Commutator estimates and Poisson bounds for
Dirichlet-to-Neumann operators
with variable coefficients
A.F.M. ter Elst1 and E.M. Ouhabaz2

Abstract

  * We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator

${\cal
A}u=-\sum_{k,l=1}^{d}\partial_{k}(c_{kl}\,\partial_{l}u)+\sum_{k=1}^{d}\Big{(}c_{k}\,\partial_{k}u-\partial_{k}(b_{k}\,u)\Big{)}+c_{0}\,u\in{\cal
D}^{\prime}(\Omega)$

with possibly complex coefficients. We study three problems:
1) Boundedness on $C^{\nu}$ and on $L_{p}$ of the commutator $[{\cal
N},M_{g}]$, where $M_{g}$ denotes the multiplication operator by a smooth
function $g$.
2) Hölder and $L_{p}$-bounds for the harmonic lifting associated with ${\cal
A}$.
3) Poisson bounds for the heat kernel of ${\cal N}$.
We solve these problems in the case where the coefficients are Hölder
continuous and the underlying domain is bounded and of class $C^{1+\kappa}$
for some $\kappa>0$. For the Poisson bounds we assume in addition that the
coefficients are real-valued. We also prove gradient estimates for the heat
kernel and the Green function $G$ of the elliptic operator with Dirichlet
boundary conditions.

Mathematics Subject Classification: 35K08, 58G11, 47B47.

Keywords: Dirichlet-to-Neumann operator, Poisson bounds, commutator estimates,
harmonic lifting, elliptic operators with complex coefficients, heat kernel
bounds, gradient estimates for Green functions.

Home institutions:
1. Department of Mathematics 2. Institut de Mathématiques de Bordeaux University of Auckland Université de Bordeaux, UMR 5251, Private bag 92019 351, Cours de la Libération Auckland 1142 33405 Talence New Zealand France<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction and the main results

Let $\Omega\subset\mathds{R}^{d}$ be an open set with $d\geq 2$ and let
$c_{kl},b_{k},c_{k},c_{0}\in L_{\infty}(\Omega)=L_{\infty}(\Omega,\mathds{C})$
for all $k,l\in\\{1,\ldots,d\\}$. We assume the usual ellipticity condition,
that is, there is a $\mu>0$ such that

$\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$

for all $\xi\in\mathds{C}^{d}$ and (a.e.) $x\in\Omega$. Define the
sesquilinear form $\gothic{a}\colon W^{1,2}(\Omega)\times
W^{1,2}(\Omega)\to\mathds{C}$ by

$\gothic{a}(u,v)=\sum_{k,l=1}^{d}\int_{\Omega}c_{kl}\,(\partial_{l}u)\,\,\overline{\partial_{k}v}+\sum_{k=1}^{d}\int_{\Omega}b_{k}\,u\,\overline{\partial_{k}v}+\sum_{k=1}^{d}\int_{\Omega}c_{k}\,(\partial_{k}u)\,\overline{v}+\int_{\Omega}c_{0}\,u\,\overline{v}.$
(1)

It is a basic fact that this form is continuous and quasi-coercive on
$W^{1,2}(\Omega)$. Its associated operator $A_{N}$ is the elliptic operator
with Neumann type boundary conditions. Consider
$\gothic{a}_{D}=\gothic{a}|_{W^{1,2}_{0}(\Omega)\times W^{1,2}_{0}(\Omega)}$,
the restriction of $\gothic{a}$ to the subspace $W^{1,2}_{0}(\Omega)$. Its
associated operator $A_{D}$ is the corresponding elliptic operator with
Dirichlet boundary conditions.

Further define ${\cal A}\colon W^{1,2}(\Omega)\to{\cal D}^{\prime}(\Omega)$ by

$\langle{\cal A}u,v\rangle_{{\cal D}^{\prime}(\Omega)\times{\cal
D}(\Omega)}=\gothic{a}(u,v).$

The operator ${\cal A}$ is given by

${\cal
A}u=-\sum_{k,l=1}^{d}\partial_{k}(c_{kl}\,\partial_{l}u)+\sum_{k=1}^{d}\Big{(}c_{k}\,\partial_{k}u-\partial_{k}(b_{k}\,u)\Big{)}+c_{0}\,u\in{\cal
D}^{\prime}(\Omega).$

A function $u\in W^{1,2}(\Omega)$ is called harmonic (or ${\cal A}$-harmonic)
if ${\cal A}u=0$.

Next suppose $\Omega$ is bounded and has Lipschitz boundary
$\Gamma=\partial\Omega$. We provide $\Gamma$ with the $(d-1)$-dimensional
Hausdorff measure. Let ${\mathop{\rm Tr\,}}\colon W^{1,2}(\Omega)\to
L_{2}(\Gamma)$ be the trace map. Note that ${\mathop{\rm
Tr\,}}W^{1,2}(\Omega)=H^{1/2}(\Gamma)$. If $0\not\in\sigma(A_{D})$, then there
exists a unique operator $\gamma\colon H^{1/2}(\Gamma)\to W^{1,2}(\Omega)$
such that ${\mathop{\rm Tr\,}}\gamma(\varphi)=\varphi$ and $\gamma(\varphi)$
is harmonic for all $\varphi\in H^{1/2}(\Gamma)$. We call $\gamma$ the
harmonic lifting.

Next we define the weak conormal derivative $\partial_{n}{a}$. If $u\in
W^{1,2}(\Omega)$ and $\psi\in L_{2}(\Gamma)$, then we say that $u\in
D(\partial_{n}{a})$ and $\partial_{n}{a}u=\psi$ if ${\cal A}u\in
L_{2}(\Omega)$ and

$\gothic{a}(u,v)-\int_{\Omega}({\cal
A}u)\,\overline{v}=\int_{\Gamma}\psi\,\overline{{\mathop{\rm Tr\,}}u}$

for all $v\in W^{1,2}(\Omega)$. It is a consequence of the Stone–Weierstraß
theorem that the function $\psi$ is unique. We call $\psi$ the (weak) conormal
derivative of $u$. If for example $c_{kl},b_{k}\in W^{1,\infty}(\Omega)$ for
all $k,l\in\\{1,\ldots,d\\}$ and the set $\Omega$ is bounded and Lipschitz,
then

$\partial_{n}{a}u=\sum_{k,l=1}^{d}n_{k}\,c_{kl}\,{\mathop{\rm
Tr\,}}(\partial_{l}u)+\sum_{k=1}^{d}n_{k}\,b_{k}\,{\mathop{\rm Tr\,}}u$

for all $u\in W^{2,2}(\Omega)$, where $n=(n_{1},\ldots,n_{d})$ is the outer
normal on $\Gamma$.

If $0\not\in\sigma(A_{D})$ and the set $\Omega$ is bounded and Lipschitz, then
we define the Dirichlet-to-Neumann operator ${\cal N}$ in $L_{2}(\Gamma)$ by

${\cal N}=\partial_{n}{a}\circ\gamma.$

So if $\varphi,\psi\in L_{2}(\Gamma)$, then $\varphi\in D({\cal N})$ and
${\cal N}\varphi=\psi$ if and only if there exists a harmonic function $u\in
W^{1,2}(\Omega)$ such that ${\mathop{\rm Tr\,}}u=\varphi$ and
$\partial_{n}{a}u=\psi$. An alternative characterisation is as follows. Let
$\varphi,\psi\in L_{2}(\Gamma)$, then $\varphi\in D({\cal N})$ and ${\cal
N}\varphi=\psi$ if and only if there exists a $u\in W^{1,2}(\Omega)$ such that
${\mathop{\rm Tr\,}}u=\varphi$ and

$\gothic{a}(u,v)=\int_{\Gamma}\psi\,\overline{{\mathop{\rm Tr\,}}v}$

for all $v\in W^{1,2}(\Omega)$.

Actually, ${\cal N}$ can also be defined as the associated operator with the
sesquilinear form $\gothic{b}\colon H^{1/2}(\Gamma)\times
H^{1/2}(\Gamma)\to\mathds{C}$ given by

$\gothic{b}(\varphi,\eta)=\gothic{a}(\gamma(\varphi),\gamma(\eta)).$

It turns out that this form is densely defined, continuous and quasi-coercive.
Therefore, as an operator associated with a sesquilinear form, ${\cal N}$ is
densely defined, quasi-accretive and $-{\cal N}$ generates a strongly
continuous holomorphic semigroup on $L_{2}(\Gamma)$. If
$c_{kl}=\overline{c_{lk}}$, $c_{k}=\overline{b_{k}}$ and $c_{0}$ is real, for
all $k,l\in\\{1,\ldots,d\\}$, then ${\cal N}$ is a self-adjoint operator. For
all these basic facts, we refer to [EO2], [EO1] and the references therein.

In this paper we are interested in a number of important properties of ${\cal
N}$, among them $L_{p}(\Gamma)$ and Hölder continuity estimates for the
commutator $[{\cal N},M_{g}]$ of ${\cal N}$ with the multiplication operator
$M_{g}$ with a function $g$. For the harmonic lifting $\gamma$ we are
interested in $L_{p}(\Gamma)\to L_{p}(\Omega)$ and Hölder continuity
estimates. Further we are interested in $L_{p}\to L_{q}$ boundedness for the
semigroup $(e^{-t{\cal N}})_{t>0}$ and Poisson upper bounds for the heat
kernel of ${\cal N}$. We emphasize that we consider these problems for non-
symmetric ${\cal N}$ with complex coefficients. No results of these types are
available in this setting in the literature. On a domain of class
$C^{1+\kappa}$ we solve some of these problems and we state now our main
results.

######  Theorem 1.1.

Let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be a bounded open set
of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$. Suppose that $0\not\in\sigma(A_{D})$. Then we have the
following assertions.

  * (a)

For all $\nu\in(0,\kappa)$ there exists a $c>0$ such that

$\|[{\cal N},M_{g}]\|_{C^{\nu}(\Gamma)\to C^{\nu}(\Gamma)}\leq c$

for all $g\in W^{2,\infty}(\mathds{R}^{d})$ such that
$\|\partial_{k}g\|_{\infty}\leq 1$ and
$\|\partial_{k}\,\partial_{l}g\|_{\infty}\leq 1$ for all
$k,l\in\\{1,\ldots,d\\}$.

  * (b)

For all $p\in(1,\infty)$ there exists a $c>0$ such that

$\|[{\cal N},M_{g}]\|_{L_{p}(\Gamma)\to L_{p}(\Gamma)}\leq c$

for all $g\in W^{2,\infty}(\mathds{R}^{d})$ such that
$\|\partial_{k}g\|_{\infty}\leq 1$ and
$\|\partial_{k}\,\partial_{l}g\|_{\infty}\leq 1$ for all
$k,l\in\\{1,\ldots,d\\}$.

  * (c)

The operator $[{\cal N},M_{g}]$ is of weak type $(1,1)$ for all $g\in
W^{2,\infty}(\mathds{R}^{d})$.

In order to be more precise, for Statement (a) we prove that for all $g\in
W^{2,\infty}(\mathds{R}^{d})$ and $\nu\in(0,\kappa)$ there is a bounded
operator $\widetilde{T}_{g}\colon C^{\nu}(\Gamma)\to C^{\nu}(\Gamma)$ such
that for all $\varphi\in D({\cal N})\cap C^{\nu}(\Gamma)$ one has
$g|_{\Gamma}\,\varphi\in D({\cal N})\cap C^{\nu}(\Gamma)$ and $[{\cal
N},M_{g}]\varphi=\widetilde{T}_{g}\varphi$. A similar formulation is valid for
the other statements. We note that even in the case where $A$ is (minus) the
Laplacian, the $C^{\nu}$-estimate for the commutator is new.

For the harmonic lifting we have the following result.

######  Theorem 1.2.

Let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be a bounded open set
of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$. Suppose that $0\not\in\sigma(A_{D})$. Then one has the
following.

  * (a)

For all $\nu\in(0,\kappa)$ the harmonic lifting extends consistently to a
continuous map from $C^{\nu}(\Gamma)$ into $C^{\nu}(\Omega)$.

  * (b)

For all $p\in[1,\infty)$ the harmonic lifting extends consistently to a
continuous map from $L_{p}(\Gamma)$ into $L_{p}(\Omega)$.

Concerning $L_{p}\to L_{q}$ boundedness and Poisson bounds, we deal with
operators with real-valued coefficients.

######  Theorem 1.3.

Let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be a bounded open set
of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$ and suppose that all these coefficients are real-valued.
Suppose also that $0\not\in\sigma(A_{D})$. Then the semigroup generated by
$-{\cal N}$ has a kernel $K$ and there exist $c>0$ and
$\omega_{0}\in\mathds{R}$ such that

$|K_{t}(z,w)|\leq\frac{c\,t^{-(d-1)}\,e^{\omega_{0}t}}{\displaystyle\Big{(}1+\frac{|z-w|}{t}\Big{)}^{d}}$

for all $z,w\in\Gamma$ and $t>0$.

Note that $L_{p}(\Gamma)\to L_{q}(\Gamma)$ estimates for $e^{-t{\cal N}}$ are
satisfied for all $1\leq p\leq q\leq\infty$ and are implicit in this theorem.
We also note that this later theorem remains valid for complex $c_{0}$. We
explain this in Section 10, where the proof of the theorem is given. We
believe that the same result holds for complex coefficients as in the first
two theorems, but this turns out to be a challenging problem which remains to
be solved.

The Dirichlet-to-Neumann map is an important operator which appears in many
mathematical problems and modelling such as electrical impedance tomography,
Calderón’s inverse problems, spectral theory, …. The famous inverse problem of
Calderón consists in determining the potential $V$ or the diffusion
coefficient $c(x)$ from the knowledge of the Dirichlet-to-Neumann operator
when ${\cal A}=-\Delta+V$ (Schrödinger operator) or ${\cal
A}=-\sum_{k=1}^{d}\partial_{k}(c(x)\partial_{k})$ for the isotropic case. This
problem is nowadays quite well understood at least when the Dirichlet-to-
Neumann operator is known on the whole boundary. The anisotropic case, i.e.,
when ${\cal A}=\sum_{k,l=1}^{d}\partial_{k}(c_{kl}\partial_{l})$, is more
complicate and one can only hope for uniqueness up to a diffeomorphism. We
refer to the papers of Lee and Uhlmann [LU] and Astala, Päivärinta and Lassas
[APL] and the references therein. For a weaker version of Calderón’s problem
with general second-order operator ${\cal A}$ as we consider here, we refer to
Behrndt and Rohleder [BR] or Ouhabaz [Ouh2]. The use of the Dirichlet-to-
Neumann operator in spectral theory is a subject which has attracted
attention. We refer the reader to the papers of Friedlander [Fri], Arendt and
Mazzeo [AM] and the more recent paper by Girouard et al. [GKLP], or for Stokes
operators as in Denis and ter Elst [DE]. It is also used in scattering theory,
see for example Behrndt, Malamud and Neidhardt [BMN]. For modelling of
electrical impedance tomography and historical notes on Calderón’s inverse
problem we refer to Uhlmann [Uhl]. The Dirichlet-to-Neumann operator with
variable coefficients plays an important role in modelling and analysis of
water waves, see e.g. Lannes [Lan]. It is also used in the theory of
homogenization and analysis of elliptic systems with rapidly oscillating
coefficients (see e.g. Kenig, Lin and Shen [KLS] and the references therein).
When $A=-\Delta$ and $g(x)=x_{k}$, then $L_{p}$-estimates for the commutator
$[{\cal N},M_{g}]$ were proved in [KLS] for $C^{1}$-domains. The result there
is also valid with $p=2$ for any bounded Lipschitz domain. This later case was
extended to the case of variable coefficients by Shen [She], i.e.,
$A=-\sum_{k,l}\partial_{k}(c_{kl}\,\partial_{l})$. It is assumed in [She] that
the coefficients are symmetric ($c_{kl}=c_{lk}$), Hölder continuous and real-
valued. The function $g$ is any smooth function on $\Gamma$. All the
assumptions on the coefficients made in [She] play an important role in the
proof. Based on this $L_{2}$-result from [She], it was proved by ter Elst and
Ouhabaz in [EO2] that for $C^{1+\kappa}$-domains, this commutator is of weak
type $(1,1)$ and it is bounded on $L_{p}(\Gamma)$ for all $p\in(1,\infty)$. In
addition it was allowed there to add a real-valued potential $V$ to the
elliptic operator $A$. No terms of order one or non-symmetric principal
$c_{kl}$, however, were considered in [EO2]. Recently, $L_{2}$-estimates for
the commutator were proved in the setting of the half-space by Hofmann and
Zhang [HZ] for complex coefficients which are a small perturbation of real
symmetric coefficients. Xu, Zhao and Zhou [XZZ] extended Shen’s commutator
result to the case of Stokes systems.

Concerning Poisson bounds we mention the results from [EO1] and [EO2]. The
first paper deals with the case of the Dirichlet-to-Neumann operator
associated with the positive Laplacian and a positive measurable potential on
a $C^{\infty}$ bounded domain. Without potential the Dirichlet-to-Neumann
operator ${\cal N}$ is a pseudo-differential operator and we used there
$L_{p}$-estimates of Coifman and Meyer [CM] for such operators. The case of
variable coefficients is treated in [EO2] for $C^{1+\kappa}$-domains and real-
valued symmetric Hölder coefficients. In [EO2] we relied on the
$L_{2}$-estimate of [She] for the commutator together with Hölder bounds for
the Green function of the elliptic operator on $L_{2}(\Omega)$ in order to
extend the commutator to $L_{p}(\Gamma)$. Once this extension was proved, we
combined the commutator estimates with several $L_{p}\to L_{q}$ estimates for
the semigroup $(e^{-t{\cal N}})_{t>0}$ in order to obtain $L_{1}\to
L_{\infty}$ estimates for iterated commutators of $e^{-t{\cal N}}$ and $M_{g}$
and then optimised over $g$. One extremely helpful fact with real coefficients
(without lower-order terms) is that the semigroup $(e^{-t{\cal N}})_{t>0}$ is
(sub-)Markovian and hence by an extrapolation argument together with the
standard Sobolev inequality on the boundary one obtains appropriate $L_{2}\to
L_{\infty}$ estimates for $e^{-t{\cal N}}$ (and hence $L_{p}\to L_{q}$
estimates for all $1\leq p\leq q\leq\infty$ by interpolation and duality).
This method is no longer applicable for operators with terms of order one or
operators with only principal part but with complex coefficients.

Considering the above problems in the setting of non-symmetric operators with
real or complex coefficients (as in the first two theorems) is not a simple
extension of the methods used in the setting of symmetric real-valued
coefficients. We had to develop new arguments since there are no available
$L_{2}$-estimates which could serve us as a starting point. We rely heavily on
Gaussian and Hölder bounds for the heat kernel of the elliptic operator
$A_{D}$ with Dirichlet boundary conditions. These bounds are then used to
prove bounds (and Hölder bounds) for the Schwartz kernel of the harmonic
lifting. They are also used to prove that the Schwartz kernel $K_{g}$ of the
commutator $[{\cal N},M_{g}]$ satisfies bounds

$|K_{g}(z,w)|\leq\frac{c}{|z-w|^{d-1}}$

and

$|K_{g}(z,w)-K_{g}(z^{\prime},w^{\prime})|\leq
c\,\frac{(|z-z^{\prime}|+|w-w^{\prime}|)^{\kappa}}{|z-w|^{d-1+\kappa}}$

for all $z,z^{\prime},w,w^{\prime}\in\Gamma$ with $z\neq w$, $z^{\prime}\neq
w^{\prime}$ and $|z-z^{\prime}|+|w-w^{\prime}|\leq\frac{1}{2}\,|z-w|$. These
bounds serve us as a keystone to prove the boundedness of $[{\cal N},M_{g}]$
on $C^{\nu}(\Gamma)$. The boundedness on $L_{2}(\Gamma)$ is then obtained by
the help of Krein’s lemma. The extension to $L_{p}(\Gamma)$ for
$p\in(1,\infty)$ is obtained by Calderón–Zygmund theory. Using techniques of
[EO4] one can transfer certain invariant closed convex sets for the semigroup
on $L_{2}(\Omega)$ generated by (minus-) the Neumann elliptic operator $A_{N}$
to an invariant closed convex set for the semigroup generated by $-{\cal N}$
on $L_{2}(\Gamma)$. In that way we show bounds $\|e^{-t{\cal
N}}\|_{L_{\infty}(\Gamma)\to L_{\infty}(\Gamma)}\leq M\,e^{\omega t}$
uniformly for $t\in(0,\infty)$ if the coefficients are real valued. As in
[EO1] and [EO2], we use commutator estimates and $L_{p}\to L_{q}$ estimates of
the semigroup $(e^{-t{\cal N}})_{t>0}$ (which are also new in this setting) to
prove $L_{1}\to L_{\infty}$ estimates for the iterated commutator
$[M_{g},[\ldots,[M_{g},e^{-t{\cal N}}]\ldots]]$ of order $d$. The Poisson
bound is then obtained easily by optimising over the function $g$.

Concerning heat kernel and Green function bounds for the elliptic operator, we
use various estimates in Morrey and Campanato spaces. This part does not
differ substantially from [EO2] although a few arguments from that paper use
self-adjointness of the operator $A_{D}$ and we have to provide different
arguments.

The outline of this paper is as follows. In Section 2 we give the definition
of Morrey and Campanato spaces and make precise some notation which will be
used throughout the paper. Section 3 is devoted to various elliptic regularity
results. Many of the results there are similar to those in [EO2], the
difference is that we deal here with possibly non self-adjoint operators with
in general complex coefficients. Also as in [EO2], we use these elliptic
regularity results to obtain Gaussian bounds, space-derivative bounds and
Hölder continuity for the heat kernel of the elliptic operator $A_{D}$. This
is done in Section 4. Section 5 is devoted to various regularity bounds for
the Green function of $A_{D}$. In Section 6 we prove pointwise bounds and
Hölder bounds for the kernel of the harmonic lifting $\gamma$. We also prove
in Theorem 6.5(d) the $L_{p}(\Gamma)\to L_{p}(\Omega)$ bounds of Theorem
1.2(b) for $\gamma$. We emphasise that several results in Sections 3, 4 and 5,
and a few in Section 6, are also valid for unbounded $\Omega$ satisfying a
local $C^{1+\kappa}$-property in a uniform manner. The Hölder continuity
bounds for $\gamma$ as stated in Theorem 1.2(a) are proved in Section 7. In
Section 8 we prove global bounds and Hölder continuity for the Schwartz kernel
of the Dirichlet-to-Neumann operator ${\cal N}$. The commutator estimates of
Theorem 1.1 are proved in Section 9. The Poisson bounds of Theorem 1.3
together with some additional results are proved in Section 10.

### Acknowledgements.

This project started in 2020 during a visit of the second named author at the
university of Auckland and continued during two visits of the first named
author at the university of Bordeaux. Both authors thank these institutions
for financial supports. The second named author wishes to thank Yuri Tomilov
who has drawn his attention to Krein’s lemma during a conference at the
university of Bordeaux several years ago.

The research of A.F.M. ter Elst was partly supported by the Marsden Fund
Council from Government funding, administered by the Royal Society of New
Zealand. The research of E.M. Ouhabaz was partly supported by the ANR project
‘Harmonic Analysis at its Boundaries’, ANR-12-BS01-0013-02.

