furthermore,

$\displaystyle
B_{q}(\varepsilon)\geq\varepsilon\sum_{k=0}^{q}\mathds{1}(\varepsilon\leq\beta(k)).$
(D.7)

Using (D.7) and (A.5), we then get:

$v\sqrt{B_{q}(\varepsilon)}\geq\upsilon\sqrt{\varepsilon}\sqrt{\sum_{k=0}^{q}\mathds{1}(\varepsilon\leq\beta(k))}\geq\lVert
h\rVert_{\phi,P}\sqrt{\Lambda_{\phi,\beta}(q)}\geq\omega_{q,h}(\varepsilon)\geq
Q_{h}(\varepsilon)\sqrt{B_{q}(\varepsilon)}\,.$

Here the first inequality follows from (D.7), second inequality from the
condition (A.5), third inequality from (D.6) and the last one from the
definition of $\omega_{q,h}$. Consequently, for any $v\geq Q_{h}(\varepsilon)$
we have:

$\lVert h\mathds{1}(|h|>v)\rVert_{L_{1}(P)}\leq\lVert
h\mathds{1}(|h|>Q_{h}(\varepsilon))\rVert_{L_{1}(P)}\,.$

The conclusion then follows by combining (D.6), (D.7), and (A.4). ∎

###### Proof of A.2.

The proof is similar to that of A.1 except that, instead of using Lemma A.2 in
(D.4), we will use the following:

$\displaystyle{\sf var}\left(\sum_{i=1}^{q}h(X_{i})\right)$
$\displaystyle=\sum_{i}{\sf var}(h(X_{i}))+2\sum_{i<j}{\sf
cov}(h(X_{i}),h(X_{j}))$ $\displaystyle\leq q\lVert
h\rVert_{\infty,\mathcal{X}}^{2}+2\sum_{i<j}\int_{0}^{\beta(|i-j|)}Q_{h}^{2}(u)\,du$
$\displaystyle\leq q\lVert h\rVert_{\infty,\mathcal{X}}^{2}+2q\lVert
h\rVert_{\infty,\mathcal{X}}^{2}\sum_{k=1}^{q-1}\beta_{k}.$ (D.8)

Here the second inequality follows from [124, Theorem 1.1] and the third
inequality follows from the fact that $\lVert
Q_{h}\rVert_{\infty,(0,1)}\leq\lVert h\rVert_{\infty,\mathcal{X}}$. The rest
of the proof follows verbatim from the proof of A.1. ∎

###### Proof of Lemma A.4.

The proof is is the same as that of Lemma A.3 except that we will replace
(D.6) with the following bound:

$\omega_{q,h}(\varepsilon)\leq\sup_{t\leq\varepsilon}\sqrt{Q_{h}^{2}(t)B_{q}(t)}\leq\lVert
h\rVert_{\infty,\mathcal{X}}\sqrt{B_{q}(1)}\leq\lVert
h\rVert_{\infty,\mathcal{X}}\sqrt{\sum_{k=0}^{q}\beta_{k}},$

for all $\varepsilon\in[0,1]$. The rest of the proof follows verbatim from the
proof of Lemma A.3. ∎

###### Proof of A.3.

The proof is similar to that of A.1 except that, instead of using Lemma A.2 in
(D.4), we will use the following:

$\displaystyle{\sf var}\left(\sum_{i=1}^{q}h(X_{i})\right)$
$\displaystyle=\sum_{i}{\sf var}(h(X_{i}))+2\sum_{i<j}{\sf
cov}(h(X_{i}),h(X_{j}))$ $\displaystyle\leq q\lVert
h\rVert_{L_{2}(P)}^{2}+2\lVert h\rVert_{L_{2}(P)}^{2}\sum_{i<j}\gamma_{|i-j|}$
$\displaystyle\leq
qb^{2-\tilde{r}}\sigma^{\tilde{r}}+2qb^{2-\tilde{r}}\sigma^{\tilde{r}}\sum_{k=1}^{q-1}\gamma_{k}.$

Here the second inequality follows from the definition of $\rho$-mixing (see
Definition 4.1) and the third inequality follows from the elementary
inequality $\int h^{2}\,dP\leq\lVert
h\rVert_{\infty,\mathcal{X}}^{2-\tilde{r}}\int|h|^{\tilde{r}}\,dP$. The rest
of the proof follows verbatim from the proof of A.1. ∎