# Homogenization estimates
for high order elliptic operators

S. E. Pastukhova

In the whole space $\mathbb{R}^{d}$, $d\geq 2$, we study homogenization of a
divergence form elliptic operator $A_{\varepsilon}$ of order $2m\geq 4$ with
measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small
parameter. For the resolvent $(A_{\varepsilon}+1)^{-1}$, we construct an
approximation with the remainder term of order $\varepsilon^{2}$ in the
operator $(L^{2}{\to}H^{m})$-norm, using the resolvent of the homogenized
operator, solutions of several auxiliary periodic problems on the unit cube,
and smoothing operators. The homogenized operator here differs from the one
commonly employed in homogenization.

##  1 Introduction

The present paper is devoted to the homogenization of high order elliptic
operators with periodic coefficients. We mean operators of an arbitrary even
order $2m\geq 4$. The first qualitative results in this topic were obtained
long ago in 70s (see, e.g., [1] and [2]). In this paper, we are interested in
estimates for the homogenization error, which are of operator type, and
continue the recent studies of [3]–[8], where the approach proposed in [9] and
[10] was applied in different situations concerning high order elliptic
operators. Here, we construct approximations for the resolvents of high order
operators with the remainder term $O(\varepsilon^{2})$ as $\varepsilon\to 0$
in the energy operator norm, i.e., in the operator $(L^{2}{\to}H^{m})$-norm.

1.1. In the whole space $\mathbb{R}^{d}$, $d{\geq}2$, we consider the
following equation of an even order $2m{\geq}4$:

$\begin{array}[]{cc}u^{\varepsilon}\in
H^{m}(\mathbb{R}^{d}),\quad(A_{\varepsilon}+1)u^{\varepsilon}=f,\quad f\in
L^{2}(\mathbb{R}^{d}),&\\\
A_{\varepsilon}=(-1)^{m}\sum\limits_{|\alpha|=|\beta|=m}D^{\alpha}(a^{\varepsilon}_{\alpha\beta}(x)D^{\beta}),&\\\
\end{array}$ (1.1)

with rapidly oscillating $\varepsilon$-periodic coefficients
$a^{\varepsilon}_{\alpha\beta}(x)=a_{\alpha\beta}(y)|_{y=\varepsilon^{-1}x},$
for small $\varepsilon{\in}(0,1)$. Here, $D^{\alpha}$ denotes the
multiderivative

$D^{\alpha}=D_{1}^{\alpha_{1}}\ldots D_{d}^{\alpha_{d}},\quad
D_{i}=\frac{\partial}{\partial x_{i}},\quad i=1,\ldots,d,$

where $\alpha{=}(\alpha_{1},\ldots,\alpha_{d})$ is the multiindex of length
$|\alpha|{=}\alpha_{1}+\ldots+\alpha_{d}$ with
$\alpha_{j}{\in}\mathbb{Z}_{\geq 0}$; the coefficients $a_{\alpha\beta}(y)$
are measurable periodic real-valued functions, and the unit cube
$Y{=}[-1/2,1/2)^{d}$ is the periodicity cell.

We assume that the following boundedness and ellipticity conditions hold:

$\displaystyle{\|a_{\alpha\beta}\|_{L^{\infty}(Y)}\leq\lambda_{1},}\atop\displaystyle{\int_{\mathbb{R}^{d}}\sum\limits_{|\alpha|=|\beta|=m}a_{\alpha\beta}(x)D^{\beta}\varphi
D^{\alpha}\varphi\,dx\geq\lambda_{0}\int_{\mathbb{R}^{d}}\sum\limits_{|\alpha|=m}|D^{\alpha}\varphi|^{2}\,dx\quad\forall\varphi\in
C_{0}^{\infty}(\mathbb{R}^{d})}$ (1.2)

for some positive constants $\lambda_{0}$, $\lambda_{1}$ and all multiindices
$\alpha$ and $\beta$ of length $m$. No symmetry conditions are imposed on the
coefficients $a_{\alpha\beta}$.

In (1.1), we use the Sobolev space $H^{m}{=}H^{m}(\mathbb{R}^{d})$, equipped
with the norm

$\|u\|^{2}_{H^{m}}=\int_{\mathbb{R}^{d}}\sum\limits_{|\alpha|\leq
m}|D^{\alpha}u|^{2}\,dx.$

As known, the set $C_{0}^{\infty}(\mathbb{R}^{d})$ of smooth compactly
supported functions is dense in $H^{m}(\mathbb{R}^{d})$ and the norm can be
equivalently introduced by a simpler way

$\|u\|^{2}_{H^{m}}=\int_{\mathbb{R}^{d}}\sum\limits_{|\alpha|=m}|D^{\alpha}u|^{2}\,dx+\int_{\mathbb{R}^{d}}|u|^{2}\,dx,$

$G$-convergence and homogenization issues for differential operators
$A_{\varepsilon}$ in (1.1) have been studied since the 70s. Even more general
operators were considered from this point of view in [2], namely, divergence
form operators

$A_{\varepsilon}=\sum\limits_{|\alpha|\leq m,|\beta|\leq
m}(-1)^{|\alpha|}D^{\alpha}(a^{\varepsilon}_{\alpha\beta}(x)D^{\beta}),$ (1.3)

with lower order terms. Moreover, the periodicity of coefficients was not
necessarily required for (1.3), instead of which the so called $N$-condition
was assumed in [2]. The well known result on homogenization of the operator
$A_{\varepsilon}$ in (1.1) means the closeness, for sufficiently small
$\varepsilon$, in the sense of the strong operator topology between the
resolvent $(A_{\varepsilon}+1)^{-1}$, regarded as an operator in
$L^{2}(\mathbb{R}^{d})$, and the resolvent $(\hat{A}+1)^{-1}f$ of the
homogenized operator $\hat{A}$. More exactly, $(A_{\varepsilon}+1)^{-1}f$
converge to $(\hat{A}+1)^{-1}f$ in the $L^{2}(\mathbb{R}^{d})$-norm as
$\varepsilon\to 0$ for any $f\in L^{2}(\mathbb{R}^{d})$. The homogenized
operator $\hat{A}$ is of the same class (1.2) as the original operator
$A_{\varepsilon}$, but much simpler:

$\hat{A}=(-1)^{m}\sum\limits_{|\alpha|=|\beta|=m}D^{\alpha}\hat{a}_{\alpha\beta}D^{\beta},$
(1.4)

where the coefficients $\hat{a}_{\alpha\beta}$ are constant and can be defined
with the help of auxiliary problems on the periodicity cell $Y$ (see
Subsection 3.1). The homogenized problem for (1.1) is written as

$u\in H^{m}(\mathbb{R}^{d}),\quad\hat{A}u+u=f,\quad f\in
L^{2}(\mathbb{R}^{d}),$ (1.5)

and the above result about the strong resolvent convergence means that the
solutions to the problems (1.1) and (1.5) are connected by the limit relation

$\lim_{\varepsilon\to 0}\|u^{\varepsilon}-u\|_{L^{2}(\mathbb{R}^{d})}=0$ (1.6)

for any right-hand side function $f\in L^{2}(\mathbb{R}^{d})$.

1.2. In [11], [3] and [4], the stronger operator convergence of
$A_{\varepsilon}$ to $\hat{A}$ was established, namely, the uniform resolvent
convergence in the operator $L^{2}(\mathbb{R}^{d})$-norm; moreover, the
convergence rate estimate with respect to the parameter $\varepsilon$ was
deduced:

$\|(A_{\varepsilon}+1)^{-1}-(\hat{A}+1)^{-1}\|_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}\leq C\varepsilon,$ (1.7)

where the constant $C$ depends only on the spatial dimension $d$ and the
constants $\lambda_{0}$ and $\lambda_{1}$ in (1.2). The main result in [3] [4]
is even more exact and stronger; it concerns approximation of the resolvent
$(A_{\varepsilon}+1)^{-1}$ in the operator $(L^{2}(\mathbb{R}^{d})\to
H^{m}(\mathbb{R}^{d}))$-norm by using the sum
$(\hat{A}+1)^{-1}+\varepsilon^{m}\mathcal{K}_{\varepsilon}$ of the resolvent
of the homogenized operator $\hat{A}$ and the correcting operator.
Furthermore,

$\|(A_{\varepsilon}+1)^{-1}-(\hat{A}+1)^{-1}-\varepsilon^{m}\mathcal{K}_{\varepsilon}\|_{L^{2}(\mathbb{R}^{d})\to
H^{m}(\mathbb{R}^{d})}\leq C\varepsilon,$ (1.8)

where the constant $C$ depends on the spatial dimension and the constants in
(1.2). The operator $\mathcal{K}_{\varepsilon}$ is determined with the help of
solutions to auxiliary problems on the periodicity cell, which are introduced
to define the coefficients of the homogenized operator $\hat{A}$. We note that

$\|\varepsilon^{m}\mathcal{K}_{\varepsilon}\|_{L^{2}(\mathbb{R}^{d})\to
H^{m}(\mathbb{R}^{d})}{\leq}c,\quad\|\mathcal{K}_{\varepsilon}\|_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}{\leq}c.$ (1.9)

Thus, the estimate (1.7) in the operator $L^{2}(\mathbb{R}^{d})$-norm may be
obtained from (1.8) by coarsening (first, by weakening the operator norm and
then transferring the term $\varepsilon^{m}\mathcal{K}_{\varepsilon}$ to the
remainder, due to the second inequality in (1.9)).

