# On Sobolev spaces of bounded subanalytic manifolds

Guillaume Valette  Instytut Matematyki Uniwersytetu Jagiellońskiego, ul. S.
Łojasiewicza 6, Kraków, Poland<EMAIL_ADDRESS>
###### Abstract.

We focus on the Sobolev spaces of bounded subanalytic submanifolds of
$\mathbb{R}^{n}$. We prove that if $M$ is such a manifold then the space
$\mathscr{C}_{0}^{\infty}(M)$ is dense in $W^{1,p}(M,\partial M)$ (the kernel
of the trace operator) for all $p\leq\mathbf{p_{{}_{M}}}$, where
$\mathbf{p_{{}_{M}}}$ is the codimension in $M$ of the singular locus of
${\overline{M}}\setminus M$ (which is always at least $2$). In the case where
$M$ is normal, i.e. when $\mathbf{B}(x_{0},\varepsilon)\cap M$ is connected
for every $x_{0}\in{\overline{M}}$ and $\varepsilon>0$ small, we show that
$\mathscr{C}^{\infty}({\overline{M}})$ is dense in $W^{1,p}(M)$ for all such
$p$. This yields some duality results between
$W^{1,p}({\Omega},\partial{\Omega})$ and $W^{-1,p^{\prime}}({\Omega})$ in the
case where $1<p\leq\mathbf{p_{{}_{\Omega}}}$ and ${\Omega}$ is a bounded
subanalytic open subset of $\mathbb{R}^{n}$. As a byproduct, we deduce
uniqueness of the (weak) solution of the Dirichlet problem associated with the
Laplace equation. We then prove a version of Sobolev’s Embedding Theorem for
subanalytic bounded manifolds, show Gagliardo-Nirenberg’s inequality (for all
$p\in[1,\infty)$), and derive some versions of Poincaré-Friedrichs’
inequality. We finish with a generalization of Morrey’s Embedding Theorem.

###### Key words and phrases:

Trace operator, density of smooth functions, Sobolev spaces, Sobolev
embeddings, Gagliardo-Nirenberg inequality, singular domains, subanalytic
sets.

###### 2020 Mathematics Subject Classification:

46E35, 32B20, 14P10

##  0\. Introduction

This article develops the theory of Sobolev spaces on subanalytic bounded
submanifolds of $\mathbb{R}^{n}$, not necessarily compact. It was shown in
[23] that Poincaré-Wirtinger inequality holds on subanalytic domains. In [27],
we focused on the Sobolev space $W^{1,p}(M)$, $M$ bounded subanalytic
manifold, in the case where $p$ is large, on which we defined a trace operator
and established that it is continuous and that the smooth compactly supported
functions are dense in its kernel. The aim is to develop the basic material to
study partial differential equations on bounded subanalytic open subsets of
$\mathbb{R}^{n}$ and more generally on subanalytic manifolds.

In the present article, we prove a version of Sobolev’s Embedding Theorem for
all $p\in[1,\infty)$ (Theorem 5.1) for bounded subanalytic manifolds,
asserting that there is a positive real number $\mu$ such that for all $p$,
the space $W^{1,p}(M)$ can be naturally compactly embedded in $L^{p+\mu}$,
where $M$ is as above. We also generalize the famous closely related
Gagliardo-Nirenberg inequality and provide some Poincaré-Friedrichs type
inequalities. We also prove, in the case where $p$ is large, that the elements
of $W^{1,p}(M)$ are Hölder continuous with respect to the inner metric
(Morrey’s embedding, Corollary 6.3). The real number $\mu$ provided by our
Sobolev Embedding Theorem is not only related to the dimension, as it is the
case on Lipschitz domains, and actually heavily depends on the Lipschitz
geometry of the singularities of the frontier of $M$. That it does not depend
on $p$ is however valuable for applications and suffices to show for instance
that any measurable function on such a manifold $M$ which is $L^{1}$ and such
that all its subsequent partial derivatives are also $L^{1}$, must coincide
almost everywhere with a function which is Lipschitz with respect to the inner
metric (Corollary 6.4). Such a result could be generalized to manifolds that
are definable in a polynomially bounded o-minimal structure [9, 5] but clearly
fails as soon as the structure is not polynomially bounded (see Example 6.5).

Although subanalytic domains do not possess the cone property, it is well
known since the original work of S. Łojasiewicz [15] that they are locally
homeomorphic to cones, and it was established that this homeomorphism can be
chosen subanalytic [5]. We rely on the more recent results of [26, 28], where
a conic structure theorem for the Lipschitz geometry at a singular point of a
subanalytic set (see Theorem 1.6) was achieved. This theorem makes it possible
to obtain some local estimates of the $L^{p}$ norm of functions that have an
$L^{p}$ derivative (see section 3.3) which, together some arguments of
stratification theory, enables us to show that the smooth compactly supported
functions are dense in the kernel of the trace operator in the case where $p$
is close to $1$ (Theorem 4.1 and Corollary 4.2). The provided real number
underneath which this density result holds, denoted $\mathbf{p_{{}_{M}}}$, is
the codimension in $M$ of the singularities of the frontier (which is always
at least $2$). In particular, our density result holds in $W^{1,2}(M)$, which
is the most important Sobolev space. Moreover, it entails some duality results
between $W^{1,p}({\Omega},\partial{\Omega})$ and $W^{-1,p^{\prime}}({\Omega})$
for $p\leq\mathbf{p_{{}_{\Omega}}}$, where $p^{\prime}$ stands for the Hölder
conjugate of $p$ and ${\Omega}$ for a bounded open subanalytic subset of
$\mathbb{R}^{n}$ (Proposition 4.11).

In [3], L. P. Bos and P. Milman prove some interesting Sobolev-Gagliardo-
Nirenberg type inequalities for the functions of the space
$\mathscr{C}^{\infty}({\overline{M}})$. Our purpose is however less to provide
new inequalities in this framework than to illustrate that the recent
developments of singularity theory make it possible to carry out a fully
satisfying theory of the much wider class constituted by the Sobolev spaces of
these manifolds, sufficiently rich to solve PDE on subanalytic domains.

As an application of our density theorem (Theorem 4.1), we also establish
(Theorem 4.13) existence and uniqueness of the solution of the Dirichlet
problem associated with the Laplace equation in $W^{1,2}({\Omega})$, if
$\Omega$ is an open bounded subanalytic set of any dimension. This result is
closely related to the question of the continuity of generalized Green
functions to which T. Kaiser answered positively on subanalytic domains of
$\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ [13] (see also [12]). The theory of
Sobolev spaces that we develop in the present article provides a versatile
framework to investigate this kind of problem. It is indeed well-known that
the generalized Dirichlet solution studied in [13] coincides with the
classical one, when the latter solution exists. Noteworthy, our result brings
new information on the class of integrability of the solution (and many
improvements of this seem to be allowed by the theory) and makes it possible
to compute effective approximations of solutions.

It is worth stressing the fact that this theory actually goes over the more
general framework of polynomially bounded o-minimal structures (expanding
$\mathbb{R}$) [5, 9]. Indeed, the arguments being essentially local, we only
need the underlying manifold $M$ to be such that every $x\in{\overline{M}}$
has a neighborhood $U$ in ${\overline{M}}$ such that $U\cap M$ is bi-Lipschitz
homeomorphic to a set definable in some polynomially bounded o-minimal
structure (that may depend on $x$), and more generally we could adapt the
statements to abstract varieties, which is useful for the complex case.
Moreover, since we could use “cut-off” functions at infinity, the density
results actually do not require the underlying manifold to be bounded.

###  0.1. Some notations

Throughout this article, $n$, $j$, and $k$ stand for integers, the letter $M$
stands for a bounded subanalytic $\mathscr{C}^{\infty}$ submanifold of
$\mathbb{R}^{n}$ and $m$ for its dimension, while the letter $\Omega$ stands
for a bounded definable open subset of $\mathbb{R}^{n}$.

The origin of $\mathbb{R}^{n}$ will be denoted $0_{\mathbb{R}^{n}}$, although
the subscript $\mathbb{R}^{n}$ will be omitted when the ambient space is
obvious from the context. We write $x\cdot y$ for the euclidean inner product
of $x$ and $y$, and $|.|$ for the euclidean norm. Given $x\in\mathbb{R}^{n}$
and $\varepsilon>0$, we respectively denote by $\mathbf{S}(x,\varepsilon)$ and
$\mathbf{B}(x,\varepsilon)$ the sphere and the open ball of radius
$\varepsilon$ that are centered at $x$ (for the euclidean norm), while
$\overline{\mathbf{B}}(x,\varepsilon)$ will stand for the corresponding closed
ball.

A mapping $h:A\to B$, $A\subset\mathbb{R}^{n},B\subset\mathbb{R}^{k}$, is
Lipschitz if there is a constant $C$ such that $|h(x)-h(x^{\prime})|\leq
C|x-x^{\prime}|$ for all $x$ and $x^{\prime}$. It is bi-Lipschitz if it is a
homeomorphism and if in addition $h$ and $h^{-1}$ are both Lipschitz. Given
two nonnegative functions $\xi$ and $\zeta$ on a set $E$ as well as a subset
$Z$ of $E$, we write “$\xi\lesssim\zeta$ on $Z$” or “$\xi(x)\lesssim\zeta(x)$
for $x\in Z$” when there is a constant $C$ such that $\xi(x)\leq C\zeta(x)$
for all $x\in Z$.

If $E\subset\mathbb{R}^{n}$ and $i\in\mathbb{N}\cup\\{\infty\\}$, we will
write $\mathscr{C}^{i}(E)$ for the space of those functions on $E$ that extend
to a $\mathscr{C}^{i}$ function on an open neighborhood of $E$ in
$\mathbb{R}^{n}$. We denote the closure of $E$ by $\overline{E}$ and set
$\delta E:=\overline{E}\setminus E$. We denote by $\mathcal{H}^{k}$ the
$k$-dimensional Hausdorff measure and by $L^{p}(E,\mathcal{H}^{k})$ the set of
$L^{p}$ functions on $E$ with respect to this measure.

By “manifold”, we will always mean submanifold of $\mathbb{R}^{n}$, and a
submanifold of $\mathbb{R}^{n}$ will always be endowed with its canonical
measure, provided by volume forms. As integrals will always be considered with
respect to this measure, we will not indicate the measure when integrating on
a submanifold of $\mathbb{R}^{n}$.

Given a measurable mapping $v:M\to\mathbb{R}^{k}$ (with respect to the
canonical measure of $M$), for each $p\in[1,\infty)$ we denote by
$||v||_{L^{p}(M)}$ the (possibly infinite) $L^{p}$ norm of $v$ to $M$. As
usual, we denote by $L^{p}(M)$ the set of measurable functions on $M$ for
which $||u||_{L^{p}(M)}$ is finite, and set

$W^{1,p}(M):=\\{u\in L^{p}(M),\;|\partial u|\in L^{p}(M)\\},$

where $\partial u$ stands for the gradient of $u$ in the sense of
distributions. As well-known, this space, equipped with the norm

$||u||_{W^{1,p}(M)}:=||u||_{L^{p}(M)}+||\partial u||_{L^{p}(M)},$

is a Banach space, in which $\mathscr{C}^{\infty}(M)$ is dense for all
$p\in[1,\infty)$.

We denote by $\mbox{\rm supp}\,u$ the support of a distribution $u$ on $M$,
and, given an open subset $Z$ of ${\overline{M}}$ containing $M$, we write
$\mbox{\rm supp}_{Z}u$ for the support of $u$ in $Z$, defined as the closure
in $Z$ of $\mbox{\rm supp}\,u$.

We will write $\mathscr{C}_{0}^{\infty}(M)$ for the space of elements of
$\mathscr{C}^{\infty}(M)$ that are compactly supported, while $W^{1,p}_{0}(M)$
will stand for the closure of this space in $W^{1,p}(M)$. We will regard
$\mathscr{C}^{\infty}({\overline{M}})$ as a subset of $W^{1,p}(M)$. In
particular, for $u\in\mathscr{C}^{\infty}({\overline{M}})$, $\mbox{\rm
supp}\,u$ will be a subset of $M$ and we set for $Z$ open in ${\overline{M}}$

$\mathscr{C}^{\infty}_{Z}({\overline{M}}):=\\{u\in\mathscr{C}^{\infty}(\overline{M}):\overline{\mbox{\rm
supp}\,u}\subset Z\\}.$

Alike $\mathscr{C}^{\infty}({\overline{M}})$, the space
$\mathscr{C}^{\infty}_{Z}({\overline{M}})$ will be regarded as a subspace of
$\mathscr{C}^{\infty}(M)$. Note that when $Z\supset M$, the space
$\mathscr{C}^{\infty}_{Z}({\overline{M}})$ is nothing but the set constituted
by the functions $u\in\mathscr{C}^{\infty}({\overline{M}})$ that are compactly
supported in $Z$, i.e., for which $\mbox{\rm supp}_{Z}u$ is compact.

Given two functions $u$ and $v$ on ${\Omega}$ such that $uv$ is $L^{1}$, we
set $<u,v>:=\int_{\Omega}uv.$ We also denote by $\partial_{i}u$ the $i^{th}$
partial derivative in the sense of distributions and, for simplicity of
statements, let $\partial_{0}u:=u$.

Given a mapping $h:M\to M$, we write $\mbox{jac}\,\,h$ for the absolute value
of the determinant of $Dh$, the derivative of $h$ (defined at the points where
$h$ is differentiable). We write $p^{\prime}$ for the Hölder conjugate of $p$,
i.e., $p^{\prime}=\frac{p}{p-1}$, and, given a topological vector space $V$,
we denote by $V^{\prime}$ its topological dual.

We shall make use of the following form of Hölder’s inequality. Let
$p_{1},\cdots,p_{\kappa}$ and $\theta_{1},\dots,\theta_{\kappa}$ be positive
real numbers and let $q\in[1,\infty)$ be such that
$\frac{1}{q}=\sum_{i=1}^{\kappa}\frac{\theta_{i}}{p_{i}}$. If
$g_{1},\dots,g_{\kappa}$ are measurable functions such that $g_{i}\in
L^{p_{i}}(M)$, for all $i$, then we have

(0.1) $||g_{1}^{\theta_{1}}\cdots
g_{\kappa}^{\theta_{\kappa}}||_{L^{q}(M)}\leq||g_{1}||^{\theta_{1}}_{L^{p_{1}}(M)}\cdots||g_{\kappa}||_{L^{p_{\kappa}}(M)}^{\theta_{\kappa}}.$

##  1\. Subanalytic sets

We refer the reader to [4, 7, 16, 28] for all the basic facts about
subanalytic geometry. Actually, [28] also gives a detailed presentation of the
Lipschitz properties of these sets, so that the reader can find there all the
needed facts to understand the result achieved in the present article.

######  Definition 1.1.

A subset $E\subset\mathbb{R}^{n}$ is called semi-analytic if it is locally
defined by finitely many real analytic equalities and inequalities. Namely,
for each $a\in\mathbb{R}^{n}$, there is a neighborhood $U$ of $a$ in
$\mathbb{R}^{n}$, and real analytic functions $f_{ij},g_{ij}$ on $U$, where
$i=1,\dots,r,j=1,\dots,s_{i}$, such that

(1.1) $E\cap U=\bigcup_{i=1}^{r}\bigcap_{j=1}^{s_{i}}\\{x\in
U:g_{ij}(x)>0\mbox{ and }f_{ij}(x)=0\\}.$

The flaw of the semi-analytic category is that it is not preserved by analytic
morphisms, even when they are proper. To overcome this problem, we prefer
working with the subanalytic sets, which are defined as the projections of the
semi-analytic sets.