##  2 Notation

Since we consider $d\geq 2$ as fixed throughout the paper, we will not write
that a given constant depends on $d$. Define

$W_{1}=\\{g\in W^{1,\infty}(\mathds{R}^{d},\mathds{R}):\|\nabla
g\|_{\infty}\leq 1\\}$

and

$W_{2}=\\{g\in W^{2,\infty}(\mathds{R}^{d},\mathds{R}):\|\nabla
g\|_{\infty}\leq 1\mbox{ and }\|\partial_{k}\,\partial_{l}g\|_{\infty}\leq
1\mbox{ for all }k,l\in\\{1,\ldots,d\\}\\}.$

Let $\Omega\subset\mathds{R}^{d}$ be an open set. Note that we do not require
$\Omega$ to be bounded. For all $\nu\in(0,1]$ define
$|||\cdot|||_{C^{\nu}(\Omega)}\colon C(\Omega)\to[0,\infty]$ by

$|||u|||_{C^{\nu}(\Omega)}=\sup_{\scriptstyle x,y\in\Omega\atop\scriptstyle
0<|x-y|\leq 1}\frac{|u(x)-u(y)|}{|x-y|^{\nu}}$

and set $C^{\nu}(\Omega)=\\{u\in
C(\Omega):|||u|||_{C^{\nu}(\Omega)}<\infty\\}$. Moreover, define

$\|u\|_{C^{\nu}(\Omega)}=|||u|||_{C^{\nu}(\Omega)}+\|u\|_{L_{\infty}(\Omega)}\in[0,\infty].$

We emphasise that elements in $C^{\nu}(\Omega)$ may be unbounded in general.

If $x\in\overline{\Omega}$ and $r>0$, then define $\Omega(x,r)=\Omega\cap
B(x,r)$. For all $\gamma\in[0,d]$, $x\in\overline{\Omega}$ and $R_{e}\in(0,1]$
define $\|\cdot\|_{M,\gamma,x,\Omega,R_{e}}\colon L_{2}(\Omega)\to[0,\infty]$
by

$\|u\|_{M,\gamma,x,\Omega,R_{e}}=\sup_{r\in(0,R_{e}]}\Big{(}r^{-\gamma}\int_{\Omega(x,r)}|u|^{2}\Big{)}^{1/2}.$

Moreover, define the Morrey norm $\|\cdot\|_{M,\gamma,R_{e}}\colon
L_{2}(\Omega)\to[0,\infty]$ by

$\|u\|_{M,\gamma,R_{e}}=\sup_{x\in\overline{\Omega}}\|u\|_{M,\gamma,x,\Omega,R_{e}}.$

Set

$M_{\gamma}(\Omega)=\\{u\in L_{2}(\Omega):\|u\|_{M,\gamma,1}<\infty\\}.$

Further, for all $\gamma\in[0,d+2]$, $x\in\overline{\Omega}$ and
$R_{e}\in(0,1]$ define $|||\cdot|||_{{\cal M},\gamma,x,\Omega,R_{e}}\colon
L_{2}(\Omega)\to[0,\infty]$ by

$|||u|||_{{\cal
M},\gamma,x,\Omega,R_{e}}=\sup_{r\in(0,R_{e}]}\Big{(}r^{-\gamma}\int_{\Omega(x,r)}|u-\langle
u\rangle_{\Omega(x,r)}|^{2}\Big{)}^{1/2}.$

We define the Campanato seminorm $|||\cdot|||_{{\cal M},\gamma,R_{e}}\colon
L_{2}(\Omega)\to[0,\infty]$ by

$|||u|||_{{\cal M},\gamma,R_{e}}=\sup_{x\in\overline{\Omega}}|||u|||_{{\cal
M},\gamma,x,\Omega,R_{e}}.$

Set

${\cal M}_{\gamma}(\Omega)=\\{u\in L_{2}(\Omega):|||u|||_{{\cal
M},\gamma,1}<\infty\\}.$

For explicit estimates between the various seminorms we refer to [ER] Lemma
3.1. If $R_{e}=1$, then we drop the symbol $R_{e}$ in the notation.

Define

$E=(-4,4)^{d}\quad\mbox{and}\quad E^{-}=(-4,4)^{d-1}\times(-4,0)$

the open cube in $\mathds{R}^{d}$ and its lower half $E^{-}$. The midplate is
$P=E\cap\\{x\in\mathds{R}^{d}:x_{d}=0\\}$. Let $\kappa\in(0,1)$, $K>0$ and
$\Omega\subset\mathds{R}^{d}$. We say that $\Omega$ is of class $C^{1+\kappa}$
with parameter $K$ if for all $x\in\partial\Omega$ there exists an open
neighbourhood $U$ of $x$ and a $C^{1+\kappa}$-diffeomorphism $\Phi$ from $U$
onto $E$ such that

  * •

$\Phi(x)=0$,

  * •

$\Phi(U\cap\Omega)=E^{-}$

  * •

$\Phi(U\cap\partial\Omega)=P$,

  * •

$K$ is larger than the Lipschitz constant for $\Phi$ and $\Phi^{-1}$, and

  * •

$|||(D\Phi)_{ij}|||_{C^{\kappa}}\leq K$ and
$|||(D(\Phi^{-1}))_{ij}|||_{C^{\kappa}}\leq K$ for all
$i,j\in\\{1,\ldots,d\\}$ where $D\Phi$ denotes the derivative of $\Phi$.

We emphasise that $\Omega$ does not have to be bounded. If $\Omega$ is of
class $C^{1+\kappa}$ with parameter $K$, then there is a $\tilde{c}>0$,
depending only on $K$, such that $|\Omega(x,r)|\geq\tilde{c}\,r^{d}$ for all
$x\in\overline{\Omega}$ and $r\in(0,1]$. Further if $\Omega$ is of class
$C^{1+\kappa}$ and $\nu\in(0,1]$, then every $u\in C^{\nu}(\Omega)$ can be
extended to an element $\tilde{u}\in C^{\nu}(\overline{\Omega})$. We will
identify $u$ and $\tilde{u}$ as no confusion is possible. In particular, we
occasionally write $u|_{\Gamma}$ for $\tilde{u}|_{\Gamma}$.

If $\gothic{a}\colon W^{1,2}(\Omega)\times W^{1,2}(\Omega)\to\mathds{C}$ is as
in (1) and $\lambda\in\mathds{R}$, then we define the shifted form
$\gothic{a}_{\lambda}\colon W^{1,2}(\Omega)\times
W^{1,2}(\Omega)\to\mathds{C}$ by

$\gothic{a}_{\lambda}(u,v)=\gothic{a}(u,v)+\lambda\,(u,v)_{L_{2}(\Omega)}.$

For a function $H$ of two variables, we use the notation $\partial^{(1)}_{k}H$
and $\partial^{(2)}_{l}H$ to denote the partial derivative with respect to the
first and second variable in the $k$-th and $l$-th direction, respectively.

Finally, for a given bounded operator from an $L_{p}(X)$ space to an
$L_{q}(Y)$ space, its norm is denoted $\|T\|_{L_{p}(X)\to L_{q}(Y)}$ or simply
$\|T\|_{p\to q}$ if $X=Y$.

##  3 Elliptic regularity

We start with some elementary estimates regarding Morrey and Campanato
seminorms.

######  Lemma 3.1.

  * (a)

Let $0<R_{1}\leq R_{2}\leq 1$. Then there is a $c>0$ such that

$\|u\|_{M,\gamma,R_{1}}\leq\|u\|_{M,\gamma,R_{2}}\leq
c\,\|u\|_{M,\gamma,R_{1}}$

for all open $\Omega\subset\mathds{R}^{d}$, $\gamma\in[0,d]$ and $u\in
L_{2}(\Omega)$.

  * (b)

If $\gamma\in[0,d+2]$, $0<R_{1}\leq R_{2}\leq 1$ and
$\Omega\subset\mathds{R}^{d}$ is open, then

$|||u|||_{{\cal M},\gamma,R_{1}}\leq|||u|||_{{\cal
M},\gamma,R_{2}}\leq|||u|||_{{\cal
M},\gamma,R_{1}}+R_{1}^{-\gamma}\,\|u\|_{M,0}$

for all $u\in L_{2}(\Omega)$.

###### Proof.

‘(a)’. By an elementary estimate there is an $N\in\mathds{N}$ such that for
all $x\in\overline{\Omega}$ there exist
$x_{1},\ldots,x_{N}\in\overline{\Omega}$ such that
$\Omega(x,1)\subset\bigcup_{n=1}^{N}\Omega(x_{n},R_{1})$. Then Statement (a)
follows.

Statement (b) is trivial. ∎

######  Lemma 3.2.

Let $\kappa\in(0,1)$, $K>0$ and $\gamma\in[0,d]$. Then there are $c>0$ and
$R_{e}\in(0,1]$ such that the following is valid.

Let $\Omega\subset\mathds{R}^{d}$ be an open set of class $C^{1+\kappa}$ with
parameter $K$. Then

$|||u|||_{{\cal M},\gamma+2,R_{e}}\leq c\,\|\nabla u\|_{M,\gamma,R_{e}}$

for all $u\in W^{1,2}(\Omega)$.

###### Proof.

It follows from [EO2] Lemma 3.12 (also if $\gamma=d$) and two coordinate
transformations that $|||u|||_{{\cal M},\gamma+2,R_{e}}\leq c\,\|\nabla
u\|_{M,\gamma,1}$ for suitable $R_{e}\in(0,1]$ and $c>0$. Then use Lemma
3.1(a). ∎

We emphasise that $\Omega$ does not have to be bounded in the next four
propositions.

######  Proposition 3.3.

Let $\kappa\in(0,1)$, $\mu,M,K>0$, $\gamma\in[0,d)$ and $\delta\in[0,2]$ with
$\gamma+\delta<d$. Then there exists a $c>0$ such that the following is valid.

Let $\Omega\subset\mathds{R}^{d}$ be an open set of class $C^{1+\kappa}$ with
parameter $K$. For all $k,l\in\\{1,\ldots,d\\}$ let $c_{kl}\in
C^{\kappa}(\Omega)\cap L_{\infty}(\Omega)$ with
$\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$. Suppose $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Let
$f_{0},f_{1},\ldots,f_{d}\in L_{2}(\Omega)$ and $u\in W^{1,2}_{0}(\Omega)$.
Suppose

$\sum_{k,l=1}^{d}\int_{\Omega}c_{kl}\,(\partial_{l}u)\,\,\overline{\partial_{k}v}=(f_{0},v)_{L_{2}(\Omega)}+\sum_{k=1}^{d}(f_{k},\partial_{k}v)_{L_{2}(\Omega)}$

for all $v\in W^{1,2}_{0}(\Omega)$. Then

$\|\nabla u\|_{M,\gamma+\delta}\leq
c\,\Big{(}\varepsilon^{2-\delta}\,\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}+\varepsilon^{-(\gamma+\delta)}\,\|\nabla
u\|_{L_{2}(\Omega)}\Big{)}$

for all $\varepsilon\in(0,1]$.

###### Proof.

This follows from [EO2] Propositions 3.9 and 3.10. ∎

Many statements in this article are formulated as in the previous proposition
in the sense that the constants in the estimates depend only on some fixed
parameters. The main reason to do so is that at some places we will use
approximation arguments. The approximation process of the coefficients by
smooth ones requires uniform estimates in order to take the limit.

######  Proposition 3.4.

Let $\kappa\in(0,1)$ and $\mu,M,K>0$. Then there exists a $c>0$ such that the
following is valid.

Let $\Omega\subset\mathds{R}^{d}$ be an open set of class $C^{1+\kappa}$ with
parameter $K$. For all $k,l\in\\{1,\ldots,d\\}$ let $c_{kl}\in
C^{\kappa}(\Omega)\cap L_{\infty}(\Omega)$, with
$\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$. Suppose $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Let
$f_{0},f_{1},\ldots,f_{d}\in L_{2}(\Omega)$ and $u\in W^{1,2}_{0}(\Omega)$.
Suppose

$\sum_{k,l=1}^{d}\int_{\Omega}c_{kl}\,(\partial_{l}u)\,\,\overline{\partial_{k}v}=(f_{0},v)_{L_{2}(\Omega)}+\sum_{k=1}^{d}(f_{k},\partial_{k}v)_{L_{2}(\Omega)}$

for all $v\in W^{1,2}_{0}(\Omega)$. Then

$\sum_{k=1}^{d}|||\partial_{k}u|||_{{\cal M},d+2\kappa}\leq
c\,\Big{(}\|f_{0}\|_{M,d+2\kappa-2}+\sum_{k=1}^{d}(|||f_{k}|||_{{\cal
M},d+2\kappa}+\|f_{k}\|_{L_{2}(\Omega)})+\|\nabla u\|_{L_{2}(\Omega)}\Big{)}.$

###### Proof.

It follows from [EO2] Proposition 3.7 (in particular the first displayed
formula on page 4240) and [EO2] Proposition 3.8 that there are
$c_{1},c_{2}>0$, depending only on $\kappa$, $\mu$, $M$, and $K$, such that

$\sum_{k=1}^{d}|||\partial_{k}u|||_{{\cal M},d+\kappa}\leq
c_{1}\,\Big{(}\|f_{0}\|_{M,d+\kappa-2}+\sum_{k=1}^{d}|||f_{k}|||_{{\cal
M},d+\kappa}+\|\nabla u\|_{M,d-\kappa}\Big{)}$ (2)

and

$\sum_{k=1}^{d}|||\partial_{k}u|||_{{\cal M},d+2\kappa}\leq
c_{2}\,\Big{(}\|f_{0}\|_{M,d+2\kappa-2}+\sum_{k=1}^{d}|||f_{k}|||_{{\cal
M},d+2\kappa}+\|\nabla u\|_{M,d}\Big{)}.$ (3)

We may assume that $\|f_{0}\|_{M,d+2\kappa-2}<\infty$ and $|||f_{k}|||_{{\cal
M},d+2\kappa}<\infty$ for all $k\in\\{1,\ldots,d\\}$. By Proposition 3.3 and
[ER] Lemma 3.1(a) it follows that $\|\nabla u\|_{M,d-\kappa}<\infty$ and then
(2) implies that $\partial_{k}u\in{\cal M}_{d+\kappa}(\Omega)\subset
C^{\kappa/2}(\Omega)$ for all $k\in\\{1,\ldots,d\\}$ by [ER] Lemma 3.1(c). By
[ER] Lemma 3.1(b) there exists a suitable $c_{3}>0$ such that

$|\langle v\rangle_{\Omega(x,\rho)}-v(x)|\leq
c_{3}\,\rho^{\kappa/2}\,|||v|||_{{\cal M},d+\kappa}$

for all $x\in\Omega$, $\rho\in(0,1]$ and $v\in C(\Omega)$. Choosing $\rho=1$
gives
$|v(x)|\leq\sqrt{\omega_{d}}\,\|v\|_{L_{2}(\Omega)}+c_{3}\,|||v|||_{{\cal
M},d+\kappa}$. Hence if $k\in\\{1,\ldots,d\\}$ and $x\in\Omega$, then

$\displaystyle|(\partial_{k}u)(x)|$ $\displaystyle\leq$
$\displaystyle\sqrt{\omega_{d}}\,\|\nabla
u\|_{L_{2}(\Omega)}+c_{3}\,c_{1}\,\Big{(}\|f_{0}\|_{M,d+\kappa-2}+\sum_{m=1}^{d}|||f_{m}|||_{{\cal
M},d+\kappa}+\|\nabla u\|_{M,d-\kappa}\Big{)}$ $\displaystyle\leq$
$\displaystyle
c_{4}\,\Big{(}\|f_{0}\|_{M,d+\kappa-2}+\sum_{m=1}^{d}(|||f_{m}|||_{{\cal
M},d+\kappa}+\|f_{m}\|_{L_{2}(\Omega)})+\|\nabla u\|_{L_{2}(\Omega)}\Big{)}$

for a suitable $c_{4}>0$, where we used Proposition 3.3 to estimate $\|\nabla
u\|_{M,d-\kappa}$ and [ER] Lemma 3.1(a) to estimate $\|f_{m}\|_{M,d-\kappa}$.
Now
$\|\partial_{k}u\|_{M,d}\leq\sqrt{\omega_{d}}\,\|\partial_{k}u\|_{L_{\infty}(\Omega)}$.
Using (3) completes the proof. ∎

We next consider operators with lower-order terms.

######  Proposition 3.5.

Let $\kappa\in(0,1)$, $\mu,M,K>0$, $\gamma\in[0,d)$ and $\delta\in[0,2]$ with
$\gamma+\delta<d$. Then there exists a $c>0$ such that the following is valid.

Let $\Omega\subset\mathds{R}^{d}$ be an open set of class $C^{1+\kappa}$ with
parameter $K$. For all $k,l\in\\{1,\ldots,d\\}$ let $c_{kl}\in
C^{\kappa}(\Omega)\cap L_{\infty}(\Omega)$ and let $b_{k},c_{k},c_{0}\in
L_{\infty}(\Omega)$, with $\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$,
$\|b_{k}\|_{L_{\infty}(\Omega)}\leq M$, $\|c_{k}\|_{L_{\infty}(\Omega)}\leq M$
and $\|c_{0}\|_{L_{\infty}(\Omega)}\leq M$. Suppose $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Define $\gothic{a}$ as in
(1).

Let $f_{0},f_{1},\ldots,f_{d}\in L_{2}(\Omega)$ and $u\in
W^{1,2}_{0}(\Omega)$. Suppose

$\gothic{a}(u,v)=(f_{0},v)_{L_{2}(\Omega)}+\sum_{k=1}^{d}(f_{k},\partial_{k}v)_{L_{2}(\Omega)}$

for all $v\in W^{1,2}_{0}(\Omega)$. Then

$\|\nabla u\|_{M,\gamma+\delta}\leq
c\,\Big{(}\varepsilon^{2-\delta}\,\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}+\varepsilon^{-(\gamma+\delta)}\,\|u\|_{W^{1,2}(\Omega)}\Big{)}$
(4)

for all $\varepsilon\in(0,1]$.

If, in addition, $\Omega$ is bounded and

$\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u),$

then there exists a $\tilde{c}>0$, depending only on $\kappa$, $\mu$, $M$,
$K$, $\gamma$, $\delta$ and $\mathop{\rm diam}\Omega$, such that

$\|\nabla
u\|_{M,\gamma+\delta}\leq\tilde{c}\,\Big{(}\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}\Big{)}.$

###### Proof.

Let $f_{0},f_{1},\ldots,f_{d}\in L_{2}(\Omega)$ and $u\in
W^{1,2}_{0}(\Omega)$. Suppose

$\gothic{a}(u,v)=(f_{0},v)_{L_{2}(\Omega)}+\sum_{k=1}^{d}(f_{k},\partial_{k}v)_{L_{2}(\Omega)}$

for all $v\in W^{1,2}_{0}(\Omega)$. Then

$\sum_{k,l=1}^{d}\int_{\Omega}c_{kl}\,(\partial_{l}u)\,\,\overline{\partial_{k}v}=(\tilde{f}_{0},v)_{L_{2}(\Omega)}+\sum_{k=1}^{d}(\tilde{f}_{k},\partial_{k}v)_{L_{2}(\Omega)}$

for all $v\in W^{1,2}_{0}(\Omega)$, where

$\tilde{f}_{0}=f_{0}-\sum_{l=1}^{d}c_{l}\,\partial_{l}u-c_{0}\,u\quad\mbox{and}\quad\tilde{f}_{k}=f_{k}-b_{k}\,u$

for all $k\in\\{1,\ldots,d\\}$.

For all $\gamma\in[0,d)$ let $P(\gamma)$ be the hypothesis

> for all $\delta\in[0,2]$ with $\gamma+\delta<d$ there exists a $c>0$,
> depending only on $\kappa$, $\mu$, $M$, $K$, $\gamma$ and $\delta$, such
> that for all $\varepsilon\in(0,1]$ one has the estimate
>
> $\|\nabla u\|_{M,\gamma+\delta}\leq
> c\,\Big{(}\varepsilon^{2-\delta}\,\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}+\varepsilon^{-(\gamma+\delta)}\,\|u\|_{W^{1,2}(\Omega)}\Big{)}.$

We start with the base case.

######  Lemma 3.6.

The hypothesis $P(0)$ is valid.

###### Proof.

Let $R_{e}\in(0,1]$ and $c>0$ be as in Lemma 3.2 with the choice $\gamma=0$.
By [ER] Lemma 3.1(a) there is a suitable $c_{1}>0$ such that

$\|u\|_{M,\delta,R_{e}}\leq c_{1}\,(|||u|||_{{\cal
M},\delta,R_{e}}+\|u\|_{L_{2}(\Omega)}).$

Then

$\displaystyle\|u\|_{M,\delta,R_{e}}$ $\displaystyle\leq$ $\displaystyle
c\,c_{1}\,\|\nabla u\|_{M,0,R_{e}}+c_{1}\,\|u\|_{L_{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle(1+c)\,c_{1}\,\|u\|_{W^{1,2}(\Omega)}.$

Hence by Lemma 3.1(a) there is a suitable $c_{2}>0$ such that

$\|u\|_{M,\delta}\leq c_{2}\,\|u\|_{W^{1,2}(\Omega)}.$

By Proposition 3.3 there exists a suitable $c_{3}>0$ such that

$\|\nabla u\|_{M,\delta}\leq
c_{3}\,\Big{(}\varepsilon^{2-\delta}\,\|\tilde{f}_{0}\|_{M,0}+\sum_{k=1}^{d}\|\tilde{f}_{k}\|_{M,\delta}+\varepsilon^{-\delta}\,\|\nabla
u\|_{L_{2}(\Omega)}\Big{)}$

for all $\varepsilon\in(0,1]$. Now

$\displaystyle\|\tilde{f}_{0}\|_{M,0}$ $\displaystyle\leq$
$\displaystyle\|f_{0}\|_{M,0}+d\,M\,\|\nabla u\|_{M,0}+M\,\|u\|_{M,0}$
$\displaystyle\leq$
$\displaystyle\|f_{0}\|_{M,0}+(d\,M+M\,c_{2})\,\|u\|_{W^{1,2}(\Omega)}$

and if $k\in\\{1,\ldots,d\\}$, then

$\|\tilde{f}_{k}\|_{M,\delta}\leq\|f_{k}\|_{M,\delta}+M\,\|u\|_{M,\delta}\leq\|f_{k}\|_{M,\delta}+c_{2}\,M\,\|u\|_{W^{1,2}(\Omega)}.$

Then the lemma follows. ∎

The induction step is in the next lemma.

######  Lemma 3.7.

Let $\gamma\in[0,d)$ and suppose that $P(\gamma)$ is valid. Let
$\tilde{\delta}\in[0,2]$ and suppose that $\gamma+\tilde{\delta}<d$. Then
$P(\gamma+\tilde{\delta})$ is valid.