In [5] and [6], estimates of type (1.8) are used in a more delicate way, which
allows to improve the $L^{2}$-estimate (1.7) with respect to the parameter
$\varepsilon$ assuming that the coefficients $a_{\alpha\beta}(y)$ in (1.1) are
real-valued and symmetric. In this case, the resolvent $(\hat{A}+1)^{-1}$ of
the homogenized operator actually approximates the resolvent
$(A_{\varepsilon}+1)^{-1}$ with a remainder of order $\varepsilon^{2}$ in the
operator $L^{2}(\mathbb{R}^{d})$-norm. Indeed, the following estimate holds

$\|(A_{\varepsilon}+1)^{-1}{-}(\hat{A}+1)^{-1}\|_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}\leq C\varepsilon^{2},$ (1.10)

where the constant $C$ depends on the spatial dimension and the constants in
(1.2). In terms of the solutions to the problems (1.1) and (1.5), this result
provides the estimate

$\|u^{\varepsilon}-u\|_{L^{2}(\mathbb{R}^{d})}\leq
C\varepsilon^{2}\|f\|_{L^{2}(\mathbb{R}^{d})}$ (1.11)

(with the same constant $C$ on the right-hand side as in (1.10)), which
specifies the convergence rate in the long-standing result (1.6) from 70s.
Without the symmetry condition on coefficients, it is shown [5] that the
approximation of $\varepsilon^{2}$ order for the resolvent
$(A_{\varepsilon}+1)^{-1}$ in the $L^{2}(\mathbb{R}^{d})$-operator norm
becomes more complicated, namely,

$(A_{\varepsilon}+1)^{-1}=(\hat{A}+1)^{-1}+\varepsilon\mathcal{K}_{1}+O(\varepsilon^{2}),$
(1.12)

where the correcting operator $\mathcal{K}_{1}$ is independent of
$\varepsilon$ by contrast with its counterpart in (1.8).

The question arises how to construct the asymptotic expansion for
$(A_{\varepsilon}+1)^{-1}$ similar to (1.12), i.e., with remainder term of
order $\varepsilon^{2}$, but in the
$(L^{2}(\mathbb{R}^{d}){\to}H^{m}(\mathbb{R}^{d}))$-norm. The answer is given
in Theorem 2.2.

1.3. We now outline the structure of the work. An exact formulation of our
results on improved $H^{m}$-approximations is given in Section 2 (see Theorems
2.1 and 2.2), while Sections 3–5 are of an auxiliary or preparatory nature. A
direct proof of the main result, that is, the estimate (2.10), is presented in
Section 6.

In what follows we systematically use the differentiation formula for the
product

$D^{\alpha}(wv)=\sum\limits_{\gamma\leq\alpha}c_{\alpha,\gamma}D^{\gamma}wD^{\alpha-\gamma}v=(D^{\alpha}w)v+\sum\limits_{\gamma<\alpha}c_{\alpha,\gamma}D^{\gamma}wD^{\alpha-\gamma}v$
(1.13)

for suitably differentiable functions $v$ and $w$ with some constants
$c_{\alpha,\gamma}$, where $c_{\alpha,0}=c_{\alpha,\alpha}=1$. The sum in
(1.13) is taken over all multiindices $\gamma$ such that $\gamma\leq\alpha$ or
$\gamma<\alpha$. We assume that $\gamma\leq\alpha$ if
$\gamma_{i}\leq\alpha_{i}$ for all $1\leq i\leq d$ and $\gamma<\alpha$ if, in
addition, for at least one index $i$ we have the strict inequality
$\gamma_{i}<\alpha_{i}$.

Throughout the paper we use the following notation. Given a 1-periodic
function $b(y)$, we denote by $b^{\varepsilon}$ or $(b)^{\varepsilon}$ the
$\varepsilon$-periodic function of the variable $x$ obtained from $b(y)$ by
substitution $y=x/\varepsilon$, i.e.,

$b^{\varepsilon}(x)=b(x/\varepsilon).$ (1.14)

For example, $N_{\alpha}^{\varepsilon}{=}N_{\alpha}(x/\varepsilon)$,
$G^{\varepsilon}_{\gamma\alpha}{=}G_{\gamma\alpha}(x/\varepsilon)$,
$(D^{\beta}G_{\gamma\alpha})^{\varepsilon}{=}(D^{\beta}G_{\gamma\alpha}(y))|_{y=x/\varepsilon}$
and so on.

##  2 Improved $H^{m}$-approximation

2.1. Our purpose is to obtain an $\varepsilon^{2}$-order approximation of the
resolvent $(A_{\varepsilon}+1)^{-1}$ in the operator
$(L^{2}(\mathbb{R}^{d}){\to}H^{m}(\mathbb{R}^{d}))$-norm. It will be the sum
of the zeroth approximation and several correctors, the number of which
increases in comparison with the approximation from (1.8). By contrast to
(1.8), one should take for the zeroth approximation, instead of
$(\hat{A}+1)^{-1}$, the resolvent $(\hat{A}_{\varepsilon}+1)^{-1}$ of the
operator $\hat{A}_{\varepsilon}$, which is more complicated than $\hat{A}$.
Namely,

$\hat{A}_{\varepsilon}=(-1)^{m}\sum\limits_{|\alpha|=m}D^{\alpha}(\sum\limits_{|\beta|=m}\hat{a}_{\alpha\beta}D^{\beta}+\varepsilon\sum\limits_{|\delta|=m+1}b_{\alpha\delta}D^{\delta}),$
(2.1)

where the constant coefficients $\hat{a}_{\alpha\beta}$, which are the same as
in (1.4), and $b_{\alpha\delta}$ are determined via solutions to cell problems
in Section 3. Clearly,

$\hat{A}_{\varepsilon}=\hat{A}+\varepsilon B,$

where the differential operator $B$ is of order $2m+1$; thereby,
$\hat{A}_{\varepsilon}$ is obtained from the commonly used homogenized
operator $\hat{A}$ by a singular perturbation.

We introduce a new version of the homogenized equation with the operator (2.1)

$(\hat{A}_{\varepsilon}+1)\hat{u}^{\varepsilon}=f,\quad f\in
L^{2}(\mathbb{R}^{d}).$ (2.2)

Since the coefficients of the equation are constant, its solution can be
obtained with the help of the Fourier transform.

Applying the Fourier transform to Equation (2.2), we get the equality

$\left(1+\Lambda(\xi)+i\varepsilon\Lambda_{0}(\xi)\right)F[\hat{u}^{\varepsilon}]=F[f],\quad
i=\sqrt{-1},$ (2.3)

where $f(x)\to F[f](\xi)$ is the Fourier transform and

$\Lambda(\xi){=}\sum_{|\alpha|=|\beta|=m}\hat{a}_{\alpha\beta}\xi^{\alpha}\xi^{\beta},\quad\Lambda_{0}(\xi){=}\sum_{|\alpha|=m,|\delta|=m+1}b_{\alpha\delta}\xi^{\alpha}\xi^{\delta},\quad\xi\in\mathbb{R}^{d}.$

Given any $\xi\in\mathbb{R}^{d}$ and multiindex
$\alpha{=}(\alpha_{1},\ldots,\alpha_{d})$, we have
$\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\ldots\xi_{d}^{\alpha_{d}}$ according to our
convention (see it after (1.1)). The coefficients $\hat{a}_{\alpha\beta}$ and
$b_{\alpha\delta}$ are real; therefore, from (2.2) the inequality

$\int_{\mathbb{R}^{d}}\left(1+\Lambda(\xi)\right)^{2}|F[\hat{u}^{\varepsilon}]|^{2}\,d\xi\leq\int_{\mathbb{R}^{d}}|F[f]|^{2}\,d\xi$
(2.4)

follows. By the ellipticity property of $\hat{a}_{\alpha\beta}$ inherited from
the $a_{\alpha\beta}(y)$ (see (1.2)) and the Plancherel identity, (2.4) yields
the $\varepsilon$-uniform estimate

$\|\hat{u}^{\varepsilon}\|_{H^{2m}(\mathbb{R}^{d})}\leq
C\|f\|_{L^{2}(\mathbb{R}^{d})},$ (2.5)

where the constant $C$ depends only on the spatial dimension and the constants
from (1.2).

2.2. We are in a position to formulate the main results of the paper. We begin
with approximations of the solution to (1.1) that have the structure of two-
scale expansions, which is usual in homogenization. An approximation for the
solution of (1.1) is taken in the form

$\tilde{u}^{\varepsilon}(x)=u^{,\varepsilon}(x)+\varepsilon^{m}\,U_{m}^{\varepsilon}(x)+\varepsilon^{m+1}\,U_{m+1}^{\varepsilon}(x)$
(2.6)

with

$\displaystyle{U_{m}^{\varepsilon}(x)=\sum\limits_{|\gamma|=m}N_{\gamma}(x/\varepsilon)D^{\gamma}u^{,\varepsilon}(x),}\atop\displaystyle{U_{m+1}^{\varepsilon}(x)=\sum\limits_{|\delta|=m+1}N_{\delta}(x/\varepsilon)D^{\delta}u^{,\varepsilon}(x),}$
(2.7)
$u^{,\varepsilon}(x)=\Theta^{\varepsilon}\,\hat{u}^{\varepsilon}(x),\quad\Theta^{\varepsilon}=S^{\varepsilon}S^{\varepsilon},$
(2.8)

where $S^{\varepsilon}$ is Steklov‘s smoothing, $N_{\gamma}(y)$,
$N_{\delta}(y)$, for all multiindices $\gamma,$ $\delta$ such that
$|\alpha|=m$ and $|\delta|=m+1$, and $\hat{u}^{\varepsilon}(x)$ are solutions
of the problems (3.2), (3.16), and (1.5), respectively. We recall that the
Steklov smoothing operator is defined by

$(S^{\varepsilon}\varphi)(x)=\int_{Y}\varphi(x-\varepsilon\omega)\,d\omega,\quad
Y=[-1/2,1/2)^{d},$ (2.9)

whenever $\varphi\in L^{1}_{loc}(\mathbb{R}^{d})$.