A subset $E\subset\mathbb{R}^{n}$ is subanalytic if each point
$x\in\mathbb{R}^{n}$ has a neighborhood $U$ such that $U\cap E$ is the image
under the canonical projection
$\pi:\mathbb{R}^{n}\times\mathbb{R}^{k}\to\mathbb{R}^{n}$ of some relatively
compact semi-analytic subset of $\mathbb{R}^{n}\times\mathbb{R}^{k}$ (where
$k$ depends on $x$).

A subset $Z$ of $\mathbb{R}^{n}$ is globally subanalytic if
$\mathcal{V}_{n}(Z)$ is a subanalytic subset of $\mathbb{R}^{n}$, where
$\mathcal{V}_{n}:\mathbb{R}^{n}\to(-1,1)^{n}$ is the homeomorphism defined by

$\mathcal{V}_{n}(x_{1},\dots,x_{n}):=(\frac{x_{1}}{\sqrt{1+|x|^{2}}},\dots,\frac{x_{n}}{\sqrt{1+|x|^{2}}}).$

We say that a mapping $f:A\to B$ is subanalytic (resp. globally subanalytic),
$A\subset\mathbb{R}^{n}$, $B\subset\mathbb{R}^{m}$ subanalytic (resp. globally
subanalytic), if its graph is a subanalytic (resp. globally subanalytic)
subset of $\mathbb{R}^{n+m}$. In the case $B=\mathbb{R}$, we say that $f$ is a
(resp. globally) subanalytic function. For simplicity globally subanalytic
sets and mappings will be referred as definable sets and mappings (this
terminology is often used by o-minimal geometers [9, 5]).

The globally subanalytic category is very well adapted to our purpose. It is
stable under intersection, union, complement, and projection. It thus
constitutes an o-minimal structure [9, 5], and consequently admits cell
decompositions and stratifications, from which it comes down that definable
sets enjoy a large number of finiteness properties (see [5, 28] for more).

### Łojasiewicz’s inequality.

Some results about $L^{p}$ functions will be valid for $p$ sufficiently large,
which means that there will be $p_{0}\in\mathbb{R}$ such that the claimed fact
will be true for all $p\in(p_{0},\infty)$ (i.e. when we do not specify that
$p$ may be infinite, it should be understood that it is finite). This number
$p_{0}$ will be provided by Łojasiewicz’s inequality, which is one of the main
tools of subanalytic geometry. It originates in the work of S. Łojasiewicz
[15], who established this inequality in order to answer a problem about
distribution theory. We shall make use of the following version (see the proof
of (1.7)):

######  Proposition 1.2.

(Łojasiewicz’s inequality) Let $f$ and $g$ be two globally subanalytic
functions on a globally subanalytic set $A$ with $\sup\limits_{x\in
A}|f(x)|<\infty$. Assume that $\lim\limits_{t\to 0}f(\gamma(t))=0$ for every
globally subanalytic arc $\gamma:(0,\varepsilon)\to A$ satisfying
$\lim\limits_{t\to 0}g(\gamma(t))=0$.

Then there exist $\nu\in\mathbb{N}$ and $C\in\mathbb{R}$ such that for any
$x\in A$:

$|f(x)|^{\nu}\leq C|g(x)|.$

See for instance [22] for a proof.

### The set $\partial M$.

We will say that $M$ is Lipschitz regular at $x\in\delta M$ if this point has
a neighborhood $U$ in $\mathbb{R}^{n}$ such that each connected component of
$U\cap M$ is the interior of a Lipschitz manifold with boundary $U\cap\delta
M$. We then let:

$\partial M:=\\{x\in\delta M:M\mbox{ is Lipschitz regular at $x$}\\}.$

This set is definable and we have

(1.2) $\dim\left(\delta M\setminus\partial M\right)\leq m-2.$

This fact follows from a famous result which is sometimes referred as Wing
Lemma by geometers (see [16, Proposition 1, section 19], [1, Proposition
9.6.13], or [28, Lemma 5.6.7]). The closure of the set $\partial M$ in
$\mathbb{R}^{n}$ is called the fence of $M$.

######  Example 1.3.

If ${\Omega}$ is a definable open subset of $\mathbb{R}^{n}$ then
$\overline{{\Omega}}$ is the union of (one or two) $\mathscr{C}^{\infty}$
submanifolds with boundary of $\mathbb{R}^{n}$ at $\mathcal{H}^{n-1}$-almost
every point of $\delta{\Omega}$. It is worth here underlying that, when $M$
has positive codimension in $\mathbb{R}^{n}$, it is not true that
${\overline{M}}$ is a union of $\mathscr{C}^{\infty}$ manifolds with boundary
at $\mathcal{H}^{m-1}$-almost every point of $\delta M$. A counterexample is
given by the graph of the function $f(x)=x^{3/2}$, $x\in[0,1]$, which is not a
$\mathscr{C}^{2}$ manifold with boundary at the origin. It is however true
that ${\overline{M}}$ is a finite union of $\mathscr{C}^{1}$ manifolds with
boundary at $\mathcal{H}^{m-1}$-almost every point of $\delta M$.

###  1.1. Stratifications.

It is a very natural idea to divide a set which is singular into
$\mathscr{C}^{\infty}$ manifolds. This is the purpose of the definition below
which, together with Theorem 1.5, will be useful to work locally at a point of
the frontier of the underlying manifold.

######  Definition 1.4.

A stratification of a subset of $\mathbb{R}^{n}$ is a finite partition of it
into definable $\mathscr{C}^{\infty}$ submanifolds of $\mathbb{R}^{n}$, called
strata. A stratification is compatible with a set if this set is the union of
some strata. A refinement of a stratification $\Sigma$ is a stratification
$\Sigma^{\prime}$ compatible with the strata of $\Sigma$.

A stratification $\Sigma$ of a set $X$ is locally bi-Lipschitz trivial if for
every $S\in\Sigma$ and $x_{0}\in S$, there are a neighborhood $W$ of $x_{0}$
in $S$ and a $\mathscr{C}^{\infty}$ definable retraction $\pi:U\to W$, with
$U$ neighborhood of $x_{0}$ in $X$, as well as a bi-Lipschitz homeomorphism

$\Lambda:U\to(\pi^{-1}(x_{0})\cap X)\times W,$

satisfying:

  1. (i)

$\pi(\Lambda^{-1}(x,y))=y$, for all $(x,y)\in(\pi^{-1}(x_{0})\cap X)\times W$.

  2. (ii)

$\Sigma_{x_{0}}:=\\{\pi^{-1}(x_{0})\cap Y:Y\in\Sigma\\}$ is a stratification
of $\pi^{-1}(x_{0})\cap X$, and $\Lambda(Y\cap U)=(\pi^{-1}(x_{0})\cap
Y)\times W$, for all $Y\in\Sigma$.

######  Theorem 1.5.

Every stratification $\Sigma$ can be refined into a stratification which is
locally bi-Lipschitz trivial.

Lipschitz stratifications, in the sense of T. Mostowski [18], are locally bi-
Lipschitz trivial. Existence of Lipschitz stratifications for subanalytic sets
was proved by A. Parusiński [20]. Stratifications that are definably bi-
Lipschitz trivial, in the sense that the trivialization $\Lambda$ can be
chosen definable are constructed in [29] for sets that are definable in a
polynomially bounded o-minimal structure, which includes all the globally
subanalytic sets.

###  1.2. Lipschitz conic structure

The main ingredient of our approach is the following result achieved in [26]
(see also [28] for a complete expository of the theory). In this theorem,
$x_{0}*(\mathbf{S}(x_{0},\varepsilon)\cap X)$ stands for the cone over
$\mathbf{S}(x_{0},\varepsilon)\cap X$ with vertex at $x_{0}$.

######  Theorem 1.6 (Lipschitz Conic Structure).

Let $X\subset\mathbb{R}^{n}$ be subanalytic and $x_{0}\in X$. For
$\varepsilon>0$ small enough, there exists a Lipschitz subanalytic
homeomorphism

$H:x_{0}*(\mathbf{S}(x_{0},\varepsilon)\cap
X)\to\overline{\mathbf{B}}(x_{0},\varepsilon)\cap X,$

satisfying $H_{|\mathbf{S}(x_{0},\varepsilon)\cap X}=Id$, preserving the
distance to $x_{0}$, and having the following metric properties:

  1. (i)

The natural retraction by deformation onto $x_{0}$

$r:[0,1]\times\overline{\mathbf{B}}(x_{0},\varepsilon)\cap
X\to\overline{\mathbf{B}}(x_{0},\varepsilon)\cap X,$

defined by

$r(s,x):=H(sH^{-1}(x)+(1-s)x_{0}),$

is Lipschitz. Indeed, there is a constant $C$ such that for every fixed
$s\in[0,1]$, the mapping $r_{s}$ defined by $x\mapsto r_{s}(x):=r(s,x)$, is
$Cs$-Lipschitz.

  2. (ii)

For each $\delta>0$, the restriction of $H^{-1}$ to $\\{x\in
X:\delta\leq|x-x_{0}|\leq\varepsilon\\}$ is Lipschitz and, for each
$s\in(0,1]$, the map $r_{s}^{-1}:\overline{\mathbf{B}}(x_{0},s\varepsilon)\cap
X\to\overline{\mathbf{B}}(x_{0},\varepsilon)\cap X$ is Lipschitz.

######  Example 1.7.

The mapping $H$ of the above theorem cannot be required to be bi-Lipschitz, as
shown by the example

$X:=\\{(x,y)\in[0,1]\times\mathbb{R}:|y|\leq x^{2}\\}.$

An explicit formula for the corresponding mapping $H$ at $x_{0}=(0,0)$ that
can help the reader to grasp the properties listed in the above theorem is
provided in [26]. It is also relevant to mention an example which shows that
this theorem fails on non polynomially bounded o-minimal structures. Let

$Y:=\\{(x,y)\in(0,1]\times\mathbb{R}:|y|\leq
e^{\frac{-1}{x}}\\}\cup\\{(0,0)\\}.$

If there were such a mapping $H$, it is not difficult to see that the points
$z:=(x,e^{\frac{-1}{x}})$ and $z^{\prime}:=(x,-e^{\frac{-1}{x}})$, for
$x\in(0,1]$, would satisfy for $s\in(0,1]$:

$|r_{s}(z)-r_{s}(z^{\prime})|\leq Ce^{\frac{-1}{sx}},$

for some constant $C$ independent of $x$ and $s$. As
$|z-z^{\prime}|=2e^{\frac{-1}{x}}$, we see that $r_{s}^{-1}$ would fail to be
Lipschitz for each $s\in(0,1)$. This example enlightens the importance of
polynomial boundedness and the key role played by Łojasiewicz’s inequality in
the theory (see also Example 6.5).

###  1.3. Lipschitz conic structure at a point of $\delta M$.

We now wish to explain in which setting we will apply the above theorem and
then mention a few facts that were established in [27]. Let us assume that
$0_{\mathbb{R}^{n}}\in\delta M$ and set for $\eta>0$:

(1.3) $M^{\eta}:=\mathbf{B}(0_{\mathbb{R}^{n}},\eta)\cap
M\;\;\quad\mbox{and}\quad\;\;N^{\eta}:=\mathbf{S}(0_{\mathbb{R}^{n}},\eta)\cap
M.$

Apply Theorem 1.6 with $X=M\cup\\{0_{\mathbb{R}^{n}}\\}$ and
$x_{0}=0_{\mathbb{R}^{n}}$. Fix $\varepsilon>0$ sufficiently small for the
statement of the theorem to hold and for $N^{\varepsilon}$ to be a
$\mathscr{C}^{\infty}$ manifold, and let $r$ denote the mapping provided by
this theorem. We will assume for convenience $\varepsilon<\frac{1}{2}$.
Observe that since $r$ is subanalytic, it is $\mathscr{C}^{\infty}$ almost
everywhere.

Since $r_{s}$ is bi-Lipschitz for every $s>0$, $\mbox{jac}\,\,r_{s}$ can only
tend to zero if $s$ is itself going to zero. Hence, by Łojasiewicz’s
inequality (see Proposition 1.2), there are a positive integer $\nu$ and a
constant $\kappa$ such that for all $s\in(0,1)$ we have for almost all $x\in
M^{\varepsilon}$:

(1.4) $\mbox{jac}\,\,r_{s}(x)\geq\frac{s^{\nu}}{\kappa}.$

A few facts. The following facts were then established in [27, section $2.1$].
There is a constant $C$ such that:

  1. (1)

For all $s\in(0,1)$ we have for almost all $x\in M^{\varepsilon}$:

(1.5) $\left|\frac{\partial r}{\partial s}(s,x)\right|\leq C|x|.$

  2. (2)

For each $v\in L^{p}(M^{\varepsilon})$, $p\in[1,\infty)$, we have for all
$\eta\in(0,\varepsilon]$:

(1.6)
$\frac{1}{C}\left(\int_{0}^{\eta}||v||_{L^{p}(N^{\zeta})}^{p}d\zeta\right)^{1/p}\leq||v||_{L^{p}(M^{\eta})}\leq
C\left(\int_{0}^{\eta}||v||_{L^{p}(N^{\zeta})}^{p}d\zeta\right)^{1/p}.$

  3. (3)

There exists $\nu\in\mathbb{N}$ such that for each $v\in
L^{p}(M^{\varepsilon})$, $p\in[1,\infty)$, $\eta\in(0,\varepsilon]$, and
$s\in(0,1)$:

(1.7) $||v\circ r_{s}||_{L^{p}(N^{\eta})}\leq
Cs^{-\nu/p}||v||_{L^{p}(N^{s\eta})}$

and

(1.8) $||v\circ r_{s}||_{L^{p}(M^{\eta})}\leq
Cs^{-\nu/p}||v||_{L^{p}(M^{s\eta})}.$

##  2\. Trace operators on subanalytic manifolds

We are going to define a trace operator $\mathbf{tr}_{\partial
M}:W^{1,p}(M)\to L^{p}_{loc}(\partial M)$ and show that it is continuous (in
section 2.2), which requires to recall (in section 2.1) a few results about
normal spaces established in [27]. We will also derive from the results of
[27] that, when $p$ is large, $\mathbf{tr}_{\partial M}u$ is actually $L^{p}$
for all $u\in W^{1,p}(M)$ (in section 2.3).

###  2.1. Normal manifolds.

The notions of normal manifold and $\mathscr{C}^{\infty}$ normalization, which
originate in [27], are natural generalizations to our framework of the notions
of normal pseudomanifold and normalization introduced in [6, 11].

######  Definition 2.1.

We say that $M$ is connected at $x\in\delta M$ if
$\mathbf{B}(x,\varepsilon)\cap M$ is connected for all $\varepsilon>0$ small
enough. We say that $M$ is normal if it is connected at each $x\in\delta M$.

We will say that it is connected along $Z\subset\delta M$ if it is connected
at each point of $Z$. The manifold $M$ will be said to be weakly normal if
there is a set $E\subset\delta M$ such that $\dim(\delta M\setminus E)\leq
m-2$, and along which $M$ is connected, which amounts to require that
$M\cup\partial M$ is a Lipschitz manifold with boundary $\partial M$. Of
course, every normal manifold is weakly normal.

A $\mathscr{C}^{\infty}$ normalization of $M$ is a definable
$\mathscr{C}^{\infty}$ diffeomorphism $h:{\check{M}}\to M$ satisfying
$\sup_{x\in{\check{M}}}|D_{x}h|<\infty$ and $\sup_{x\in
M}|D_{x}h^{-1}|<\infty$, with ${\check{M}}$ normal $\mathscr{C}^{\infty}$
submanifold of $\mathbb{R}^{k}$, for some $k$. We will sometimes say that
${\check{M}}$ is a normalization of $M$.