###### Proof.

Let $\delta\in[0,2]$ and suppose that $\gamma+\tilde{\delta}+\delta<d$. By
hypothesis $P(\gamma)$ there is a suitable $c_{1}>0$ such that

$\|\nabla u\|_{M,\gamma+\delta}\leq
c_{1}\,\Big{(}\varepsilon^{2-\delta}\,\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}+\varepsilon^{-(\gamma+\delta)}\,\|u\|_{W^{1,2}(\Omega)}\Big{)}$

for all $\varepsilon\in(0,1]$. By Lemma 3.2 there are $c_{2}>0$ and
$R_{e}\in(0,1]$ such that

$|||v|||_{{\cal M},\gamma+\delta+2,R_{e}}\leq c_{2}\,\|\nabla
v\|_{M,\gamma+\delta,R_{e}}$

for all $v\in W^{1,2}(\Omega)$. By [ER] Lemma 3.1(a) there is a suitable
$c_{3}>0$ such that

$\|u\|_{M,\gamma+\delta+\tilde{\delta},R_{e}}\leq c_{3}\,(|||u|||_{{\cal
M},\gamma+\delta+\tilde{\delta},R_{e}}+\|u\|_{L_{2}(\Omega)}).$

Then

$\displaystyle\|u\|_{M,\gamma+\delta+\tilde{\delta},R_{e}}$
$\displaystyle\leq$ $\displaystyle c_{3}\,(|||u|||_{{\cal
M},\gamma+\delta+2,R_{e}}+\|u\|_{L_{2}(\Omega)})$ $\displaystyle\leq$
$\displaystyle c_{3}\,c_{2}\,\|\nabla
u\|_{M,\gamma+\delta,R_{e}}+c_{3}\,\|u\|_{L_{2}(\Omega)}$ $\displaystyle\leq$
$\displaystyle
c_{3}\,(1+c_{2}\,c_{1})\,\Big{(}\varepsilon^{2-\delta}\,\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}+\varepsilon^{-(\gamma+\delta)}\,\|u\|_{W^{1,2}(\Omega)}\Big{)}$

for all $\varepsilon\in(0,1]$. Hence by Lemma 3.1(a) there is a suitable
$c_{4}>0$ such that

$\|u\|_{M,\gamma+\delta+\tilde{\delta}}\leq
c_{4}\,\Big{(}\varepsilon^{2-\delta}\,\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}+\varepsilon^{-(\gamma+\delta)}\,\|u\|_{W^{1,2}(\Omega)}\Big{)}$
(5)

for all $\varepsilon\in(0,1]$. By Proposition 3.3 there exists a suitable
$c_{5}>0$ such that

$\|\nabla u\|_{M,\gamma+\tilde{\delta}+\delta}\leq
c_{5}\,\Big{(}\varepsilon^{2-\delta}\,\|\tilde{f}_{0}\|_{M,\gamma+\tilde{\delta}}+\sum_{k=1}^{d}\|\tilde{f}_{k}\|_{M,\gamma+\tilde{\delta}+\delta}+\varepsilon^{-(\gamma+\tilde{\delta}+\delta)}\,\|\nabla
u\|_{L_{2}(\Omega)}\Big{)}$ (6)

for all $\varepsilon\in(0,1]$. Moreover, the hypothesis $P(\gamma)$ applied
with $\tilde{\delta}$ instead of $\delta$ and with $\varepsilon=1$ gives that
there is a suitable $c_{6}>0$ such that

$\|\nabla u\|_{M,\gamma+\tilde{\delta}}\leq
c_{6}\,\Big{(}\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\tilde{\delta}}+\|u\|_{W^{1,2}(\Omega)}\Big{)}.$

Now

$\displaystyle\|\tilde{f}_{0}\|_{M,\gamma+\tilde{\delta}}$ $\displaystyle\leq$
$\displaystyle\|f_{0}\|_{M,\gamma+\tilde{\delta}}+d\,M\,\|\nabla
u\|_{M,\gamma+\tilde{\delta}}+M\,\|u\|_{M,\gamma+\tilde{\delta}}$ (7)
$\displaystyle\leq$
$\displaystyle\|f_{0}\|_{M,\gamma+\tilde{\delta}}+d\,M\,c_{6}\,\Big{(}\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\tilde{\delta}}+\|u\|_{W^{1,2}(\Omega)}\Big{)}$
$\displaystyle{}+c_{4}\,M\,\Big{(}\varepsilon^{2-\delta}\,\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}+\varepsilon^{-(\gamma+\delta)}\,\|u\|_{W^{1,2}(\Omega)}\Big{)}$
$\displaystyle\leq$ $\displaystyle
c_{7}\,\Big{(}\|f_{0}\|_{M,\gamma+\tilde{\delta}}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\tilde{\delta}+\delta}+\|u\|_{W^{1,2}(\Omega)}\Big{)},$

where $c_{7}=1+c_{6}\,d\,M+c_{4}\,M$. If $k\in\\{1,\ldots,d\\}$, then

$\displaystyle\|\tilde{f}_{k}\|_{M,\gamma+\tilde{\delta}+\delta}$
$\displaystyle\leq$
$\displaystyle\|f_{k}\|_{M,\gamma+\tilde{\delta}+\delta}+M\,\|u\|_{M,\gamma+\tilde{\delta}+\delta}$
(8) $\displaystyle\leq$
$\displaystyle\|f_{k}\|_{M,\gamma+\tilde{\delta}+\delta}$ $\displaystyle\hskip
28.45274pt{}+c_{4}\,M\,\Big{(}\varepsilon^{2-\delta}\,\|f_{0}\|_{M,\gamma}+\sum_{m=1}^{d}\|f_{m}\|_{M,\gamma+\delta}+\varepsilon^{-(\gamma+\delta)}\,\|u\|_{W^{1,2}(\Omega)}\Big{)}.$

Inserting (7) and (8) into (6) gives the lemma. ∎

Now we complete the proof of Proposition 3.5.

The estimate (4) follows from Lemma 3.6 and iteration of Lemma 3.7.

Now suppose in addition that $\Omega$ is bounded and

$\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u).$

Then there exists an $N\in\mathds{N}$, depending only on $\mathop{\rm
diam}\Omega$, such that $\overline{\Omega}$ can be covered by at most $N$
balls of radius 1 with centre in $\Omega$. Therefore

$\displaystyle\|u\|_{W^{1,2}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\frac{2}{\mu}\,(\|f_{0}\|_{L_{2}(\Omega)}+\sum_{k=1}^{d}\|f_{k}\|_{L_{2}(\Omega)})$
$\displaystyle\leq$
$\displaystyle\frac{2N}{\mu}\,(\|f_{0}\|_{M,0}+\sum_{k=1}^{d}\|f_{k}\|_{M,0})$
$\displaystyle\leq$
$\displaystyle\frac{2N}{\mu}\,(\|f_{0}\|_{M,\gamma}+\sum_{k=1}^{d}\|f_{k}\|_{M,\gamma+\delta}).$

The proof is complete. ∎

For a similar estimate in the Campanato region we need an explicit bound on
products of two Hölder continuous functions.

######  Lemma 3.8.

Let $\gamma\in(d,d+2]$, $\Omega\subset\mathds{R}^{d}$ open, $u\in
C^{(\gamma-d)/2}(\Omega)\cap L_{\infty}(\Omega)$ and $v\in L_{2}(\Omega)$.
Then

$\displaystyle|||u\,v|||_{{\cal M},\gamma}^{2}$ $\displaystyle\leq$
$\displaystyle 2\|u\|_{L_{\infty}(\Omega)}^{2}\,|||v|||_{{\cal
M},\gamma}^{2}+2^{1+\gamma-d}\,\omega_{d}\,\|v\|_{L_{\infty}(\Omega)}^{2}\,|||u|||_{C^{(\gamma-d)/2}(\Omega)}^{2}$
$\displaystyle\hskip
28.45274pt{}+2^{\gamma}\,\omega_{d}\,\|u\|_{L_{\infty}(\Omega)}^{2}\,\|v\|_{L_{\infty}(\Omega)}^{2}.$

###### Proof.

Let $x\in\Omega$ and $r\in(0,\frac{1}{2}]$. Then

$\displaystyle\int_{\Omega(x,r)}|u\,v-\langle u\,v\rangle_{\Omega(x,r)}|^{2}$
$\displaystyle\leq$ $\displaystyle
2\int_{\Omega(x,r)}\Big{|}\frac{1}{|\Omega(x,r)|}\,\int_{\Omega(x,r)}u(y)\,(v(y)-v(z))\,dz\Big{|}^{2}\,dy$
$\displaystyle\hskip
28.45274pt{}+2\int_{\Omega(x,r)}\Big{|}\frac{1}{|\Omega(x,r)|}\,\int_{\Omega(x,r)}v(z)\,(u(y)-u(z))\,dz\Big{|}^{2}\,dy$
$\displaystyle\leq$ $\displaystyle
2\|u\|_{L_{\infty}(\Omega)}^{2}\,r^{\gamma}\,|||v|||_{{\cal M},\gamma}^{2}$
$\displaystyle\hskip
28.45274pt{}+2|\Omega(x,r)|\,\Big{(}\|v\|_{L_{\infty}(\Omega)}\,|||u|||_{C^{(\gamma-d)/2}(\Omega)}\,(2r)^{(\gamma-d)/2}\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\Big{(}2\|u\|_{L_{\infty}(\Omega)}^{2}\,|||v|||_{{\cal
M},\gamma}^{2}+2^{1+\gamma-d}\,\omega_{d}\,\|v\|_{L_{\infty}(\Omega)}^{2}\,|||u|||_{C^{(\gamma-d)/2}(\Omega)}^{2}\Big{)}\,r^{\gamma}.$

Then the lemma follows easily. ∎

######  Proposition 3.9.

Let $\kappa\in(0,1)$ and $\mu,M,K>0$. Then there exists a $c>0$ such that the
following is valid.

Let $\Omega\subset\mathds{R}^{d}$ be an open set of class $C^{1+\kappa}$ with
parameter $K$. For all $k,l\in\\{1,\ldots,d\\}$ let $c_{kl},b_{k}\in
C^{\kappa}(\Omega)\cap L_{\infty}(\Omega)$, and let $c_{k},c_{0}\in
L_{\infty}(\Omega)$, with $\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$,
$\|b_{k}\|_{C^{\kappa}(\Omega)}\leq M$, $\|c_{k}\|_{L_{\infty}(\Omega)}\leq M$
and $\|c_{0}\|_{L_{\infty}(\Omega)}\leq M$. Suppose $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Define $\gothic{a}$ as in
(1). Let $f_{0},f_{1},\ldots,f_{d}\in L_{2}(\Omega)$ and $u\in
W^{1,2}_{0}(\Omega)$. Suppose

$\gothic{a}(u,v)=(f_{0},v)_{L_{2}(\Omega)}+\sum_{k=1}^{d}(f_{k},\partial_{k}v)_{L_{2}(\Omega)}$

for all $v\in W^{1,2}_{0}(\Omega)$. Then

$\sum_{k=1}^{d}|||\partial_{k}u|||_{{\cal M},d+2\kappa}\leq
c\,\Big{(}\|f_{0}\|_{M,d+2\kappa-2}+\sum_{k=1}^{d}(|||f_{k}|||_{{\cal
M},d+2\kappa}+\|f_{k}\|_{L_{2}(\Omega)})+\|u\|_{W^{1,2}(\Omega)}\Big{)}.$

If, in addition, $\Omega$ is bounded and

$\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u),$ (9)

then there exists a $\tilde{c}>0$, depending only on $\kappa$, $\mu$, $M$, $K$
and $\mathop{\rm diam}\Omega$, such that

$\sum_{k=1}^{d}\|\partial_{k}u\|_{C^{\kappa}(\Omega)}\leq\tilde{c}\,\Big{(}\|f_{0}\|_{M,d+2\kappa-2}+\sum_{k=1}^{d}\|f_{k}\|_{C^{\kappa}(\Omega)}\Big{)}.$

###### Proof.

We proceed as in the proof of Proposition 3.5. Again

$\sum_{k,l=1}^{d}\int_{\Omega}c_{kl}\,(\partial_{l}u)\,\,\overline{\partial_{k}v}=(\tilde{f}_{0},v)_{L_{2}(\Omega)}+\sum_{k=1}^{d}(\tilde{f}_{k},\partial_{k}v)_{L_{2}(\Omega)}$

for all $v\in W^{1,2}_{0}(\Omega)$, where

$\tilde{f}_{0}=f_{0}-\sum_{l=1}^{d}c_{l}\,\partial_{l}u-c_{0}\,u\quad\mbox{and}\quad\tilde{f}_{k}=f_{k}-b_{k}\,u$

for all $k\in\\{1,\ldots,d\\}$. By Proposition 3.4 there is a suitable
$c_{1}>0$ such that

$\sum_{k=1}^{d}|||\partial_{k}u|||_{{\cal M},d+2\kappa}\leq
c_{1}\,\Big{(}\|\tilde{f}_{0}\|_{M,d+2\kappa-2}+\sum_{k=1}^{d}(|||\tilde{f}_{k}|||_{{\cal
M},d+2\kappa}+\|\tilde{f}_{k}\|_{L_{2}(\Omega)})+\|\nabla
u\|_{L_{2}(\Omega)}\Big{)}.$

Now

$\|\tilde{f}_{0}\|_{M,d+2\kappa-2}\leq\|f_{0}\|_{M,d+2\kappa-2}+d\,M\,\|\nabla
u\|_{M,d+2\kappa-2}+M\,\|u\|_{M,d+2\kappa-2}.$

Since $d+2\kappa-2<d$ it follows from Proposition 3.5 that there is a suitable
$c_{2}>0$ such that

$\|\nabla u\|_{M,d+2\kappa-2}\leq
c_{2}\,(\|f_{0}\|_{M,d+2\kappa-2}+\sum_{k=1}^{d}\|f_{k}\|_{M,d+2\kappa-2}+\|u\|_{W^{1,2}(\Omega)}).$

Arguing as in (5) there is a suitable $c_{3}>0$ such that

$\|u\|_{M,d+2\kappa-2}\leq
c_{3}\,(\|f_{0}\|_{M,d+2\kappa-2}+\sum_{k=1}^{d}\|f_{k}\|_{M,d+2\kappa-2}+\|u\|_{W^{1,2}(\Omega)}).$

If $k\in\\{1,\ldots,d\\}$, then

$|||\tilde{f}_{k}|||_{{\cal M},d+2\kappa}\leq|||f_{k}|||_{{\cal
M},d+2\kappa}+|||b_{k}\,u|||_{{\cal M},d+2\kappa}.$

By Lemma 3.8 and [ER] Lemma 3.1(c) there is a suitable $c_{4}>0$ such that

$|||b_{k}\,u|||_{{\cal M},d+2\kappa}\leq
c_{4}\,\|b_{k}\|_{C^{\kappa}(\Omega)}\,(|||u|||_{{\cal
M},d+2\kappa}+\|u\|_{L_{\infty}(\Omega)}).$

By Lemma 3.2 there are suitable $c_{5}>0$ and $R_{e}\in(0,1]$ such that

$\displaystyle|||u|||_{{\cal M},d+2\kappa,R_{e}}$ $\displaystyle\leq$
$\displaystyle c_{5}\,\|\nabla u\|_{M,d+2\kappa-2}$ $\displaystyle\leq$
$\displaystyle
c_{2}\,c_{5}(\|f_{0}\|_{M,d+2\kappa-2}+\sum_{k=1}^{d}\|f_{k}\|_{M,d+2\kappa-2}+\|u\|_{W^{1,2}(\Omega)}).$

By [ER] Lemma 3.1(b) there is a suitable $c_{6}>0$ such that

$|\langle u\rangle_{\Omega(x,\rho)}-u(x)|\leq
c_{6}\,\rho^{\kappa}\,|||u|||_{{\cal M},d+2\kappa,R_{e}}$

for all $\rho\in(0,R_{e}]$ and $x\in\Omega$. Therefore

$\|u\|_{L_{\infty}(\Omega)}\leq\sqrt{\omega_{d}}\,R_{e}^{d/2}\,\|u\|_{L_{2}(\Omega)}+c_{6}\,R_{e}^{\kappa}\,|||u|||_{{\cal
M},d+2\kappa,R_{e}}.$ (10)

Estimating as before gives the first part of the proposition.

If $\Omega$ is bounded and (9) is valid, then one can bound
$\|u\|_{W^{1,2}(\Omega)}$ as at the end of the proof of Proposition 3.5. The
$C^{\kappa}(\Omega)$-norm of $\partial_{k}u$ can be bounded by a suitable
multiple of $|||\partial_{k}u|||_{{\cal
M},d+2\kappa}+\|\partial_{k}u\|_{L_{\infty}(\Omega)}$ and then as in (10) by
$|||\partial_{k}u|||_{{\cal M},d+2\kappa}+\|\partial_{k}u\|_{L_{2}(\Omega)}$.
Further one can bound the Campanato seminorm $|||f_{k}|||_{{\cal
M},d+2\kappa}$ by a suitable multiple of $\|f_{k}\|_{C^{\kappa}(\Omega)}$. ∎

##  4 Hölder-Gaussian bounds

This section is devoted to bounds and regularity of the heat kernel of the
elliptic operator $A_{D}$ subject to Dirichlet boundary conditions. We prove
Gaussian bounds, Hölder regularity as well as $L_{p}$ estimates on the
boundary.

The set $\Omega$ is allowed to be unbounded in the next theorem.

######  Theorem 4.1.

Let $\kappa,\tau^{\prime}\in(0,1)$ and $\mu,M,\tau,K>0$. Then there exist
$a,b>0$ and $\omega\in\mathds{R}$ such that the following is valid.

Let $\Omega\subset\mathds{R}^{d}$ be an open set of class $C^{1+\kappa}$ with
parameter $K$. For all $k,l\in\\{1,\ldots,d\\}$ let $c_{kl},b_{k},c_{k}\in
C^{\kappa}(\Omega)\cap L_{\infty}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$, with $\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$,
$\|b_{k}\|_{C^{\kappa}(\Omega)}\leq M$, $\|c_{k}\|_{C^{\kappa}(\Omega)}\leq M$
and $\|c_{0}\|_{L_{\infty}(\Omega)}\leq M$. Suppose $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Then there exists a function
$(t,x,y)\mapsto H_{t}(x,y)$ from $(0,\infty)\times\Omega\times\Omega$ into
$\mathds{C}$ such that the following is valid.

  * (a)

The function $(t,x,y)\mapsto H_{t}(x,y)$ is continuous from
$(0,\infty)\times\Omega\times\Omega$ into $\mathds{C}$.

  * (b)

For all $t\in(0,\infty)$ the function $H_{t}$ is the kernel of the operator
$e^{-tA_{D}}$.

  * (c)

For all $t\in(0,\infty)$ the function $H_{t}$ is once differentiable in each
variable and the derivative with respect to one variable is differentiable in
the other variable. Moreover, for every multi-index $\alpha,\beta$ with
$0\leq|\alpha|,|\beta|\leq 1$ one has

$|(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,H_{t})(x,y)|\leq
a\,t^{-d/2}\,t^{-(|\alpha|+|\beta|)/2}\,e^{-b\,\frac{|x-y|^{2}}{t}}\,e^{\omega
t}$ (11)

and

$\displaystyle|(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,H_{t})(x+h,y+k)-(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,H_{t})(x,y)|$
$\displaystyle\leq$ $\displaystyle
a\,t^{-d/2}\,t^{-(|\alpha|+|\beta|)/2}\,\left(\frac{|h|+|k|}{\sqrt{t}+|x-y|}\right)^{\kappa}\,e^{-b\,\frac{|x-y|^{2}}{t}}\,e^{\omega
t}$

for all $x,y\in\Omega$ and $h,k\in\mathds{R}^{d}$ with $x+h,y+k\in\Omega$ and
$|h|+|k|\leq\tau\,\sqrt{t}+\tau^{\prime}\,|x-y|$.

  * (d)

$\|\partial^{\alpha}\,e^{-tA_{D}}\|_{L_{p}(\Omega)\to L_{q}(\Omega)}\leq
a\,t^{-|\alpha|/2}\,t^{-\frac{d}{2}(\frac{1}{p}-\frac{1}{q})}\,e^{\omega t}$
for all $t>0$, multi-index $\alpha$ and $p,q\in[1,\infty]$ with $p\leq q$ and
$|\alpha|\leq 1$.

  * (e)

$\|\partial^{\alpha}\,e^{-tA_{D}}\|_{L_{p}(\Omega)\to C^{\nu}(\Omega)}\leq
a\,t^{-|\alpha|/2}t^{-\frac{d}{2p}}\,t^{-\nu/2}\,e^{\omega t}$ for all $t>0$,
$p\in[1,\infty]$, multi-index $\alpha$ and $\nu\in(0,\kappa]$ with
$|\alpha|\leq 1$.

  * (f)

If, in addition,

$\frac{\mu}{2}\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm Re}\gothic{a}(u,u)$

for all $u\in W^{1,2}(\Omega)$, then one can choose $\omega=-\frac{\mu}{4}<0$.

###### Proof.

‘(a)–(c)’. If $b_{k}=c_{k}=0$, then this is Theorem 3.1 in [EO2]. It is stated
there for bounded $\Omega$ but we proved in the previous section all the
necessary results which allow us to adopt the proof of [EO2] to the setting of
this theorem. Cf. Theorem 3.16 in [EO2].

‘(f)’. Note that the coercivity bound implies that
$\|e^{-tA_{D}}\|_{L_{2}(\Omega)\to L_{2}(\Omega)}^{2}\leq e^{-\mu t}$ for all
$t>0$. We can repeat the same arguments to obtain uniform bounds (with respect
to $t$) for $A_{D}+\omega$ with $\omega=-\frac{\mu}{4}$.

‘(d)’. The Gaussian bounds (11) give

$\|\partial^{\alpha}\,e^{-tA_{D}}\|_{L_{p}(\Omega)\to L_{p}(\Omega)}\leq
a\,t^{-|\alpha|/2}\,e^{\omega
t}\,\int_{\mathds{R}^{d}}t^{-d/2}\,e^{-b\,\frac{|x-y|^{2}}{t}}\,dx=a\,(\pi/b)^{d/2}\,t^{-|\alpha|/2}\,e^{\omega
t}$

for all $p\in[1,\infty]$. Using Hölder’s estimate or Young’s inequality one
estimates similarly

$|(\partial^{\alpha}\,e^{-tA_{D}}u)(x)|\leq a\,\Big{(}(\pi/b)^{d/2}\vee
1\Big{)}\,t^{-|\alpha|/2}\,t^{-\frac{d}{2p}}\,e^{\omega
t}\,\|u\|_{L_{p}(\Omega)}$

for all $p\in[1,\infty]$, $u\in L_{p}(\Omega)$ and $x\in\Omega$. Then

$\|\partial^{\alpha}\,e^{-tA_{D}}\|_{L_{p}(\Omega)\to L_{q}(\Omega)}\leq
a\,\,\Big{(}(\pi/b)^{d/2}\vee
1\Big{)}\,t^{-|\alpha|/2}\,t^{-\frac{d}{2}(\frac{1}{p}-\frac{1}{q})}\,e^{\omega
t}$

by interpolation for all $p,q\in[1,\infty]$ with $p\leq q$ and $|\alpha|\leq
1$. Note that the constants $a$, $b$ and in particular $\omega$ are the same
as in Statement (c).

‘(e)’. Choosing $\tau=1$, then the bounds in Statement (c) are valid with
certain values of $a$, $b$ and $\omega$ for all $h$ whenever
$|h|\leq\sqrt{t}$. Hence if $x,x^{\prime}\in\Omega$ with
$|x-x^{\prime}|\leq\sqrt{t}$, then as in the proof of Statement (d) the Hölder
Gaussian bounds give

$|(\partial^{\alpha}e^{-tA_{D}}u)(x)-(\partial^{\alpha}e^{-tA_{D}}u)(x^{\prime})|\leq
a\,\Big{(}(\pi/b)^{d/2}\vee
1\Big{)}\,t^{-|\alpha|/2}\,t^{-\frac{d}{2p}}\,\Big{(}\frac{|x-x^{\prime}|}{\sqrt{t}}\Big{)}^{\kappa}\,e^{\omega
t}\,\|u\|_{L_{p}(\Omega)}.$

Alternatively, if $|x-x^{\prime}|\geq\sqrt{t}$, then

$\displaystyle|(\partial^{\alpha}e^{-tA_{D}}u)(x)-(\partial^{\alpha}e^{-tA_{D}}u)(x^{\prime})|$
$\displaystyle\leq$ $\displaystyle
2\Big{(}\frac{|x-x^{\prime}|}{\sqrt{t}}\Big{)}^{\kappa}\,\|\partial^{\alpha}e^{-tA_{D}}u\|_{L_{\infty}(\Omega)}$
$\displaystyle\leq$ $\displaystyle 2a\,\Big{(}(\pi/b)^{d/2}\vee
1\Big{)}\,t^{-|\alpha|/2}\,t^{-\frac{d}{2p}}\,\Big{(}\frac{|x-x^{\prime}|}{\sqrt{t}}\Big{)}^{\kappa}\,e^{\omega
t}\,\|u\|_{L_{p}(\Omega)}$

and the bound from $L_{p}(\Omega)$ into $C^{\kappa}(\Omega)$ follows. Finally,
the bounds in Statement (c) are obviously valid with $\nu$ instead of $\kappa$
for any $\nu\in(0,\kappa)$. Thus we obtain the bound from $L_{p}(\Omega)$ into
$C^{\nu}(\Omega)$. ∎

For bounded $\Omega$ one also has estimates for the semigroup from
$L_{p}(\Omega)$ into $L_{p}(\Gamma)$.