###### Theorem 2.1

For the difference of the solution to (1.1) and the function given in
(2.6)–(2.8), the estimate

$\|u^{\varepsilon}-\tilde{u}^{\varepsilon}\|_{H^{m}(\mathbb{R}^{d})}\leq
C\varepsilon^{2}\|f\|_{L^{2}(\mathbb{R}^{d})}$ (2.10)

holds, where the constant $C$ depends only on the dimension $d$ and the
constants from (1.2).

The proof is in Section 6.

Together with $\tilde{u}^{\varepsilon}$, we consider the following
approximation of a simpler structure:

$v^{\varepsilon}(x){=}\hat{u}^{\varepsilon}{+}\varepsilon^{m}\,\sum\limits_{|\gamma|=m}N_{\gamma}(\frac{x}{\varepsilon})D^{\gamma}S^{\varepsilon}\hat{u}^{\varepsilon}(x){+}\varepsilon^{m+1}\sum\limits_{|\delta|=m+1}N_{\delta}(\frac{x}{\varepsilon})D^{\delta}S^{\varepsilon}S^{\varepsilon}\hat{u}^{\varepsilon}(x),$
(2.11)

where the zeroth approximation does not involve smoothing operators, and only
the first and the second correctors contain the Steklov smoothing operator
$S^{\varepsilon}$ defined in (2.9) or its iteration
$S^{\varepsilon}S^{\varepsilon}$ respectively. Using only the properties of
smoothing (see Section 4), we derive from Theorem 2.1 the estimate

$\|u^{\varepsilon}-{v}^{\varepsilon}\|_{H^{m}(\mathbb{R}^{d})}\leq
C\varepsilon^{2}\|f\|_{L^{2}(\mathbb{R}^{d})},$ (2.12)

where the constant $C$ is of the same type as in (2.10).

It is possible to rewrite estimates (2.10) and (2.12) in the operator terms,
i.e., for the difference between the resolvent $(A_{\varepsilon}+1)^{-1}$ and
its approximations. For example, from (2.11) and (2.12), we deduce

$\|(\hat{A}_{\varepsilon}+1)^{-1}+\varepsilon^{m}\mathcal{K}_{2}(\varepsilon)+\varepsilon^{m{+}1}\mathcal{K}_{3}(\varepsilon)-(A_{\varepsilon}+1)^{-1}\|_{L^{2}(\mathbb{R}^{d})\to
H^{m}(\mathbb{R}^{d})}\leq C\varepsilon^{2},$ (2.13)

where the structure of $\mathcal{K}_{2}(\varepsilon)$ and
$\mathcal{K}_{3}(\varepsilon)$ is restored from the view of the correctors in
(2.11):

$\displaystyle{\mathcal{K}_{2}(\varepsilon)f(x)=\sum\limits_{|\gamma|=m}N_{\gamma}(\frac{x}{\varepsilon})S^{\varepsilon}D^{\gamma}\hat{u}^{\varepsilon}(x),\,\hat{u}^{\varepsilon}(x)=(\hat{A}_{\varepsilon}{+}1)^{-1}f(x),}\atop\displaystyle{\mathcal{K}_{3}(\varepsilon)f(x){=}\\!\sum\limits_{|\delta|=m+1}N_{\delta}(\frac{x}{\varepsilon})D^{\delta}S^{\varepsilon}S^{\varepsilon}\hat{u}^{\varepsilon}(x),\,\hat{u}^{\varepsilon}(x){=}(\hat{A}_{\varepsilon}+1)^{-1}f(x),}$
(2.14)

###### Theorem 2.2

Let $\hat{A}_{\varepsilon}$, $\mathcal{K}_{2}(\varepsilon)$, and
$\mathcal{K}_{3}(\varepsilon)$ be operators defined in (2.1) and (2.14). Then
the sum

$(\hat{A}_{\varepsilon}+1)^{-1}+\varepsilon^{m}\mathcal{K}_{2}(\varepsilon)+\varepsilon^{m{+}1}\mathcal{K}_{3}(\varepsilon)$
(2.15)

approximates the resolvent $(A_{\varepsilon}+1)^{-1}$ of the operator in (1.1)
in the energy operator norm with the estimate (2.13).

2.3. A few words about the statement of the above theorems and the technique
we use. It is not at once apparent that the functions given in (2.6) and
(2.11) belong to the Sobolev space $H^{m}(\mathbb{R}^{d})$. More exactly, it
is not clear why the correctors in (2.6) and (2.11), together with their
gradients of order up to $m$, belong to $L^{2}(\mathbb{R}^{d})$ at first
sight. But this is true thanks to the properties of the smoothing operators
included inside the correctors. Similarly, it is not quite evident that the
correcting operators $\mathcal{K}_{2}(\varepsilon)$ and
$\mathcal{K}_{3}(\varepsilon)$ send functions $f\in L^{2}(\mathbb{R}^{d})$ to
elements of $H^{m}(\mathbb{R}^{d})$. In fact, it holds that

$\|\varepsilon^{m}\mathcal{K}_{2}(\varepsilon)\|_{L^{2}(\mathbb{R}^{d})\to
H^{m}(\mathbb{R}^{d})}{\leq}C,\quad\|\varepsilon^{m}\mathcal{K}_{3}(\varepsilon)\|_{L^{2}(\mathbb{R}^{d})\to
H^{m}(\mathbb{R}^{d})}{\leq}C$ (2.16)

with the constant $C$ depending only on the spatial dimension and the
constants in (1.2). Being the counterparts of (1.9)1, these inequalities show
that, generally, the correctors $\varepsilon^{m}\mathcal{K}_{2}(\varepsilon)$
and $\varepsilon^{m{+}1}\mathcal{K}_{3}(\varepsilon)$ may not be transferred
to the remainder term, for its operator norms are of irrelevant order with
respect to $\varepsilon$. Besides, by (2.16), the orders of smallness for the
correctors in (2.15) are not alike.

We pay attention to the fact that the iterated Steklov smoothing operator is
employed in (2.7) and (2.14); moreover, one cannot rule it out in these
formulas at all and treat only the operator $S^{\varepsilon}$ everywhere. The
necessity of the Steklov smoothing operator $S^{\varepsilon}$ and its
iterations will be clarified in Section 4.

##  3 Cell problems

3.1. On the set of smooth 1-periodic functions $u{\in}C_{\rm per}^{\infty}(Y)$
with zero mean

$\langle u\rangle=\int_{Y}u(y)\,dy=0,$

the norm is determined by the expression

$\left(\int_{Y}\sum_{|\alpha|=m}|D^{\alpha}u|^{2}\,dy\right)^{1/2}.$

The completion of this set in this norm is denoted by $\mathcal{W}$.

Since the coefficients $\\{a_{\alpha\beta}(x)\\}$ are periodic, then the
inequality (1.2) on smooth compactly supported functions yields a similar
inequality on periodic functions:

$\int_{Y}\sum\limits_{|\alpha|=|\beta|=m}a_{\alpha\beta}(y)D^{\beta}uD^{\alpha}u\,dy\geq\lambda_{0}\int_{Y}\sum\limits_{|\alpha|=m}|D^{\alpha}u|^{2}\,dy\quad\forall
u\in C_{\rm per}^{\infty}(Y),$ (3.1)

(see Lemma 3.1 in [3]). The inequality (3.1) can be extended by closure to
$u{\in}\mathcal{W}$; thereby, the operator

$A=(-1)^{m}\sum_{|\alpha|=|\beta|=m}D^{\alpha}(a_{\alpha\beta}(y)D^{\beta})$

acting from $\mathcal{W}$ to its dual $\mathcal{W}^{\prime}$ is coercive.

For any multiindex $\gamma$, $|\gamma|{=}m$, we consider the problem on the
cell

$N_{\gamma}\in\mathcal{W},\quad\sum_{|\alpha|=|\beta|=m}D^{\alpha}(a_{\alpha\beta}(y)D^{\beta}N_{\gamma}(y))=-\sum_{|\alpha|=m}D^{\alpha}(a_{\alpha\gamma}(y)).$
(3.2)

The right-hand side of (3.2) naturally defines the functional $F_{\gamma}$ on
$\mathcal{W}$, and the equation can be written as

$AN_{\gamma}{=}F_{\gamma}\quad(N_{\gamma}{\in}\mathcal{W}).$ (3.3)

Therefore, the unique solvability of (3.2) with the estimate for the solution

$\|N_{\gamma}\|_{\mathcal{W}}\leq c,\quad c=const(\lambda_{0},\lambda_{1}),$
(3.4)

is guaranteed by the following well known abstract assertion for operators
acting from a real reflexive Banach space $V$ to its dual $V^{\prime}$.