It can be seen [27] that if $h:{\check{M}}\to M$ is a normalization and
$x_{0}\in\delta M$ then the number of points of $h^{-1}(x_{0})$ coincides with
the number of connected components of $\mathbf{B}(x_{0},\varepsilon)\cap M$,
$\varepsilon>0$ small. The following proposition gathers Propositions $3.3$
and $3.4$ of [27], yielding existence and uniqueness of $\mathscr{C}^{\infty}$
normalizations.

######  Proposition 2.2.

  1. (1)

Every bounded definable $\mathscr{C}^{\infty}$ manifold admits a
$\mathscr{C}^{\infty}$ normalization.

  2. (2)

Every $\mathscr{C}^{\infty}$ normalization $h:{\check{M}}\to M$ extends
continuously to a mapping from $\overline{{\check{M}}}$ to ${\overline{M}}$.
Moreover, if $h_{1}:{\check{M}}_{1}\to M$ and $h_{2}:{\check{M}}_{2}\to M$ are
two $\mathscr{C}^{\infty}$ normalizations of $M$, then $h_{2}^{-1}h_{1}$
extends to a homeomorphism between $\overline{{\check{M}}_{1}}$ and
$\overline{{\check{M}}_{2}}$.

###  2.2. The trace on $\partial M$.

In [27] we have defined some trace operators on $W^{1,p}(M)$ in the case where
$p$ is large (this is recalled in the next section). We explain in this
section that we can always (i.e. for any $p$) define a trace which is locally
$L^{p}$ in $\partial M$. As ${\overline{M}}$ is a finite union of Lipschitz
manifolds with boundary at every point of $\partial M$, this actually follows
from the classical theory. A similar trace operator was defined in [24] on
open subsets of $\mathbb{R}^{n}$ that are definable in an o-minimal structure.

For $p\in[1,\infty)$, we say that a sequence of measurable functions $u_{i}\in
W^{1,p}(M)$ converges to $u\in W^{1,p}(M)$ in the strong $W^{1,p}_{loc}$
topology if every $x\in M\cup\partial M$ has a neighborhood $U$ in
$\mathbb{R}^{n}$ such that $u_{i|U\cap M}$ and $\partial u_{i|U\cap M}$
respectively tend to $u_{|U\cap M}$ and $\partial u_{|U\cap M}$ in the $L^{p}$
norm (we say “strong” because it is local in $M\cup\partial M$, not only in
$M$).

### The case of a normal manifold.

We first define our trace in the case where $M$ is normal (weakly normal is
actually enough for what we do in the proof below). In this case,
$M\cup\partial M$ is a Lipschitz manifold with boundary.

######  Proposition 2.3.

If $M$ is normal then for any $p\in[1,\infty)$, the mapping

(2.1) $\mathscr{C}^{\infty}({\overline{M}})\to L^{p}_{loc}(\partial
M),\quad\varphi\mapsto\varphi_{|\partial M}\,,$

extends to a mapping

$\mathbf{tr}_{\partial M}:W^{1,p}(M)\to L^{p}_{loc}(\partial M)$

which is continuous in the strong $W^{1,p}_{loc}$ topology.

###### Proof.

The classical theory on Lipschitz manifolds with boundary implies that for
every compact subset $K$ of $\partial M$ and every open subset $V$ of
$M\cup\partial M$ containing $K$, there is a constant $C$ (depending on $K$
and $V$) such that for all $\varphi\in\mathscr{C}^{\infty}(M\cup\partial M)$
we have

$||\varphi||_{L^{p}(K,\mathcal{H}^{m-1})}\leq C||\varphi||_{W^{1,p}(V)}.$

As $M\cup\partial M$ is a manifold with boundary, the space
$\mathscr{C}^{\infty}(M\cup\partial M)$ is dense in $W^{1,p}(M)$ in the strong
$W^{1,p}_{loc}$ topology, and the mapping defined in (2.1) extends to a
continuous mapping on $W^{1,p}(M)$. ∎

In the case of non necessarily normal manifolds, we are going to define a
trace operator which is “multi-valued” (since $M$ may have several connected
components locally at a generic frontier point).

### General definition of the trace.

There are definable subsets $S_{1},\dots,S_{k}$ of $\partial M$ satisfying
$\mathcal{H}^{m-1}(\partial M\setminus\bigcup_{i=1}^{k}S_{i})=0$ and such that
each $S_{i}$ has a neighborhood $U_{i}$ in $\mathbb{R}^{n}$ for which each
connected component of $U_{i}\cap M$ is the interior of a manifold with
boundary $S_{i}$ (such a covering is induced for instance by the images of the
simplices of a triangulation of $M$ compatible with $\partial M$). We denote
by $Z_{i,1},\dots,Z_{i,l_{i}}$ these connected components.

We first define the trace on $S_{i}$, denoted $\mathbf{tr}_{S_{i}}$,
$i\in\\{1,\dots,k\\}$. Let $\mathbf{tr}^{j}_{i}:W^{1,p}(Z_{i,j})\to
L^{p}_{loc}(S_{i},\mathcal{H}^{m-1})$ denote the operator obtained by applying
Proposition 2.3 to $Z_{i,j}$ (which is normal if $U_{i}$ is sufficiently
small), and then define

$\mathbf{tr}_{S_{i}}:W^{1,p}(M)\to
L^{p}_{loc}(S_{i},\mathcal{H}^{m-1})^{l},\qquad
l=\underset{i=1,\dots,k}{\max}l_{i}\,,$

by setting for $u\in W^{1,p}(M)$ and $x\in S_{i}$:

(2.2)
$\mathbf{tr}_{S_{i}}u(x):=(\mathbf{tr}_{i}^{1}u_{|Z_{i,1}}(x),\dots,\mathbf{tr}_{i}^{l_{i}}u_{|Z_{i,l_{i}}}(x),0,\dots,0).$

Here, we add $(l-l_{i})$ times the zero function because it will be convenient
that the trace has the same number of components for all $i$. Note that, as
$M\cup\partial M$ is a finite union of manifolds with boundary at every point
of $\partial M$, $\mathbf{tr}_{S_{i}}u(x)$ is actually $L^{p}$ on a
neighborhood of every point of $\overline{S_{i}}\cap\partial M$.

This mapping of course depends on the way the elements
$Z_{i,1},\dots,Z_{i,l_{i}}$ are enumerated. However, up to permutation of the
$l_{i}$ first components, $\mathbf{tr}_{S_{i}}u(x)$ is unique, and it only
depends on the germ of $u$ near $S_{i}$. In particular, the kernel of this
mapping is independent of any choice.

We now can define $\mathbf{tr}_{\partial M}u$ as the function which coincides
with $\mathbf{tr}_{S_{i}}u$ on $S_{i}\subset\partial M$, for each $i$. Since
$\partial M\setminus\bigcup_{i=1}^{k}S_{i}$ is $\mathcal{H}^{m-1}$-negligible,
this defines a function (almost everywhere) on $\partial M$. Moreover, as the
germ of $\mathbf{tr}_{i}^{j}u_{|Z_{i,j}}$ at a point of
$\overline{S_{i}}\cap\partial M$ is $L^{p}$ for all $i$ and $j$, it is clear
that this defines an element of $L^{p}_{loc}(\partial M)^{l}$. The above
proposition yields that the mapping

$\mathbf{tr}_{\partial M}:W^{1,p}(M)\to L^{p}_{loc}(\partial M)^{l}$

is continuous in the strong $W^{1,p}_{loc}$ topology.

For $p\in[1,\infty)$, we then set for an arbitrary open subset $Z$ of
$\partial M$:

$W^{1,p}(M,Z):=\\{u\in W^{1,p}(M):\mathbf{tr}_{\partial M}u\equiv 0\mbox{ on
}Z\\}.$

We will see that $W^{1,p}(M,\partial M)$ and $W^{1,p}_{0}(M)$ coincide for
small values of $p$ (Corollary 4.2). We deal with this issue in the case “$p$
large” in the next section.

######  Remark 2.4.

It directly follows from our local definition of the trace that a function
$u\in W^{1,p}({\Omega})$ belongs to $W^{1,p}({\Omega},\partial{\Omega})$ if
and only if it satisfies for all
$\varphi\in\mathscr{C}^{\infty}_{{\Omega}\cup\partial{\Omega}}(\overline{{\Omega}})$
and all $i=1,\dots,n$:

(2.3) $<u,\partial_{i}\varphi>=-<\partial_{i}u,\varphi>.$

######  Example 2.5.

[24] It is easy to produce examples of definable domains $\Omega$ admitting a
function $u\in W^{1,p}({\Omega})$ such that $\mathbf{tr}_{\partial{\Omega}}u$
is not $L^{p}$ on the boundary. Let for $k>2$,

$\Omega_{k}:=\\{(x,y)\in(0,1)^{2}:y<x^{k}\\},$

and let $u(x,y):=\frac{1}{x}.$ Clearly, $u\in W^{1,p}({\Omega}_{k})$, for each
$p\in[1,\frac{k}{2}]$, while $\mathbf{tr}_{\partial\Omega_{k}}u$ is not
$L^{p}$ on $\partial{\Omega}_{k}$.

###  2.3. The trace when $p$ is large.

We now wish to explain that in the case where $p$ is large, the results of
[27] yield that the mapping $\mathbf{tr}_{\partial M}$, defined in the
preceding section, is continuous in the $L^{p}$ norm (see Corollary 2.7 below
and Example 2.5 just above).

######  Theorem 2.6.

[27] Assume that $M$ is normal and let $A$ be any definable subset of $\delta
M$. For all $p\in[1,\infty)$ sufficiently large, we have:

  1. (i)

$\mathscr{C}^{\infty}(\overline{M})$ is dense in $W^{1,p}(M)$.

  2. (ii)

The linear operator

$\mathscr{C}^{\infty}(\overline{M})\ni\varphi\mapsto\varphi_{|A}\in
L^{p}(A,\mathcal{H}^{k}),\qquad k:=\dim A,$

is continuous in the norm $||\cdot||_{W^{1,p}(M)}$ and thus extends to a
mapping $\mathbf{tr}_{A}:W^{1,p}(M)\to L^{p}(A,\mathcal{H}^{k})$.

  3. (iii)

If $\mathscr{S}$ is a stratification of $A$, then
$\mathscr{C}^{\infty}_{\overline{M}\setminus\overline{A}}({\overline{M}})$ is
a dense subspace of $\bigcap\limits_{Y\in\mathscr{S}}\ker\mathbf{tr}_{Y}$.

Using a normalization, it is then not difficult extend the notion of trace to
the case of a non necessarily normal manifold (see [27] for details). Alike in
the previous section, this trace operator takes values in
$L^{p}(A,\mathcal{H}^{k})^{l},$ for some $l\geq 1$. We then have [27,
Corollary 3.7]:

######  Corollary 2.7.

Let $A\subset\delta M$ be a subanalytic set of dimension $k$. For
$p\in[1,\infty)$ sufficiently large, the linear operator

$\mathbf{tr}_{A}:W^{1,p}(M)\to L^{p}(A,\mathcal{H}^{k})^{l},$

is continuous.

Of course, since the trace is induced by restriction of functions to $\partial
M$, in the case $A=\partial M$, this operator coincides with the trace
operator defined in the previous section. We thus get that the mapping
$\mathbf{tr}_{\partial M}:W^{1,p}(M)\to L^{p}(\partial M)^{l}$ is continuous
for large values of $p\in[1,\infty)$. The situation is however here a bit more
complicated than on Lipschitz manifolds with boundary, as it is not true that
$\mathscr{C}_{0}^{\infty}(M)$ is dense in $\ker\mathbf{tr}_{\partial M}$,
i.e., as $W_{0}^{1,p}(M)$ can be strictly smaller than $W^{1,p}(M,\partial
M)$:

######  Example 2.8.

If $M$ is the open unit disk in $\mathbb{R}^{2}$ centered at $0$ from which
was taken off the center then every element of $\mathscr{C}_{0}^{\infty}(M)$
must vanish in the vicinity of the origin. For every sufficiently large value
of $p$, $\mathbf{tr}_{\\{0\\}}:W^{1,p}(M)\to L^{p}(M)$ is well-defined (by
Corollary 2.7), and the closure of $\mathscr{C}_{0}^{\infty}(M)$ is included
in the kernel of this linear mapping. Therefore, a smooth function $u$ which
is $1$ near the origin and that has support in the disk of radius
$\frac{1}{2}$ belongs to $W^{1,p}(M,\partial M)$ and not to $W_{0}^{1,p}(M)$.

We however have [27, Corollary $3.9$]:

######  Corollary 2.9.

If $\Sigma$ is a stratification of a dense subset of $\delta M$ then
$\mathscr{C}^{\infty}_{0}(M)$ is dense in
$\underset{S\in\Sigma}{\bigcap}\ker\mathbf{tr}_{S}$ for all $p\in[1,\infty)$
sufficiently large.

In the light of Example 2.8 and Corollary 2.9, we understand the only problem
for $\mathscr{C}_{0}^{\infty}(M)$ to be dense in $\ker\mathbf{tr}_{\partial
M}$ (for $p$ large): there might be points of $\delta M$ at which this set has
dimension less than $(m-1)$. This leads us to the following natural notion.

######  Definition 2.10.

We say that $M$ is fenced if $\delta M=\overline{\partial M}$.

When this occurs, by Corollary 2.9, we have for all $p$ sufficiently large:

(2.4) $W^{1,p}(M,\partial M)=W_{0}^{1,p}(M).$

######  Remark 2.11.

For an open subset ${\Omega}$ of $\mathbb{R}^{n}$, the set

(2.5) $\tilde{\Omega}:=\overline{\Omega}\setminus\overline{\partial{\Omega}}$

is an open bounded definable subset of $\mathbb{R}^{n}$ which is always
fenced. In addition, $\Omega$ is fenced if and only if
$\tilde{\Omega}=\Omega$. In particular,
$\tilde{\tilde{\Omega}}=\tilde{\Omega}$.

##  3\. Some preliminary local estimates

In this section, we will be working in the setting of section 1.3, with
$\varepsilon$, $r_{s}$, $M^{\eta}$, and $N^{\eta}$, as defined there (see
(1.3)).

###  3.1. The retraction $R_{t}^{\eta}$.

Let us remark that the definition of $r_{s}(x)$ that we gave in Theorem 1.6
actually makes sense for each $(s,x)\in[0,\infty)\times{M^{\varepsilon}}$
satisfying $s\leq\frac{\varepsilon}{|x|}$. We therefore can define a mapping
$R:P\to M$, where $P=\\{(t,x)\in\mathbb{R}\times M:1\leq
t\leq\frac{\varepsilon}{|x|}\\}$, by

$R(t,x):=H(tH^{-1}(x)).$

We will denote by $R^{\eta}:[1,\frac{\varepsilon}{\eta}]\times N^{\eta}\to M$
and $R^{\eta}_{t}:N^{\eta}\to N^{t\eta}$ the respective restrictions of $R$
and $R(t,\cdot)$.

We do not use the notation $r(s,x)$ when $s>1$, since the properties of $R$
will be very different from the properties of $r$ listed in Theorem 1.6. The
mapping $R$ should rather be regarded as an inverse mapping, as it indeed
directly follows from the definitions that the mapping
$R_{t}^{\eta}:N^{\eta}\to N^{t\eta}$, induced by $R$, is the inverse of the
mapping $r_{\frac{1}{t}}^{t\eta}:N^{t\eta}\to N^{\eta}$ induced by $r$.
Although $R$ is not Lipschitz, we have the following estimates of its first
derivative.