######  Proposition 4.2.

Let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be a bounded open set
of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$. Suppose there is a $\mu>0$ such that $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose that
$0\not\in\sigma(A_{D})$. Then there exist $a,\omega>0$ such that

$\|\partial_{k}\,e^{-tA_{D}}\|_{L_{p}(\Omega)\to L_{p}(\Gamma)}\leq
a\,t^{-\frac{1}{2}-\frac{1}{2p}}\,e^{\omega t}$

for all $k\in\\{1,\ldots,d\\}$, $t>0$ and $p\in[1,\infty]$.

###### Proof.

By Theorem 4.1 the semigroup $(e^{-tA_{D}})_{t>0}$ has a kernel
$(H_{t})_{t>0}$ which is differentiable in the first entry and there are
$a,b,\omega>0$ such that

$|(\partial^{(1)}_{k}H_{t})(x,y)|\leq
a\,t^{-d/2}\,t^{-1/2}\,e^{-b\,\frac{|x-y|^{2}}{t}}\,e^{\omega t}$

for all $t>0$, $x,y\in\Omega$ and $k\in\\{1,\ldots,d\\}$.

Let $k\in\\{1,\ldots,d\\}$. We use that
$\|\partial_{k}\,e^{-tA_{D}}\|_{L_{1}(\Omega)\to
L_{1}(\Gamma)}\leq\sup_{y\in\Omega}\int_{\Gamma}|(\partial^{(1)}_{k}H_{t})(z,y)|\,dz$.
Let $z_{0}\in\Gamma$. After a suitable rotation and translation we may assume
that there are $\delta>0$ and a $C^{1}_{b}$-function
$\tau\colon\mathds{R}^{d-1}\to\mathds{R}$ such that $z_{0}=0$ and
$\Gamma\cap[-\delta,\delta]^{d}=\\{(\xi,\tau(\xi)):\xi\in[-\delta,\delta]^{d-1}\\}$.
Let $y=(\tilde{y},y_{d})\in\Omega$. Then

$\displaystyle\int_{\Gamma\cap[-\delta,\delta]^{d}}|(\partial^{(1)}_{k}H_{t})(z,y)|\,dz$
$\displaystyle\leq$
$\displaystyle\int_{[-\delta,\delta]^{d-1}}a\,t^{-d/2}\,t^{-1/2}\,e^{-b\,\frac{|\xi-\tilde{y}|^{2}}{t}}\,e^{-b\,\frac{|\tau(\xi)-y_{d}|^{2}}{t}}\,e^{\omega
t}\,\sqrt{1+|(\nabla\tau)(\xi)|^{2}}\,d\xi$ $\displaystyle\leq$
$\displaystyle\int_{\mathds{R}^{d-1}}a\,t^{-d/2}\,t^{-1/2}\,e^{-b\,\frac{|\xi-\tilde{y}|^{2}}{t}}\,e^{\omega
t}\,\sqrt{1+\|\nabla\tau\|_{\infty}^{2}}\,d\xi$ $\displaystyle=$
$\displaystyle
a\,t^{-1}\,(\pi/b)^{(d-1)/2}\,\sqrt{1+\|\nabla\tau\|_{\infty}^{2}}\,e^{\omega
t}.$

Covering $\Gamma$ with a finite number of these type of sets, one deduces that
there is an $a_{1}>0$ such that

$\|\partial_{k}\,e^{-tA_{D}}\|_{L_{1}(\Omega)\to
L_{1}(\Gamma)}\leq\sup_{y\in\Omega}\int_{\Gamma}|(\partial^{(1)}_{k}H_{t})(z,y)|\,dz\leq
a_{1}\,t^{-1}\,e^{\omega t}$

for all $t>0$.

Much easier one obtains

$\|\partial_{k}\,e^{-tA_{D}}\|_{L_{\infty}(\Omega)\to
L_{\infty}(\Gamma)}\leq\sup_{z\in\Gamma}\,\int_{\Omega}|(\partial^{(1)}_{k}H_{t})(z,y)|\,dy\leq
a\,(\pi/b)^{d/2}t^{-1/2}\,e^{\omega t}$

for all $t>0$. Then the proposition follows by interpolation. ∎

The idea of using Morrey and Campanato spaces in order to obtain Gaussian
upper bounds and Hölder regularity for heat kernels of elliptic operators on
$\mathds{R}^{d}$ was used in [Aus]. We refer to [EO2] for more details and
other references on this subject.

##  5 Hölder bounds for the Green function

We consider now the Green function associated with the elliptic operator. We
prove upper bounds and Hölder continuity. We present two versions. The first
is valid for possibly unbounded $\Omega$, but requires an additional
coercivity condition. The second version does not require the additional
coercivity condition, but then we use that $\Omega$ is bounded.

######  Theorem 5.1.

Let $\kappa\in(0,1)$ and $\mu,M,K>0$. Then there exists a $c>0$ such that the
following is valid.

Let $\Omega\subset\mathds{R}^{d}$ be an open set of class $C^{1+\kappa}$ with
parameter $K$. For all $k,l\in\\{1,\ldots,d\\}$ let $c_{kl},b_{k},c_{k}\in
C^{\kappa}(\Omega)\cap L_{\infty}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$ with $\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$,
$\|b_{k}\|_{C^{\kappa}(\Omega)}\leq M$, $\|c_{k}\|_{C^{\kappa}(\Omega)}\leq M$
and $\|c_{0}\|_{L_{\infty}(\Omega)}\leq M$. Suppose $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose that

$\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm Re}\gothic{a}(u,u)$
(12)

for all $u\in W^{1,2}(\Omega)$.

Then the operator $A_{D}^{-1}$ has a kernel
$G\colon\\{(x,y)\in\Omega\times\Omega:x\neq y\\}\to\mathds{C}$, which is
differentiable in each variable and the derivative is differentiable in the
other variable. Moreover, for every multi-index $\alpha,\beta$ with
$0\leq|\alpha|,|\beta|\leq 1$ the function
$\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,G$ extends to a continuous
function on $\\{(x,y)\in\overline{\Omega}\times\overline{\Omega}:x\neq y\\}$
with estimates

$|(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,G)(x,y)|\leq\left\\{\begin{array}[]{ll}c\,|x-y|^{-(d-2+|\alpha|+|\beta|)}&\mbox{if
}d-2+|\alpha|+|\beta|\neq 0\\\\[5.0pt] c\,\log(1+\frac{1}{|x-y|})&\mbox{if
}d-2+|\alpha|+|\beta|=0\end{array}\right.$

and

$|(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,G)(x^{\prime},y^{\prime})-(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,G)(x,y)|\leq
c\,\frac{(|x^{\prime}-x|+|y^{\prime}-y|)^{\kappa}}{|x-y|^{d-2+|\alpha|+|\beta|+\kappa}}$

for all $x,x^{\prime},y,y^{\prime}\in\Omega$ with $x\neq y$, $x^{\prime}\neq
y^{\prime}$ and $|x-x^{\prime}|+|y-y^{\prime}|\leq\frac{1}{2}\,|x-y|$.

###### Proof.

The additional assumption (12) is the condition of Theorem 4.1(f). Choose
$\tau=1$ and $\tau^{\prime}=\frac{1}{2}$ in Theorem 4.1. Therefore one can
choose $\omega<0$ in Theorem 4.1(c). Then the estimates follow easily by the
Laplace transform as in the proof of Theorem 4.1 in [EO2]. ∎

If merely $0\not\in\sigma(A_{D})$ one can also obtain fairly uniform Green
kernel bounds.

######  Theorem 5.2.

Let $\kappa\in(0,1)$ and $\mu,M,K,\widetilde{K},R>0$. Then there exists a
$c>0$ such that the following is valid.

Let $\Omega\subset\mathds{R}^{d}$ be a open bounded set of class
$C^{1+\kappa}$ with parameter $K$ and $\mathop{\rm diam}\Omega\leq R$. For all
$k,l\in\\{1,\ldots,d\\}$ let $c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)\cap
L_{\infty}(\Omega)$, and let $c_{0}\in L_{\infty}(\Omega)$ with
$\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$, $\|b_{k}\|_{C^{\kappa}(\Omega)}\leq
M$, $\|c_{k}\|_{C^{\kappa}(\Omega)}\leq M$ and
$\|c_{0}\|_{L_{\infty}(\Omega)}\leq M$. Suppose

$\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$

for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose
$0\notin\sigma(A_{D})$ and $\|A_{D}^{-1}\|_{2\to 2}\leq\widetilde{K}$.

Then the operator $A_{D}^{-1}$ has a kernel
$G\colon\\{(x,y)\in\Omega\times\Omega:x\neq y\\}\to\mathds{C}$, which is
differentiable in each variable and the derivative is differentiable in the
other variable. Moreover, for every multi-index $\alpha,\beta$ with
$0\leq|\alpha|,|\beta|\leq 1$ the function
$\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,G$ extends to a continuous
function on $\\{(x,y)\in\overline{\Omega}\times\overline{\Omega}:x\neq y\\}$
with estimates

$|(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,G)(x,y)|\leq\left\\{\begin{array}[]{ll}c\,|x-y|^{-(d-2+|\alpha|+|\beta|)}&\mbox{if
}d-2+|\alpha|+|\beta|\neq 0\\\\[5.0pt] c\,\log(1+\frac{1}{|x-y|})&\mbox{if
}d-2+|\alpha|+|\beta|=0\end{array}\right.$ (13)

and

$|(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,G)(x^{\prime},y^{\prime})-(\partial_{x}^{\alpha}\,\partial_{y}^{\beta}\,G)(x,y)|\leq
c\,\frac{(|x^{\prime}-x|+|y^{\prime}-y|)^{\kappa}}{|x-y|^{d-2+|\alpha|+|\beta|+\kappa}}$

for all $x,x^{\prime},y,y^{\prime}\in\Omega$ with $x\neq y$, $x^{\prime}\neq
y^{\prime}$ and $|x-x^{\prime}|+|y-y^{\prime}|\leq\frac{1}{2}\,|x-y|$.

###### Proof.

Let $a,b>0$ and $\omega\in\mathds{R}$ be as in Theorem 4.1 with the choice
$\tau=1$ and $\tau^{\prime}=\frac{1}{2}$. Choose $\lambda=\omega+2$ and
consider the operator $A_{D}+\lambda\,I$.

The resolvent equation gives

$A_{D}^{-1}-(A_{D}+\lambda\,I)^{-1}=\lambda\,(A_{D}+\lambda\,I)^{-1}\,A_{D}^{-1}.$

Hence by iteration

$\displaystyle A_{D}^{-1}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{2d-1}\lambda^{k-1}\,(A_{D}+\lambda\,I)^{-k}+\lambda^{2d-1}\,(A_{D}+\lambda\,I)^{-2d}\,A_{D}^{-1}$
(14) $\displaystyle=$
$\displaystyle\sum_{k=1}^{2d-1}\lambda^{k-1}\,(A_{D}+\lambda\,I)^{-k}+\lambda^{2d-1}\,(A_{D}+\lambda\,I)^{-d}\,A_{D}^{-1}\,(A_{D}+\lambda\,I)^{-d}.$

If $m\in\mathds{N}$, then

$(A_{D}+\lambda\,I)^{-m}=\frac{1}{(m-1)!}\int_{0}^{\infty}t^{m-1}\,e^{-t(A_{D}+\lambda\,I)}\,dt.$

Since $t^{m-1}\leq(m-1)!\,\varepsilon^{-(m-1)}\,e^{\varepsilon t}$ for all
$\varepsilon>0$, one can estimate the kernel of $(A_{D}+\lambda\,I)^{-m}$ as
in the previous theorem.

It remains to bound the last term in (14). Recall that

$\|\partial^{\alpha}\,e^{-tA_{D}}\|_{L_{2}(\Omega)\to C^{\kappa}(\Omega)}\leq
a\,t^{-|\alpha|/2}\,t^{-d/4}\,t^{-\kappa/2}\,e^{\omega t}$

for all $t>0$ and for every multi-index $\alpha$ with $|\alpha|\leq 1$ by
Theorem 4.1(e). Hence

$\displaystyle\|\partial^{\alpha}\,(A_{D}+\lambda\,I)^{-d}\|_{L_{2}(\Omega)\to
C^{\kappa}(\Omega)}$ $\displaystyle=$
$\displaystyle\frac{1}{(d-1)!}\,\Big{\|}\int_{0}^{\infty}t^{d-1}\,e^{-\lambda
t}\,\partial^{\alpha}\,e^{-tA_{D}}\,dt\Big{\|}_{L_{2}(\Omega)\to
C^{\kappa}(\Omega)}$ $\displaystyle\leq$ $\displaystyle
c_{\alpha}=\frac{1}{(d-1)!}\,\int_{0}^{\infty}t^{d-1}\,\,a\,t^{-|\alpha|/2}\,t^{-d/4}\,t^{-\kappa/2}\,e^{-2t}\,dt.$

Similarly, Theorem 4.1(d) gives a bound

$\|\partial^{\alpha}\,(A_{D}+\lambda\,I)^{-d}\|_{L_{2}(\Omega)\to
C_{b}(\Omega)}\leq\widetilde{c}_{\alpha}=\frac{1}{(d-1)!}\,\int_{0}^{\infty}t^{d-1}\,\,a\,t^{-|\alpha|/2}\,t^{-d/4}\,e^{-2t}\,dt.$

Without loss of generality we may assume that

$\|\partial^{\alpha}\,(A_{D}^{*}+\lambda\,I)^{-d}\|_{L_{2}(\Omega)\to
C^{\kappa}(\Omega)}\leq c_{\alpha}$

and

$\|\partial^{\alpha}\,(A_{D}^{*}+\lambda\,I)^{-d}\|_{L_{2}(\Omega)\to
C_{b}(\Omega)}\leq\tilde{c}_{\alpha}$

for the two adjoint operators.

Next let $\alpha,\beta$ be two multi-indices with $|\alpha|,|\beta|\leq 1$. By
the Riesz representation theorem there exists a function
$K^{\alpha}\colon\Omega\times\Omega\to\mathds{C}$ such that
$K^{\alpha}(x,\cdot)\in L_{2}(\Omega)$ and

$(\partial^{\alpha}\,(A_{D}+\lambda\,I)^{-d}\,A_{D}^{-1}u)(x)=(u,\overline{K^{\alpha}(x,\cdot)})_{L_{2}(\Omega)}$

for all $u\in L_{2}(\Omega)$ and $x\in\Omega$. Then

$(\partial^{\alpha}\,(A_{D}+\lambda\,I)^{-d}\,A_{D}^{-1}u)(x)=\int_{\Omega}K^{\alpha}(x,y)\,u(y)\,dy$

for all $u\in L_{2}(\Omega)$ and $x\in\Omega$. Similarly, there exists a
function $K_{\beta}\colon\Omega\times\Omega\to\mathds{C}$ such that
$K_{\beta}(x,\cdot)\in L_{2}(\Omega)$ and

$(\partial^{\beta}\,(A_{D}^{*}+\lambda\,I)^{-d}u)(x)=\int_{\Omega}K_{\beta}(x,y)\,u(y)\,dy$

for all $u\in L_{2}(\Omega)$ and $x\in\Omega$. Then

$((\partial^{\beta}\,(A_{D}^{*}+\lambda\,I)^{-d})^{*}u)(x)=\int_{\Omega}\overline{K_{\beta}(y,x)}\,u(y)\,dy$

for all $u\in L_{2}(\Omega)$ and $x\in\Omega$. Define
$K^{(\alpha,\beta)}\colon\Omega\times\Omega\to\mathds{C}$ by

$K^{(\alpha,\beta)}(x,y)=(-1)^{|\beta|}\,\int_{\Omega}K^{\alpha}(x,z)\,\overline{K_{\beta}(y,z)}\,dz=(-1)^{|\beta|}\,(K^{\alpha}(x,\cdot),K_{\beta}(y,\cdot))_{L_{2}(\Omega)}.$

Then

$\Big{(}\partial^{\alpha}\,(A_{D}+\lambda\,I)^{-d}\,A_{D}^{-1}\,(A_{D}+\lambda\,I)^{-d}\,\partial^{\beta}u\Big{)}(x)=\int_{\Omega}K^{(\alpha,\beta)}(x,y)\,u(y)\,dy$

for all $u\in L_{2}(\Omega)$ (see also (1) in [AE2]). By the Cauchy–Schwarz
inequality,

$|K^{(\alpha,\beta)}(x,y)|\leq\tilde{c}_{\alpha}\,\tilde{c}_{\beta}\,\|A_{D}^{-1}\|_{2\to
2}$

for all $x,y\in\Omega$, which gives immediately uniform bounds as in (13).
Here the diameter of $\Omega$ is involved. Next let
$x,x^{\prime},y,y^{\prime}\in\Omega$ with $|x-x^{\prime}|\leq 1$ and
$|y-y^{\prime}|\leq 1$. Then

$\displaystyle|K^{(\alpha,\beta)}(x,y)-K^{(\alpha,\beta)}(x^{\prime},y^{\prime})|$
$\displaystyle\leq$
$\displaystyle|(K^{\alpha}(x,\cdot),K_{\beta}(y,\cdot)-K_{\beta}(y^{\prime},\cdot))_{L_{2}(\Omega)}|$
$\displaystyle\hskip
28.45274pt{}+|(K^{\alpha}(x,\cdot)-K^{\alpha}(x^{\prime},\cdot),K_{\beta}(y^{\prime},\cdot))_{L_{2}(\Omega)}|$
$\displaystyle\leq$ $\displaystyle
c_{\beta}\,\tilde{c}_{\alpha}\,\|A_{D}^{-1}\|_{2\to
2}\,|y-y^{\prime}|^{\kappa}+c_{\alpha}\,\tilde{c}_{\beta}\,\|A_{D}^{-1}\|_{2\to
2}\,|x-x^{\prime}|^{\kappa}.$

Since $\Omega$ is bounded, there is a suitable $c_{1}>0$ such that

$|K^{(\alpha,\beta)}(x,y)-K^{(\alpha,\beta)}(x^{\prime},y^{\prime})|\leq
c_{1}\,\frac{(|x^{\prime}-x|+|y^{\prime}-y|)^{\kappa}}{|x-y|^{d-2+|\alpha|+|\beta|+\kappa}}$

for all $x,x^{\prime},y,y^{\prime}\in\Omega$ with $x\neq y$, $x^{\prime}\neq
y^{\prime}$ and $|x-x^{\prime}|+|y-y^{\prime}|\leq\frac{1}{2}\,|x-y|$. Also
here the diameter of $\Omega$ is involved. This proves the theorem. ∎

##  6 The harmonic lifting on $L_{p}$

The semigroup $(e^{-tA_{D}})_{t>0}$ on $L_{2}(\Omega)$ extends consistently to
a semigroup on $L_{p}(\Omega)$ for all $p\in[1,\infty]$, which is a
$C_{0}$-semigroup if $p\in[1,\infty)$. We denote by $-A_{D}^{(p)}$ the
generator of this semigroup if confusion is possible.

######  Proposition 6.1.

Let $\kappa\in(0,1)$ and $K>0$. Let $\Omega\subset\mathds{R}^{d}$ be an open
set of class $C^{1+\kappa}$ with parameter $K$. For all
$k,l\in\\{1,\ldots,d\\}$ let $c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)\cap
L_{\infty}(\Omega)$, and let $c_{0}\in L_{\infty}(\Omega)$. Suppose there is a
$\mu>0$ such that $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose that
$0\not\in\sigma(A_{D})$. Then one has the following.

  * (a)

If $p\in[1,\infty)$, then $D(A_{D}^{(p)})\subset W^{1,p}(\Omega)$ and the map
$\partial^{\alpha}\,(A_{D}^{(p)})^{-1}$ is continuous from $L_{p}(\Omega)$
into $L_{p}(\Omega)$ for every multi-index $\alpha$ with $|\alpha|\leq 1$.

  * (b)

If $\nu\in(0,\kappa]$, $p\in(\frac{d}{1-\nu},\infty]$ and $\alpha$ is a multi-
index with $|\alpha|\leq 1$, then $\partial^{\alpha}\,A_{D}^{-1}$ maps
$L_{p}(\Omega)$ continuously into $C^{\nu}(\Omega)\cap L_{\infty}(\Omega)$.

  * (c)

If $p\in(1,\infty]$, $\Omega$ is bounded and $\alpha$ is a multi-index with
$|\alpha|\leq 1$, then the operator
$u\mapsto(\partial^{\alpha}\,A_{D}^{-1}u)|_{\Gamma}$ from $L_{\infty}(\Omega)$
into $C(\Gamma)$ extends consistently to a bounded operator from
$L_{p}(\Omega)$ into $L_{p}(\Gamma)$.

  * (d)

If $\nu\in(0,\kappa]$, $p\in(\frac{d}{1-\nu},\infty]$ and $\alpha$ is a multi-
index with $|\alpha|\leq 1$, then
$u\mapsto(\partial^{\alpha}\,A_{D}^{-1}u)|_{\Gamma}$ is continuous from
$L_{p}(\Omega)$ into $C^{\nu}(\Gamma)\cap L_{\infty}(\Gamma)$.

###### Proof.

‘(a)’. There exists a $\lambda>0$ such that

$\frac{\mu}{2}\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u)+\lambda\|u\|_{L_{2}(\Omega)}^{2}$

for all $u\in W^{1,2}(\Omega)$. Then for all $p\in[1,\infty]$ the function
$t\mapsto\partial^{\alpha}\,e^{-t(A_{D}+\lambda\,I)}$ from $(0,\infty)$ into
${\cal L}(L_{p}(\Omega))$ is integrable by Theorem 4.1(d), see also Statement
(f) of that theorem. Hence $D(A_{D}^{(p)}+\lambda\,I)\subset W^{1,p}(\Omega)$
and the map $\partial^{\alpha}\,(A_{D}^{(p)}+\lambda\,I)^{-1}$ is continuous
from $L_{p}(\Omega)$ into $L_{p}(\Omega)$ for all $k\in\\{1,\ldots,d\\}$. Then
also
$\partial^{\alpha}\,(A_{D}^{(p)})^{-1}=\partial^{\alpha}\,(A_{D}^{(p)}+\lambda\,I)^{-1}(A_{D}+\lambda\,I)\,A_{D}^{-1}$
is bounded.

The proof of Statement (b) is similar using Theorem 4.1(e) and for Statement
(c) use Proposition 4.2. Statement (d) follows from Statement (b). ∎

In the proof of the next proposition we use the uniform bounds that we derived
in Section 4.