###### Theorem 3.1

Let $L:V\to V^{\prime}$ be a continuous linear operator satisfying the
coercitivity condition: $\langle Lv,v\rangle\geq\lambda\|v\|^{2}_{V}$ for any
$v{\in}V$. Then the equation $Lv{=}f\,\,(v{\in}V)$ is uniquely solvable for
any $f{\in}V^{\prime}$, and the solution satisfies the estimate
$\|v\|_{V}\leq\lambda^{-1}\|f\|_{V^{\prime}}.$

The coefficients of the homogenized operator $\hat{A}$ defined in (1.4) are
found through the solutions to the problems (3.2) as follows:

$\hat{a}_{\alpha\beta}=\langle
a_{\alpha\beta}(\cdot)+\sum\limits_{|\gamma|=m}a_{\alpha\gamma}(\cdot)D^{\gamma}N_{\beta}(\cdot)\rangle,\quad|\alpha|=m,|\beta|=m.$
(3.5)

We set

$e_{\alpha\beta}=\left\\{\begin{array}[]{rcl}1,\text{ if }\alpha=\beta,\\\
0,\text{ if }\alpha\neq\beta.\\\ \end{array}\right.$

Then

$\hat{a}_{\alpha\beta}=\langle\sum\limits_{|\gamma|=m}a_{\alpha\gamma}(\cdot)(e_{\gamma\beta}+D^{\gamma}N_{\beta}(\cdot))\rangle,$
(3.6)

or

$\hat{a}_{\alpha\beta}=\langle\tilde{a}_{\alpha\beta}\rangle,\quad\tilde{a}_{\alpha\beta}(y)=\sum\limits_{|\gamma|=m}a_{\alpha\gamma}(y)(e_{\gamma\beta}+D^{\gamma}N_{\beta}(y)).$
(3.7)

It is known (see, e.g., [3, Lemma 3.2]) that the matrix of the homogenized
coefficients (3.5) belongs to the class (1.2). This fact is essential in the
proof of the elliptic estimate (2.5).

Setting

$g_{\alpha\beta}(y)=\tilde{a}_{\alpha\beta}(y)-\hat{a}_{\alpha\beta}\quad\forall\alpha,\beta,$
(3.8)

we obtain the relations

$\langle
g_{\alpha\beta}\rangle=0,\quad\sum\limits_{|\alpha|=m}D^{\alpha}g_{\alpha\beta}=0\quad\forall\beta,$
(3.9)

which follow from the statement of the problem (3.2) and the definitions (3.8)
and (3.7). The equality (3.9)2 can be understood in the sense of the integral
identity on periodic functions

$\sum\limits_{|\alpha|=m}\int_{Y}g_{\alpha\beta}D^{\alpha}\varphi\,dy=0\quad\forall\varphi\in
C^{\infty}_{\rm per}(Y),$ (3.10)

or in the sense of distributions in $\mathbb{R}^{d}$. The integral identity
(3.10) is extended by closure to functions $\varphi\in\mathcal{W}$.

3.2. We recall that periodic solenoidal vectors with zero mean admit
representations in terms of the divergence of a skew-symmetric matrix. More
exactly, if $g{\in}L^{2}_{\rm per}(Y)^{d}$, ${\rm div}\,g{=}0$, and $\langle
g\rangle{=}0$, then there exists a skew-symmetric matrix $G\in H^{1}_{\rm
per}(Y)^{d\times d}$ such that

${\rm div}\,G{=}g,\quad\|G\|_{H^{1}_{\rm per}(Y)^{d\times d}}\leq
c\|g\|_{H^{1}_{\rm per}(Y)^{d}},\,c{=}const(d)$

(see, e.g., [12], Chapter 1). The property ${\rm div}\,g{=}0$ is understood in
the sense of the integral identity

$\langle g\cdot\nabla\varphi\rangle=0\quad\forall\varphi\in C^{\infty}_{\rm
per}(Y),$ (3.11)

or in the sense of distributions in $\mathbb{R}^{d}$.

The following lemma extends the above assertion on solenoidal vectors (in the
classical sense) to the case where the orthogonality property of type (3.11)
holds with the gradient $\nabla^{m}$ of order $m\geq 2$; here,
$\nabla^{m}\varphi=\\{D^{\alpha}\varphi\\}_{\alpha},$ $|\alpha|{=}m$. Let $p$
be the number of multiindices of length $m$, then $\nabla^{m}\varphi$ is
regarded as a vector with $p$ components.

###### Lemma 3.2

Assume that $\\{g_{\alpha}\\}_{|\alpha|=m}\in L_{\rm per}^{2}(Y)^{p}$ and

$\langle g_{\alpha}\rangle=0,\quad\sum_{|\alpha|=m}D^{\alpha}g_{\alpha}=0.$
(3.12)

Then there exists a matrix $\\{G_{\gamma\alpha}\\}_{|\alpha|=|\gamma|=m}$ from
$H^{m}_{\rm per}(Y)^{p\times p}$ such that for all miltiindices $\alpha$,
$\gamma$

$G_{\gamma\alpha}=-G_{\gamma\alpha},\quad\sum_{|\gamma|=m}D^{\gamma}G_{\gamma\alpha}=g_{\alpha},$
(3.13) $\|G_{\gamma\alpha}\|_{H^{m}(Y)}\leq
c\sum_{|\alpha|=m}\|g_{\alpha}\|_{L^{2}(Y)},\quad c=const({d,m}).$ (3.14)

The proof is given in [3] (see also [4] and [13]).

For every fixed $\beta$ the vector $\\{g_{\alpha\beta}\\}_{\alpha}$,
$|\alpha|=m$, in (3.8) satisfies the assumptions of Lemma 3.2. Consequently,
there is a matrix $\\{G_{\gamma\alpha\beta}\\}_{\gamma,\alpha}\in H^{m}$,
${|\alpha|{=}|\gamma|{=}m}$, such that identities of the form (3.13) hold
componentwise, i.e.,

$G_{\alpha\gamma\beta}=-G_{\gamma\alpha\beta},\quad
g_{\alpha\beta}=\sum_{|\gamma|=m}D^{\gamma}G_{\gamma\alpha\beta},$ (3.15)

and $G_{\gamma\alpha\beta}$ satisfies $H^{m}$-estimate of kind (3.14).

We are ready now to introduce, for any multiindex $\delta$,
$|\delta|{=}m{+}1$, the problem on the cell, which depends on the solutions
$N_{\gamma}$ to (3.2) and the functions $G_{\gamma\alpha\beta}$ from the
representation (3.15). This problem is a kind of (3.3), namely,

$N_{\delta}\in\mathcal{W},\quad\sum_{|\alpha|=|\beta|=m}D^{\alpha}(a_{\alpha\beta}D^{\beta}N_{\delta}(y))=-\sum_{|\alpha|=m}D^{\alpha}F_{\alpha,\delta}.$
(3.16)

The right-hand side in (3.16) defines the functional on $\mathcal{W}$ if we
set

$F_{\alpha,\delta}=\sum_{|\gamma|=|\beta|=m,\,\beta<\delta}c_{\gamma,\beta+\gamma-\delta}(a_{\alpha\gamma}(y)D^{\beta+\gamma-\delta}N_{\beta}+D^{\beta+\gamma-\delta}G_{\alpha\gamma\beta})$
(3.17)

with the constants $c_{\beta,\beta+\gamma-\delta}$ from the product rule
(1.13).

Related to (3.16) and (3.17), the matrix

$\tilde{g}_{\alpha\delta}(y)=\sum_{|\beta|=m}a_{\alpha\beta}(y)D^{\beta}N_{\delta}(y)+F_{\alpha,\delta}-b_{\alpha\delta},$
(3.18)

where

$b_{\alpha\delta}=\langle\sum_{|\beta|=m}a_{\alpha\beta}(y)D^{\beta}N_{\delta}(y)+F_{\alpha,\delta}\rangle,$
(3.19)

turns to be the counterpart of $g_{\alpha\beta}(y)$. In fact, for every fixed
$\delta$, $|\delta|{=}m{+}1$, the vector
$\\{\tilde{g}_{\alpha\delta}\\}_{\alpha}$, $|\alpha|{=}m$, in (3.18) satisfies
the assumptions of Lemma 3.2, namely,

$\langle\tilde{g}_{\alpha\delta}\rangle=0,\quad\sum_{|\alpha|=m}D^{\alpha}\tilde{g}_{\alpha\delta}=0.$
(3.20)

Consequently, there is a matrix
$\\{\tilde{G}_{\gamma\alpha\delta}\\}_{\gamma,\alpha}$,
${|\alpha|{=}|\gamma|{=}m}$, such that

$\tilde{G}_{\alpha\gamma\delta}=-\tilde{G}_{\gamma\alpha\delta},\quad\tilde{g}_{\alpha\delta}=\sum_{|\gamma|=m}D^{\gamma}\tilde{G}_{\gamma\alpha\delta},$
(3.21)

and $\tilde{G}_{\gamma\alpha\delta}$ satisfies $H^{m}$-estimate of kind
(3.14).

###### Remark 3.3

Taking into account the structure of the functions $F_{\alpha,\delta}$ defined
in (3.17), we can simplify the formula for the coefficients $b_{\alpha\delta}$
in such a way that $b_{\alpha\delta}$ will be expressed only in terms of the
solutions to (3.2) and (3.16).

###### Remark 3.4

We can rewrite equation (3.16) for $N_{\delta}$, $|\delta|=m+1$, using firstly
(3.15)2, then (3.8) and (3.7), thus, excluding in the right-hand side firstly
the functions $G_{\alpha\gamma\beta}$ and then $g_{\alpha\beta}$. As a result,
the right-hand side in (3.16) will be determined only by the functions
$N_{\gamma}$, $|\gamma|=m$, its derivatives, and the coefficients
$a_{\alpha\beta}$. For further calculations, we prefer the equation for
$N_{\delta}$, $|\delta|=m+1$, in display (3.16).