###  3.2. Some properties of $R_{t}^{\eta}$.

There is a constant $C$ such that for all $\eta\in(0,\varepsilon)$, we have
for all $(t,x)\in[1,\frac{\varepsilon}{\eta}]\times N^{\eta}$:

(3.1) $\mbox{jac}\,R_{t}^{\eta}(x)\geq\frac{t^{m-1}}{C},$

(3.2) $\left|\frac{\partial R^{\eta}}{\partial t}(t,x)\right|\leq C\eta.$

###### Proof.

Since for all $s\in(0,1)$ and $\eta\in(0,\varepsilon)$, the induced mapping
$r_{s}^{\eta}:N^{\eta}\to N^{s\eta}$ is $Cs$-Lipschitz for some constant $C$
independent of $s$ and $\eta$, we have

(3.3) $\mbox{jac}\,r_{s}^{\eta}\leq C^{m-1}s^{m-1}.$

Hence, for $t\in[1,\frac{\varepsilon}{\eta}]$, we can write for $x\in
N^{\eta}$:

$\mbox{jac}\,R^{\eta}_{t}(x)=\mbox{jac}\,(r_{\frac{1}{t}}^{t\eta})^{-1}(x)=\frac{1}{\mbox{jac}\,r_{\frac{1}{t}}^{t\eta}\big{(}(r_{\frac{1}{t}}^{t\eta})^{-1}(x)\big{)}}\overset{(\ref{eq_jac_1})}{\geq}\frac{t^{m-1}}{C^{m-1}},$

which yields (3.1). To show the second inequality, observe that, by definition
of $R$, we have $\frac{\partial R}{\partial
t}(t,x)=D_{tH^{-1}(x)}H(H^{-1}(x))$, where $H$ is provided by Theorem 1.6 (at
$x_{0}=0$). As a matter of fact, (3.2) immediately comes down from the facts
that $H^{-1}$ preserves the distance to the origin and $H$ has bounded
derivative. ∎

###  3.3. Local estimates of the $L^{p}$ norm.

In [27], local estimates were obtained in the case where $p$ is large by
integrating along the trajectories of $r_{s}$ (see (3.13) below). We are now
going to make use of the isotopy $R_{t}$ that we just introduced in a similar
way so as to get estimates that are valid for all $p$.

######  Lemma 3.1.

For every $p\in[1,\infty)$ there is a constant $C$ such that for all
$u\in\mathscr{C}^{1}(M)$ with $\mbox{\rm
supp}\,u\subset\mathbf{B}(0,\varepsilon)$ as well as $|\partial u|\in
L^{p}(M)$, and all $\eta\in(0,\varepsilon]$, we have:

  1. (i)

If $p<m$ then

$||u||_{L^{p}(N^{\eta})}\leq
C\eta^{\frac{p-1}{p}}\left(\eta^{\frac{m-p}{4p}}||\partial
u||_{L^{p}(M)}+||\partial u||_{L^{p}(M^{\eta^{1/2}})}\right).$

  2. (ii)

If $p=m$ then

$||u||_{L^{p}(N^{\eta})}\leq
C\cdot\eta^{\frac{m-1}{m}}\cdot\big{(}\ln\frac{1}{\eta}\big{)}^{\frac{m-1}{m}}\cdot||\partial
u||_{L^{p}(M)},$

  3. (iii)

If $p>m$ then

$||u||_{L^{p}(N^{\eta})}\leq C\cdot\eta^{\frac{m-1}{p}}\cdot||\partial
u||_{L^{p}(M)}.$

###### Proof.

We first focus on $(i)$. By the fundamental theorem of calculus, we have for
all $p\in[1,\infty)$, $u\in\mathscr{C}^{1}(M)$ with $\mbox{\rm
supp}\,u\subset\mathbf{B}(0,\varepsilon)$ and $|\partial u|\in L^{p}(M)$, as
well as $\eta\in(0,\varepsilon]$, and $x\in N^{\eta}$:

$u(x)=-\int_{1}^{\varepsilon/\eta}\frac{\partial(u\circ R^{\eta})}{\partial
t}(t,x)dt,$

from which it follows that:

(3.4) $\displaystyle||u||_{L^{p}(N^{\eta})}$ $\displaystyle\leq$
$\displaystyle\int_{1}^{\varepsilon/\eta}\left(\int_{N^{\eta}}\left|\frac{\partial(u\circ
R^{\eta})}{\partial t}\right|^{p}(t,x)dx\right)^{1/p}dt\qquad\mbox{(by
Minkowski's inequality)}$ $\displaystyle=$
$\displaystyle\int_{1}^{\varepsilon/\eta}t^{-\frac{l}{p}}\left(\int_{N^{\eta}}\left|\frac{\partial(u\circ
R^{\eta})}{\partial t}\right|^{p}(t,x)t^{l}dx\right)^{1/p}dt,\quad\mbox{with
$l:=\frac{m+p}{2}-1$},$ $\displaystyle\leq$
$\displaystyle\left(\int_{1}^{\varepsilon/\eta}t^{-\frac{lp^{\prime}}{p}}dt\right)^{1/p^{\prime}}\left(\int_{1}^{\varepsilon/\eta}\int_{N^{\eta}}\left|\frac{\partial(u\circ
R^{\eta})}{\partial t}\right|^{p}(t,x)t^{l}\,dx\,dt\right)^{1/p},$

by Hölder’s inequality (for $p>1$; if $p=1$ then simply notice that
$t^{-l/p}\leq 1$ since $l\geq 0$). As, for $p\in(1,m)$, we have

$\frac{lp^{\prime}}{p}=\frac{\frac{m+p}{2}-1}{p-1}>1,$

raising the preceding estimate to the power $p$, we get for $p\in[1,m)$:

(3.5) $\displaystyle||u||_{L^{p}(N^{\eta})}^{p}$ $\displaystyle\lesssim$
$\displaystyle\int^{\varepsilon/\eta}_{1}\int_{N^{\eta}}\left|\frac{\partial(u\circ
R^{\eta})}{\partial t}(t,x)\right|^{p}t^{l}dx\,dt$ $\displaystyle\leq$
$\displaystyle\int^{\varepsilon/\eta}_{1}\int_{N^{\eta}}|\partial
u(R^{\eta}_{t}(x))|^{p}\left|\frac{\partial R^{\eta}}{\partial
t}(t,x)\right|^{p}t^{l}dx\,dt$
$\displaystyle\overset{(\ref{eq_par})}{\lesssim}$
$\displaystyle\eta^{p}\int^{\varepsilon/\eta}_{1}t^{l}\int_{N^{\eta}}|\partial
u(R^{\eta}_{t}(x))|^{p}dx\,dt$
$\displaystyle\overset{(\ref{eq_jad_reta})}{\lesssim}$
$\displaystyle\eta^{p}\int^{\varepsilon/\eta}_{1}t^{l-m+1}\int_{N^{\eta}}|\partial
u(R^{\eta}_{t}(x))|^{p}\,\mbox{jac}\,R^{\eta}_{t}(x)\,dx\,dt$ $\displaystyle=$
$\displaystyle\eta^{p}\int^{\varepsilon/\eta}_{1}t^{l-m+1}||\partial
u||_{L^{p}(N^{t\eta})}^{p}dt.$

Observe now that $l-m+1=\frac{p-m}{2}$, which is negative for $p$ in $[1,m)$,
so that:

(3.6) $\int^{\eta^{-1/2}}_{1}t^{\frac{p-m}{2}}||\partial
u||_{L^{p}(N^{t\eta})}^{p}dt\leq\int^{\eta^{-1/2}}_{1}||\partial
u||_{L^{p}(N^{t\eta})}^{p}dt\overset{(\ref{eq_coarea_sph})}{\lesssim}\frac{1}{\eta}||\partial
u||_{L^{p}(M^{\eta^{1/2}})}^{p}\;.$

Moreover, if $t\geq\eta^{-1/2}$ then
$t^{\frac{p-m}{2}}\leq\eta^{\frac{m-p}{4}}$ (since $\frac{p-m}{2}<0$), which
entails that for $\eta^{-\frac{1}{2}}<\frac{\varepsilon}{\eta}$:

$\int^{\varepsilon/\eta}_{\eta^{-1/2}}t^{\frac{p-m}{2}}||\partial
u||_{L^{p}(N^{t\eta})}^{p}dt\leq\eta^{\frac{m-p}{4}}\int^{\varepsilon/\eta}_{\eta^{-1/2}}||\partial
u||_{L^{p}(N^{t\eta})}^{p}dt\overset{(\ref{eq_coarea_sph})}{\lesssim}\eta^{\frac{m-p}{4}-1}||\partial
u||_{L^{p}(M)}^{p}.$

Together with (3.5) and (3.6), this yields $(i)$. In order to show now
$(iii)$, observe first that for all $p>m$ we have:

(3.7)
$\left(\int_{1}^{\varepsilon/\eta}t^{-\frac{(m-1)p^{\prime}}{p}}dt\right)^{1/p^{\prime}}\lesssim\eta^{\frac{m}{p}-1},$

so that, by (3.4) for $l=m-1$, we get

$\displaystyle||u||_{L^{p}(N^{\eta})}^{p}\lesssim\eta^{m-p}\int^{\varepsilon/\eta}_{1}\int_{N^{\eta}}\left|\frac{\partial(u\circ
R^{\eta})}{\partial t}(t,x)\right|^{p}t^{l}dx\,dt.$

This double integral can be estimated by the computation carried out in (3.5)
(still for $l=m-1$). We thus obtain:

$\displaystyle||u||_{L^{p}(N^{\eta})}^{p}\overset{(\ref{eq_u_neta})}{\lesssim}\eta^{m-p}\cdot\eta^{p}\int^{\varepsilon/\eta}_{1}||\partial
u||_{L^{p}(N^{t\eta})}^{p}dt\overset{(\ref{eq_coarea_sph})}{\lesssim}\eta^{m-1}||\partial
u||_{L^{p}(M)}^{p},$

as required. To show $(ii)$, just replace (3.7) in the proof of $(iii)$
(assuming now $p=m$) with (for $m>1$):

$\left(\int_{1}^{\varepsilon/\eta}t^{-\frac{(m-1)p^{\prime}}{p}}dt\right)^{1/p^{\prime}}\lesssim\;\big{(}\ln\frac{1}{\eta}\big{)}^{\frac{m-1}{m}},$

and proceed in the same way (if $p=m=1$, simply notice that $t^{-(m-1)}\equiv
1$). ∎

######  Remark 3.2.

  1. (a)

In $(i)$ and $(iii)$ of the lemma, the constant $C$ can be bounded
independently of $p$ if this number stays bounded away from $m$.

  2. (b)

Compiling together $(i)$, $(ii)$, and $(iii)$ of the above lemma, we can get
that for any $p\in[1,\infty)$ we have for $u\in\mathscr{C}^{1}(M)$ satisfying
$\mbox{\rm supp}\,u\subset\mathbf{B}(0,\varepsilon)$ and $|\partial u|\in
L^{p}(M)$, as well as $\eta\leq\varepsilon$:

(3.8)
$||u||_{L^{p}(N^{\eta})}\lesssim\eta^{\frac{a-1}{p}}\cdot\big{(}\ln\frac{1}{\eta}\big{)}^{\frac{m-1}{m}}\cdot||\partial
u||_{L^{p}(M)},$

with $a:=\min(m,p)$.

######  Example 3.3.

The estimate provided by $(ii)$ of the latter lemma is sharp in the sense that
we cannot lower the exponents. Let $M:=\\{(x,y)\in\mathbb{R}^{2}:y>x>0\\}$ and
let $u(x,y):=\ln^{\alpha}\frac{1}{y}$. We have $\frac{\partial u}{\partial
y}(x,y)=\frac{-\alpha}{y\ln^{1-\alpha}\frac{1}{y}}$ so that $||\partial
u||_{W^{1,2}(N^{\eta})}^{2}\in L^{2}([0,\varepsilon])$ if and only if
$\alpha<\frac{1}{2}$, while for $\eta>0$

$\eta^{\frac{1}{2}}\ln^{\alpha}\frac{1}{\eta}\lesssim||u||_{L^{2}(N^{\eta})}.$

######  Lemma 3.4.

For every $p\in[1,\infty)$, there is a constant $C$ such that for any
distribution $u$ on $M$ satisfying $\mbox{\rm
supp}\,u\subset\mathbf{B}(0,\varepsilon)$ and $|\partial u|\in L^{p}(M)$, we
have for all $\eta\in(0,\varepsilon]$:

  1. (1)

If $p\in[1,m)$ then

(3.9) $||u||_{L^{p}(M^{\eta})}\leq
C\,\eta\left(\eta^{\frac{m-p}{4p}}||\partial u||_{L^{p}(M)}+||\partial
u||_{L^{p}(M^{\eta^{1/2}})}\right).$

  2. (2)

If $p=m$ then

(3.10) $||u||_{L^{p}(M^{\eta})}\leq
C\,\eta\cdot\big{(}\ln\frac{1}{\eta}\big{)}^{\frac{m-1}{m}}\cdot||\partial
u||_{L^{p}(M)},$

  3. (3)

If $p>m$ then

(3.11) $||u||_{L^{p}(M^{\eta})}\leq C\,\eta^{\frac{m}{p}}\cdot||\partial
u||_{L^{p}(M)}.$

###### Proof.

If $u$ is a $\mathscr{C}^{1}$ function, it suffices to integrate with respect
to $\eta$ the estimates of the preceding lemma raised to the power $p$, and to
apply (1.6).

To prove the result in the case where $u$ may fail to be smooth, note that
such a distribution $u$ is an $L^{p}_{loc}$ function [17, Ch. 1]. Using some
regularization operators on manifolds [21], we can construct a sequence
$u_{i}\in\mathscr{C}^{\infty}(M)$ such that $u_{i}$ tends to $u$ in the
$L^{p}_{loc}$ topology and $\partial u_{i}$ tends to $\partial u$ in the
$L^{p}$ norm (the $L^{p}$ convergence of the regularizations is proved in
[10]). The desired estimates then follow from Fatou Lemma. ∎

In the case where the function $u$ tends to zero at the origin and $p$ is
large, inequality (3.13) below will provide an estimate which is better than
(3.11). The advantage of (3.11) is on the other hand to be valid for all
$p>m$.

######  Remark 3.5.

As in Remark 3.2, compiling together these estimates, we get that for any
$p\in[1,\infty)$, we have for distributions $u$ on $M$ satisfying $\mbox{\rm
supp}\,u\subset\mathbf{B}(0,\varepsilon)$ and $|\partial u|\in L^{p}(M)$, and
$\eta\leq\varepsilon$:

(3.12)
$||u||_{L^{p}(M^{\eta})}\lesssim\eta^{\frac{a}{p}}\cdot\big{(}\ln\frac{1}{\eta}\big{)}^{\frac{m-1}{m}}\cdot||\partial
u||_{L^{p}(M)},$

where, as in the latter remark, $a:=\min(m,p)$.

We end this section with an estimate devoted to the large values of $p$
((3.15) below) that we shall need to establish the continuity of Morrey’s
embedding in section 6.