######  Proposition 6.2.

Let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be an open bounded set
of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$. Suppose there is a $\mu>0$ such that $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose that
$0\not\in\sigma(A_{D})$. Let $\nu\in(0,\kappa]$ and
$p\in(\frac{d}{1-\nu},\infty]$. Let $u\in L_{p}(\Omega)$. Then $A_{D}^{-1}u$
has a weak conormal derivative and

$\partial_{n}{a}\,A_{D}^{-1}u=\sum_{k,l=1}^{d}n_{k}\,(c_{kl}\,\partial_{l}\,A_{D}^{-1}u)|_{\Gamma}+\sum_{k=1}^{d}n_{k}\,(b_{k}\,A_{D}^{-1}u)|_{\Gamma}.$

Moreover, $\partial_{n}{a}\,A_{D}^{-1}u\in C^{\nu}(\Gamma)$ and the map
$u\mapsto\partial_{n}{a}\,A_{D}^{-1}u$ is continuous from $L_{p}(\Omega)$ into
$C^{\nu}(\Gamma)$.

###### Proof.

Let $M>0$ be such that $\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$,
$\|b_{k}\|_{C^{\kappa}(\Omega)}\leq M$, $\|c_{k}\|_{C^{\kappa}(\Omega)}\leq M$
and $\|c_{0}\|_{L_{\infty}(\Omega)}\leq M$. There exists a $\lambda>0$,
depending only on $\mu$ and $M$, such that

$\frac{3\mu}{4}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u)+\lambda\|u\|_{L_{2}(\Omega)}^{2}$

for all $u\in W^{1,2}(\Omega)$. One can regularise the coefficients. First
extend the $c_{kl}$ to functions $\tilde{c}_{kl}\in
C^{\kappa}(\mathds{R}^{d})$ and similarly to the $b_{k}$ and $c_{k}$. Then
obtain coefficients $c^{(n)}_{kl}$, etc., and operators $A_{D}^{(n)}$. Without
loss of generality

$\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}^{(n)}(u,u)+\lambda\|u\|_{L_{2}(\Omega)}^{2}$

for all $u\in W^{1,2}(\Omega)$ and $n\in\mathds{N}$. By Theorem 4.1(e) and (f)
for all $k\in\\{1,\ldots,d\\}$ the function
$t\mapsto\partial_{k}\,e^{-t(A_{D}^{(n)}+\lambda\,I)}$ from $(0,\infty)$ into
${\cal L}(L_{p}(\Omega),C^{\nu}(\Omega))$ is integrable with a uniform bound
in $n\in\mathds{N}$. Now we prove the desired equality for the operator
$A_{D}^{(n)}$ with $C^{\infty}$ coefficients and then let $n\to\infty$. The
arguments are exactly the same as in Step 2 of the proof of Proposition 5.3 in
[EO2]. We then deduce that for all $u\in L_{p}(\Omega)$ the function
$(A_{D}+\lambda\,I)^{-1}u$ has a weak conormal derivative and

$\partial_{n}{a}(A_{D}+\lambda\,I)^{-1}u=\sum_{k,l=1}^{d}n_{k}\,{\mathop{\rm
Tr\,}}(c_{kl}\,\partial_{l}\,(A_{D}+\lambda\,I)^{-1}u)+\sum_{k=1}^{d}n_{k}\,{\mathop{\rm
Tr\,}}(b_{k}\,(A_{D}+\lambda\,I)^{-1}u).$

Finally replace $u$ by $(A_{D}+\lambda\,I)\,A_{D}^{-1}u$ to obtain the
proposition. ∎

Every element of the domain $D(A_{D})$ of the elliptic operator has a conormal
derivative. The next proposition is an extension of [BE] Proposition 4.2(a).

######  Proposition 6.3.

Let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be an open bounded set
of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$. Suppose there is a $\mu>0$ such that $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose that
$0\not\in\sigma(A_{D})$. Then $D(A_{D})\subset D(\partial_{n}{a})$.

###### Proof.

Define $R\colon L_{\infty}(\Omega)\to C(\Gamma)$ by

$Ru=\sum_{k,l=1}^{d}n_{k}\,(c_{kl}\,\partial_{l}\,A_{D}^{-1}u)|_{\Gamma}+\sum_{k=1}^{d}n_{k}\,(b_{k}\,A_{D}^{-1}u)|_{\Gamma}.$

By Proposition 6.1(c) the operator $R$ extends to a bounded operator $R_{2}$
from $L_{2}(\Omega)$ into $L_{2}(\Gamma)$. Let $v\in W^{1,2}(\Omega)$. Next
let $u\in L_{\infty}(\Omega)$. It follows from Proposition 6.2 that
$A_{D}^{-1}u\in D(\partial_{n}{a})$ and $Ru=\partial_{n}{a}\,A_{D}^{-1}u$.
Hence

$(Ru,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}=(\partial_{n}{a}\,A_{D}^{-1}u,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}=\gothic{a}(A_{D}^{-1}u,v)-(u,v)_{L_{2}(\Omega)}.$

Since $A_{D}^{-1}$ is continuous from $L_{2}(\Omega)$ into $W^{1,2}(\Omega)$
by the closed graph theorem, and $L_{\infty}(\Omega)$ is dense in
$L_{2}(\Omega)$, one deduces that

$(R_{2}u,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}=\gothic{a}(A_{D}^{-1}u,v)-(u,v)_{L_{2}(\Omega)}$

for all $u\in L_{2}(\Omega)$. Hence $D(A_{D})\subset D(\partial_{n}{a})$. ∎

There is a remarkable relation between the harmonic lifting, conormal
derivative $\partial_{n}^{\gothic{a}^{*}}$ with respect of the dual form
$\gothic{a}^{*}$ and the resolvent of the Dirichlet operator.

######  Lemma 6.4.

Let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be an open bounded set
of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$. Suppose there is a $\mu>0$ such that $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose that
$0\not\in\sigma(A_{D})$. Let $v\in L_{2}(\Omega)$ and $\varphi\in
H^{1/2}(\Gamma)$. Then

$(\gamma(\varphi),v)_{L_{2}(\Omega)}=-(\varphi,\partial_{n}^{\gothic{a}^{*}}\,(A_{D}^{*})^{-1}v)_{L_{2}(\Gamma)}.$

###### Proof.

Since $(A_{D}^{*})^{-1}v\in W_{0}^{1,2}(\Omega)$, we have by the definition of
$\gamma(\varphi)$

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\overline{\gothic{a}(\gamma(\varphi),(A_{D}^{*})^{-1}v)}$
$\displaystyle=$
$\displaystyle\gothic{a}^{*}((A_{D}^{*})^{-1}v,\gamma(\varphi))$
$\displaystyle=$
$\displaystyle\int_{\Omega}(A_{D}^{*}\,(A_{D}^{*})^{-1}v)\,\overline{\gamma(\varphi)}+\int_{\Gamma}\Big{(}\partial_{n}^{\gothic{a}^{*}}\,(A_{D}^{*})^{-1}v\Big{)}\,\overline{\varphi},$

where we used the fact that the conormal derivative
$\partial_{n}^{\gothic{a}^{*}}\,(A_{D}^{*})^{-1}v$ exists by Proposition 6.3.
∎

If one considers $\gamma$ as a densely defined operator from $L_{2}(\Gamma)$
into $L_{2}(\Omega)$, then we just proved that
$\gamma^{*}=-(\partial_{n}^{\gothic{a}^{*}}\,(A_{D}^{*})^{-1})^{*}$. Note that
Theorem 1.2(b) is a consequence of Statement (d) in the next theorem.

######  Theorem 6.5.

Let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be an open bounded set
with a $C^{1+\kappa}$-boundary. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$. Suppose there is a $\mu>0$ such that $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose that
$0\not\in\sigma(A_{D})$. Let $G\colon\\{(x,y)\in\Omega\times\Omega:x\neq
y\\}\to\mathds{C}$ be as in Theorem 5.1. Then $G$ is differentiable on
$\\{(x,y)\in\Omega\times\Omega:x\neq y\\}$ by Theorem 5.1 and the derivative
extends to a continuous function on
$\\{(x,y)\in\overline{\Omega}\times\overline{\Omega}:x\neq y\\}$. Define the
function $K_{\gamma}\colon\Omega\times\Gamma\to\mathds{C}$ by

$K_{\gamma}(x,z)=-\sum_{k,l=1}^{d}n_{k}(z)\,c_{lk}(z)\,(\partial^{(2)}_{l}G)(x,z)-\sum_{k=1}^{d}n_{k}(z)\,c_{k}(z)\,G(x,z).$

Then one has the following.

  * (a)

The map $K_{\gamma}$ is continuous.

Define $T\colon L_{1}(\Gamma)\to C(\Omega)$ by

$(T\varphi)(x)=\int_{\Gamma}K_{\gamma}(x,z)\,\varphi(z)\,dz.$

  * (b)

If $\varphi\in H^{1/2}(\Gamma)$, then $\gamma(\varphi)=T\varphi$ a.e. In
particular, $T$ is an extension of the harmonic map $\gamma$.

  * (c)

There exists a $\tilde{c}>0$ such that

$|K_{\gamma}(x,z)|\leq\frac{\tilde{c}}{|x-z|^{d-1}}\quad\mbox{and}\quad|K_{\gamma}(x^{\prime},z^{\prime})-K_{\gamma}(x,z)|\leq\tilde{c}\,\frac{(|x^{\prime}-x|+|z^{\prime}-z|)^{\kappa}}{|x-z|^{d-1+\kappa}}$

for all $x,x^{\prime}\in\Omega$ and $z,z^{\prime}\in\Gamma$ with
$|x^{\prime}-x|+|z^{\prime}-z|\leq\frac{1}{2}\,|x-z|$.

  * (d)

If $p\in[1,\infty)$ and $\varphi\in L_{p}(\Gamma)$, then $T\varphi\in
L_{p}(\Omega)$ and the map $T\colon L_{p}(\Gamma)\to L_{p}(\Omega)$ is
bounded. In particular $\gamma$ extends to a bounded map from $L_{2}(\Gamma)$
into $L_{2}(\Omega)$.

  * (e)

Let $T_{2}=T|_{L_{2}(\Gamma)}\colon L_{2}(\Gamma)\to L_{2}(\Omega)$. Then for
all $p\in(\frac{d}{1-\kappa},\infty]$ there exists a $c>0$ such that
$T_{2}^{*}u\in C^{\kappa}(\Gamma)$ and
$\|T_{2}^{*}u\|_{C^{\kappa}(\Gamma)}\leq c\,\|u\|_{L_{p}(\Omega)}$ for all
$u\in L_{p}(\Omega)$.

###### Proof.

Statements (a)–(c) and also the $p=1$ version of Statement (d) follow as in
the proof of Proposition 5.5 in [EO2], where we now use Lemma 6.4 and
Proposition 6.2.

‘(d)’. Let $p\in(1,\infty)$ and let $q\in(1,\infty)$ be the dual exponent.
Define $R\colon L_{\infty}(\Omega)\to C(\Gamma)$ by

$Ru=\sum_{k,l=1}^{d}n_{k}\,(\overline{c_{lk}}\,\partial_{l}\,(A_{D}^{*})^{-1}u)|_{\Gamma}+\sum_{k=1}^{d}n_{k}\,(\overline{c_{k}}\,(A_{D}^{*})^{-1}u)|_{\Gamma}.$

Then by Proposition 6.1(c) the map $R$ extends to a bounded operator $R_{q}$
from $L_{q}(\Omega)$ into $L_{q}(\Gamma)$. Let $\varphi\in H^{1/2}(\Gamma)$
and $u\in L_{\infty}(\Omega)$. Then Statement (b), Lemma 6.4 and Proposition
6.2 give

$(T\varphi,u)_{L_{2}(\Omega)}=(\gamma(\varphi),u)_{L_{2}(\Omega)}=-(\varphi,\partial_{n}^{\gothic{a}^{*}}\,(A_{D}^{*})^{-1}u)_{L_{2}(\Gamma)}=-(\varphi,Ru)_{L_{2}(\Gamma)}.$

Hence

$\int_{\Omega}(T\varphi)\,\overline{u}=-\int\varphi\,\overline{Ru}$ (15)

first for all $\varphi\in H^{1/2}(\Gamma)$ and then by continuity and density
for all $\varphi\in L_{1}(\Gamma)$.
Now let $\varphi\in L_{p}(\Gamma)$. Then

$\Big{|}\int_{\Omega}(T\varphi)\,\overline{u}\Big{|}=|\int\varphi\,\overline{Ru}|\leq\|\varphi\|_{L_{p}(\Gamma)}\,\|Ru\|_{L_{q}(\Gamma)}\leq\|\varphi\|_{L_{p}(\Gamma)}\,\|R_{q}\|_{L_{q}(\Omega)\to
L_{q}(\Gamma)}\,\|u\|_{L_{q}(\Omega)}$

for all $u\in L_{\infty}(\Omega)$. Hence $T\varphi\in L_{p}(\Omega)$ and
$\|T\varphi\|_{L_{p}(\Omega)}\leq\|R_{q}\|_{L_{q}(\Omega)\to
L_{q}(\Gamma)}\,\|\varphi\|_{L_{p}(\Gamma)}$.

‘(e)’. It follows from (15) that
$(T_{2}\varphi,u)_{L_{2}(\Omega)}=-(\varphi,R_{2}u)_{L_{2}(\Gamma)}$ first for
all $\varphi\in L_{2}(\Gamma)$ and $u\in L_{\infty}(\Omega)$, and then by
density for all $\varphi\in L_{2}(\Gamma)$ and $u\in L_{2}(\Omega)$. Hence
$T_{2}^{*}=-R_{2}$. If $u\in L_{p}(\Omega)$, then $T_{2}^{*}u=-R_{2}u=R_{p}u$.
Now the statement follows from Proposition 6.1(d). This completes the proof of
the proposition. ∎

If no confusion is possible, then we will identify $\gamma$ with
$T|_{L_{2}(\Gamma)}\colon L_{2}(\Gamma)\to L_{2}(\Omega)$.

##  7 The harmonic lifting on $C^{\nu}$

Throughout this section let $\kappa\in(0,1)$ and $\Omega\subset\mathds{R}^{d}$
a bounded open set of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$
let $c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$ and $c_{0}\in
L_{\infty}(\Omega)$. Let the form $\gothic{a}$ and operator $A_{D}$ be as in
the introduction. We assume that $0\not\in\sigma(A_{D})$. Let ${\cal N}$ be
the associated Dirichlet-to-Neumann operator. In this section we prove Hölder
continuity estimates for the harmonic lifting as stated in Theorem 1.2(a).

Let $M>0$ be such that $\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$,
$\|b_{k}\|_{C^{\kappa}(\Omega)}\leq M$, $\|c_{k}\|_{C^{\kappa}(\Omega)}\leq M$
and $\|c_{0}\|_{L_{\infty}(\Omega)}\leq M$. There exists a $\lambda>0$,
depending only on $\mu$ and $M$, such that

$\frac{\mu}{2}\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u)+\lambda\|u\|_{L_{2}(\Omega)}^{2}$ (16)

for all $u\in W^{1,2}(\Omega)$. Let $\gamma_{\lambda}$ and ${\cal
N}_{\lambda}$ be the harmonic lifting and Dirichlet-to-Neumann operator with
respect to the operator with coefficient $c_{0}+\lambda\,\mathds{1}_{\Omega}$
instead of $c_{0}$. These correspond to the form $\gothic{a}_{\lambda}$.

First we need a family of suitable cut-off functions.

######  Lemma 7.1.

There exists a $\hat{c}\geq 1$ and for all $r\in(0,\infty)$ there exists a
function $\chi_{r}\in C_{c}^{\infty}(\mathds{R}^{d})$ such that
$0\leq\chi_{r}\leq 1$, $\chi_{r}(x)=1$ for all $x\in B(0,2r)$, $\mathop{\rm
supp}\chi_{r}\subset B(0,3r)$ and
$\|\partial^{\alpha}\chi_{r}\|_{\infty}\leq\frac{\hat{c}}{r^{|\alpha|}}$ for
every multi-index with $1\leq|\alpha|\leq 2$.

###### Proof.

Fix $\chi\in C_{c}^{\infty}(\mathds{R}^{d})$ such that $0\leq\chi\leq 1$,
$\chi(x)=1$ for all $x\in B(0,2)$ and $\mathop{\rm supp}\chi\subset B(0,3)$.
Let $r\in(0,\infty)$. Define $\chi_{r}\in C_{c}^{\infty}(\mathds{R}^{d})$ by
$\chi_{r}(x)=\chi(r^{-1}\,x)$. ∎

The proof of Theorem 1.2(a) heavily depends on the following estimate.

######  Lemma 7.2.

Let $\hat{c}\geq 1$ and the $\chi_{r}$ be as in Lemma 7.1. Then there exists a
$c^{\prime}>0$ such that the following is valid.

Let $r\in(0,\mathop{\rm diam}(\Omega)]$ and $y\in\Omega$. Define
$\Psi\colon\overline{\Omega}\to\mathds{R}$ by $\Psi(x)=\chi_{r}(x-y)$. Then
$|(\gamma_{\lambda}(\Psi|_{\Gamma}))(y)|\leq c^{\prime}$.

At the moment, for the proof of Lemma 7.2, we need only the first three parts
of the next technical lemma. In order to avoid too much repetition in the
proof of the commutator estimates, we already include some results for the
Dirichlet-to-Neumann operator. Recall the convention that we identify
$C^{\nu}(\Omega)$ with $C^{\nu}(\overline{\Omega})$.

######  Lemma 7.3.

There exists an $\widetilde{M}>0$ such that the following is valid.

Let $r\in(0,\mathop{\rm diam}\Omega]$, $\Psi\in C^{1+\kappa}(\Omega)$,
$y\in\overline{\Omega}$ and $c\geq 0$. Suppose that $\mathop{\rm
supp}\Psi\subset B(y,3r)$,

$\|\Psi\|_{L_{\infty}(\Omega)}\leq
c\,r,\quad\|\partial_{k}\Psi\|_{L_{\infty}(\Omega)}\leq
c\quad\mbox{and}\quad|||\partial_{k}\Psi|||_{C^{\kappa}(\Omega)}\leq
c\,r^{-\kappa}$

for all $k\in\\{1,\ldots,d\\}$. Then $\gamma_{\lambda}(\Psi|_{\Gamma})\in
C^{1+\kappa}(\Omega)$, $\Psi|_{\Gamma}\in D({\cal N}_{\lambda})$ and ${\cal
N}_{\lambda}(\Psi|_{\Gamma})\in C^{\kappa}(\Gamma)$. Moreover,

  * (a)

$\|\nabla\gamma_{\lambda}(\Psi|_{\Gamma})\|_{L_{\infty}(\Omega)}\leq\widetilde{M}\,c$,

  * (b)

$|||\nabla\gamma_{\lambda}(\Psi|_{\Gamma})|||_{C^{\kappa}(\Omega)}\leq\widetilde{M}\,c\,r^{-\kappa}$,

  * (c)

$|\Psi(x)-(\gamma_{\lambda}(\Psi|_{\Gamma}))(x)|\leq\widetilde{M}\,c\,d(x,\Gamma)$
for all $x\in\Omega$,

  * (d)

$\|{\cal N}_{\lambda}(\Psi|_{\Gamma})\|_{C(\Gamma)}\leq\widetilde{M}\,c$, and

  * (e)

$|||{\cal
N}_{\lambda}(\Psi|_{\Gamma})|||_{C^{\kappa}(\Gamma)}\leq\widetilde{M}\,c\,r^{-\kappa}$.

###### Proof.

There is a $\hat{c}>0$ such that $|\Omega(x,\rho)|\geq\hat{c}\,\rho^{d}$ for
all $x\in\overline{\Omega}$ and $\rho\in(0,\mathop{\rm diam}(\Omega)]$.

Define $\tau\in W^{1,2}_{0}(\Omega)$ by
$\tau=\Psi-\gamma_{\lambda}(\Psi|_{\Gamma})$. Recall that $\lambda>0$ is
introduced in (16). Then

$\gothic{a}_{\lambda}(\tau,v)=\gothic{a}_{\lambda}(\Psi,v)=\sum_{k=1}^{d}(f_{k},\partial_{k}v)_{L_{2}(\Omega)}+(f_{0},v)_{L_{2}(\Omega)}$

for all $v\in W^{1,2}_{0}(\Omega)$, where

$f_{k}=b_{k}\,\Psi+\sum_{l=1}^{d}c_{kl}\,\partial_{l}\Psi\quad\mbox{and}\quad
f_{0}=(c_{0}+\lambda)\,\Psi+\sum_{l=1}^{d}c_{l}\,\partial_{l}\Psi$

for all $k\in\\{1,\ldots,d\\}$. Note that $\mathop{\rm
supp}f_{k}\subset\Omega(y,3r)$ for all $k\in\\{0,1,\ldots,d\\}$. Then there is
an $M_{1}>0$, depending only on $M$, $\mu$ and $\Omega$, but independent of
$\Psi$, $y$, $r$ and $c$, such that

$\|f_{0}\|_{L_{\infty}(\Omega)}\leq
M_{1}\,c,\quad\|f_{k}\|_{L_{\infty}(\Omega)}\leq
M_{1}\,c\quad\mbox{and}\quad|||f_{k}|||_{C^{\kappa}(\Omega)}\leq
M_{1}\,c\,r^{-\kappa}$

for all $k\in\\{1,\ldots,d\\}$. Hence
$\|f_{0}\|_{M,d+2\kappa-2}\leq\|f_{0}\|_{M,d}\leq\sqrt{\omega_{d}}\,\|f_{0}\|_{L_{\infty}(\Omega)}\leq\sqrt{\omega_{d}}\,M_{1}\,c$
and $\|f_{k}\|_{C^{\kappa}(\Omega)}\leq 2M_{1}\,c\,r^{-\kappa}$ for all
$k\in\\{1,\ldots,d\\}$. Let $\tilde{c}>0$ be as in Proposition 3.9. Then

$\sum_{k=1}^{d}\|\partial_{k}\tau\|_{C^{\kappa}(\Omega)}\leq
c\,\tilde{c}\,M_{1}\,(1+2d)\,r^{-\kappa}.$ (17)

On the other hand,

$\|f_{k}\|_{L_{2}(\Omega)}\leq\|f_{k}\|_{L_{\infty}(\Omega)}\,|\Omega(y,3r)|^{1/2}\leq\sqrt{\omega_{d}}\,3^{d/2}\,M_{1}\,c\,r^{d/2}$

for all $k\in\\{0,1,\ldots,d\\}$. Then coercivity (16) implies that

$\|\nabla\tau\|_{L_{2}(\Omega)}\leq\frac{2}{\mu}\,\sum_{k=0}^{d}\|f_{k}\|_{L_{2}(\Omega)}\leq\frac{2}{\mu}\,(d+1)\,\sqrt{\omega_{d}}\,3^{d/2}\,M_{1}\,c\,r^{d/2}.$

Let $x\in\Omega$ and $k\in\\{1,\ldots,d\\}$. If $\tilde{x}\in\Omega(x,r\wedge
1)$, then
$|(\partial_{k}\tau)(x)|\leq|(\partial_{k}\tau)(\tilde{x})|+|||\partial_{k}\tau|||_{C^{\kappa}(\Omega)}\,(r\wedge
1)^{\kappa}$. Integrating over $\tilde{x}\in\Omega(x,r\wedge 1)$ and dividing
by $|\Omega(x,r\wedge 1)|$ gives

$\displaystyle|(\partial_{k}\tau)(x)|$ $\displaystyle\leq$
$\displaystyle|||\partial_{k}\tau|||_{C^{\kappa}(\Omega)}\,(r\wedge
1)^{\kappa}+\langle|\partial_{k}\tau|\rangle_{\Omega(x,r\wedge 1)}$
$\displaystyle\leq$
$\displaystyle|||\partial_{k}\tau|||_{C^{\kappa}(\Omega)}\,(r\wedge
1)^{\kappa}+|\Omega(x,r\wedge
1)|^{-1/2}\,\|\partial_{k}\tau\|_{L_{2}(\Omega)}\leq M_{2}\,c,$

where
$M_{2}=\tilde{c}\,M_{1}\,(1+2d)+\hat{c}^{-1/2}\,\frac{2}{\mu}\,(d+1)\,\sqrt{\omega_{d}}\,3^{d/2}\,M_{1}(1\vee(\mathop{\rm
diam}\Omega)^{d/2})$. This proves Statements (a) and (b), since
$\gamma_{\lambda}(\Psi)|_{\Gamma})=\Psi-\tau$.