3.3. As a corollary of Lemma 3.2, we have the next

###### Lemma 3.5

Let a 1-periodic vector $\\{g_{\alpha}\\}_{|\alpha|=m}\in L_{\rm
per}^{2}(Y)^{p}$ satisfy the assumptions of Lemma 3.2, and let a 1-periodic
matrix $\\{G_{\gamma\alpha}(y)\\}_{|\alpha|=|\gamma|=m}$ be the associated
matrix potential such that (3.13) and (3.14) hold. Then

$\displaystyle{g_{\alpha}(x/\varepsilon)\Phi(x)}\atop\displaystyle{=\sum_{|\gamma|=m}D^{\gamma}(\varepsilon^{m}G^{\varepsilon}_{\gamma\alpha}\Phi)-\sum_{|\gamma|=m}\sum_{\mu<\gamma}\varepsilon^{m-|\mu|}c_{\gamma,\mu}(D^{\mu}G_{\gamma\alpha})^{\varepsilon}D^{\gamma-\mu}\Phi}$
(3.22)

for any $\Phi\in C^{\infty}_{0}(\mathbb{R}^{d})$ and all indices $\alpha$,
$|\alpha|=m$, where the constants $c_{\gamma,\mu}$ are from (1.13).
Furthermore, the vector

$\\{M_{\alpha}(x)\\}_{|\alpha|=m},\quad
M_{\alpha}(x)=\sum_{|\gamma|=m}D^{\gamma}(G^{\varepsilon}_{\gamma\alpha}\Phi),$
(3.23)

satisfies the relation of kind (3.12)2, that is,

$\sum_{|\alpha|=m}D^{\alpha}M_{\alpha}=0\quad(\mbox{in the sense of
distributions on}\quad\mathbb{R}^{d}).$ (3.24)

Proof. By assumption,

$g_{\alpha}(y)=\sum_{|\gamma|=m}D^{\gamma}G_{\gamma\alpha}(y),\quad
g_{\alpha}(x/\varepsilon)\Phi(x)=\sum_{|\gamma|=m}D^{\gamma}(\varepsilon^{m}G_{\gamma\alpha}(x/\varepsilon))\Phi(x).$

Hence, to obtain (3.22), it suffices to recall the product rule (1.13) taking
into account the two-scale nature of the expression
$g_{\alpha}(x/\varepsilon)\Phi(x)$ and using the notation (1.14).

The relation (3.24) holds for the vector (3.23) because of the integral
identity

$\int_{\mathbb{R}^{d}}\varphi\sum_{|\alpha|=m}D^{\alpha}M_{\alpha}\,dx=0\quad\forall\varphi\in
C^{\infty}_{0}(\mathbb{R}^{d})$

obtained from the chain of integral identities

$\int_{\mathbb{R}^{d}}\varphi\sum_{|\alpha|=m}D^{\alpha}M_{\alpha}\,dx=\int_{\mathbb{R}^{d}}\varphi\sum_{|\gamma|=|\alpha|=m}D^{\gamma}D^{\alpha}(G^{\varepsilon}_{\gamma\alpha}\Phi)\,dx$
$=\int_{\mathbb{R}^{d}}\sum_{|\gamma|=|\alpha|=m}(D^{\gamma}D^{\alpha}\varphi)G^{\varepsilon}_{\gamma\alpha}\Phi\,dx=\int_{\mathbb{R}^{d}}\Phi\sum_{|\gamma|=|\alpha|=m}(D^{\gamma}D^{\alpha}\varphi)G^{\varepsilon}_{\gamma\alpha}\,dx,$

and the pointwise identity

$\sum_{|\gamma|=|\alpha|=m}(D^{\gamma}D^{\alpha}\varphi)G^{\varepsilon}_{\gamma\alpha}=0$

since the matrix $G_{\gamma\alpha}$ is skew-symmetric (see (3.13)1).

##  4 Smoothing operators

In this paper, we prove homogenization estimates by method, coming from [9]
and [10] (see also the overview [13]), where the difficulties connected with
the lack of regularity in data are overcome by using smoothing operators,
among which we highlight out Steklov‘s smoothing operator $S^{\varepsilon}$
defined in (2.9). We start with the simplest properties of the operator
$S^{\varepsilon}$:

$\|S^{\varepsilon}\varphi\|\leq\|\varphi\|,$ (4.1)
$\|S^{\varepsilon}\varphi-\varphi\|\leq(\sqrt{d}/2)\varepsilon\|\nabla\varphi\|\quad\forall\varphi\in
H^{1}(\mathbb{R}^{d}).$ (4.2)

Here and throughout this Section, $\|\cdot\|$ denotes, for brevity, the norm
in $L^{2}(\mathbb{R}^{d})$. We mention also the evident property
$S^{\varepsilon}(D^{\alpha}\varphi)=D^{\alpha}(S^{\varepsilon}\varphi)$ for
any derivative $D^{\alpha}$, which is exploited systematically in the sequel.

The following property of the Steklov smoothing operator $S^{\varepsilon}$ to
interrelate with $\varepsilon$-periodic factors is pivotal for our method (see
its simple proof, e.g., in [10] or [13]).

###### Lemma 4.1

If $\varphi{\in}L^{2}(\mathbb{R}^{d})$, $b{\in}L^{2}_{\rm per}(Y)$, and
$b^{\varepsilon}(x){=}b(\varepsilon^{-1}x)$, then

$\|b^{\varepsilon}S^{\varepsilon}\varphi\|\leq\langle
b^{2}\rangle^{1/2}\|\varphi\|.$ (4.3)

The above estimate (4.2) can be improved under higher regularity conditions.
For example,

$\|S^{\varepsilon}\varphi-\varphi\|\leq
C\varepsilon^{2}\|\nabla^{2}\varphi\|\quad\forall\varphi\in
H^{2}(\mathbb{R}^{d}),\quad C=const(d).$ (4.4)

By duality, from (4.4) we obtain

$\|S^{\varepsilon}\varphi-\varphi\|_{H^{-2}(\mathbb{R}^{d})}\leq
C\varepsilon^{2}\|\varphi\|_{L^{2}(\mathbb{R}^{d})}\quad\forall\varphi\in
L^{2}(\mathbb{R}^{d}),\quad C=const(d).$ (4.5)

Instead of Steklov‘s smoothing operator, we can often consider the general
smoothing operator with an arbitrary kernel. This is

$\Theta^{\varepsilon}\varphi(x)=\int_{\mathbb{R}^{d}}\varphi(x-\varepsilon\omega)\theta(\omega)\,d\omega,$
(4.6)

where $\theta\in L^{\infty}(\mathbb{R}^{d})$ is compactly supported,
$\theta\geq 0$, and $\int_{\mathbb{R}^{d}}\theta(x)dx{=}1$.

The estimates (4.1)–(4.3) formulated for the Steklov smoothing operator remain
valid for the general smoothig operator (4.6) with only one note that right-
hand side contains constants depending not only on the dimension $d$, but also
on the kernel $\theta$. If, in addition, the kernel $\theta$ is even, then
$\Theta^{\varepsilon}$ possesses the properties of type (4.4) and (4.5).

The following useful properties of the operator (4.6) are highlighted in [14]
and [8].

###### Lemma 4.2

Assume that $\theta$ is piecewise $C^{k}$-smooth, $k$ is a natural number,
$b\in L^{2}_{\rm per}(Y)$, $b^{\varepsilon}(x)=b(x/\varepsilon)$, and
$\varphi\in L^{2}(\mathbb{R}^{d})$. Then

$\|\Theta^{\varepsilon}\nabla^{k}\varphi\|\leq
C\varepsilon^{-k}\|\varphi\|,\quad C=const(\theta),$ (4.7)
$\|b^{\varepsilon}\Theta^{\varepsilon}\nabla^{k}\varphi\|\leq
C\varepsilon^{-k}\langle b^{2}\rangle^{1/2}\|\varphi\|,\quad C=const(\theta).$
(4.8)

It is obvious that the Steklov smoothing operator $S^{\varepsilon}$ can be
defined by (4.6) if the smoothing kernel $\theta_{1}(x)$ is the characteristic
function of the cube $Y{=}[-{1}/{2},{1}/{2})^{d}$. The double Steklov
smoothing operator $(S^{\varepsilon})^{2}=S^{\varepsilon}S^{\varepsilon}$ has
the form (4.6) with the smoothing kernel equal to the convolution
$\theta_{2}=\theta_{1}*\theta_{1}$. The kernels $\theta_{2}$ (see computations
for it in [8]) turns to be piecewise $C^{1}$-smooth. As a consequence, the
properties (4.7) and (4.8) with $k{=}1$ hold for the smoothing operator
$\Theta^{\varepsilon}=(S^{\varepsilon})^{2}$. In addition, since $\theta_{2}$
is even, the operator $(S^{\varepsilon})^{2}$ possesses the properties of type
(4.4) and (4.5).

##  5 Preliminary calculations

Before we proceed to the direct proof of the estimate (2.10), we make some
formal calculations under strong regularity assumptions which will be later
taken off. Let

$v^{\varepsilon}(x)=u(x)+\varepsilon^{m}\sum\limits_{|\gamma|=m}N^{\varepsilon}_{\gamma}(x)D^{\gamma}u(x)+\varepsilon^{m+1}\sum\limits_{|\delta|=m+1}N_{\delta}^{\varepsilon}(x)D^{\delta}u(x),$
(5.1)

where $N_{\gamma}$, $|\gamma|=m$, and $N_{\delta}$, $|\delta|=m+1$, are the
solutions to the cell problems (3.2) and (3.16) respectively.