######  Lemma 3.6.

There is a constant $C$ such that for all $p\in[1,\infty)$ sufficiently large
and all $u\in W^{1,p}(M^{\varepsilon})$ satisfying
$\mathbf{tr}_{\\{0_{\mathbb{R}^{n}}\\}}u=0$ we have for almost every
$\eta<\varepsilon$:

(3.13)
$||u||_{L^{p}(N^{\eta})}\lesssim\eta^{\frac{p-1}{p}}\,||u||_{W^{1,p}(M^{\eta})}.$

###### Proof.

This inequality follows from $(2.8)$ of [27], where it is stated with the
operator $\Theta^{M}$ introduced there. If
$\mathbf{tr}_{\\{0_{\mathbb{R}^{n}}\\}}u=0$ then $u=\Theta^{M}u$ almost
everywhere, by Lemma $2.4$ of the latter article. ∎

Inequalities (3.13) and (1.6) yield that for all $p$ large enough, $u\in
W^{1,p}({M^{\varepsilon}})$ satisfying
$\mathbf{tr}_{\\{0_{\mathbb{R}^{n}}\\}}u=0$, and $\eta\in(0,\varepsilon)$, we
have:

(3.14) $||u||_{L^{p}(M^{\eta})}\lesssim\eta||u||_{W^{1,p}(M^{\eta})}.$

This estimate enables us to establish:

######  Lemma 3.7.

There is a constant $C$ such that for all $p$ sufficiently large and all $u\in
W^{1,p}({M^{\varepsilon}})$ satisfying
$\mathbf{tr}_{\\{0_{\mathbb{R}^{n}}\\}}u=0$, we have for all
$\eta\leq\varepsilon$ and all $s\in(0,1)$:

(3.15) $||u\circ r_{s}||_{W^{1,p}(M^{\eta})}\leq
Cs^{1-\frac{\nu}{p}}||u||_{W^{1,p}(M^{s\eta})},$

where $\nu$ is given by (1.4).

###### Proof.

For $\eta\in[0,\varepsilon]$ and almost every $s\in(0,1)$, we have for all
$u\in W^{1,p}({M^{\varepsilon}})$:

(3.16) $\displaystyle||\partial(u\circ r_{s})||_{L^{p}(M^{\eta})}^{p}$
$\displaystyle=$ $\displaystyle\int_{M^{\eta}}\left|{}^{\bf
t}D_{x}r_{s}\left(\partial
u(r_{s}(x))\right)\right|^{p}dx\lesssim\int_{M^{\eta}}\left|\partial
u(r_{s}(x))\right|^{p}s^{p}dx$
$\displaystyle\overset{(\ref{eq_jacr_s})}{\lesssim}$
$\displaystyle\int_{M^{\eta}}\left|\partial
u(r_{s}(x))\right|^{p}\cdot\mbox{jac}\,r_{s}^{\eta}(x)\cdot
s^{p-\nu}dx\lesssim s^{p-\nu}||\partial u||_{L^{p}(M^{s\eta})}^{p}.$

Moreover, whenever $u$ satisfies $\mathbf{tr}_{\\{0_{\mathbb{R}^{n}}\\}}u=0$,
we have for $p$ large:

$||u\circ
r_{s}||_{L^{p}(M^{\eta})}\overset{(\ref{lnm})}{\lesssim}s^{-\frac{\nu}{p}}||u||_{L^{p}(M^{s\eta})}\overset{(\ref{theta_contm})}{\lesssim}s^{1-\frac{\nu}{p}}\cdot\eta\cdot||u||_{W^{1,p}(M^{s\eta})}.$

Together with (3.16), this yields (3.15). ∎

##  4\. Density of smooth functions and duality between Sobolev spaces

We focus in this section on the “small” values of $p$ and partially generalize
the results of [27]. We then derive duality results between
$W^{1,p}({\Omega},\partial{\Omega})$ and $W^{-1,p^{\prime}}({\Omega})$.

###  4.1. Density of smooth functions.

We adopt the convention $\dim\emptyset=-\infty$.

######  Theorem 4.1.

Let $Z$ be a definable open subset of $\partial M$ and let $A$ be a definable
subset of $\delta M$ containing $\delta M\setminus\partial M$ and $\delta Z$,
with $\dim A\leq m-2$. If $M$ is connected along $\partial
M\setminus(Z\cup\overline{A})$ then for all $p\in[1,m-\dim A]$ not infinite,
$\mathscr{C}^{\infty}_{{\overline{M}}\setminus(Z\cup\overline{A})}({\overline{M}})$
is dense in $W^{1,p}(M,Z)$.

###### Proof.

If $A$ is empty then so is $\delta M\setminus\partial M$ and $Z$ is closed
(and open in $\delta M$ by assumption), which means that ${\overline{M}}$ is a
Lipschitz manifold with boundary $\partial M$ at each point of this set, and a
finite union of manifolds with boundary at each point of $Z$, in which cases
it is well-known that we can find such approximations (as we can use a
partition of unity, it is enough to prove that we can approximate the germ of
a function $u\in W^{1,p}(M,Z)$ at a point of $\delta M$).

We thus assume $A$ to be nonempty and fix a locally bi-Lipschitz trivial
stratification $\Sigma$ of $\overline{M}$ compatible with $\delta
M,Z,\overline{A}$, and $\partial M$. Given $k\in\\{0,\dots,c\\}$, where
$c:=\dim A$, let $A_{k}$ denote the reunion of all the strata
$S\subset\overline{A}$ of $\Sigma$ satisfying $\dim S\leq k$, and set
$A_{-1}:=\emptyset$. For $u\in W^{1,p}(M)$, let us define

(4.1) $\kappa_{u}:=\max\left\\{k\in\\{0,\dots,c+1\\}\,:A_{k-1}\cap\mbox{\rm
supp}_{\overline{M}}u=\emptyset\right\\}.$

We will prove that we can approximate such a function $u$ by decreasing
induction on $\kappa_{u}\in\\{0,\dots,c+1\\}$.

Assume first $\kappa_{u}=c+1$, take $u\in W^{1,p}(M,Z)$, and notice that in
this case, $\mbox{\rm supp}_{\overline{M}}u$ is a compact subset of
${\overline{M}}\setminus A_{c}$. It suffices to show that every point
${x_{0}}$ of $\overline{\mbox{\rm supp}u}\cap\delta
M\subset{\overline{M}}\setminus\overline{A}$ has a neighborhood on which $u$
admits approximations. If ${x_{0}}\in Z$, then ${\overline{M}}$ is, in the
vicinity of ${x_{0}}$, a finite union of Lipschitz manifolds with common
boundary $Z$, in which case, since $u$ has zero trace on $Z$, the result is
well-known [2]. If ${x_{0}}\in\delta M\setminus Z$, then as
${x_{0}}\notin\overline{A}$, which contains $\delta M\setminus\partial M$ and
$\delta Z$, we must have ${x_{0}}\in\partial
M\setminus(\overline{Z}\cup\overline{A})$, and our assumptions entail that $M$
is connected at ${x_{0}}$, which means that ${\overline{M}}$ is a Lipschitz
manifold with boundary at ${x_{0}}$, and that the result is again well-known.

To perform the induction step, we fix an integer $k\leq c$, assume that the
claimed statements hold for all $v\in W^{1,p}(M)$ satisfying $\kappa_{v}>k$,
and take a function $u\in W^{1,p}(M)$ with $\kappa_{u}\geq k$.

Every point $x_{0}\in\overline{M}$ admits a neighborhood $U_{x_{0}}$ in
$\overline{M}$ on which we have a bi-Lipschitz trivialization of $\Sigma$. As
we can use a partition of unity, we may assume that $\mbox{\rm
supp}_{\overline{M}}u$ fits in $U_{x_{0}}$, for some $x_{0}\in\overline{M}$.

Let $S$ denote the stratum of $\Sigma$ that contains the point $x_{0}$.
Observe that if $\dim S<\kappa_{u}$ then the function $u$ is zero near $S$, so
that in such a situation, choosing $U_{x_{0}}$ smaller if necessary, we can
require the function $u$ to be zero on $U_{x_{0}}$. We thus will suppose
$\kappa_{u}\leq\dim S$.

Moreover, as $\mbox{\rm supp}_{\overline{M}}u\subset U_{x_{0}}$ which does not
meet any stratum of dimension less than $\dim S$, we see that if $\dim S$ is
bigger than $k$ then $\kappa_{u}>k$, and the result follows from the induction
hypothesis. We thus can assume $\dim S\leq k$.

We conclude from the reductions of the two above paragraphs that we can assume
$\dim S\leq k\leq\kappa_{u}\leq\dim S$, which means that it is no loss of
generality to only deal with the case $\dim S=\kappa_{u}=k$. We will do the
proof for $x_{0}=0_{\mathbb{R}^{n}}\in\delta M$.

By definition of bi-Lipschitz triviality, there is a bi-Lipschitz
homeomorphism
$\Lambda:U_{0}\to(\pi^{-1}(0_{\mathbb{R}^{n}})\cap\overline{M})\times W_{0}$
(here $U_{0}=U_{0_{\mathbb{R}^{n}}}$), where $\pi:U_{0}\to S$ is a
$\mathscr{C}^{\infty}$ retraction and $W_{0}$ is a neighborhood of the origin
in $S$ (here $k$ may be $0$, in which case $\Lambda$ is the identity).

Let

(4.2) $F^{\eta}:=\pi^{-1}(0_{\mathbb{R}^{n}})\cap M^{\eta}\quad\mbox{and}\quad
V^{\eta}:=F^{\eta}\times W_{0}.$

We may assume the set $F^{\varepsilon}$, $\varepsilon>0$ small, to be a
$\mathscr{C}^{\infty}$ manifold of dimension $(m-k)\geq(m-c)$ (see Remark
$1.6$ of [27]). We postpone the case $p=m-c$, assuming for the moment $p<m-c$.

Thanks to (3.9) for the manifold $F^{\varepsilon}$, we thus see that for $v\in
W^{1,p}(F^{\varepsilon})$ with $\mbox{\rm
supp}\,v\subset\mathbf{B}(0,\varepsilon)$, $\eta\leq\varepsilon$, we have (as
$p\in[1,m-c)\subset[1,m-k)$):

$||v||_{L^{p}(F^{\eta})}\lesssim\eta\left(\eta^{\frac{m-k-p}{4p}}||v||_{W^{1,p}(F^{\varepsilon})}+||v||_{W^{1,p}(F^{\eta^{1/2}})}\right).$

For simplicity, we will identify $U_{0}\cap M$ with its image under $\Lambda$
(which is a bi-Lipschitz homeomorphism) and work as if $u$ were a function on
this set. Given $y\in W_{0}$ and $x\in F^{\varepsilon}$, set
$v^{y}(x):=u(x,y)$. Applying the just above inequality to $v^{y}$, raising it
to the power $p$, and integrating with respect to $y\in W_{0}$, we easily
derive that for $\eta\leq\varepsilon$ and $p\in[1,m-c)$ (assuming $u\in
W^{1,p}(V^{\varepsilon})$ and $\overline{\mbox{\rm
supp}\,u}\subset\mathbf{B}(0,\varepsilon)\times W_{0}$):

(4.3)
$||u||_{L^{p}(V^{\eta})}\lesssim\eta\left(\eta^{\frac{m-k-p}{4p}}||u||_{W^{1,p}(V^{\varepsilon})}+||u||_{W^{1,p}(V^{\eta^{1/2}})}\right).$

Let now $\psi:\mathbb{R}\to[0,1]$ be a $\mathscr{C}^{\infty}$ function such
that $\psi\equiv 0$ on $(-\infty,\frac{1}{2})$ and $\psi\equiv 1$ near
$[1,+\infty)$, and set for $\eta$ positive and $(x,y)\in V^{\varepsilon}$,

(4.4) $\psi_{\eta}(x,y):=\psi\left(\frac{|x|^{2}}{\eta^{2}}\right).$

We claim that $v_{\eta}:=\psi_{\eta}u$ tends to $u$ in $W^{1,p}(M)$ as
$\eta\to 0$. Clearly, $||v_{\eta}-u||_{L^{p}(M)}$ and
$||(\psi_{\eta}-1)\partial u||_{L^{p}(M)}$ both tend to zero as $\eta$ tends
to zero. Consequently, if we write:

(4.5) $||\partial(\psi_{\eta}u)-\partial
u||_{L^{p}(M)}\leq||u\,\partial\psi_{\eta}||_{L^{p}(M)}+||(\psi_{\eta}-1)\,\partial
u||_{L^{p}(M)},$

we see that it suffices to check that $||u\,\partial\psi_{\eta}||_{L^{p}(M)}$
tends to zero. But since $\sup|\partial\psi_{\eta}|\lesssim\frac{1}{\eta}$ and
$\mbox{\rm supp}\,|\partial\psi_{\eta}|\subset V^{\eta}$, we have:

(4.6)
$||u\,\partial\psi_{\eta}||_{L^{p}(M)}\lesssim\;\frac{||u||_{L^{p}(V^{\eta})}}{\eta},$

which, due to (4.3), must tend to zero as $\eta\to 0$, which yields our claim.
As $\kappa_{v_{\eta}}>k$, we now see that the result (in the case $p<m-c$)
follows from our induction hypothesis (since $\mathbf{tr}_{\partial
M}v_{\eta}=\psi_{\eta}\cdot\mathbf{tr}_{\partial M}u$).

It therefore only remains to carry out the induction step in the case $p=m-c$
(assuming $A\neq\emptyset$, since otherwise $p=+\infty$ which is excluded in
the statement of the theorem). We will rely on the same argument, just
replacing the just above function $\psi_{\eta}$ with an adequate function
$\psi^{\prime}_{\eta}$, and the proof that we are going to perform actually
goes for each $p\in(1,m-c]$.

We thus first construct, for every $\eta>0$ small, a function
$\psi^{\prime}_{\eta}\in W^{1,\infty}(\mathbb{R}^{n})$ such that

  1. (1)

$\psi_{\eta}^{\prime}\equiv 0$ near the origin and $\psi_{\eta}^{\prime}\equiv
1$ on $\mathbb{R}^{n}\setminus\mathbf{B}(0,\eta)$.

  2. (2)

For almost all $\zeta\in(0,\eta]$

(4.7)
$||\partial\psi^{\prime}_{\eta}||_{L^{\infty}(\mathbf{S}(0,\zeta))}\leq\frac{1}{\zeta\cdot\ln\frac{1}{\zeta}\cdot\ln\left(\ln\frac{1}{\zeta}\right)}.$

Let for this purpose, given $\zeta\in(0,\frac{1}{e})$,

$g(\zeta):=\ln\left(\ln\left(\ln\frac{1}{\zeta}\right)\right).$

As $g$ is a smooth function that tends to infinity at $0$, for every
$\eta\in(0,\frac{1}{e})$ there is $a_{\eta}<\eta$ such that
$g(a_{\eta})=1+g(\eta)$. Let then

$\phi_{\eta}(\zeta)=\begin{cases}1\quad&\mbox{ if }\;\zeta\geq\eta\\\
1+g(\eta)-g(\zeta)\quad&\mbox{ if }\;a_{\eta}\leq\zeta\leq\eta\\\
0\quad&\mbox{ if }\;\zeta\leq a_{\eta}\;,\end{cases}$

and finally set $\psi^{\prime}_{\eta}(x):=\phi_{\eta}(|x|)$.

The function $\psi^{\prime}_{\eta}$ is continuous, piecewise smooth, and
constant on every $\mathbf{S}(0,\zeta)$, $\zeta>0$. It is not difficult to
check that $\psi_{\eta}^{\prime}\in W^{1,\infty}(\mathbb{R}^{n})$ and that for
$a_{\eta}<|x|<\eta$ we have

(4.8)
$\partial\psi^{\prime}_{\eta}(x)=\phi^{\prime}_{\eta}(|x|)\cdot\frac{x}{|x|}=g^{\prime}(|x|)\cdot\frac{x}{|x|}=\frac{-1}{|x|\cdot\ln\frac{1}{|x|}\cdot\ln\left(\ln\frac{1}{|x|}\right)}\cdot\frac{x}{|x|},$

which clearly yields (4.7).