Before we turn to the proof of Statement (c), we derive a bound for
$\|\tau\|_{L_{\infty}(\Omega)}$ which is not optimal in $r$, but suffices for
the proof of Statement (c). Coercivity (16) gives as before

$\|\tau\|_{L_{2}(\Omega)}\leq\frac{2}{\mu}\,(d+1)\,\sqrt{\omega_{d}}\,3^{d/2}\,M_{1}\,c\,r^{d/2}.$

Using Proposition 3.5 there exists an $M_{2}>0$, independent of $\Psi$, $y$,
$r$ and $c$, such that

$\|\nabla\tau\|_{M,d-1}\leq
M_{2}\sum_{k=0}^{d}\|f_{k}\|_{L_{\infty}(\Omega)}\leq(d+1)\,M_{1}\,M_{2}\,c.$

Then Lemma 3.2 gives a bound for $|||\tau|||_{{\cal M},d+1}$. Next use [ER]
Lemma 3.1(c) to bound the $C^{1/2}(\Omega)$-seminorm by the Campanato norm.
Hence there is an $M_{3}>0$, independent of $\Psi$, $r$ and $c$, such that
$|||\tau|||_{C^{1/2}(\Omega)}\leq M_{3}\,c$. Let $x\in\Omega$. Then

$|\tau(x)|\leq|||\tau|||_{C^{1/2}(\Omega)}\,(r\wedge
1)^{1/2}+\langle|\tau|\rangle_{\Omega(x,r\wedge 1)}\leq M_{4}\,c,$

where
$M_{4}=M_{3}+\frac{2}{\mu}\,(d+1)\,\sqrt{\omega_{d}}\,3^{d/2}\,M_{1}\,(1\vee(diam\Omega)^{d/2})$.

Now we prove Statement (c). Let $x\in\Omega$. Then there is a $z\in\Gamma$
such that $d(x,\Gamma)=|x-z|$. Obviously the straight line segment from $x$ to
$z$ is in $\Omega$, except of course for the point $z$. Since $\tau(z)=0$, the
estimate in Statement (c) follows from the mean value theorem and the bound of
Statement (a).

Finally we turn to the Dirichlet-to-Neumann operator. First assume that
$c_{kl},b_{k},c_{k},c_{0}\in C^{\infty}_{b}(\Omega)$ for all
$k,l\in\\{1,\ldots,d\\}$ and there is a $\Phi\in C^{\infty}(\mathds{R}^{d})$
such that $\Psi=\Phi|_{\overline{\Omega}}$. Then $\tau\in C^{\infty}(\Omega)$
by interior regularity. It follows as in the proof of Lemma 5.2 in [EO2] that
$\Psi-\tau$ has a weak conormal derivative and

$\displaystyle\partial_{n}^{\gothic{a}_{\lambda}}(\Psi-\tau)$ $\displaystyle=$
$\displaystyle\sum_{k,l=1}^{d}n_{k}\,(c_{kl}\,\partial_{l}(\Psi-\tau))|_{\Gamma}+\sum_{k=1}^{d}n_{k}\,(b_{k}\,(\Psi-\tau))|_{\Gamma}$
$\displaystyle=$
$\displaystyle\sum_{k,l=1}^{d}n_{k}\,(c_{kl}\,\partial_{l}(\Psi-\tau))|_{\Gamma}+\sum_{k=1}^{d}n_{k}\,(b_{k}\,\Psi)|_{\Gamma},$

where we used in the last step that $\tau|_{\Gamma}=0$ pointwise because
$\tau\in C(\overline{\Omega})\cap H^{1}_{0}(\Omega)$ and $\Omega$ is
Lipschitz.

So $\gamma_{\lambda}(\Psi|_{\Gamma})=\Psi-\tau\in
D(\partial_{n}^{\gothic{a}_{\lambda}})$ and $\Psi|_{\Gamma}\in D({\cal
N}_{\lambda})$. Moreover,

$\displaystyle{\cal N}_{\lambda}(\Psi|_{\Gamma})$ $\displaystyle=$
$\displaystyle\partial_{n}^{\gothic{a}_{\lambda}}\,\gamma_{\lambda}(\Psi)|_{\Gamma})=\partial_{n}^{\gothic{a}_{\lambda}}\,(\Psi-\tau)=\sum_{k,l=1}^{d}n_{k}\,\Big{(}c_{kl}\,\partial_{l}(\Psi-\tau)\Big{)}|_{\Gamma}+\sum_{k=1}^{d}n_{k}\,(b_{k}\,\Psi)|_{\Gamma}.$

Hence

$\gothic{a}_{\lambda}(\gamma_{\lambda}(\Psi|_{\Gamma}),v)=\int_{\Gamma}\sum_{k,l=1}^{d}n_{k}\,\Big{(}c_{kl}\,\partial_{l}(\Psi-\tau)\Big{)}|_{\Gamma}\,\overline{{\mathop{\rm
Tr\,}}v}+\sum_{k=1}^{d}n_{k}\,(b_{k}\,\Psi)|_{\Gamma}\,\overline{{\mathop{\rm
Tr\,}}v}$ (18)

for all $v\in W^{1,2}(\Omega)$.

Next we consider Hölder continuous coefficients. We proceed again by
approximation. First extend the $c_{kl}$ to functions $\tilde{c}_{kl}\in
C^{\kappa}(\mathds{R}^{d})$ and similarly to the $b_{k}$ and $c_{k}$. Then one
obtains coefficients $c^{(n)}_{kl}$, etc., operator $A_{D}^{(n)}$, form
$\gothic{a}^{(n)}$, harmonic lifting $\gamma_{\lambda}^{(n)}$ and Dirichlet-
to-Neumann operator ${\cal N}_{\lambda}^{(n)}$ for all $n\in\mathds{N}$. Note
that (16) remains valid for $\gothic{a}^{(n)}$ instead of $\gothic{a}$ with
the same $\lambda$ if $n\in\mathds{N}$ is large enough and without loss of
generality for all $n\in\mathds{N}$. We still assume that there is a $\Phi\in
C^{\infty}(\mathds{R}^{d})$ such that $\Psi=\Phi|_{\Omega}$. For all
$n\in\mathds{N}$ define
$\tau_{n}=\Psi-\gamma_{\lambda}^{(n)}(\Psi|_{\Gamma})$. Then
$\sup_{n\in\mathds{N}}\|\partial_{k}\tau_{n}\|_{C^{\kappa}(\Omega)}<\infty$ by
(17). By the Arzelà–Ascoli theorem there exists a $\tilde{\tau}\in
C^{1+\kappa}(\Omega)$ such that, after passing to a subsequence if necessary,
$\lim\tau_{n}=\tilde{\tau}$ in $C^{1+\kappa}(\Omega)$. Then $\tilde{\tau}\in
W^{1,2}_{0}(\Omega)$. If $v\in C_{c}^{\infty}(\Omega)$, then

$\gothic{a}_{\lambda}^{(n)}(\tau_{n},v)=\gothic{a}_{\lambda}^{(n)}(\Psi,v)$

for all $n\in\mathds{N}$. Taking the limit $n\to\infty$ gives

$\gothic{a}_{\lambda}(\tilde{\tau},v)=\gothic{a}_{\lambda}(\Psi,v).$ (19)

Then by density, (19) is valid for all $v\in W^{1,2}_{0}(\Omega)$. Hence
$\tilde{\tau}=\tau$. Moreover,

$\gamma_{\lambda}(\Psi|_{\Gamma})=\Psi-\tau=\lim(\Psi-\tau_{n})=\lim\gamma_{\lambda}^{(n)}(\Psi|_{\Gamma})$

in $W^{1,2}(\Omega)$. Now $\lim c^{(n)}_{kl}=c_{kl}$ weakly∗ in
$L_{\infty}(\Omega)$ and similarly for the other coefficients (including
$c_{0}$). Let $v\in W^{1,2}(\Omega)$. Then (18) is applicable for
$\gothic{a}_{\lambda}^{(n)}$ and gives

$\gothic{a}_{\lambda}^{(n)}(\gamma_{\lambda}^{(n)}(\Psi|_{\Gamma}),v)=\int_{\Gamma}\sum_{k,l=1}^{d}n_{k}\,\Big{(}c_{kl}^{(n)}\,\partial_{l}(\Psi-\tau_{n})\Big{)}|_{\Gamma}\,\overline{{\mathop{\rm
Tr\,}}v}+\sum_{k=1}^{d}n_{k}\,(b_{k}^{(n)}\,\Psi)|_{\Gamma}\,\overline{{\mathop{\rm
Tr\,}}v}$

for all $n\in\mathds{N}$. Taking the limits $n\to\infty$ implies that (18) is
valid for $\gothic{a}_{\lambda}$. We deduce that (18) is valid whenever there
is a $\Phi\in C^{\infty}(\mathds{R}^{d})$ such that $\Psi=\Phi|_{\Omega}$, but
without any additional smoothness assumptions on the coefficients.

We now also drop the additional smoothness assumption on $\Psi$. Extend $\Psi$
to an element $\widetilde{\Psi}\in C^{1+\kappa}(\mathds{R}^{d})$. Regularise
to obtain $\widetilde{\Psi}_{n}\in C^{\infty}(\mathds{R}^{d})$ for all
$n\in\mathds{N}$ with the same bounds as for $\Psi$ and
$\lim\widetilde{\Psi}_{n}=\widetilde{\Psi}$ in $C^{1+\kappa}(\mathds{R}^{d})$.
Let $\Psi_{n}=\widetilde{\Psi}_{n}|_{\Omega}$ for all $n\in\mathds{N}$. For
all $n\in\mathds{N}$ define
$\tau_{n}=\Psi_{n}-\gamma_{\lambda}(\Psi_{n})|_{\Gamma})$. Then
$\sup_{n\in\mathds{N}}\|\partial_{k}\tau_{n}\|_{C^{\kappa}(\Omega)}<\infty$ by
(17) again. One can repeat almost the same argument and deduces that (18) is
valid without any additional smoothness assumption.

Hence in the general case one obtains that $\Psi|_{\Gamma}\in D({\cal
N}_{\lambda})$ and

${\cal
N}_{\lambda}(\Psi|_{\Gamma})=\sum_{k,l=1}^{d}n_{k}\,\Big{(}c_{kl}\,\partial_{l}(\Psi-\tau)\Big{)}|_{\Gamma}+\sum_{k=1}^{d}n_{k}\,(b_{k}\,\Psi)|_{\Gamma}.$

Therefore ${\cal N}_{\lambda}(\Psi|_{\Gamma})\in C^{\kappa}(\Gamma)$ and the
bounds of Statements (d) and (e) follow by the estimates at the beginning of
the proof. ∎

###### Proof of Lemma 7.2..

Let $\widetilde{M}>0$ be as in Lemma 7.3. It follows from the bounds of Lemma
7.1 that there is a $c>0$, independent of $r$ and $y$, such that

$\|r\,\Psi\|_{L_{\infty}(\Omega)}\leq
c\,r,\quad\|\partial_{k}(r\,\Psi)\|_{L_{\infty}(\Omega)}\leq
c\quad\mbox{and}\quad|||\partial_{k}(r\,\Psi)|||_{C^{\kappa}(\Omega)}\leq
c\,r^{-\kappa}$

for all $k\in\\{1,\ldots,d\\}$. Write
$\tau=\Psi-\gamma_{\lambda}(\Psi|_{\Gamma})$. We distinguish two cases.

Case 1. Suppose $3r<d(y,\Gamma)$. Then $\Psi|_{\Gamma}=0$ and hence
$\gamma_{\lambda}(\Psi|_{\Gamma})=0$ and trivially
$|(\gamma_{\lambda}(\Psi|_{\Gamma}))(y)|=0$.

Case 2. Suppose $3r\geq d(y,\Gamma)$. It follows from Lemma 7.3(c) that

$|r\,\tau(y)|\leq\widetilde{M}\,c\,d(y,\Gamma)\leq 3\widetilde{M}\,c\,r.$

So $|\tau(y)|\leq 3\widetilde{M}\,c$ and
$|(\gamma_{\lambda}(\Psi|_{\Gamma}))(y)|\leq 1+3\widetilde{M}\,c$. ∎

We will need some integral estimates. They are rather elementary and we state
them in the following lemma.

######  Lemma 7.4.

  * (a)

There exists a $c_{1}>0$ such that

$\int_{\\{w\in\Gamma:|w-z|\geq
r\\}}\frac{1}{|w-z|^{d-1+\beta}}\,dw\leq\frac{c_{1}}{1-2^{-\beta}}\,\frac{1}{r^{\beta}}$

for all $\beta\in(0,\infty)$, $r\in(0,\infty)$ and $z\in\Gamma$.

  * (b)

There exists a $c_{2}>0$ such that

$\int_{\\{w\in\Gamma:|w-z|\leq
r\\}}\frac{1}{|w-z|^{d-1-\beta}}\,dw\leq\frac{c_{2}}{1-2^{-\beta}}\,r^{\beta}$

for all $\beta\in(0,d-1]$, $r\in(0,\infty)$ and $z\in\Gamma$.

  * (c)

There exists a $c_{3}>0$ such that

$\int_{\\{w\in\Gamma:|w-x|\geq
r\\}}\frac{1}{|w-x|^{d-1+\beta}}\,dw\leq\frac{4^{\beta}\,c_{3}}{1-2^{-\beta}}\,\frac{1}{r^{\beta}}$

for all $\beta\in(0,\infty)$, $r\in(0,\infty)$ and $x\in\Omega$.

  * (d)

There exists a $c_{4}>0$ such that

$\int_{\\{w\in\Gamma:|w-x|\leq
r\\}}\frac{1}{|w-x|^{d-1-\beta}}\,dw\leq\frac{c_{4}}{1-2^{-\beta}}\,r^{\beta}$

for all $\beta\in(0,d-1]$, $r\in(0,\infty)$ and $x\in\Omega$.

###### Proof.

There exists a $c>0$ such that the surface measure satisfies
$\sigma(\Gamma\cap B(z,\rho))\leq c\,\rho^{d-1}$ for all $z\in\Gamma$ and
$\rho\in(0,\infty)$.

‘(a)’. Let $\beta\in(0,\infty)$, $r\in(0,\infty)$ and $z\in\Gamma$. For all
$n\in\mathds{N}$ define the annulus
$I_{n}=\\{w\in\Gamma:2^{n-1}\,r\leq|w-z|\leq 2^{n}\,r\\}$. Then

$\int_{\\{w\in\Gamma:|w-z|\geq
r\\}}\frac{1}{|w-z|^{d-1+\beta}}\,dw\leq\sum_{n=1}^{\infty}\frac{1}{(2^{n-1}\,r)^{d-1+\beta}}\cdot
c\,(2^{n}\,r)^{d-1}=\frac{2^{d-1}c}{1-2^{-\beta}}\,\frac{1}{r^{\beta}}$

as required.

‘(b)’. Let $\beta\in(0,\infty)$, $r\in(0,\infty)$ and $z\in\Gamma$. For all
$n\in\mathds{N}_{0}$ define the annulus
$I_{n}=\\{w\in\Gamma:2^{-n-1}\,r\leq|w-z|\leq 2^{-n}\,r\\}$. Then

$\int_{\\{w\in\Gamma:|w-z|\leq
r\\}}\frac{1}{|w-z|^{d-1-\beta}}\,dw\leq\sum_{n=0}^{\infty}\frac{1}{(2^{-n-1}\,r)^{d-1-\beta}}\cdot
c\,(2^{-n}\,r)^{d-1}=\frac{2^{d-1}\,c}{2^{\beta}-1}\,r^{\beta},$

where we used that $d-1-\beta\geq 0$ in the inequality.

‘(c)’. Let $\beta\in(0,\infty)$, $r\in(0,\infty)$ and $x\in\Omega$. Let
$z\in\Gamma$ be such that $d(x,\Gamma)=|x-z|$. If $w\in\Gamma$, then
$|x-z|=d(x,\Gamma)\leq|x-w|$ and hence $|w-z|\leq|w-x|+|x-z|\leq 2|w-x|$. We
consider two cases. Let $c_{1}>0$ be as in Statement (a).

Case 1.  Suppose $r\geq 2d(x,\Gamma)$. Let $w\in\Gamma$ with $|w-x|\geq r$.
Then $|w-z|\geq|w-x|-|x-z|\geq r-\frac{1}{2}\,r=\frac{1}{2}\,r$. Therefore

$\displaystyle\int_{\\{w\in\Gamma:|w-x|\geq
r\\}}\frac{1}{|w-x|^{d-1+\beta}}\,dw$ $\displaystyle\leq$
$\displaystyle\int_{\\{w\in\Gamma:|w-z|\geq
r/2\\}}\Big{(}\frac{2}{|w-z|}\Big{)}^{d-1+\beta}\,dw$ $\displaystyle\leq$
$\displaystyle
2^{d-1+\beta}\,\frac{c_{1}}{1-2^{-\beta}}\,\frac{1}{(r/2)^{\beta}}=\frac{2^{d-1}\,4^{\beta}\,c_{1}}{1-2^{-\beta}}\,\frac{1}{r^{\beta}}$

as required.

Case 2.  Suppose $r\leq 2d(x,\Gamma)$. We split the integral in two parts: the
first is over $\\{w\in\Gamma:r\leq|w-x|<2d(x,\Gamma)\\}$ and the second is
over $\\{w\in\Gamma:|w-x|\geq 2d(x,\Gamma)\\}$. The second is covered in Case
1, so we have to deal with the first part. Let $w\in\Gamma$ and suppose that
$|w-x|<2d(x,\Gamma)$. Then $|w-z|\leq|w-x|+|x-z|\leq 3|x-z|$. So

$\displaystyle\int_{\\{w\in\Gamma:|w-x|<2d(x,\Gamma)\\}}\frac{1}{|w-x|^{d-1+\beta}}\,dw$
$\displaystyle\leq$
$\displaystyle\int_{\\{w\in\Gamma:|w-z|<3|x-z|\\}}\frac{1}{|z-x|^{d-1+\beta}}\,dw$
$\displaystyle=$ $\displaystyle\frac{\sigma(\Gamma\cap
B(z,3|x-z|))}{|z-x|^{d-1+\beta}}\leq\frac{c\,(3|x-z|)^{d-1}}{|z-x|^{d-1+\beta}}$
$\displaystyle=$
$\displaystyle\frac{3^{d-1}c}{|z-x|^{\beta}}\leq\frac{2^{\beta}\,3^{d-1}c}{r^{\beta}}.$

Therefore both contributions together give

$\displaystyle\int_{\\{w\in\Gamma:|w-x|\geq
r\\}}\frac{1}{|w-x|^{d-1+\beta}}\,dw$ $\displaystyle\leq$
$\displaystyle\frac{2^{\beta}\,3^{d-1}c}{r^{\beta}}+\frac{2^{d-1}\,4^{\beta}\,c_{1}}{1-2^{-\beta}}\,\frac{1}{(2d(x,\Gamma))^{\beta}}$
$\displaystyle\leq$
$\displaystyle\frac{2^{\beta}\,3^{d-1}c}{r^{\beta}}+\frac{2^{d-1}\,4^{\beta}\,c_{1}}{1-2^{-\beta}}\,\frac{1}{r^{\beta}}$

as required.

‘(d)’. Let $\beta\in(0,d-1]$, $r\in(0,\infty)$ and $x\in\Omega$. If
$d(x,\Gamma)>r$, then the inequality is trivial. So we may assume that
$d(x,\Gamma)\leq r$. Let $z\in\Gamma$ be such that $d(x,\Gamma)=|x-z|$. If
$w\in\Gamma$ and $|w-x|\leq r$, then $|w-z|\leq|w-x|+|x-z|\leq
r+d(x,\Gamma)\leq 2r$. So $\\{w\in\Gamma:|w-x|\leq
r\\}\subset\\{w\in\Gamma:|w-z|\leq 2r\\}$. We distinguish two cases. Let
$c_{2}>0$ be as in Statement (b).

Case 1.  Suppose $r\leq 2d(x,\Gamma)$. Let $w\in\Gamma$ and suppose $|w-x|\leq
r$. Then $|w-z|\leq 2r\leq 4d(x,\Gamma)\leq 4|x-w|$. Hence

$\displaystyle\int_{\\{w\in\Gamma:|w-x|\leq
r\\}}\frac{1}{|w-x|^{d-1-\beta}}\,dw$ $\displaystyle\leq$
$\displaystyle\int_{\\{w\in\Gamma:|w-z|\leq
2r\\}}\Big{(}\frac{4}{|w-z|}\Big{)}^{d-1-\beta}\,dw$ $\displaystyle\leq$
$\displaystyle
4^{d-1-\beta}\,\frac{c_{2}}{1-2^{-\beta}}\,(2r)^{\beta}=\frac{4^{d-1}\,c_{2}\,2^{-\beta}}{1-2^{-\beta}}\,r^{\beta}$

as required.

Case 2.  Suppose $r\geq 2d(x,\Gamma)$. We split the integral in two parts: the
first is over $\\{w\in\Gamma:|w-x|\leq 2d(x,\Gamma)\\}$ and the second is over
$\\{w\in\Gamma:2d(x,\Gamma)<|w-x|\leq r\\}$. The first is covered in Case 1,
so we have to deal with the second part. If $w\in\Gamma$ and
$2d(x,\Gamma)<|w-x|$, then
$|w-z|\leq|w-x|+|x-z|\leq|w-x|+\frac{1}{2}\,|w-x|\leq 2|w-x|$. Hence

$\displaystyle\int_{\\{w\in\Gamma:2d(x,\Gamma)<|w-x|\leq
r\\}}\frac{1}{|w-x|^{d-1-\beta}}\,dw$ $\displaystyle\leq$
$\displaystyle\int_{\\{w\in\Gamma:|w-z|\leq
2r\\}}\Big{(}\frac{2}{|w-z|}\Big{)}^{d-1-\beta}\,dw$ $\displaystyle\leq$
$\displaystyle 2^{d-1-\beta}\,\frac{c_{2}}{1-2^{-\beta}}\,(2r)^{\beta}$
$\displaystyle=$
$\displaystyle\frac{2^{d-1}\,c_{2}}{1-2^{-\beta}}\,r^{\beta}.$

Therefore

$\displaystyle\int_{\\{w\in\Gamma:|w-x|\leq
r\\}}\frac{1}{|w-x|^{d-1-\beta}}\,dw$ $\displaystyle\leq$
$\displaystyle\frac{4^{d-1}\,c_{2}\,2^{-\beta}}{1-2^{-\beta}}\,(2d(x,\Gamma))^{\beta}+\frac{2^{d-1}\,c_{2}}{1-2^{-\beta}}\,r^{\beta}$
$\displaystyle\leq$
$\displaystyle\frac{4^{d-1}\,c_{2}}{1-2^{-\beta}}\,r^{\beta}+\frac{2^{d-1}\,c_{2}}{1-2^{-\beta}}\,r^{\beta}$

as required. ∎

We need the following extension for Hölder continuous functions.