We first suppose that the function $u(x)$ in (5.1) is infinitely
differentiable and, together with its derivatives, is decreasing at infinity
sufficiently rapidly, so that $v^{\varepsilon}\in H^{m}(\mathbb{R}^{d})$.
Therefore, the discrepancy of the function $v^{\varepsilon}$ in (1.1), that is
$(A_{\varepsilon}+1)v^{\varepsilon}-f$, can be calculated. But let the right-
hand side function $f$ be of the form $f=(\hat{A}_{\varepsilon}+1)u$, where
$\hat{A}_{\varepsilon}$ is the homogenized operator defined in (2.1) and the
function $u(x)$ is the same as in (5.1). Thus,

$\displaystyle{(A_{\varepsilon}+1)v^{\varepsilon}-f=(A_{\varepsilon}+1)v^{\varepsilon}-(\hat{A}_{\varepsilon}+1)u=A_{\varepsilon}v^{\varepsilon}-\hat{A}_{\varepsilon}u+(v^{\varepsilon}-u)}\atop\displaystyle{\stackrel{{\scriptstyle(\ref{1}),(\ref{2.1})}}{{=}}\sum_{|\alpha|=m}(-1)^{m}D^{\alpha}(\Gamma_{\alpha}(v^{\varepsilon},A_{\varepsilon})-\Gamma_{\alpha}(u,\hat{A}_{\varepsilon}))+(v^{\varepsilon}-u),}$
(5.2)

where we introduce the generalized gradients

$\displaystyle{\Gamma_{\alpha}(v^{\varepsilon},A_{\varepsilon})=\sum\limits_{|\beta|=m}a^{\varepsilon}_{\alpha\beta}D^{\beta}v^{\varepsilon},}\atop\displaystyle{\Gamma_{\alpha}(u,\hat{A}_{\varepsilon}){=}\Gamma_{\alpha}(u,\hat{A}){+}\Gamma_{\alpha}(u,\varepsilon
B){=}\\!\sum\limits_{|\beta|=m}\hat{a}_{\alpha\beta}D^{\beta}u{+}\\!\sum\limits_{|\delta|=m+1}\varepsilon
b_{\alpha\delta}D^{\delta}u}$ (5.3)

for any index $\alpha$, $|\alpha|=m$.

In view of (5.2), we need to compare the generalized gradients. A simple
calculation shows that, for any $\beta$, $|\beta|=m,$

$D^{\beta}(\varepsilon^{m}N^{\varepsilon}_{\gamma}D^{\gamma}u)=(D^{\beta}N_{\gamma})^{\varepsilon}D^{\gamma}u+\sum_{\mu<\beta}\varepsilon^{m-|\mu|}c_{\beta,\mu}(D^{\mu}N_{\gamma})^{\varepsilon}D^{\beta+\gamma-\mu}u,$

by the product rule (1.13). Thus, due to (5.3)1 and (5.1), we have

$\Gamma_{\alpha}(v^{\varepsilon},A_{\varepsilon})=\sum_{|\beta|=m}a^{\varepsilon}_{\alpha\beta}D^{\beta}(u+\varepsilon^{m}\\!\\!\sum_{|\gamma|=m}N_{\gamma}^{\varepsilon}D^{\gamma}u+\varepsilon^{m+1}\\!\\!\sum\limits_{|\delta|{=}m{+}1}N_{\delta}^{\varepsilon}D^{\delta}u)$
$=\\!\sum_{|\beta|=m}(a^{\varepsilon}_{\alpha\beta}D^{\beta}u{+}\sum_{|\gamma|=m}a^{\varepsilon}_{\alpha\gamma}(D^{\gamma}N_{\beta})^{\varepsilon}D^{\beta}u)+\varepsilon\sum_{|\gamma|=m}\sum\limits_{|\delta|{=}m{+}1}\\!a^{\varepsilon}_{\alpha\gamma}(D^{\gamma}N_{\delta})^{\varepsilon}D^{\delta}u$
$+\varepsilon\sum_{|\gamma|=|\beta|=m}\,\sum_{\mu<\gamma,|\mu|=m-1}a^{\varepsilon}_{\alpha\gamma}c_{\gamma,\mu}(D^{\mu}N_{\beta})^{\varepsilon}D^{\beta+\gamma-\mu}u+w^{\varepsilon}_{\alpha},$

where the sum

$\displaystyle{w^{\varepsilon}_{\alpha}=\sum_{|\gamma|=|\beta|=m}\,\sum_{\mu<\gamma,|\mu|<m-1}\varepsilon^{m-|\mu|}a^{\varepsilon}_{\alpha\gamma}c_{\gamma,\mu}(D^{\mu}N_{\beta})^{\varepsilon}D^{\beta+\gamma-\mu}u}\atop\displaystyle{+\sum_{|\gamma|=m}\sum\limits_{|\delta|{=}m{+}1}\sum_{\mu<\gamma,|\mu|<m}\varepsilon^{m+1-|\mu|}a^{\varepsilon}_{\alpha\gamma}c_{\gamma,\mu}(D^{\mu}N_{\delta})^{\varepsilon}D^{\delta+\gamma-\mu}u}$
(5.4)

collects all the terms containing powers $\varepsilon^{n}$, $n\geq 2$, as
factors.

Using the notation from (3.6)–(3.8), we transform the first sum in the above
representation of $\Gamma_{\alpha}(v^{\varepsilon},A_{\varepsilon})$ as
follows:

$\sum_{|\beta|=m}(a^{\varepsilon}_{\alpha\beta}D^{\beta}u{+}\sum_{|\gamma|=m}a^{\varepsilon}_{\alpha\gamma}(D^{\gamma}N_{\beta})^{\varepsilon}D^{\beta}u)=\\!\sum_{|\beta|=|\gamma|=m}a^{\varepsilon}_{\alpha\gamma}(e_{\gamma\beta}+(D^{\gamma}N_{\beta})^{\varepsilon})D^{\beta}u$
$\stackrel{{\scriptstyle(\ref{c4}),(\ref{5.3})_{2}}}{{=}}\sum_{|\beta|=m}(\tilde{a}^{\varepsilon}_{\alpha\beta}-\hat{a}_{\alpha\beta})D^{\beta}u+\Gamma_{\alpha}(u,\hat{A})=\Gamma_{\alpha}(u,\hat{A})+\sum_{|\beta|=m}g^{\varepsilon}_{\alpha\beta}D^{\beta}u.$

Consequently, we rewrite the representation

$\Gamma_{\alpha}(v^{\varepsilon},A_{\varepsilon})=\Gamma_{\alpha}(u,\hat{A})+\sum_{|\beta|=m}g^{\varepsilon}_{\alpha\beta}D^{\beta}u+\varepsilon\sum_{|\gamma|=m}\sum\limits_{|\delta|{=}m{+}1}a^{\varepsilon}_{\alpha\gamma}(D^{\gamma}N_{\delta})^{\varepsilon}D^{\delta}u$
$+\varepsilon\sum_{|\gamma|=|\beta|=m}\,\sum_{\mu<\gamma,|\mu|=m-1}a^{\varepsilon}_{\alpha\gamma}c_{\gamma,\mu}(D^{\mu}N_{\beta})^{\varepsilon}D^{\beta+\gamma-\mu}u+w^{\varepsilon}_{\alpha}.$
(5.5)

Applying Lemma 3.5 to the term $g^{\varepsilon}_{\alpha\beta}D^{\beta}u$ (note
that the vector $\\{g_{\alpha\beta}\\}_{\alpha}$, with $\beta$ fixed,
satisfies the assumptions of Lemma 3.5), we obtain

$\displaystyle{g^{\varepsilon}_{\alpha\beta}D^{\beta}u=\sum_{|\gamma|=m}(D^{\gamma}G_{\gamma\alpha\beta})^{\varepsilon}D^{\beta}u}\atop\displaystyle{=\sum_{|\gamma|=m}D^{\gamma}(\varepsilon^{m}G^{\varepsilon}_{\gamma\alpha\beta}D^{\beta}u)-\sum_{|\gamma|=m}\sum_{\mu<\gamma}\varepsilon^{m-|\mu|}c_{\gamma,\mu}(D^{\mu}G_{\gamma\alpha\beta})^{\varepsilon}D^{\gamma-\mu}D^{\beta}u.}$
(5.6)

Setting

$M_{\alpha}:=\sum_{|\gamma|=m}D^{\gamma}(G^{\varepsilon}_{\gamma\alpha\beta}D^{\beta}u),$

we get the vector $\\{M_{\alpha}(x)\\}_{|\alpha|=m}$ with the property (3.24).
Therefore,

$\displaystyle{\sum_{|\alpha|=m}D^{\alpha}(g^{\varepsilon}_{\alpha\beta}D^{\beta}u)\stackrel{{\scriptstyle(\ref{5.6})}}{{=}}{-}\\!\sum_{|\alpha|=|\gamma|=m}D^{\alpha}\sum_{\mu<\gamma}\varepsilon^{m-|\mu|}c_{\gamma,\mu}(D^{\mu}G_{\gamma\alpha\beta})^{\varepsilon}D^{\beta+\gamma-\mu}u}\atop\displaystyle{=\\!\sum_{|\alpha|=|\gamma|=m}D^{\alpha}\,\sum_{\mu<\gamma,|\mu|=m-1}\varepsilon
c_{\gamma,\mu}(D^{\mu}G_{\alpha\gamma\beta})^{\varepsilon}D^{\beta+\gamma-\mu}u+\sum_{|\alpha|=m}D^{\alpha}{w}^{\varepsilon}_{\alpha\beta},}$
(5.7)

where the sum

${w}^{\varepsilon}_{\alpha\beta}=\sum_{|\gamma|=m}\sum_{\mu<\gamma,|\mu|<m-1}\varepsilon^{m-|\mu|}c_{\gamma,\mu}(D^{\mu}G_{\alpha\gamma\beta})^{\varepsilon}D^{\beta+\gamma-\mu}u$
(5.8)

collects the terms with powers $\varepsilon^{n}$, $n\geq 2$, as a factor. To
change the sign in (5.7), the skew-symmetry property of
$G_{\alpha\gamma\beta}$ is used, i.e.,
$G_{\alpha\gamma\beta}=-G_{\gamma\alpha\beta}$.