We are now ready to perform the proof in the case $p=m-c$. Let

$G_{y}^{\eta}:=G^{\eta}\times\\{y\\}\quad\mbox{and}\quad
F_{y}^{\eta}:=F^{\eta}\times\\{y\\}.$

Thanks to $(i)$ and $(ii)$ of Lemma 3.1 for the manifold $F^{\varepsilon}$, we
see that for almost every $y\in W_{0}$ (as $\mbox{\rm supp}\,u\subset
V^{\varepsilon}$), we have (as $p=m-c\leq m-k$) for almost every
$\zeta\leq\varepsilon$:

(4.9)
$||u||_{L^{p}(G_{y}^{\zeta})}^{p}\lesssim\zeta^{p-1}\cdot\ln^{p-1}\zeta\cdot||u||_{W^{1,p}(F_{y}^{\varepsilon})}^{p}.$

We claim that the function $v_{\eta}^{\prime}:=\psi^{\prime}_{\eta}u$ tends to
$u$ in $W^{1,p}(F^{\varepsilon}\times W_{0})$ as $\eta\to 0$, where we regard
the above function $\psi^{\prime}_{\eta}:\mathbb{R}^{n}\to\mathbb{R}$ as a
function on $F^{\varepsilon}\times W_{0}$, constant with respect to the second
variable. As in the case $p<m-c$, it suffices to check that
$||u\,\partial\psi_{\eta}^{\prime}||_{L^{p}(M)}$ tends to zero. For almost
every $y\in W_{0}$, by (4.7) and (4.9), we have for $\eta>0$ small

(4.10)
$||u\partial\psi^{\prime}_{\eta}||_{L^{p}(G_{y}^{\zeta})}^{p}\lesssim\frac{\zeta^{p-1}\cdot\ln^{p-1}\zeta}{\zeta^{p}\cdot\ln^{p}\zeta\cdot\ln^{p}\left(\ln\frac{1}{\zeta}\right)}\cdot||u||_{W^{1,p}(F_{y}^{\varepsilon})}^{p}=\frac{1}{\zeta\cdot\ln\zeta\cdot\ln^{p}\left(\ln\frac{1}{\zeta}\right)}\cdot||u||_{W^{1,p}(F_{y}^{\varepsilon})}^{p}$

so that, since $\mbox{\rm supp}\,|\partial\psi^{\prime}_{\eta}|\subset
F^{\eta}\times W_{0}$, we then can write

$\displaystyle||u\partial\psi^{\prime}_{\eta}||_{L^{p}(F_{y}^{\varepsilon})}^{p}$
$\displaystyle\overset{(\ref{eq_coarea_sph})}{\lesssim}$
$\displaystyle\int_{0}^{\eta}||u\partial\psi^{\prime}_{\eta}||_{L^{p}(G_{y}^{\zeta})}^{p}d\zeta$
$\displaystyle\overset{(\ref{eq_u_pa_psi_eta_N})}{\lesssim}$
$\displaystyle\int_{0}^{\eta}\frac{d\zeta}{\zeta\cdot\ln\zeta\cdot\ln^{p}\left(\ln\frac{1}{\zeta}\right)}\cdot||u||_{W^{1,p}(F_{y}^{\varepsilon})}^{p}$
$\displaystyle\lesssim$
$\displaystyle\ln^{1-p}\left(\ln\frac{1}{\eta}\right)\cdot||u||_{W^{1,p}(F_{y}^{\varepsilon})}^{p}\;.$

As the constant of this estimate is independent of $y$, we get after
integrating with respect to $y\in W_{0}$ for $\eta>0$ small

$||u\partial\psi^{\prime}_{\eta}||_{L^{p}(M)}^{p}\lesssim\ln^{1-p}\left(\ln\frac{1}{\eta}\right)\cdot||u||_{W^{1,p}(F^{\varepsilon}\times
W_{0})}^{p},$

which tends to zero as $\eta$ tends to zero (as $p>1$), yielding our claim.

Consequently, as $\kappa_{v^{\prime}_{\eta}}>k$, we can see that the result
follows from our induction hypothesis (since $\mathbf{tr}_{\partial
M}v^{\prime}_{\eta}=\psi^{\prime}_{\eta}\cdot\mathbf{tr}_{\partial M}u$). ∎

Let now

$\mathbf{p_{{}_{M}}}:=m-\dim\,(\delta M\setminus\partial
M)\overset{(\ref{eq_pa1})}{\geq}2.$

with, as above, the convention $\dim\emptyset=-\infty$.

Observe now that, thanks to (1.2), Theorem 4.1 gives in the case $A=\delta
M\setminus\partial M$ and $Z=\partial M$ or $\emptyset$:

######  Corollary 4.2.

For all $p\in[1,\mathbf{p_{{}_{M}}}]$ (not infinite), the following assertions
are true:

  1. (i)

$\mathscr{C}_{0}^{\infty}(M)$ is dense in $W^{1,p}(M,\partial M)$, which
amounts to

$W^{1,p}(M,\partial M)=W^{1,p}_{0}(M).$

  2. (ii)

If $M$ is weakly normal then $\mathscr{C}^{\infty}_{M\cup\partial
M}({\overline{M}})$ is dense in $W^{1,p}(M)$.

######  Remark 4.3.

It is easy to see that the conclusion of (ii) fails as soon as $M$ is not
weakly normal (see the proof of Corollary $3.10$ of [27]).

We now are going to give consequences of these results in the case where the
underlying manifold is an open set $\Omega$.

######  Remark 4.4.

We recall that
$\tilde{\Omega}=\overline{{\Omega}}\setminus\overline{\partial{\Omega}}$ (see
Remark 2.11). Since $\dim\tilde{\Omega}\setminus{\Omega}<n-1$, this set is
“removable”, i.e. $W^{1,p}({\Omega})=W^{1,p}(\tilde{\Omega})$ for all
$p\in[1,\infty)$ (see for instance [17, section 1.1.18]). As the inclusion map
$W^{1,p}({\Omega})\hookrightarrow W^{1,p}(\tilde{\Omega})$ identifies the
respective kernels of the trace operators, we deduce that:

$W^{1,p}({\Omega},\partial{\Omega})=W^{1,p}(\tilde{\Omega},\partial\tilde{\Omega}).$

A proof of these facts (for all $p\in[1,\infty)$) can also be carried out by
applying Theorem 4.1 (with $p=1$) to a $\mathscr{C}^{\infty}$ normalization of
${\Omega}$. Note also that Corollary 4.2 and (2.4) then entail that we have
for all $p$ in $[1,\mathbf{p_{{}_{\Omega}}}]$ or sufficiently large (not
infinite):

$W^{1,p}({\Omega},\partial{\Omega})=W_{0}^{1,p}(\tilde{\Omega}).$

This remark, and Corollary 4.2 $(\ref{item_avide_ker})$, yield:

######  Proposition 4.5.

  1. (1)

For each $p\in[1,\mathbf{p_{{}_{\Omega}}}]$ or sufficiently large (not
infinite), if $u\in W^{1,p}({\Omega},\partial{\Omega})$ then $\overline{u}$,
defined as $\overline{u}(x)=u(x)$ if $x\in{\Omega}$ and $\overline{u}(x)=0$
otherwise, belongs to $W^{1,p}(\mathbb{R}^{n})$.

  2. (2)

For any $p\in[1,\mathbf{p_{{}_{\Omega}}}]$ (not infinite), every distribution
$T:\mathscr{C}_{0}^{\infty}({\Omega})\to\mathbb{R}$ which is continuous in the
norm $||\cdot||_{W^{1,p}({\Omega})}$ uniquely extends continuously (in this
norm) to $\mathscr{C}_{0}^{\infty}(\tilde{\Omega})$.

######  Example 4.6.

Proposition 4.5 $(\ref{item_ext_dist})$ is not true for $p$ large. For
instance, if $\Omega$ coincides with the manifold $M$ introduced in Example
2.8 then the Dirac measure concentrated at the origin is bounded for
$||\cdot||_{W^{1,p}({\Omega})}$ for $p$ large, is zero on
$\mathscr{C}_{0}^{\infty}({\Omega})$, but nonzero on
$\mathscr{C}_{0}^{\infty}(\tilde{\Omega})$.

######  Proposition 4.7.

For all $p$ in $[1,\mathbf{p_{{}_{\Omega}}}]$ or sufficiently large (not
infinite), we have for each function $u\in W^{1,p}({\Omega},\partial{\Omega})$
and $v\in W^{1,p^{\prime}}(\Omega)$, and all $i\in\\{1,\dots,n\\}$:

(4.11) $<u,\partial_{i}v>=-<\partial_{i}u,v>.$

###### Proof.

In the case where $p\in[1,\mathbf{p_{{}_{\Omega}}}]$, by Corollary 4.2
$(\ref{item_avide_ker})$, we see that it indeed suffices to prove this
identity for $u\in\mathscr{C}^{\infty}_{0}(\Omega)$ and
$v\in\mathscr{C}^{\infty}(\Omega)$, for which the result follows after
integration by parts.

In the case where $p$ is large, by Remark 4.4, we can assume that
$\Omega=\tilde{\Omega}$, or equivalently (see Remark 2.11), that $\Omega$ is
fenced, so that thanks to (2.4), we can also suppose that
$u\in\mathscr{C}^{\infty}_{0}(\Omega)$ and $v\in\mathscr{C}^{\infty}(\Omega)$,
and the result can again be established by integrating by parts. ∎

######  Remark 4.8.

It is possible to prove that this proposition actually also holds in the case
$p=\infty$.

###  4.2. Duality in Sobolev spaces.

As usual, we let for $p\in[1,\infty]$:

$W^{-1,p}(\Omega):=\\{T\in\mathscr{C}_{0}^{\infty}({\Omega})^{\prime}:\exists\,f_{0},\dots,f_{n}\in
L^{p}(\Omega),\;\;T=\sum_{i=0}^{n}\partial_{i}f_{i}\\},$

that we endow with the norms:

$||T||_{W^{-1,p}(\Omega)}:=\inf\left(\sum_{i=0}^{n}||f_{i}||_{L^{p}(\Omega)}^{p}\right)^{1/p},\quad\mbox{for
each }p\in[1,\infty),$

and

$||T||_{W^{-1,\infty}(\Omega)}:=\inf\left(\max_{i\in\\{0,\dots,n\\}}||f_{i}||_{L^{\infty}(\Omega)}\right),\quad\mbox{for
}p=\infty,$

where the infimum is taken among all the $f_{0},\dots,f_{n}$ in
$L^{p}(\Omega)$ satisfying $T=\sum_{i=0}^{n}\partial_{i}f_{i}$.

Observe that, due to Proposition 4.5 (2), we have for all
$p\in[1,\mathbf{p_{{}_{\Omega}}}]$ (not infinite):

(4.12) $W^{-1,p^{\prime}}(\tilde{\Omega})=W^{-1,p^{\prime}}({\Omega}).$

These two spaces can however be different for $p$ large (see Example 4.6).

######  Remark 4.9.

For every $p\in(1,\infty)$, the linear mapping
$\mathbf{A}_{p}:W^{1,p}({\Omega})\to L^{p}({\Omega})^{n+1}$ defined as
$\mathbf{A}_{p}(u):=(u,\partial u)$, is continuous, injective, and has closed
image. Its dual mapping
$\mathbf{A}_{p}^{\prime}:L^{p^{\prime}}({\Omega})^{n+1}\to
W^{1,p}({\Omega})^{\prime}$ is therefore onto. As a matter of fact, every
continuous linear functional $T:W^{1,p}({\Omega})\to\mathbb{R}$ can be written
$\alpha+\partial^{*}\beta$ for some $\alpha\in L^{p^{\prime}}({\Omega})$ and
$\beta\in L^{p^{\prime}}({\Omega})^{n}$.

### The morphisms $\Lambda_{p}$ and $\Lambda^{\prime}_{p}$.

As well-known, since the elements of $W^{-1,p^{\prime}}({\Omega})$ are by
definition continuous functionals on the space
$(\mathscr{C}_{0}^{\infty}({\Omega}),||\cdot||_{W^{1,p}({\Omega})})$, which is
dense in $W_{0}^{1,p}(\Omega)$, we can define two mappings

$\Lambda_{p}:W^{1,p}_{0}(\Omega)\to
W^{-1,p^{\prime}}({\Omega})^{\prime}\quad\mbox{and}\quad\Lambda_{p}^{\prime}:W^{-1,p^{\prime}}({\Omega})\to
W^{1,p}_{0}(\Omega)^{\prime}$

by setting for $T\in W^{-1,p^{\prime}}(\Omega)$ and $u\in
W^{1,p}_{0}(\Omega)$:

$\Lambda_{p}u(T)=\Lambda^{\prime}_{p}T(u)=T(u).$

The following lemma provides some well-known facts that are not specific to
subanalytic domains, and that will be of service to derive our duality result.

######  Lemma 4.10.

For all $p\in(1,\infty)$, $\Lambda_{p}$ and $\Lambda^{\prime}_{p}$ are
isomorphisms, and we have

(4.13) $W^{1,p}_{0}({\Omega})=\\{u\in W^{1,p}({\Omega}):\forall\,\beta\in
W^{1,p^{\prime}}_{\nabla}({\Omega}),\;\;<u,\nabla\beta>=-<\partial
u,\beta>\\},$

where $W^{1,p^{\prime}}_{\nabla}({\Omega}):=\\{\beta\in
L^{p^{\prime}}({\Omega})^{n}:\nabla u\in L^{p^{\prime}}({\Omega})\\}$,
$\nabla$ denoting the divergence operator.

###### Proof.

As $\Lambda_{p}$ and $\Lambda^{\prime}_{p}$ are clearly injective, it suffices
to show that they are onto. The ontoness of $\Lambda^{\prime}_{p}$ is a direct
consequence of Remark 4.9.

Let us show (4.13). Let $Z_{p}$ denote the set which stands on the right-hand-
side of this equality. As we clearly have $W^{1,p}_{0}({\Omega})\subset
Z_{p}$, let us focus on the proof of the reversed inclusion, for which we will
show that every continuous linear functional
$T:W^{1,p}({\Omega})\to\mathbb{R}$ that vanishes on
$\mathscr{C}_{0}^{\infty}({\Omega})$ is identically zero on $Z_{p}$. By Remark
4.9, such a functional $T$ can be written $\alpha+\partial^{*}\beta$ for some
$\alpha\in L^{p^{\prime}}({\Omega})$ and $\beta\in
L^{p^{\prime}}({\Omega})^{n}$. As for every
$\varphi\in\mathscr{C}_{0}^{\infty}({\Omega})$, we have
$<\alpha,\varphi>+<\beta,\partial\varphi>=0$, we see that $\alpha=\nabla\beta$
in the sense of distribution, which means that $\beta\in
W^{1,p^{\prime}}_{\nabla}({\Omega})$, so that

$T(u)=<\nabla\beta,u>+<\beta,\partial u>=0,$

for every $u\in Z_{p}$, as required.