######  Lemma 7.5.

Let $\nu\in(0,1)$ and $\varphi\in C^{\nu}(\Gamma)$. Then there exists a
$\Phi\in C^{\nu}(\mathds{R}^{d})$ such that $\Phi|_{\Gamma}=\varphi$,

$|\Phi(x)-\Phi(y)|\leq 2\sqrt{2}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|x-y|^{\nu}$

for all $x,y\in\mathds{R}^{d}$ and

$\|\,\Phi|_{\Omega}\,\|_{C^{\nu}(\Omega)}\leq\Big{(}2+2\sqrt{2}(\mathop{\rm
diam}\Omega)^{\nu}\Big{)}\,\|\varphi\|_{C^{\nu}(\Gamma)}.$

###### Proof.

First note that $|\varphi(w)-\varphi(z)|\leq
2\|\varphi\|_{C^{\nu}(\Gamma)}\,|w-z|^{\nu}$ for all $w,z\in\Gamma$, also if
$|w-z|>1$. Then the existence and Hölder estimate follows from [McS] Theorem
1. The factor $\sqrt{2}$ arises since we work over the complex numbers.

Fix $z\in\Gamma$. If $x\in\overline{\Omega}$, then

$\displaystyle|\Phi(x)|$ $\displaystyle\leq$
$\displaystyle|\Phi(x)-\Phi(z)|+|\varphi(z)|$ $\displaystyle\leq$
$\displaystyle\sqrt{2}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|x-z|^{\nu}+\|\varphi\|_{L_{\infty}(\Gamma)}\leq\Big{(}2\sqrt{2}(\mathop{\rm
diam}\Omega)^{\nu}+1\Big{)}\,\|\varphi\|_{C^{\nu}(\Gamma)}$

and the norm estimate follows. ∎

###### Proof of Theorem 1.2(a)..

Let $T_{\lambda}\colon L_{1}(\Gamma)\to C(\Omega)$ be the extension of
$\gamma_{\lambda}$ as in Theorem 6.5. For simplicity we first drop the
subscript $\lambda$.

Recall that $T|_{H^{1/2}(\Gamma)}=\gamma$. Let $\tilde{c}$ be as in Theorem
6.5(c). Let $c_{3}$ be as in Lemma 7.4(c). Let $c_{4}$ be as in Lemma 7.4(d).
Let $\hat{c}$ be as in Lemma 7.1. Let $c^{\prime}>0$ be as in Lemma 7.2.

Fix $\varphi\in C^{\nu}(\Gamma)$. Let $\Phi\in C^{\nu}(\overline{\Omega})$ be
an extension of $\varphi$ as in Lemma 7.5. Let $x,x_{0}\in\Omega$ with
$|x-x_{0}|\leq 1$. Then

$\displaystyle(T\varphi)(x)-(T\varphi)(x_{0})$ $\displaystyle=$
$\displaystyle\int_{\Gamma}\Big{(}K_{\gamma}(x,z)-K_{\gamma}(x_{0},z)\Big{)}\varphi(z)\,dz$
$\displaystyle=$
$\displaystyle\int_{\Gamma}\Big{(}K_{\gamma}(x,z)-K_{\gamma}(x_{0},z)\Big{)}(\varphi(z)-\Phi(x_{0}))\,dz$
$\displaystyle{}+\Phi(x_{0})\Big{(}(\gamma(\mathds{1}_{\Gamma}))(x)-(\gamma(\mathds{1}_{\Gamma}))(x_{0})\Big{)}$
$\displaystyle=$
$\displaystyle\int_{\Gamma}\Big{(}K_{\gamma}(x,z)-K_{\gamma}(x_{0},z)\Big{)}\chi_{r}(x-z)\,(\varphi(z)-\Phi(x_{0}))\,dz$
$\displaystyle{}+\int_{\Gamma}\Big{(}K_{\gamma}(x,z)-K_{\gamma}(x_{0},z)\Big{)}(1-\chi_{r}(x-z))\,(\varphi(z)-\Phi(x_{0}))\,dz$
$\displaystyle{}+\Phi(x_{0})\Big{(}(\gamma(\mathds{1}_{\Gamma}))(x)-(\gamma(\mathds{1}_{\Gamma}))(x_{0})\Big{)}$
$\displaystyle=$ $\displaystyle I_{1}+I_{2}+I_{3},$

where $\chi_{r}$ is the cut-off function of Lemma 7.1 for which we take
$r=|x-x_{0}|$.

We first deal with the third term $I_{3}$. It follows from Lemma 7.3 with
$r=\mathop{\rm diam}\Omega$ that
$\gamma(\mathds{1}_{\Gamma})=\gamma(\mathds{1}|_{\Gamma})\in
C^{1+\kappa}(\Omega)$ Hence $\gamma(\mathds{1}_{\Gamma})\in C^{\nu}(\Omega)$
and so

$|\Phi(x_{0})\Big{(}(\gamma(\mathds{1}_{\Gamma}))(x)-(\gamma(\mathds{1}_{\Gamma}))(x_{0})\Big{)}|\leq(2+2\sqrt{2}(\mathop{\rm
diam}\Omega)^{\nu})\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|||\gamma(\mathds{1}_{\Gamma})|||_{C^{\nu}(\Omega)}\,|x-x_{0}|^{\nu}$

by the bounds of Lemma 7.5.

Next we consider $I_{2}$. By the definition of $\chi_{r}$, the integral in
$I_{2}$ concerns the region $|z-x|>2|x-x_{0}|$. Hence by the Hölder bounds of
$K_{\gamma}$ in Theorem 6.5(c) one estimates

$\displaystyle|I_{2}|$ $\displaystyle=$
$\displaystyle|\int_{\\{z\in\Gamma:|x-z|>2|x-x_{0}|\\}}\Big{(}K_{\gamma}(x,z)-K_{\gamma}(x_{0},z)\Big{)}(1-\chi_{r}(x-z))(\varphi(z)-\Phi(x_{0}))\,dz|$
$\displaystyle\leq$
$\displaystyle\sqrt{2}\,\tilde{c}\,|||\varphi|||_{C^{\nu}(\Gamma)}\int_{\\{z\in\Gamma:|x-z|>2|x-x_{0}|\\}}\frac{|x-x_{0}|^{\kappa}}{|x-z|^{d-1+\kappa}}\,|x_{0}-z|^{\nu}\,dz$
$\displaystyle\leq$ $\displaystyle
4\tilde{c}\,|||\varphi|||_{C^{\nu}(\Gamma)}\int_{\\{z\in\Gamma:|x-z|>2|x-x_{0}|\\}}\frac{|x-x_{0}|^{\kappa}}{|x-z|^{d-1+\kappa}}\,|x-z|^{\nu}\,dz$
$\displaystyle\leq$
$\displaystyle\frac{16\tilde{c}\,c_{3}}{2^{\kappa-\nu}-1}\,|||\varphi|||_{C^{\nu}(\Gamma)}\,|x-x_{0}|^{\nu},$

where we used the triangle inequality $|x_{0}-z|\leq|x_{0}-x|+|x-z|\leq
2|x-z|$ and Lemma 7.4(c) with $\beta=\kappa-\nu$.

Now we deal with $I_{1}$. We decompose it as follows

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\int_{\Gamma}-K_{\gamma}(x_{0},z)\,\chi_{r}(x-z)\,(\varphi(z)-\Phi(x_{0}))\,dz+\int_{\Gamma}K_{\gamma}(x,z)\,\chi_{r}(x-z)\,(\varphi(z)-\Phi(x))\,dz$
$\displaystyle{}+\int_{\Gamma}K_{\gamma}(x,z)\,\chi_{r}(x-z)\,(\Phi(x)-\Phi(x_{0}))\,dz$
$\displaystyle=$ $\displaystyle I_{1a}+I_{1b}+I_{1c}.$

We use the kernel bound of Theorem 6.5(c) for $I_{1a}$. If $z\in\Gamma$ and
$\chi_{r}(x-z)\neq 0$, then $|x-z|\leq 3r=3|x-x_{0}|$ and $|z-x_{0}|\leq
4|x-x_{0}|$. Therefore

$\displaystyle|I_{1a}|$ $\displaystyle\leq$
$\displaystyle\sqrt{2}\,\tilde{c}\,|||\varphi|||_{C^{\nu}(\Gamma)}\int_{\\{z\in\Gamma:|z-x_{0}|\leq
4|x-x_{0}|\\}}\frac{1}{|x_{0}-z|^{d-1}}\,|x_{0}-z|^{\nu}\,dz$
$\displaystyle\leq$
$\displaystyle\sqrt{2}\,\tilde{c}\,|||\varphi|||_{C^{\nu}(\Gamma)}\frac{c_{4}}{1-2^{-\nu}}\,(4|x-x_{0}|)^{\nu}.$

The term $I_{1b}$ is estimated exactly in the same way.

It remains to estimate $I_{1c}$. Clearly,

$|I_{1c}|\leq\sqrt{2}\,|||\varphi|||_{C^{\nu}(\Gamma)}\,|x-x_{0}|^{\nu}\,|\gamma(\Psi|_{\Gamma})(x)|\leq\sqrt{2}\,c^{\prime}\,|||\varphi|||_{C^{\nu}(\Gamma)}\,|x-x_{0}|^{\nu},$

where $\Psi\colon\mathds{R}^{d}\to\mathds{R}$ is defined by
$\Psi(y)=\chi_{r}(x-y)$ and we used Lemma 7.2.

Together we proved that there is a $c_{5}>0$, independent of $\varphi$, $x$
and $x_{0}$, such that

$|(T\varphi)(x)-(T\varphi)(x_{0})|\leq
c_{5}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|x-x_{0}|^{\nu}.$

Next we provide an $L_{\infty}$-bound for $T\varphi$. Let $x\in\Omega$. Then

$(T\varphi)(x)=\int_{\Gamma}K_{\gamma}(x,z)\,\varphi(z)\,dz=\int_{\Gamma}K_{\gamma}(x,z)\,(\varphi(z)-\Phi(x))\,dz+\Phi(x)\,(\gamma(\mathds{1}_{\Gamma}))(x).$

Using the kernel bound of Theorem 6.5(c), Lemma 7.5 and Lemma 7.4(d) one
estimates

$\displaystyle|\int_{\Gamma}K_{\gamma}(x,z)\,(\varphi(z)-\Phi(x))\,dz|$
$\displaystyle\leq$
$\displaystyle\int_{\Gamma}\frac{\tilde{c}}{|x-z|^{d-1}}\cdot
2\sqrt{2}\,\|\varphi\|_{C^{\nu}(\Omega)}\,|z-x|^{\nu}\,dz$ $\displaystyle\leq$
$\displaystyle 2\sqrt{2}\,\tilde{c}\,\frac{c_{4}}{1-2^{-\nu}}\,(\mathop{\rm
diam}\Omega))^{\nu}\,\|\varphi\|_{C^{\nu}(\Gamma)},$

and

$|\Phi(x)\,(\gamma(\mathds{1}_{\Gamma}))(x)|\leq(2+2\sqrt{2}\,(\mathop{\rm
diam}\Omega)^{\nu})\,\|\gamma(\mathds{1}_{\Gamma})\|_{L_{\infty}(\Omega)}\,\|\varphi\|_{C^{\nu}(\Gamma)}.$

Restoring the notation with $\lambda$, we proved that $T_{\lambda}$ maps
$C^{\nu}(\Gamma)$ into $C^{\nu}(\Omega)$ and there exists a $c_{6}>0$ such
that $\|T_{\lambda}\varphi\|_{C^{\nu}(\Omega)}\leq
c_{6}\,\|\varphi\|_{C^{\nu}(\Gamma)}$.

Finally, $\gamma=\gamma_{\lambda}+\lambda\,A_{D}^{-1}\,\gamma_{\lambda}$ and
hence by continuity and density
$T=T_{\lambda}+\lambda\,A_{D}^{-1}\,T_{\lambda}$. Now $T_{\lambda}$ maps
$(C^{\nu}(\Gamma),\|\cdot\|_{C^{\nu}(\Gamma)})$ continuously into
$(C^{\nu}(\Omega),\|\cdot\|_{C^{\nu}(\Omega)})$ and $L_{\infty}(\Omega)$, and
$A_{D}^{-1}$ maps $L_{\infty}(\Omega)$ continuously into
$(C^{\nu}(\Omega),\|\cdot\|_{C^{\nu}(\Omega)})$ by Proposition 6.1(b). This
completes the proof of Theorem 1.2(a). ∎

##  8 The Schwarz kernel of the Dirichlet-to-Neumann operator

We next turn to the Dirichlet-to-Neumann operator. Throughout this short
section let $\kappa\in(0,1)$. Let $\Omega\subset\mathds{R}^{d}$ be an open
bounded set of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$, and let $c_{0}\in
L_{\infty}(\Omega)$. Suppose there is a $\mu>0$ such that $\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$
for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Further suppose that
$0\not\in\sigma(A_{D})$.

Let $G\colon\\{(x,y)\in\Omega\times\Omega:x\neq y\\}\to\mathds{C}$ be as in
Theorem 5.1. Then $G$ is differentiable on
$\\{(x,y)\in\Omega\times\Omega:x\neq y\\}$ by Theorem 5.1 and the derivatives
extend to continuous functions on
$\\{(x,y)\in\overline{\Omega}\times\overline{\Omega}:x\neq y\\}$. Define
$\widetilde{K}_{\cal N}\colon\\{(z,w)\in\Gamma\times\Gamma:z\neq
w\\}\to\mathds{C}$ by

$\displaystyle\widetilde{K}_{\cal N}(z,w)$ $\displaystyle=$
$\displaystyle-\sum_{k,l,k^{\prime},l^{\prime}=1}^{d}n_{k}(z)\,c_{kl}(z)\,n_{k^{\prime}}(w)\,c_{l^{\prime}k^{\prime}}(w)\,(\partial^{(1)}_{l}\partial^{(2)}_{l^{\prime}}G)(z,w)$
$\displaystyle{}-\sum_{k,k^{\prime},l^{\prime}=1}^{d}n_{k}(z)\,b_{k}(z)\,n_{k^{\prime}}(w)\,c_{l^{\prime}k^{\prime}}(w)\,(\partial^{(2)}_{l^{\prime}}G)(z,w)$
$\displaystyle{}-\sum_{k,k^{\prime},l=1}^{d}n_{k}(z)\,c_{kl}(z)\,n_{k^{\prime}}(w)\,c_{k^{\prime}}(w)\,(\partial^{(1)}_{l}G)(z,w)$
$\displaystyle{}-\sum_{k,k^{\prime}=1}^{d}n_{k}(z)\,b_{k}(z)\,n_{k^{\prime}}(w)\,c_{k^{\prime}}(w)\,G(z,w).$

######  Proposition 8.1.

Suppose that $\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u)$ for all $u\in W^{1,2}(\Omega)$. Then $\widetilde{K}_{\cal
N}$ is the Schwartz kernel of ${\cal N}$.

###### Proof.

This follows as for the proof of Corollary 6.3 in [EO2]. Because of the
coercivity condition
$\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm Re}\gothic{a}(u,u)$
for all $u\in W^{1,2}(\Omega)$, we can regularise the coefficients and the
uniform estimates as in [EO2] Proposition 6.6 are also valid in the current
situation because of Theorem 5.1. ∎

For the general setting, that is without the coercivity condition, we need the
following lemma.

######  Lemma 8.2.

Let $\lambda\in\mathds{R}$. Suppose that $0\not\in\sigma(A_{D}+\lambda\,I)$.
Let $\gamma\colon H^{1/2}(\Gamma)\to H^{1}(\Omega)$ be the harmonic lifting
and let $\gamma_{\lambda}^{\\#}\colon H^{1/2}(\Gamma)\to H^{1}(\Omega)$ be the
harmonic lifting with respect to the form $\gothic{a}_{\lambda}^{*}$. We
identify $\gamma_{\lambda}^{\\#}$ with the bounded extension from
$L_{2}(\Gamma)$ into $L_{2}(\Omega)$ (cf. Theorem 6.5(d)). Then

${\cal N}={\cal N}_{\lambda}-\lambda\,(\gamma_{\lambda}^{\\#})^{*}\,\gamma.$

###### Proof.

Let $\varphi\in D({\cal N})$ and $\psi\in D({\cal N}_{\lambda}^{*})$. Write
$u=\gamma(\varphi)$ and $v=\gamma_{\lambda}^{\\#}(\psi)$. Then $({\cal
N}_{\lambda}^{*}\psi,\varphi)_{L_{2}(\Gamma)}=\gothic{a}_{\lambda}^{*}(v,u)=\overline{\gothic{a}_{\lambda}(u,v)}$
by [AE1] Remark 2.8. So

$\displaystyle({\cal N}\varphi,\psi)_{L_{2}(\Gamma)}-(\varphi,{\cal
N}_{\lambda}^{*}\psi)_{L_{2}(\Gamma)}$ $\displaystyle=$
$\displaystyle\gothic{a}(u,v)-\gothic{a}_{\lambda}(u,v)$ $\displaystyle=$
$\displaystyle-\lambda\,(u,v)_{L_{2}(\Omega)}=-\lambda\,((\gamma_{\lambda}^{\\#})^{*}\,\gamma(\varphi),\psi)_{L_{2}(\Gamma)}.$

By the boundedness of $(\gamma_{\lambda}^{\\#})^{*}\,\gamma$ from
$L_{2}(\Gamma)$ into $L_{2}(\Gamma)$ one deduces that $\varphi\in D({\cal
N}_{\lambda}^{**})=D({\cal N}_{\lambda})$. Hence

$(({\cal N}-{\cal
N}_{\lambda})\varphi,\psi)_{L_{2}(\Gamma)}=-\lambda\,((\gamma_{\lambda}^{\\#})^{*}\,\gamma(\varphi),\psi)_{L_{2}(\Gamma)}.$

Since $D({\cal N}_{\lambda}^{*})$ is dense in $L_{2}(\Gamma)$ and similarly
$D({\cal N}_{\lambda})\subset D({\cal N})$, the lemma follows. ∎

In the next theorem we show that ${\cal N}$ has a Schwartz kernel and this
kernel satisfies appropriate bounds.

######  Theorem 8.3.

The Dirichlet-to-Neumann operator ${\cal N}$ has a Schwartz kernel $K_{\cal
N}$. Moreover, there exists a $c>0$ such that

$|K_{{\cal N}}(z,w)|\leq\frac{c}{|z-w|^{d}}$

and

$|K_{{\cal N}}(z,w)-K_{{\cal N}}(z^{\prime},w^{\prime})|\leq
c\,\frac{(|z-z^{\prime}|+|w-w^{\prime}|)^{\kappa}}{|z-w|^{d+\kappa}}$

for all $z,z^{\prime},w,w^{\prime}\in\Gamma$ with $z\neq w$, $z^{\prime}\neq
w^{\prime}$ and $|z-z^{\prime}|+|w-w^{\prime}|\leq\frac{1}{2}\,|z-w|$.

###### Proof.

There exists a $\lambda>0$ such that
$\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u)+\lambda\,\|u\|_{L_{2}(\Omega)}^{2}$ for all $u\in
W^{1,2}(\Omega)$. Let $\gamma_{\lambda}^{\\#}\colon H^{1/2}(\Gamma)\to
H^{1}(\Omega)$ be the harmonic lifting with respect to the form
$\gothic{a}_{\lambda}^{*}$. Then ${\cal N}={\cal
N}_{\lambda}-\lambda\,(\gamma_{\lambda}^{\\#})^{*}\,\gamma$ by Lemma 8.2. Now
$(\gamma_{\lambda}^{\\#})^{*}\,\gamma$ has a Schwartz kernel $K_{Q}$ given by

$K_{Q}(z,w)=\int_{\Omega}\overline{K_{\gamma_{\lambda}^{\\#}}(x,z)}\,K_{\gamma}(x,w)\,dx$

for all $z,w\in\Gamma$ with $z\neq w$, where we use the notation of Theorem
6.5. Hence $\widetilde{K}_{{\cal N}_{\lambda}}-\lambda\,K_{Q}$ is the Schwartz
kernel for ${\cal N}$.

It remains to show the kernel bounds and it suffices to show that
$\widetilde{K}_{{\cal N}_{\lambda}}$ and $K_{Q}$ satisfy the appropriate
kernel bounds. Using the bounds in Theorem 5.2 for the Green function we
obtain the desired bounds for the kernel $\widetilde{K}_{{\cal N}_{\lambda}}$
as in the proof of [EO2] Proposition 6.4. Using the bounds in Theorem 6.5 for
the kernel of the harmonic lifting we obtain the bounds for $K_{Q}$ as in the
proof of [EO2] Proposition 6.5. ∎

##  9 Commutator estimates

Throughout this section let $\kappa\in(0,1)$ and $\Omega\subset\mathds{R}^{d}$
a bounded open set of class $C^{1+\kappa}$. For all $k,l\in\\{1,\ldots,d\\}$
let $c_{kl},b_{k},c_{k}\in C^{\kappa}(\Omega)$ and $c_{0}\in
L_{\infty}(\Omega)$. Let the form $\gothic{a}$ and operator $A_{D}$ be as in
the introduction. We assume that $0\not\in\sigma(A_{D})$. Let ${\cal N}$ be
the associated Dirichlet-to-Neumann operator. In this section we prove the
commutator estimates of Theorem 1.1.

We start with removing the shift in the last two statements of Lemma 7.3.

######  Lemma 9.1.

There exists an $\widetilde{M}>0$ such that the following is valid.

Let $r\in(0,\mathop{\rm diam}\Omega]$, $\Psi\in C^{1+\kappa}(\Omega)$,
$y\in\overline{\Omega}$ and $c\geq 0$. Suppose that $\mathop{\rm
supp}\Psi\subset B(y,3r)$,

$\|\Psi\|_{L_{\infty}(\Omega)}\leq
c\,r,\quad\|\partial_{k}\Psi\|_{L_{\infty}(\Omega)}\leq
c\quad\mbox{and}\quad|||\partial_{k}\Psi|||_{C^{\kappa}(\Omega)}\leq
c\,r^{-\kappa}$

for all $k\in\\{1,\ldots,d\\}$. Then $\Psi|_{\Gamma}\in D({\cal N})$ and
${\cal N}(\Psi|_{\Gamma})\in C^{\kappa}(\Gamma)$. Moreover,

  * (a)

$\|{\cal N}(\Psi|_{\Gamma})\|_{C(\Gamma)}\leq\widetilde{M}\,c$, and

  * (b)

$|||{\cal
N}(\Psi|_{\Gamma})|||_{C^{\kappa}(\Gamma)}\leq\widetilde{M}\,c\,r^{-\kappa}$.

######  Proof.

Let $M>0$ be such that $\|c_{kl}\|_{C^{\kappa}(\Omega)}\leq M$,
$\|b_{k}\|_{C^{\kappa}(\Omega)}\leq M$, $\|c_{k}\|_{C^{\kappa}(\Omega)}\leq M$
and $\|c_{0}\|_{L_{\infty}(\Omega)}\leq M$. There exists a $\lambda>0$,
depending only on $\mu$ and $M$, such that

$\frac{\mu}{2}\,\|u\|_{W^{1,2}(\Omega)}^{2}\leq\mathop{\rm
Re}\gothic{a}(u,u)+\lambda\,\|u\|_{L_{2}(\Omega)}^{2}$

for all $u\in W^{1,2}(\Omega)$. Let $\gamma_{\lambda}$ and ${\cal
N}_{\lambda}$ be the harmonic lifting and Dirichlet-to-Neumann operator with
respect to the operator with coefficient $c_{0}+\lambda\,\mathds{1}_{\Omega}$
instead of $c_{0}$. Let $\widetilde{M}$ be as in Lemma 7.3. Then $\|{\cal
N}_{\lambda}(\Psi|_{\Gamma})\|_{C(\Gamma)}\leq\widetilde{M}\,c$ and $|||{\cal
N}_{\lambda}(\Psi|_{\Gamma})|||_{C^{\kappa}(\Gamma)}\leq\widetilde{M}\,c\,r^{-\kappa}$.
Let $\gamma_{\lambda}^{\\#}$ be as in Lemma 8.2. Then ${\cal N}={\cal
N}_{\lambda}-\lambda\,(\gamma_{\lambda}^{\\#})^{*}\,\gamma$ by Lemma 8.2. It
remains to show that $(\gamma_{\lambda}^{\\#})^{*}\,\gamma$ can be considered
as a perturbation on $L_{\infty}(\Gamma)$ and $C^{\kappa}(\Gamma)$.