From (5.3), (5.5), and (5.7) it follows that

$R_{\varepsilon}:=\sum_{|\alpha|=m}D^{\alpha}(\Gamma_{\alpha}(v^{\varepsilon},A_{\varepsilon})-\Gamma_{\alpha}(u,\hat{A}_{\varepsilon}))$
(5.9)
$=\sum_{|\alpha|=m}D^{\alpha}(\Gamma_{\alpha}(v^{\varepsilon},A_{\varepsilon})-\Gamma_{\alpha}(u,\hat{A})-\Gamma_{\alpha}(u,\varepsilon
B))$
$=\varepsilon\sum_{|\alpha|=|\gamma|=m}D^{\alpha}\sum\limits_{|\delta|{=}m{+}1}a^{\varepsilon}_{\alpha\gamma}(D^{\gamma}N_{\delta})^{\varepsilon}D^{\delta}u$
$+\varepsilon\sum_{|\alpha|=|\gamma|=|\beta|=m}D^{\alpha}(\sum_{\mu<\gamma,|\mu|=m-1}a^{\varepsilon}_{\alpha\gamma}c_{\gamma,\mu}(D^{\mu}N_{\beta})^{\varepsilon}D^{\beta+\gamma-\mu}u$
$+\sum_{\mu<\gamma,|\mu|=m-1}c_{\gamma,\mu}(D^{\mu}G_{\alpha\gamma\beta})^{\varepsilon}D^{\beta+\gamma-\mu}u)-\varepsilon\sum\limits_{|\alpha|=m}D^{\alpha}\sum\limits_{|\delta|=m+1}b_{\alpha\delta}D^{\delta}u$
$+\sum_{|\alpha|=m}D^{\alpha}w^{\varepsilon}_{\alpha}+\sum_{|\alpha|=|\beta|=m}D^{\alpha}{w}^{\varepsilon}_{\alpha\beta},$

where $w^{\varepsilon}_{\alpha}$ and ${w}^{\varepsilon}_{\alpha\beta}$ defined
in (5.4) and (5.8) are of order $O(\varepsilon^{2})$ for small $\varepsilon$.
We can shortly write

$R_{\varepsilon}=\varepsilon
R_{\varepsilon,1}+\sum_{|\alpha|=m}D^{\alpha}w^{\varepsilon}_{\alpha}+\sum_{|\alpha|=|\beta|=m}D^{\alpha}{w}^{\varepsilon}_{\alpha\beta},$

where the summand $\varepsilon R_{\varepsilon,1}$ absorbs all the terms
containing the factor $\varepsilon$ in the above representation. It holds
$D^{\delta}{=}D^{\beta+\gamma-\mu}$ if the exponents coincide, i.e.,
$\delta{=}\beta+\gamma-\mu$ or $\mu{=}\beta+\gamma-\delta$. In this case,
$\mu{<}\gamma\Leftrightarrow\beta{<}\delta$. Rearranging the summation in the
above representation, we get

$\displaystyle{\varepsilon
R_{\varepsilon,1}=\varepsilon\sum_{|\alpha|=m}D^{\alpha}\sum_{\delta=m+1}D^{\delta}u[\sum_{|\gamma|=m}(a_{\alpha\gamma}D^{\gamma}N_{\delta}}\atop\displaystyle{+\sum_{|\beta|=m,\beta<\delta}c_{\gamma,\beta+\gamma-\delta}(a_{\alpha\gamma}D^{\beta+\gamma-\delta}N_{\beta}+D^{\beta+\gamma-\delta}G_{\alpha\gamma\beta}))^{\varepsilon}-b_{\alpha\delta}].}$

The oscillating $\varepsilon$-periodic multiplier standing above in the square
brackets can be written as $\tilde{g}^{\varepsilon}_{\alpha\delta}$, and the
corresponding 1-periodic function $\tilde{g}_{\alpha\delta}$ is just the same
as in (3.18) according to (3.17)–(3.19).

We summarize in a short way what is obtained by this moment:

$R_{\varepsilon}=\varepsilon\sum_{|\alpha|=m}D^{\alpha}\sum_{\delta=m+1}\tilde{g}^{\varepsilon}_{\alpha\delta}D^{\delta}u+\sum_{|\alpha|=m}D^{\alpha}w^{\varepsilon}_{\alpha}+\sum_{|\alpha|=|\beta|=m}D^{\alpha}{w}^{\varepsilon}_{\alpha\beta}.$
(5.10)

For each fixed multiindex $\delta$, $|\delta|{=}m+1$, by properties of
$\tilde{g}_{\alpha\delta}$ (see (3.20) and (3.21)) and Lemma 3.5, it holds

$\displaystyle{\tilde{g}_{\alpha\delta}(x/\varepsilon)\Phi(x)}\atop\displaystyle{=\sum_{|\gamma|=m}D^{\gamma}(\varepsilon^{m}\tilde{G}^{\varepsilon}_{\gamma\alpha\delta}\Phi)-\sum_{|\gamma|=m}\sum_{\mu<\gamma}\varepsilon^{m-|\mu|}c_{\gamma,\mu}(D^{\mu}\tilde{G}_{\gamma\alpha\delta})^{\varepsilon}D^{\gamma-\mu}\Phi}$

with $\Phi=D^{\delta}u$ and the matrix
$\\{\tilde{G}_{\gamma\alpha\delta}\\}_{\gamma,\alpha}$,
${|\alpha|{=}|\gamma|{=}m}$, satisfying (3.21), where

$\sum_{|\alpha|=m}D^{\alpha}\sum_{|\gamma|=m}D^{\gamma}(\varepsilon^{m}\tilde{G}^{\varepsilon}_{\gamma\alpha\delta}\Phi)=0.$

Finally, we transform (5.10) as follows:

$\displaystyle{R_{\varepsilon}=-\varepsilon\sum_{|\alpha|=|\gamma|=m}D^{\alpha}\sum_{\delta=m+1}\,\sum_{\mu<\gamma}\varepsilon^{m-|\mu|}c_{\gamma,\mu}(D^{\mu}\tilde{G}_{\gamma\alpha\delta})^{\varepsilon}D^{\gamma-\mu+\delta}u}\atop\displaystyle{+\sum_{|\alpha|=m}D^{\alpha}w^{\varepsilon}_{\alpha}+\sum_{|\alpha|=|\beta|=m}D^{\alpha}{w}^{\varepsilon}_{\alpha\beta}.}$
(5.11)

Thus, each term in the right-hand side of (5.11) includes some power
$\varepsilon^{n}$, $n\geq 2$, as a factor (see the definitions (5.4) and (5.8)
for $w^{\varepsilon}_{\alpha}$ and ${w}^{\varepsilon}_{\alpha\beta}$). The
same is true for the the entire right-hand side of (5.2) if we recall the
notation (5.9) and the equality

$v^{\varepsilon}-u\stackrel{{\scriptstyle(\ref{5.1})}}{{=}}\varepsilon^{m}\sum\limits_{|\gamma|=m}N^{\varepsilon}_{\gamma}D^{\gamma}u+\varepsilon^{m+1}\sum\limits_{|\delta|=m+1}N_{\delta}^{\varepsilon}D^{\delta}u,$

where $m\geq 2$. This yields the final representation for the discrepancy of
the function (5.1) with equation (1.1) as the sum of several terms of the same
type

$(A_{\varepsilon}+1)v^{\varepsilon}{-}f{=}\sum_{j}\varepsilon^{n_{j}}b_{j}(x/\varepsilon)\Phi_{j}(x){+}\\!\sum_{|\alpha|=m}D^{\alpha}\sum_{j}\varepsilon^{n_{j}}\tilde{b}_{j}(x/\varepsilon)\tilde{\Phi}_{j}(x).$
(5.12)

Here, any exponent $n_{j}\geq 2$ and all $\varepsilon$-periodic functions
$b_{j}(x/\varepsilon)$ and $\tilde{b}_{j}(x/\varepsilon)$ are formed of their
1-periodic counterparts from the list

$a_{\alpha\beta},\,N_{\gamma},\,D^{\mu}N_{\gamma},\,G_{\gamma\alpha\beta},\,D^{\mu}G_{\gamma\alpha\beta},\,\tilde{G}_{\gamma\alpha\beta},\,D^{\mu}\tilde{G}_{\gamma\alpha\beta}$
(5.13)

consisting of the coefficients of (1.1), the solutions to the cell problems
(3.2) and (3.16) together with the derivatives up to the order $m$, the
components of the matrix potentials from (3.15) and (3.21) together with the
derivatives up to the order $m$. The functions $\Phi_{j}$ and
$\tilde{\Phi}_{j}$ in (5.12) coincide with the function $u$ or its derivatives
$D^{\nu}u$ up to the order $2m+1$, where $(\hat{A}_{\varepsilon}+1)u=f$.

Since
$({A}_{\varepsilon}{+}1)u^{\varepsilon}{=}f{=}(\hat{A}_{\varepsilon}{+}1)u$
for the solution $u^{\varepsilon}$ to (1.1), from (5.12) we get

$(A_{\varepsilon}+1)(v^{\varepsilon}-u^{\varepsilon}){=}O(\varepsilon^{2}),$

whereof, by the energy estimate, we derive the estimate

$\|v^{\varepsilon}-u^{\varepsilon}\|_{H^{m}(\mathbb{R}^{d})}=O(\varepsilon^{2}),$

which is not the same as the desired estimate (2.10). We remind that our
preceding calculations and the final estimate make sense only under strong
regularity assumptions. Furthermore, the majorant in the extracted estimate
cannot be guaranteed to be of the type given in (2.10). We will show how to
cope with these difficulties in the next Section.