It remains to check that $\Lambda_{p}$ is onto. Let
$\mathbf{B}_{p}:L^{p^{\prime}}({\Omega})^{n+1}\to W^{-1,p^{\prime}}({\Omega})$
be the continuous linear mapping defined by
$\mathbf{B}_{p}(\alpha,\beta):=\alpha+\partial^{*}\beta$ for $\alpha\in
L^{p^{\prime}}({\Omega})$ and $\beta\in L^{p^{\prime}}({\Omega})^{n}$. Take
$T\in W^{-1,p^{\prime}}({\Omega})^{\prime}$ and observe that, as the pull-back
$\mathbf{B}_{p}^{*}T$ is a continuous functional on
$L^{p^{\prime}}({\Omega})^{n+1}$, there are $u\in L^{p}({\Omega})$ and $v\in
L^{p}({\Omega})^{n}$ such that

(4.14) $\mathbf{B}_{p}^{*}T(\alpha,\beta)=<u,\alpha>+<v,\beta>,$

for all $\alpha\in L^{p^{\prime}}({\Omega})$ and $\beta\in
L^{p^{\prime}}({\Omega})^{n}$. Since $\mathbf{B}_{p}^{*}T$ vanishes on
$\ker\mathbf{B}_{p}$, we deduce that $v=\partial u$ in the distribution sense,
which means that $u\in W^{1,p}({\Omega})$. Moreover, we also see that for all
$\beta\in W^{1,p^{\prime}}_{\nabla}({\Omega})$ we have (as
$(\nabla\beta,\beta)\in\ker\mathbf{B}_{p}$)

$<u,\nabla\beta>+<\partial u,\beta>=\mathbf{B}_{p}^{*}T(\nabla\beta,\beta)=0,$

which yields that $u\in Z_{p}\overset{(\ref{eq_w0})}{=}W^{1,p}_{0}({\Omega})$.
Note that, since $v=\partial u$, (4.14) establishes that
$\mathbf{B}_{p}^{*}T(\alpha,\beta)=\mathbf{B}_{p}^{*}\Lambda_{p}u(\alpha,\beta)$
for all $\alpha\in L^{p^{\prime}}({\Omega})$ and $\beta\in
L^{p^{\prime}}({\Omega})^{n}$. Since $\mathbf{B}_{p}$ is surjective, this
shows that $T=\Lambda_{p}u$, yielding that $\Lambda_{p}$ is onto. ∎

This leads us to the following result which naturally extends the duality that
holds on Lipschitz domains.

######  Proposition 4.11.

For every $p\in(1,\mathbf{p_{{}_{\Omega}}}]$ (not infinite), $\Lambda_{p}$ and
$\Lambda^{\prime}_{p}$ induce isomorphisms

$W^{1,p}({\Omega},\partial{\Omega})\simeq
W^{-1,p^{\prime}}({\Omega})^{\prime}\quad\mbox{and}\quad
W^{-1,p^{\prime}}({\Omega})\simeq
W^{1,p}({\Omega},\partial{\Omega})^{\prime}.$

If ${\Omega}$ is fenced, this remains true for each $p\in[1,\infty)$
sufficiently large.

###### Proof.

In virtue of Lemma 4.10, $\Lambda_{p}$ and $\Lambda_{p}^{\prime}$ are
isomorphisms. The first sentence thus follows from $(\ref{item_avide_ker})$ of
Corollary 4.2, and the second one is a consequence of (2.4). ∎

######  Remark 4.12.

The assumption “fenced” in the last sentence of this proposition can be
avoided by applying this proposition to $\tilde{\Omega}$. Indeed, it is worthy
of notice that thanks to Remark 4.4, we then get for all
$p\in(1,\mathbf{p_{{}_{\Omega}}}]$ or sufficiently large (not infinite):

$W^{1,p}({\Omega},\partial{\Omega})\simeq
W^{-1,p^{\prime}}(\tilde{\Omega})^{\prime}\quad\mbox{and}\quad
W^{-1,p^{\prime}}(\tilde{\Omega})\simeq
W^{1,p}({\Omega},\partial{\Omega})^{\prime}.$

###  4.3. Applications the Dirichlet problem.

As promised, we end this article with an application of the material that we
have developed to the study of the Dirichlet problem associated with the
Laplace equation.

We denote by $\Delta:W^{1,2}({\Omega})\to W^{-1,2}({\Omega})$ the linear
mapping induced by the Laplace operator $\sum_{i=1}^{n}\partial_{i}^{2}$.

######  Theorem 4.13.

For every $f\in W^{-1,2}({\Omega})$, the equation

$\begin{cases}\Delta u=f\qquad\mbox{on }{\Omega},\\\ u=0\qquad\mbox{on
}\partial{\Omega}\end{cases}$

has a unique (weak) solution in $W^{1,2}({\Omega})$.

###### Proof.

Since $\mathbf{p_{{}_{\Omega}}}\geq 2$, Corollary 4.2 $(i)$ yields that $f$
induces a functional on $W^{1,2}({\Omega},\partial{\Omega})$. Observe in
addition that the density of $\mathscr{C}_{0}^{\infty}({\Omega})$ in
$W^{1,2}({\Omega},\partial{\Omega})$ also entails that
$||u||_{L^{2}({\Omega})}\lesssim||\partial u||_{L^{2}({\Omega})}$ for $u\in
W^{1,2}({\Omega},\partial{\Omega})$ (one could also make use here of
Proposition 5.3 below or of the Poincaré inequality proved in [24]), which, by
Lax-Milgram’s Theorem, establishes existence and uniqueness of the solution. ∎

######  Remark 4.14.

The problem of the continuity of the solution at boundary points is
investigated in [13]. Theorem 6.2 below is likely to bring new information on
this issue.

##  5\. Sobolev embeddings and Poincaré inequality

In this section, we derive some inequalities that come down from the local
estimates of section 3.3. We start with a generalization of Sobolev’s
Embedding Theorem:

######  Theorem 5.1.

There is a positive real number $\mu$ such that for all $p$ and $q$ in
$[1,\infty)$ with $p\leq q\leq p+\mu$, there are $\alpha_{p,q}\in[0,1)$ and
$C>0$ such that for all $u\in W^{1,p}(M)$ we have:

(5.1) $||u||_{L^{q}(M)}\leq
C||u||_{L^{p}(M)}^{1-\alpha_{p,q}}\cdot||u||_{W^{1,p}(M)}^{\alpha_{p,q}}.$

Moreover, the embedding $W^{1,p}(M)\hookrightarrow L^{q}(M)$ is compact for
all such $p$ and $q$, and we have:

(5.2) $\alpha_{p,q}\leq c|p-q|.$

for some constant $c$ independent of $p$ and $q$.

###### Proof.

As $\mathscr{C}^{\infty}(M)$ is dense in $W^{1,p}(M)$, it suffices show (5.1)
for a $\mathscr{C}^{\infty}$ function $u$. The compactness of this embedding
will be established afterwards. We argue by induction on $m$, the case $m=0$
being vacuous.

Since we can use a partition of unity, it is enough to prove that every point
$x_{0}\in\delta M$ has a neighborhood $U_{x_{0}}$ such that $(\ref{ineq})$
holds for every function $u\in W^{1,p}(M)$ satisfying $\mbox{\rm
supp}\,u\subset U_{x_{0}}$, and we can assume $x_{0}=0$.

Let $r_{s}$, $\varepsilon$, $M^{\eta}$, and $N^{\eta}$ (for
$\eta\leq\varepsilon$), be as in section 1.3, and apply the induction
hypothesis to $N^{\varepsilon}$. This provides a constant $\tilde{\mu}$ such
that for every $p$ and $q$ satisfying $p\leq q\leq p+\tilde{\mu}$, there is
$\tilde{\alpha}_{p,q}\in[0,1)$ such that for all $v\in
W^{1,p}({N^{\varepsilon}})$:

(5.3)
$||v||_{L^{q}({N^{\varepsilon}})}\lesssim||v||_{L^{p}({N^{\varepsilon}})}^{1-\tilde{\alpha}_{p,q}}\cdot||v||_{W^{1,p}({N^{\varepsilon}})}^{\tilde{\alpha}_{p,q}}.$

Fix $p\in[1,\infty)$ and set $\tilde{p}:=p+\tilde{\mu}$, as well as
$\beta:={\tilde{\alpha}}_{p,\tilde{p}}$. Observe now that, as $r$ is
Lipschitz, we have for $u\in\mathscr{C}^{\infty}(M)$ and $s\in(0,1)$:

$\displaystyle||u||_{L^{\tilde{p}}(N^{s\varepsilon})}\lesssim||u\circ
r_{s}||_{L^{\tilde{p}}(N^{\varepsilon})}\overset{(\ref{ineq_hr})}{\lesssim}||u\circ
r_{s}||_{L^{p}(N^{\varepsilon})}^{1-\beta}||u\circ
r_{s}||_{W^{1,p}(N^{\varepsilon})}^{\beta},$

which by (1.7) (applied to both $u\circ r_{s}$ and $|\partial u|\circ r_{s}$,
as $|\partial(u\circ r_{s})|\lesssim|\partial u|\circ r_{s}$) gives

(5.4) $||u||_{L^{\tilde{p}}(N^{s\varepsilon})}\lesssim
s^{-\frac{\nu}{p}}\cdot||u||_{L^{p}(N^{s\varepsilon})}^{1-\beta}\cdot||u||_{W^{1,p}(N^{s\varepsilon})}^{\beta}.$

Set now $\mu:=\frac{\tilde{\mu}}{(\tilde{\mu}+1)(2\nu+\tilde{\mu}+\beta)}$,
fix $q\in(p,p+\mu]$, and set

(5.5)
$\theta:=\frac{\tilde{p}(q-p)}{q(\tilde{p}-p)}=\frac{\tilde{p}(q-p)}{q\tilde{\mu}},$

in order to have $\frac{1}{q}=\frac{\theta}{\tilde{p}}+\frac{(1-\theta)}{p}$.
Note that as $(q-p)\leq\mu$ and $\frac{\tilde{p}}{q}<\tilde{\mu}+1$, it
follows from the definitions of $\mu$ and $\theta$ that

(5.6) $\theta<\frac{1}{2\nu+\tilde{\mu}+\beta}.$

As $\frac{1}{q}=\frac{\theta}{\tilde{p}}+\frac{(1-\theta)}{p}$, by Hölder’s
inequality (0.1), we have for all $u\in\mathscr{C}^{\infty}(M)$ and all
$s\in(0,1)$

$||u||_{L^{q}(N^{s\varepsilon})}\leq||u||_{L^{p}(N^{s\varepsilon})}^{(1-\theta)}\cdot||u||_{L^{\tilde{p}}(N^{s\varepsilon})}^{\theta}\;,$

so that, substituting (5.4) in the right-hand-side, we get:

$\displaystyle||u||_{L^{q}(N^{s\varepsilon})}$ $\displaystyle\lesssim$
$\displaystyle||u||_{L^{p}(N^{s\varepsilon})}^{(1-\theta)}\cdot\left(s^{-\frac{\nu}{p}}\cdot||u||_{L^{p}(N^{s\varepsilon})}^{1-\beta}\cdot||u||_{W^{1,p}(N^{s\varepsilon})}^{\beta}\right)^{\theta}$
$\displaystyle=$ $\displaystyle
s^{-\frac{\nu\theta}{p}}\cdot||u||_{L^{p}(N^{s\varepsilon})}^{(1-\beta\theta)}\cdot||u||_{W^{1,p}(N^{s\varepsilon})}^{\beta\theta}\;.$

If $u$ satisfies $\mbox{\rm
supp}\,u\subset\mathbf{B}(0_{\mathbb{R}^{n}},\varepsilon)$, via (3.8), this
entails that if we set

(5.7) $k:=2\nu+\tilde{\mu}+\beta$

then we get (since $k\in(\beta,\frac{1}{\theta})$, see (5.6))

(5.8) $||u||_{L^{q}(N^{s\varepsilon})}\lesssim
s^{-\frac{\nu\theta}{p}}\cdot\ln\frac{1}{s\varepsilon}\cdot||u||_{L^{p}(N^{s\varepsilon})}^{(1-k\theta)}\cdot||u||_{W^{1,p}(N^{s\varepsilon})}^{\beta\theta}\cdot||u||_{W^{1,p}(M^{\varepsilon})}^{(k-\beta)\theta}\;,$

and consequently:

(5.9)
$||u||_{L^{q}(M^{\varepsilon})}\overset{(\ref{eq_coarea_sph})}{\lesssim}\left(\int_{0}^{1}||u||_{L^{q}(N^{s\varepsilon})}^{q}ds\right)^{1/q}\overset{(\ref{eq_ulq_avant_h2})}{\lesssim}||g||_{L^{q}([0,1])}\cdot||u||_{W^{1,p}(M^{\varepsilon})}^{(k-\beta)\theta}\;,$

where

$g(s):=s^{-\frac{\nu\theta}{p}}\cdot\ln\frac{1}{s\varepsilon}\cdot||u||_{L^{p}(N^{s\varepsilon})}^{(1-k\theta)}\cdot||u||_{W^{1,p}(N^{s\varepsilon})}^{\beta\theta}.$

To estimate the $L^{q}$ norm of $g$, we wish to apply (0.1) with $\kappa=3$
and

$p_{1}=p_{2}:=p,\qquad
g_{1}(s):=||u||_{L^{p}(N^{s\varepsilon})},\quad\theta_{1}:=(1-k\theta),$

and

$g_{2}(s):=||u||_{W^{1,p}(N^{s\varepsilon})},\qquad\theta_{2}:=\beta\theta,$

as well as with

$g_{3}(s):=s^{-\frac{\nu\theta}{p}}\cdot\ln\frac{1}{s\varepsilon},\qquad\theta_{3}:=1,$

with $p_{3}$ such that

(5.10)
$\frac{\theta_{1}}{p}+\frac{\theta_{2}}{p}+\frac{1}{p_{3}}=\frac{1}{q}.$

This requires to check that such $p_{3}>0$ exists and, for the value of
$p_{3}$ that we will determine, that $g_{3}$ is $L^{p_{3}}$, which amounts to
check that $\frac{\nu\theta p_{3}}{p}<1$.

To see this, observe that as (by (5.7))
$\theta_{1}+\theta_{2}=1-(2\nu+\tilde{\mu})\theta$, we must have (due to
(5.10), $\frac{1}{q}=\frac{\theta}{\tilde{p}}+\frac{1-\theta}{p}$, (5.5), and
$\tilde{p}=p+\tilde{\mu}$)

(5.11)
$p_{3}=\frac{\tilde{\mu}pq}{(q-p)(2\nu\tilde{p}+\tilde{\mu}(\tilde{p}-1))}>0,$

and, by (5.5), we then also have

(5.12) $\frac{\nu\theta
p_{3}}{p}=\frac{\nu\tilde{p}}{2\nu\tilde{p}+\tilde{\mu}(\tilde{p}-1)}\leq\frac{1}{2}$

(since $\tilde{p}\geq 1$), yielding the required fact. By Hölder’s inequality
(0.1), we therefore can conclude for $q\in[p,p+\mu]$:

$||g||_{L^{q}((0,1))}\lesssim||u||_{L^{p}(M^{\varepsilon})}^{(1-k\theta)}\cdot||u||_{W^{1,p}(M^{\varepsilon})}^{\beta\theta}.$

Plugging this inequality into (5.9), we get for such $q$:

$||u||_{L^{q}({M^{\varepsilon}})}\lesssim||u||_{L^{p}(M^{\varepsilon})}^{1-k\theta}\cdot||u||_{W^{1,p}(M^{\varepsilon})}^{k\theta},$

which yields (5.1), with $\alpha_{p,q}:=k\theta$, so that (5.2) then
immediately follows from (5.5) and (5.7).

It remains to check the compactness of the embeddings. Fix any
$p\in[1,\infty)$ and remark first that, thanks to (5.1), it suffices to check
it for $q=p$.

Let us fix a bounded subset $\mathcal{F}$ of $W^{1,p}(M)$ and note that as the
set

$K_{\eta}:=\\{x\in M:\forall y\in\delta M,\;|x-y|\geq\eta\\}$

is compact (for each $\eta>0$), it can be covered by finitely many coordinate
systems of $M$ that are bi-Lipschitz mappings. It therefore follows from the
compactness of classical Sobolev embeddings that every bounded sequence of
$W^{1,p}(M)$ has, for every $\eta$, a subsequence whose restrictions to
$K_{\eta}$ constitute a convergent sequence of $L^{p}(K_{\eta})$. We thus only
have to check the sometimes called equivanishing condition i.e., that if we
set $Z_{\eta}:=M\setminus K_{\eta}$ then $||u||_{L^{p}(Z_{\eta})}$ tends to
zero as $\eta$ goes to zero uniformly in $u\in\mathcal{F}$.