Fix $p\in(\frac{d}{1-\kappa},\infty)$. The inclusion $L_{\infty}(\Gamma)\to
L_{p}(\Gamma)$ is bounded. The map $\gamma$ is bounded from $L_{p}(\Gamma)$
into $L_{p}(\Omega)$ by Theorem 6.5(d). The map $(\gamma_{\lambda}^{\\#})^{*}$
is bounded from $L_{p}(\Omega)$ into
$(C^{\kappa}(\Gamma),\|\cdot\|_{C^{\kappa}(\Gamma)})$ by Theorem 6.5(e). The
inclusion from $(C^{\kappa}(\Gamma),\|\cdot\|_{C^{\kappa}(\Gamma)})$ into
$L_{\infty}(\Gamma)$ is clearly bounded. Hence by composition the map
$\lambda\,(\gamma_{\lambda}^{\\#})^{*}\,\gamma$ is bounded from
$L_{\infty}(\Gamma)$ into $L_{\infty}(\Gamma)$. Let
$c_{1}=\|\lambda\,(\gamma_{\lambda}^{\\#})^{*}\,\gamma\|_{L_{\infty}(\Gamma)\to
L_{\infty}(\Gamma)}$. Then

$\|\lambda\,(\gamma_{\lambda}^{\\#})^{*}\,\gamma(\Psi|_{\Gamma})\|_{L_{\infty}(\Gamma)}\leq
c_{1}\,\|\Psi|_{\Gamma}\|_{L_{\infty}(\Gamma)}\leq c_{1}\,c\,r\leq
c_{1}\,c\,\mathop{\rm diam}(\Omega)$

and Statement (a) follows.

The proof of Statement (b) is similar. The inclusion $C^{\kappa}(\Gamma)\to
L_{p}(\Gamma)$ is bounded, and again $\gamma$ is bounded from $L_{p}(\Gamma)$
into $L_{p}(\Omega)$ and $(\gamma_{\lambda}^{\\#})^{*}$ is bounded from
$L_{p}(\Omega)$ into $(C^{\kappa}(\Gamma),\|\cdot\|_{C^{\kappa}(\Gamma)})$. ∎

A principle part of the proof of Theorem 1.1 is given in the next lemma.

######  Lemma 9.2.

Let $\hat{c}\geq 1$ and the $\chi_{r}$ be as in Lemma 7.1. Then there exists a
$c^{\prime}>0$ such that the following is valid.

Let $r\in(0,\mathop{\rm diam}(\Omega)]$ and $z\in\Gamma$. Define
$\Psi\colon\mathds{R}^{d}\to\mathds{R}$ by $\Psi(x)=\chi_{r}(x-z)$. Then
$(g\,\Psi)|_{\Gamma}\in D({\cal N})$, ${\cal N}((g\,\Psi)|_{\Gamma})\in
C(\Gamma)$ and $\|{\cal N}((g\,\Psi)|_{\Gamma})\|_{L_{\infty}(\Gamma)}\leq
c^{\prime}$ for all $g\in W_{2}$ with $g(z)=0$.

###### Proof.

Note that $\mathop{\rm supp}(g\,\Psi)\subset B(z,3r)$. If $x\in B(z,3r)$ and
$k,l\in\\{1,\ldots,d\\}$, then $|g(x)|=|g(x)-g(z)|\leq|x-z|\leq 3r$,
$|(g\,\Psi)(x)|\leq 3r$,

$|(\partial_{l}(g\,\Psi))(x)|\leq|(\partial_{l}g)(x)|\,\Psi(x)+|g(x)|\,|(\partial_{l}\Psi)(x)|\leq
1\cdot 1+3r\cdot\frac{\hat{c}}{r}\leq 4\hat{c}$

and

$\displaystyle|(\partial_{k}\,\partial_{l}(g\,\Psi))(x)|$ $\displaystyle\leq$
$\displaystyle|(\partial_{k}\,\partial_{l}g)(x)|\,\Psi(x)+|(\partial_{l}g)(x)|\,|(\partial_{k}\Psi)(x)|$
$\displaystyle\hskip
56.9055pt{}+|(\partial_{k}g)(x)|\,|(\partial_{l}\Psi)(x)|+|g(x)|\,|(\partial_{k}\,\partial_{l}\Psi)(x)|$
$\displaystyle\leq$ $\displaystyle 1\cdot 1+2\cdot
1\cdot\frac{\hat{c}}{r}+3r\cdot\frac{\hat{c}}{r^{2}}\leq\frac{5\hat{c}+\mathop{\rm
diam}(\Omega)}{r}.$

Here we used a second derivative of $g$. Obviously these estimates are also
valid if $x\in\mathds{R}^{d}\setminus B(z,3r)$. Hence if
$x,y\in\mathds{R}^{d}$, then

$|(\partial_{l}(g\,\Psi))(x)-(\partial_{l}(g\,\Psi))(y)|\leq
d^{1/2}\,\frac{5\hat{c}+\mathop{\rm diam}(\Omega)}{r}\,|x-y|$

and trivially $|(\partial_{l}(g\,\Psi))(x)-(\partial_{l}(g\,\Psi))(y)|\leq
8\hat{c}$, so

$|(\partial_{l}(g\,\Psi))(x)-(\partial_{l}(g\,\Psi))(y)|\leq(8\hat{c})^{1-\kappa}\,d^{\kappa/2}\,(5\hat{c}+\mathop{\rm
diam}(\Omega))^{\kappa}\,\frac{|x-y|^{\kappa}}{r^{\kappa}}$

for all $l\in\\{1,\ldots,d\\}$. Now the lemma follows from Lemma 9.1, applied
with $(g\,\Psi)|_{\Omega}$. ∎

######  Proposition 9.3.

There exists a $c^{\prime\prime}>0$ such that $g|_{\Gamma}\in D({\cal N})$ and
$\|{\cal N}(g|_{\Gamma})\|_{C^{\kappa}(\Gamma)}\leq c^{\prime\prime}$ for all
$g\in W_{2}$.

###### Proof.

This follows from Lemma 9.1(a) and (b) with the choice $r=\mathop{\rm
diam}\Omega$ and $\Psi=g$. ∎

Since $\mathds{1}_{\mathds{R}^{d}}\in W_{2}$ one obtains the next corollary.

######  Corollary 9.4.

$\mathds{1}_{\Gamma}\in D({\cal N})$ and ${\cal N}\mathds{1}_{\Gamma}\in
C^{\kappa}(\Gamma)$.

Even more general one deduces that many functions are in the domain of ${\cal
N}$.

######  Corollary 9.5.

Let $\Psi\in W^{2,\infty}(\mathds{R}^{d})$. Then $\Psi|_{\Gamma}\in D({\cal
N})$ and ${\cal N}(\Psi|_{\Gamma})\in C^{\kappa}(\Gamma)$.

Next we show that the Schwartz kernel of the commutator $[{\cal N},M_{g}]$
satisfies Calderón–Zygmund estimates.

######  Lemma 9.6.

There exists a $c>0$ such that for all $g\in W_{1}$ the operator $[{\cal
N},M_{g}]$ has a Schwartz kernel $K_{g}$ such that

$|K_{g}(z,w)|\leq\frac{c}{|z-w|^{d-1}}$

and

$|K_{g}(z,w)-K_{g}(z^{\prime},w^{\prime})|\leq
c\,\frac{(|z-z^{\prime}|+|w-w^{\prime}|)^{\kappa}}{|z-w|^{d-1+\kappa}}$

for all $z,z^{\prime},w,w^{\prime}\in\Gamma$ with $z\neq w$, $z^{\prime}\neq
w^{\prime}$ and $|z-z^{\prime}|+|w-w^{\prime}|\leq\frac{1}{2}\,|z-w|$.

###### Proof.

By Theorem 8.3 the Dirichlet-to-Neumann operator ${\cal N}$ has a Schwartz
kernel $K_{{\cal N}}$ and there exists a $c>0$ such that

$|K_{{\cal N}}(z,w)|\leq\frac{c}{|z-w|^{d}}$

and

$|K_{{\cal N}}(z,w)-K_{{\cal N}}(z^{\prime},w^{\prime})|\leq
c\,\frac{(|z-z^{\prime}|+|w-w^{\prime}|)^{\kappa}}{|z-w|^{d+\kappa}}$

for all $z,z^{\prime},w,w^{\prime}\in\Gamma$ with $z\neq w$, $z^{\prime}\neq
w^{\prime}$ and $|z-z^{\prime}|+|w-w^{\prime}|\leq\frac{1}{2}\,|z-w|$.

Now let $g\in W_{1}$. Then $[{\cal N},M_{g}]$ has a Schwartz kernel $K_{g}$
given by $K_{g}(z,w)=(g(w)-g(z))\,K_{{\cal N}}(z,w)$. Then
$|K_{g}(z,w)|\leq|w-z|\,\frac{c}{|z-w|^{d}}=\frac{c}{|z-w|^{d-1}}$ for all
$z,w\in\Gamma$ with $z\neq w$. Next let $z,z^{\prime},w,w^{\prime}\in\Gamma$
with $z\neq w$, $z^{\prime}\neq w^{\prime}$ and
$|z-z^{\prime}|+|w-w^{\prime}|\leq\frac{1}{2}\,|z-w|$. Then
$|w^{\prime}-z^{\prime}|\leq|w^{\prime}-w|+|w-z|+|z-z^{\prime}|\leq 2|z-w|$.
Therefore

$\displaystyle|K_{g}(z,w)-K_{g}(z^{\prime},w^{\prime})|$ $\displaystyle=$
$\displaystyle|(g(w)-g(z))\,K_{{\cal
N}}(z,w)-(g(w^{\prime})-g(z^{\prime}))\,K_{{\cal N}}(z^{\prime},w^{\prime})|$
$\displaystyle=$
$\displaystyle|(g(w)-g(w^{\prime})+g(z^{\prime})-g(z))\,K_{{\cal
N}}(z,w)+(g(w^{\prime})-g(z^{\prime}))\,(K_{{\cal N}}(z,w)-K_{{\cal
N}}(z^{\prime},w^{\prime}))|$ $\displaystyle\leq$
$\displaystyle(|z-z^{\prime}|+|w-w^{\prime}|)\,\frac{c}{|z-w|^{d}}+|w^{\prime}-z^{\prime}|\,\frac{c\,(|z-z^{\prime}|+|w-w^{\prime}|)^{\kappa}}{|z-w|^{d+\kappa}}$
$\displaystyle\leq$
$\displaystyle(|z-z^{\prime}|+|w-w^{\prime}|)^{\kappa}\,|z-w|^{1-\kappa}\,\frac{c}{|z-w|^{d}}+2|w-z|\,\frac{c\,(|z-z^{\prime}|+|w-w^{\prime}|)^{\kappa}}{|z-w|^{d+\kappa}}$
$\displaystyle=$ $\displaystyle
3c\,\frac{(|z-z^{\prime}|+|w-w^{\prime}|)^{\kappa}}{|z-w|^{d-1+\kappa}}$

as required. ∎

For all $g\in W_{1}$ let $K_{g}$ be as in Lemma 9.6. By the definition of
Schwartz kernel,

$([{\cal N},M_{g}]\varphi)(z)=\int_{\Gamma}K_{g}(z,w)\,\varphi(w)\,dw$

for all $z\in\Gamma\setminus\mathop{\rm supp}\varphi$. For the sequel we need
to surpass this support condition. To do so, we introduce the following
operator.

For all $g\in W_{2}$ define $T_{g}\colon\\{\Phi|_{\Gamma}:\Phi\in
C^{\infty}(\mathds{R}^{d})\\}\to C(\Gamma)$ by

$(T_{g}\varphi)(z)=\int_{\Gamma}K_{g}(z,w)\,\Big{(}\varphi(w)-\varphi(z)\Big{)}\,dw+\varphi(z)\,([{\cal
N},M_{g}]\mathds{1}_{\Gamma})(z).$

Note that the integral converges and that $\mathds{1}_{\Gamma}\in D([{\cal
N},M_{g}])$ by Corollary 9.5. Moreover, if $\lambda\in\mathds{R}$ and
$\tilde{g}=g-\lambda\,\mathds{1}_{\mathds{R}^{d}}$, then
$T_{g}=T_{\tilde{g}}$.

######  Proposition 9.7.

Let $g\in W_{2}$ and $\varphi\in\\{\Phi|_{\Gamma}:\Phi\in
C^{\infty}(\mathds{R}^{d})\\}$. Then

$T_{g}\varphi=[{\cal N},M_{g}]\varphi.$

###### Proof.

Fix $z\in\Gamma$. We prove that

$(T_{g}\varphi)(z)=([{\cal N},M_{g}]\varphi)(z).$ (20)

The proof is given in several steps.

Step 1 Suppose $z\not\in\mathop{\rm supp}\varphi$. Then (20) is valid.
This follows from $\varphi(z)=0$ and the definition of the Schwartz kernel
$K_{g}$ of $[{\cal N},M_{g}]$.

Step 2 Suppose $\varphi(z)=g(z)=0$. Then (20) is valid.
Let $\hat{c}\geq 1$ and the $\chi_{r}$ be as in Lemma 7.1. For all
$n\in\mathds{N}$ define $\tau_{n}\colon\mathds{R}^{d}\to\mathds{R}$ by
$\tau_{n}(y)=\chi_{1/n}(y-z)$. Then

$(T_{g}(((\mathds{1}-\tau_{n})\,\varphi)|_{\Gamma}))(z)=([{\cal
N},M_{g}]((\mathds{1}-\tau_{n})\,\varphi)|_{\Gamma})(z)$

for all $n\in\mathds{N}$ by Step 9. Let $c>0$ be as in Lemma 9.6. There is a
$\tilde{c}>0$ such that $|\varphi(w)-\varphi(z)|\leq\tilde{c}\,|w-z|$ for all
$w\in\Gamma$. Then

$|(T_{g}((\tau_{n}\,\varphi)|_{\Gamma}))(z)|\leq\int_{\\{w\in\Gamma:|w-z|\leq
3/n\\}}\frac{c}{|w-z|^{d-1}}\,\tilde{c}\,|w-z|\,dw$

for all $n\in\mathds{N}$ and hence
$\lim_{n\to\infty}(T_{g}((\tau_{n}\,\varphi)|_{\Gamma}))(z)=0$.

It remains to show that $\lim_{n\to\infty}([{\cal
N},M_{g}]((\tau_{n}\,\varphi)|_{\Gamma}))(z)=0$. Let $\Phi\in
C^{\infty}(\mathds{R}^{d})$ be such that $\Phi|_{\Gamma}=\varphi$. Then
$\Phi(z)=0$. Note that $([{\cal
N},M_{g}]((\tau_{n}\,\varphi)|_{\Gamma}))(z)=({\cal
N}((g\,\tau_{n}\,\Phi)|_{\Gamma}))(z)$ for all $n\in\mathds{N}$. It follows as
in the proof of Lemma 9.2 that there is a $\check{c}>0$ such that

$\|(g\,\tau_{n}\,\Phi)|_{\Omega}\|_{L_{\infty}(\Omega)}\leq\frac{\check{c}}{n}\,\frac{1}{n},\quad\|\partial_{k}(g\,\tau_{n}\,\Phi)|_{\Omega}\|_{L_{\infty}(\Omega)}\leq\frac{\check{c}}{n}\quad\mbox{and}\quad\|\partial_{k}(g\,\tau_{n}\,\Phi)|_{\Omega}\|_{C^{\kappa}(\Omega)}\leq\frac{\check{c}}{n}\,n^{\kappa}$

for all $k\in\\{1,\ldots,d\\}$ and $n\in\mathds{N}$. Let $\widetilde{M}>0$ be
as in Lemma 9.1. Then Lemma 9.1(a) gives

$\|{\cal
N}((g\,\tau_{n}\,\Phi)|_{\Gamma})\|_{C(\Gamma)}\leq\widetilde{M}\,\frac{\check{c}}{n}$

for all large $n\in\mathds{N}$. In particular $\lim_{n\to\infty}({\cal
N}((g\,\tau_{n}\,\Phi)|_{\Gamma}))(z)=0$ and the proof of Step 9 is complete.

Step 3 Suppose $\varphi(z)=0$. Then (20) is valid.
Define $\tilde{g}\colon\mathds{R}^{d}\to\mathds{R}$ by
$\tilde{g}=g-g(z)\,\mathds{1}_{\mathds{R}^{d}}$. Then Step 9 gives

$(T_{g}\varphi)(z)=(T_{\tilde{g}}\varphi)(z)=([{\cal
N},M_{\tilde{g}}]\varphi)(z)=([{\cal N},M_{g}]\varphi)(z)$

as required.

Step 4 Now we prove the proposition.
Define $\tilde{\varphi}\colon\Gamma\to\mathds{C}$ by
$\tilde{\varphi}=\varphi-\varphi(z)$. Then Step 9 gives

$\displaystyle(T_{g}\varphi)(z)-\varphi(z)\,([{\cal
N},M_{g}]\mathds{1}_{\Gamma})(z)$ $\displaystyle=$
$\displaystyle(T_{g}\varphi)(z)-\varphi(z)\,(T_{g}\mathds{1}_{\Gamma})(z)$
$\displaystyle=$ $\displaystyle(T_{g}\tilde{\varphi})(z)=([{\cal
N},M_{g}]\tilde{\varphi})(z)$ $\displaystyle=$ $\displaystyle([{\cal
N},M_{g}]\varphi)(z)-\varphi(z)\,([{\cal N},M_{g}]\mathds{1}_{\Gamma})(z).$

This completes the proof. ∎

Now we are able to prove the commutator estimates of Theorem 1.1.

###### Proof of Theorem 1.1.

We give the proof in four steps.

Step 1 Let $\nu\in(0,\kappa)$. We prove that there is a $c>0$ such that

$\|T_{g}\varphi\|_{C^{\nu}(\Gamma)}\leq c\,\|\varphi\|_{C^{\nu}(\Gamma)}$

for all $\varphi\in\\{\Phi|_{\Gamma}:\Phi\in C^{\infty}(\mathds{R}^{d})\\}$
and $g\in W_{2}$.

Let $c>0$ be as in Lemma 9.6. Further, let $\hat{c}\geq 1$ and the $\chi_{r}$
be as in Lemma 7.1. Moreover, let $c_{1}$ and $c_{2}$ be as in Lemma 7.4, let
$c^{\prime}$ be as in Lemma 9.2 and let $c^{\prime\prime}$ be as in
Proposition 9.3.

Let $g\in W_{2}$ and let $K_{g}$ be the Schwartz kernel of $[{\cal N},M_{g}]$.
Let $\varphi\in\\{\Phi|_{\Gamma}:\Phi\in C^{\infty}(\mathds{R}^{d})\\}$. Let
$z,z_{0}\in\Gamma$ with $|z-z_{0}|\leq 1$. We may assume that $z\neq z_{0}$.
Further without loss of generality we may assume that $g(z)=0$. Choose
$r=|z-z_{0}|$. Then

$\displaystyle(T_{g}\varphi)(z)-(T_{g}\varphi)(z_{0})$ $\displaystyle=$
$\displaystyle\int_{\Gamma}K_{g}(z,w)\,\Big{(}\varphi(w)-\varphi(z)\Big{)}\,dw-\int_{\Gamma}K_{g}(z_{0},w)\,\Big{(}\varphi(w)-\varphi(z_{0})\Big{)}\,dw$
$\displaystyle\hskip 28.45274pt{}+\varphi(z)\,([{\cal
N},M_{g}]\mathds{1}_{\Gamma})(z)-\varphi(z_{0})\,([{\cal
N},M_{g}]\mathds{1}_{\Gamma})(z_{0})$ $\displaystyle=$
$\displaystyle\int_{\Gamma}K_{g}(z,w)\,\chi_{r}(w-z)\,\Big{(}\varphi(w)-\varphi(z)\Big{)}\,dw$
$\displaystyle\hskip
28.45274pt{}-\int_{\Gamma}K_{g}(z_{0},w)\,\chi_{r}(w-z)\,\Big{(}\varphi(w)-\varphi(z_{0})\Big{)}\,dw$
$\displaystyle\hskip
28.45274pt{}+\int_{\Gamma}(K_{g}(z,w)-K_{g}(z_{0},w))\,(1-\chi_{r}(w-z))\,(\varphi(w)-\varphi(z_{0}))\,dw$
$\displaystyle\hskip
28.45274pt{}+\int_{\Gamma}K_{g}(z,w)\,(1-\chi_{r}(w-z))\,\Big{(}\varphi(z_{0})-\varphi(z)\Big{)}\,dw$
$\displaystyle\hskip 28.45274pt{}+\Big{(}\varphi(z)\,([{\cal
N},M_{g}]\mathds{1}_{\Gamma})(z)-\varphi(z_{0})\,([{\cal
N},M_{g}]\mathds{1}_{\Gamma})(z_{0})\Big{)}$ $\displaystyle=$ $\displaystyle
I_{1}+I_{2}+I_{3}+I_{4}+I_{5}.$

We estimate the five terms separately.

First using Lemma 7.4(a) with $\beta=\nu$ we estimate

$\displaystyle|I_{1}|$ $\displaystyle\leq$
$\displaystyle\int_{\\{w\in\Gamma:|w-z|\leq
3|z-z_{0}|\\}}c\,\frac{1}{|w-z|^{d-1}}\,2\|\varphi\|_{C^{\nu}(\Gamma)}\,|w-z|^{\nu}$
$\displaystyle\leq$ $\displaystyle
2c\,\|\varphi\|_{C^{\nu}(\Gamma)}\,c_{2}\,\frac{1}{1-2^{-\nu}}\,(3|z-z_{0}|)^{\nu}=2\frac{3^{\nu}\,c\,c_{2}}{1-2^{-\nu}}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|z-z_{0}|^{\nu}.$

For the second term, note that if $\chi_{r}(w-z)\neq 0$, then $|w-z|\leq
3|z-z_{0}|$ and hence $|w-z_{0}|\leq|w-z|+|z-z_{0}|\leq 4|z-z_{0}|$. Hence

$|I_{2}|\leq
2\frac{4^{\nu}\,c\,c_{2}}{1-2^{-\nu}}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|z-z_{0}|^{\nu}.$

For the third term, if $(1-\chi_{r}(w-z))\neq 0$, then $|w-z|\geq 2r$. Hence
$|w-z_{0}|\leq|w-z|+|z-z_{0}|\leq\frac{3}{2}\,|w-z|$. Therefore
$|\varphi(w)-\varphi(z_{0})|\leq
2\|\varphi\|_{C^{\nu}(\Gamma)}\,|w-z_{0}|^{\nu}\leq
4\|\varphi\|_{C^{\nu}(\Gamma)}\,|w-z|^{\nu}$. Then Lemma 7.4(a) with
$\beta=\kappa-\nu$ gives

$\displaystyle|I_{3}|$ $\displaystyle\leq$ $\displaystyle
4\|\varphi\|_{C^{\nu}(\Gamma)}\int_{\\{w\in\Gamma:|w-z|\geq
2|z-z_{0}|\\}}c\,\frac{|z-z_{0}|^{\kappa}}{|z-w|^{d-1+\kappa}}\,|w-z|^{\nu}\,dw$