##  6 $H^{m}$-estimate of order $\varepsilon^{2}$

In this Section, we will prove the $H^{m}$-estimate (2.10) for the difference
between the solution $u^{\varepsilon}$ to (1.1) and its approximation defined
in (2.6).

6.1. Firstly, let us verify that the function (2.6) belongs to the space
$H^{m}(\mathbb{R}^{d})$. It suffices to show that every term in the both
correctors (2.7), together with its derivatives up to order $m$, belongs to
the space $L^{2}(\mathbb{R}^{d})$. For this we apply lemmas 4.1 and 4.2. The
latter one is employed to handle the products

$\varepsilon^{m+1}N_{\delta}(x/\varepsilon)D^{\alpha+\delta}u^{,\varepsilon}(x),\quad|\alpha|=m,\,|\delta|=m+1,$
(6.1)

that emerge from differentiation of order $m$ of the second corrector in
(2.7). In this case,

$N_{\delta}(x/\varepsilon)D^{\alpha+\delta}u^{,\varepsilon}(x)\stackrel{{\scriptstyle(\ref{2.8})}}{{=}}N_{\delta}(x/\varepsilon)D^{\alpha+\delta}\Theta^{\varepsilon}\,\hat{u}^{\varepsilon}(x)=N_{\delta}(x/\varepsilon)\Theta^{\varepsilon}D_{j}\varphi(x),$

where $\varphi=D^{\nu}\hat{u}^{\varepsilon}$ for some multiindex $\nu$,
$|\nu|=2m$, and thereby, $\varphi\in L^{2}(\mathbb{R}^{d})$ with the estimate

$\|\varphi\|_{L^{2}(\mathbb{R}^{d})}=\|D^{\nu}\hat{u}^{\varepsilon}\|_{L^{2}(\mathbb{R}^{d})}\stackrel{{\scriptstyle(\ref{2.5})}}{{\leq}}C\|f\|_{L^{2}(\mathbb{R}^{d})}.$
(6.2)

It remains to note that the smoothing operator
$\Theta^{\varepsilon}=S^{\varepsilon}S^{\varepsilon}$ has the smoothing kernel
which is piecewise $C^{1}$-smooth (see discussion in the end of Section 4).
Thus, applying (4.8) yields

$\|N_{\delta}^{\varepsilon}D^{\alpha+\delta}u^{,\varepsilon}\|_{L^{2}(\mathbb{R}^{d})}=\|N_{\delta}^{\varepsilon}\Theta^{\varepsilon}D_{j}\varphi\|_{L^{2}(\mathbb{R}^{d})}\leq
C\varepsilon^{-1}\langle
N_{\delta}^{2}\rangle^{1/2}\|\varphi\|_{L^{2}(\mathbb{R}^{d})};$

therefore, in view of (6.2) and the bound of type (3.4) for $N_{\delta}$, we
derive

$\|\varepsilon^{m+1}N_{\delta}^{\varepsilon}D^{\alpha+\delta}u^{,\varepsilon}\|_{L^{2}(\mathbb{R}^{d})}\leq
C\varepsilon^{m}\|f\|_{L^{2}(\mathbb{R}^{d})},\quad m\geq 2,$ (6.3)

where the constant $C$ depends on $d$ and the constants in (1.2). We have just
proved more than it was required, that is, the $L^{2}(\mathbb{R}^{d})$-bound
for the term (6.1); but we have found the majorant of special structure for
the latter (see (6.3)). Here is the first place we need to employ the
iteration of Steklov‘s smoothing operator while mere Steklov‘s smoothing is
not enough.

The differentiation of order $m$ of the correctors in (2.7) produces also the
products other than those of (6.1), namely,

$\varepsilon^{m-|\mu|}(D^{\mu}N_{\gamma})^{\varepsilon}D^{\gamma+\alpha-\mu}u^{,\varepsilon},\quad
0\leq|\mu|\leq m,\,|\alpha|=|\gamma|=m,$ (6.4)

and

$\varepsilon^{m+1-|\mu|}(D^{\mu}N_{\delta})^{\varepsilon}D^{\delta+\alpha-\mu}u^{,\varepsilon},\quad
0<|\mu|\leq m,\,|\alpha|=m,\,|\delta|=m+1.$ (6.5)

We handle the terms of types (6.4) and (6.5) by straightforward applying of
Lemma 4.1. We omit details which are similar to those used to obtain (6.3).

6.2. We pass to the proof of (2.10). The discrepancy of the function (2.6) in
equation (1.1) admits the representation

$\displaystyle{A_{\varepsilon}\tilde{u}^{\varepsilon}{+}\tilde{u}^{\varepsilon}{-}f=(A_{\varepsilon}{+}1)\tilde{u}^{\varepsilon}{-}(\hat{A}_{\varepsilon}{+}1)u^{,\varepsilon}{+}(f^{,\varepsilon}{-}f)}\atop\displaystyle{=(A_{\varepsilon}\tilde{u}^{\varepsilon}{-}\hat{A}_{\varepsilon}u^{,\varepsilon}){+}(\tilde{u}^{\varepsilon}{-}u^{,\varepsilon}){+}(f^{,\varepsilon}{-}f),}$
(6.6)

where at the first step we take into account the equality

$(\hat{A}_{\varepsilon}+1)u^{,\varepsilon}=f^{,\varepsilon},\quad
f^{,\varepsilon}=\Theta^{\varepsilon}f,$

obtained by applying the smoothing operator $\Theta^{\varepsilon}$ to the both
parts of (2.2) if the notation
$\Theta^{\varepsilon}\hat{u}^{\varepsilon}=u^{,\varepsilon}$ is used (see
(2.8)).

Comparing (2.6) with (5.1) and, in parallel, (6.6) with (5.2), we see that
$\tilde{u}^{\varepsilon}$ correlates to $u^{,\varepsilon}$ in (2.6) as
$v^{\varepsilon}$ to $u$ in (5.1). The calculations accomplished for the pair
$v^{\varepsilon}$ and $u$ in Section 5 can be repeated for the pair
$\tilde{u}^{\varepsilon}$ and $u^{,\varepsilon}$. By this process all
expressions and all transitions make sense, which can be justified with the
help of arguments similar to those used in Subsection 6.1. We rely in an
essential way on lemmas 4.1 and 4.2. Thus, similarly as in (5.12) (see also
(5.2)), we obtain

$\displaystyle{(A_{\varepsilon}{+}1)\tilde{u}^{\varepsilon}{-}(\hat{A}_{\varepsilon}{+}1)u^{,\varepsilon}}\atop\displaystyle{=\sum_{j}\varepsilon^{n_{j}}b_{j}(x/\varepsilon)\Phi_{j}(x)+\sum_{|\alpha|=m}D^{\alpha}\sum_{j}\varepsilon^{n_{j}}\tilde{b}_{j}(x/\varepsilon)\tilde{\Phi}_{j}(x)=:F_{\varepsilon},\,n_{j}{\geq}2,}$
(6.7)

with the only difference that the functions $\Phi_{j}$ and $\tilde{\Phi}_{j}$
in (6.7) coincide with the function $u^{,\varepsilon}$ (defined in (2.8)) or
the derivatives $D^{\nu}u^{,\varepsilon}$ up to the order $2m+1$. The right-
hand side of (6.7) denoted by $F_{\varepsilon}$ has the estimate

$\|F_{\varepsilon}\|_{H^{-m}(\mathbb{R}^{d})}\leq
C\varepsilon^{2}\|f\|_{L^{2}(\mathbb{R}^{d})}.$ (6.8)

As for the remaining term $f^{,\varepsilon}{-}f$ in the right-hand side of
(6.6), it also possesses the desired estimate, due to (4.5), because, for
$m\geq 2$, it holds

$\|f^{,\varepsilon}{-}f\|_{H^{-m}(\mathbb{R}^{d})}\leq\|f^{,\varepsilon}{-}f\|_{H^{-2}(\mathbb{R}^{d})}\leq
C\varepsilon^{2}\|f\|_{L^{2}(\mathbb{R}^{d})}.$ (6.9)

From (6.6) and the last two estimates, we readily come to (2.10). In fact,

$A_{\varepsilon}\tilde{u}^{\varepsilon}{+}\tilde{u}^{\varepsilon}{-}f=A_{\varepsilon}\tilde{u}^{\varepsilon}{+}\tilde{u}^{\varepsilon}-(A_{\varepsilon}{u}^{\varepsilon}{+}{u}^{\varepsilon})=(A_{\varepsilon}+1)(\tilde{u}^{\varepsilon}-{u}^{\varepsilon});$

therefore, using the notation from (6.7), we have

$(A_{\varepsilon}+1)(\tilde{u}^{\varepsilon}-{u}^{\varepsilon})=F_{\varepsilon}+(f^{,\varepsilon}{-}f),$

and it suffices to recall the energy estimate for this equation together with
(6.8) and (6.9).

6.3. We conclude with some remarks.

###### Remark 6.1

The approximation (2.6) can be taken with arbitrary smoothing operator (4.6),
provided that the smoothing kernel is even and, at least, piecewise
$C^{1}$-smooth. In this case, the estimate (2.10) remains valid.

###### Remark 6.2

There is another approach to prove operator type estimates in homogenization.
It relies on the Floquet–Bloch transform and spectral expansions; thus, it is
often called the spectral method. This method is tightly linked with periodic
setting because of the Floquet–Bloch transform which fits only this setting.
One of the first applications of the spectral approach for homogenization
estimates can be found in [15] where diffusion problems were studied. As for
high order operators, this method was applied, e.g., in [11], [16], and [17].

###### Remark 6.3

The method demonstrated to prove estimates (2.10) and (2.13) can be extended
to more general situations when operators are matrix-valued. We choose to
consider here the scalar problem only for brevity of exposition.

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