Remark that it follows from (5.1) that

(5.13) $C:=\sup\\{||u||_{L^{p+\mu}(M)}:u\in\mathcal{F}\\}<\infty.$

Moreover, it easily follows from Cauchy-Crofton’s formula that
$\mathcal{H}^{m}(Z_{\eta})$ goes to zero as $\eta$ tends to zero (see [28,
Proposition $5.3.4$]). If $b\in[1,\infty)$ is such that
$\frac{1}{p+\mu}+\frac{1}{b}=\frac{1}{p}$, by Hölder’s inequality, we
therefore have:

$||u||_{L^{p}(Z_{\eta})}\leq\mathcal{H}^{m}(Z_{\eta})^{\frac{1}{b}}\cdot||u||_{L^{p+\mu}(M)}\overset{(\ref{eq_f_bounded_lq})}{\leq}C\mathcal{H}^{m}(Z_{\eta})^{\frac{1}{b}},$

from which we can conclude that $||u||_{L^{p}(Z_{\eta})}$ tends to zero,
uniformly in $u\in\mathcal{F}$. ∎

######  Remark 5.2.

It seems that the constant $C$ of this theorem could be required to be
independent of $p$ and $q$. Moreover, alike in the case of Lipschitz manifolds
with boundary, it is possible to prove some embedding theorems for $L^{p}(M)$
into $W^{-1,q}(M)$ for some $q>p$. For technical reasons, we have decided to
study $W^{-1,p}(M)$ in the only case where $M$ is an open (definable) subset
of $\mathbb{R}^{n}$, in which case this kind of theorems is easy to derive
from the classical Sobolev’s embeddings, as $L^{p}(M)$ is easy to embed in
$L^{p}(\mathbb{R}^{n})$ in this case.

######  Proposition 5.3.

(Generalized Poincaré inequality) Assume that $M$ is connected let $U$ be a
nonempty open subset of $\partial M$. For every $p\in[1,\infty)$, there is a
constant $C$ such that we have for each $u\in W^{1,p}(M)$:

(5.14) $||u||_{L^{p}(M)}\leq C||\partial
u||_{L^{p}(M)}+C||\mathbf{tr}_{\partial M}\,u||_{L^{p}(U)}.$

###### Proof.

We derive it from the compactness of Sobolev’s embedding using a very
classical argument. Assume that (5.14) fails. There is a sequence $u_{i}\in
W^{1,p}(M)$ satisfying $||u_{i}||_{L^{p}(M)}\equiv 1$ and
$\lim||\mathbf{tr}_{\partial M}u_{i}||_{L^{p}(U)}=\lim||\partial
u_{i}||_{L^{p}(M)}=0$. By the compactness of Sobolev’s embedding (for $p=q$),
extracting a subsequence if necessary, we can assume this sequence to be
convergent in $W^{1,p}(M)$. Its limit $u_{\infty}\in L^{p}(M)$ then clearly
satisfies $\partial u_{\infty}\equiv 0$. As $M$ is connected, $u_{\infty}$
must be constant (almost everywhere), and since $\mathbf{tr}_{\partial
M}u_{\infty}\equiv 0$ on $U\neq\emptyset$, this constant must be zero. ∎

######  Remark 5.4.

This estimate continues to hold for a definable subset $U\subset\delta M$ of
possibly empty interior if $p$ is sufficiently large. Then, of course, one has
to write $||\mathbf{tr}_{U}u||_{L^{p}(U,\mathcal{H}^{k})}$, where $k:=\dim U$,
in the estimate.

The combination of (5.1) with (5.14) gives the famous Gagliardo-Nirenberg-
Sobolev inequality:

######  Corollary 5.5.

If $M$ is connected then, given a nonempty open subset $U$ of $\partial M$,
for each $p$ there is a constant $C$ such that for all $u\in W^{1,p}(M,U)$:

$||u||_{L^{p+\mu}(M)}\leq C||\partial u||_{L^{p}(M)},$

where $\mu$ is provided by Theorem 5.1.

In the same spirit, we have the following result, which can be proved with
exactly the same proof as Proposition 5.3:

######  Proposition 5.6.

Assume that $M$ is connected and let $Z\subset M$ be of nonempty interior (in
$M$). For every $p\in[1,\infty)$ there is a constant $C$ such that we have for
each $u\in W^{1,p}(M)$:

(5.15) $||u||_{L^{p}(M)}\leq C||u||_{L^{p}(Z)}+C||\partial u||_{L^{p}(M)}.$

This leads us to the following theorem which is another famous feature of
Lipschitz domains [8, 17] that we extend to the category of bounded
subanalytic manifolds.

######  Theorem 5.7.

Let $u$ be a distribution on $M$ and let $p\in[1,\infty)$. If $|\partial u|$
is $L^{p}$ then so is $u$.

###### Proof.

We can assume that $M$ is a connected set. Note that such a distribution $u$
is an $L^{p}_{loc}$ function [17, Ch. 1] or [8]. Choose a bi-Lipschitz trivial
stratification $\Sigma$ of ${\overline{M}}$ of which $M$ is a stratum, and
denote by $\Sigma_{k}$ the reunion of all the strata of dimension less than or
equal to $k$, with by convention $\Sigma_{-1}=\emptyset$.

The idea is one more time to argue by induction on the dimension of the strata
that touch the domain of the considered function. Set for this purpose, given
$A\subset{\overline{M}}$,

$\kappa(A):=\max\\{k:\Sigma_{k-1}\cap A=\emptyset\\}.$

We shall show the following statements by decreasing induction on $k$:

$\mathbf{(H_{k})}$. Let $A$ be a closed subset of ${\overline{M}}$ satisfying
$\kappa(A)\leq k$. If a distribution $u$ on $\mathcal{O}\cap M$ satisfies
$\partial u\in L^{p}(\mathcal{O}\cap M)^{n}$, where $\mathcal{O}$ is some open
neighborhood of $A$ in ${\overline{M}}$, then $u\in
L^{p}(\mathcal{O}^{\prime}\cap M)$, for some open neighborhood
$\mathcal{O}^{\prime}$ of $A$ in ${\overline{M}}$.

If $\kappa(A)=m$ then $A$ does not meet $\delta M$, and, since such a function
$u$ is $L^{p}_{loc}$, it must be $L^{p}$ on some neighborhood of $A$. Let us
thus fix $k<m$, and, assuming $\mathbf{(H_{k+1})}$, take $u$ and $A$ like in
the statement of $\mathbf{(H_{k})}$.

As $A$ is compact, it suffices to prove that every point of $A$ has a
neighborhood $U$ in ${\overline{M}}$ such that $u\in L^{p}(U\cap M)$. Take any
$x_{0}\in A$ and let $S\in\Sigma$ denote the stratum containing this point.
Observe that $\dim S\geq k$, and, if $\dim S>k$ then the result comes down
from $\mathbf{(H_{k+1})}$. We thus can assume $\dim S=k$ and will work near
$x_{0}={0_{\mathbb{R}^{n}}}$.

Let $U_{0}$, $W_{0}$, and $\Lambda$ be as defined just before (4.2) and let,
for $\eta>0$ small, $F^{\eta}$ and $V^{\eta}$ be as in (4.2). As $\Lambda$ is
bi-Lipschitz, we can work as if $u$ were defined on $\Lambda(U_{0}\cap M)$.

Let now $\psi:\mathbb{R}\to[0,1]$ be a $\mathscr{C}^{\infty}$ function such
that $\psi\equiv 1$ on $(-\infty,\frac{1}{4})$ and $\psi\equiv 0$ on
$[\frac{1}{2},+\infty)$, and set for $(x,y)\in\mathbf{B}(0,\varepsilon)\times
W_{0}$,

$\phi(x,y):=\psi\left(\frac{|x|^{2}}{\varepsilon^{2}}\right).$

It suffices to show that both $u\phi$ and $(1-\phi)u$ are $L^{p}$ in the
vicinity of ${0_{\mathbb{R}^{n}}}$. As $(1-\phi)\equiv 0$ on

$Z:=\mathbf{B}(0,\frac{\varepsilon}{2})\times W_{0}\,,$

and since $\kappa(\,\overline{A\cap V^{\varepsilon}\setminus Z}\,)>k$, the
assumption $\mathbf{(H_{k+1})}$ entails that $(1-\phi)u$ is $L^{p}$ on
$V^{\varepsilon}$. To establish that so is $\phi u$, observe that for each
$y\in W_{0}$ the function

$v_{y}(x):=\phi(x,y)u(x,y),\qquad x\in F^{\varepsilon},$

satisfies $|\partial v_{y}|\in L^{p}(F^{\varepsilon})$ and vanishes for $|x|$
close to $\varepsilon$. As a matter of fact, by (3.12), for each $y\in W_{0}$,
$v_{y}$ is $L^{p}$ and there is a constant $C$ independent of $y$ such that:

$||v_{y}||_{L^{p}(F^{\varepsilon})}\overset{(\ref{eq_link_forallpm})}{\leq}C||\partial
v_{y}||_{L^{p}(F^{\varepsilon})}\leq C||\partial(\phi u)||_{L^{p}(F_{y})},$

where $F_{y}:=F^{\varepsilon}\times\\{y\\}$. Raising it to the power $p$ and
integrating with respect to $y$ we get:

$||u||_{L^{p}(V^{\varepsilon})}\leq C||\partial(\phi
u)||_{L^{p}(V^{\varepsilon})}\leq
C||u\partial\phi||_{L^{p}(V^{\varepsilon})}+C||\phi\partial
u||_{L^{p}(V^{\varepsilon})}.$

By assumption, $\phi|\partial u|$ is $L^{p}$, and since we have
$\partial\phi\equiv 0$ on $Z$ as well as $\kappa(\overline{A\cap
V^{\varepsilon}\setminus Z})>k$, the induction hypothesis yields that so is
$u\partial_{i}\phi$ on $V^{\varepsilon}$, for all $1\leq i\leq n$. ∎

##  6\. Morrey’s embedding

We recall that a function $u$ is said to be Hölder continuous (with exponent
$\alpha>0$) on a set $P\subset\mathbb{R}^{n}$ when there is a constant $C$
such that for all $x$ and $y$ in $P$

(6.1) $|u(x)-u(y)|\leq C|x-y|^{\alpha}.$

Given $x$ and $y$ in the same connected component of a definable set $A$, let:

$d_{A}(x,y):=\inf\\{lg(\gamma):\gamma:[0,1]\to A,\mbox{ $\mathscr{C}^{0}$
definable arc joining }x\mbox{ and }y\\},$

where $lg(\gamma)$ stands for the length of the definable arc $\gamma$ (which
is piecewise $\mathscr{C}^{1}$). It will be convenient to extend this function
by setting $d_{A}(x,y)=1$ if $x$ and $y$ lie in different connected
components. This defines a metric on $A$, generally referred as the inner
metric of $A$.

A function $u$ on $A$ is Hölder continuous with respect to the inner metric
(with exponent $\alpha>0$) if there is a constant $C$ such that for all $x$
and $y$ in $A$

$|u(x)-u(y)|\leq C\;d_{A}(x,y)^{\alpha}.$

When $\alpha=1$, $u$ will be said to be Lipschitz with respect to the inner
metric.

To avoid any confusion, a function which is Hölder continuous in the sense of
(6.1) will sometimes be said to be Hölder continuous with respect to the outer
metric. We denote by $\mathscr{C}^{0,\alpha}(A)$ (resp.
$\check{\mathscr{C}}^{0,\alpha}(A)$) the space of the functions on $A$ that
are Hölder continuous with respect to the outer (resp. inner) metric with
exponent $\alpha$. It is worthy of notice that each element of
$W^{1,\infty}(M)$ coincides almost everywhere with an element of
$\check{\mathscr{C}}^{0,1}(M)$.

######  Remark 6.1.

Of course, if a function is Hölder continuous (with respect to the outer
metric), it is Hölder continuous with respect to the inner metric with the
same exponent, i.e.,
$\mathscr{C}^{0,\alpha}(M)\subset\check{\mathscr{C}}^{0,\alpha}(M)$, for all
$\alpha>0$. On the other hand, it easily follows from Łojasiewicz’s inequality
and some basic Lipschitz properties of definable sets, that if $M$ is normal
(and bounded) then
$\check{\mathscr{C}}^{0,\alpha}(M)\subset\mathscr{C}^{0,\theta\alpha}(M)$, for
some $\theta>0$ depending on $M$.

Given $u\in\mathscr{C}^{0,\alpha}(M)$ we set

$|u|_{\mathscr{C}^{0,\alpha}(M)}:=\sup\big{\\{}\frac{|u(x)-u(y)|}{|x-y|^{\alpha}}:x,y\in
M,x\neq y\big{\\}},$

as well as

$||u||_{\mathscr{C}^{0,\alpha}(M)}:=||u||_{L^{\infty}(M)}+|u|_{\mathscr{C}^{0,\alpha}(M)}.$

It is well-known that $||\cdot||_{\mathscr{C}^{0,\alpha}(M)}$ is a norm that
makes of $\mathscr{C}^{0,\alpha}(M)$ a Banach space. The norm
$||\cdot||_{\check{\mathscr{C}}^{0,\alpha}(M)}$ is defined analogously.

######  Theorem 6.2.

If $M$ is normal then there is a positive real number $\alpha$ such that for
all $p$ sufficiently large, the embedding

$(\mathscr{C}^{\infty}({\overline{M}}),||\cdot||_{W^{1,p}(M)})\hookrightarrow(\mathscr{C}^{0,\alpha}(M),||\cdot||_{\mathscr{C}^{0,\alpha}(M)})$

is continuous and consequently extends to a compact embedding
$W^{1,p}(M)\hookrightarrow\mathscr{C}^{0,\alpha}(M)$.

Proof. We shall show that the embedding is continuous. That it extends to
$W^{1,p}(M)$ then follows from Theorem 2.6 $(i)$ (for $p$ sufficiently large),
and its compactness comes down from Ascoli-Arzela’s Theorem.

Let $\Sigma$ be a locally bi-Lipschitz trivial stratification of
${\overline{M}}$ compatible with $M$ of which $M$ is a stratum. Given $k\leq
m$, we denote by $\Sigma_{k}$ the union of the strata of dimension less than
or equal to $k$, and, given an open subset $U$ of $M$, we let

$\kappa(U):=\min\\{k:\Sigma_{k}\cap\overline{U}\neq\emptyset\\}.$

We are going to prove the following statements by downward induction on
$k\in\\{0,\dots,m\\}$:

$(\mathbf{A}_{k})$. For every $U\subset M$ such that $\kappa(U)\geq k$, there
are positive real numbers $p_{0}$ and $\alpha$ such that the linear mapping
$\Psi_{U}:(\mathscr{C}^{\infty}({\overline{M}}),||\cdot||_{W^{1,p}(M)})\to\mathscr{C}^{0,\alpha}(U)$,
$u\mapsto u_{|U}$ is continuous for all $p>p_{0}$.

The assertion $(\mathbf{A}_{m-1})$ is well-known and the theorem follows from
$(\mathbf{A}_{0})$. Assume the result true for some $(k+1)$,
$k\in\\{0,\dots,m-1\\}$, and let us prove it for $k$. Fix for this purpose an
open subset $U$ of $M$ satisfying $\kappa(U)=k$.

To summarize, we have to show, assuming $(\mathbf{A}_{k+1})$, that we can find
$\alpha>0$ such that we have for all $p$ sufficiently large and $u\in
W^{1,p}(M)$:

(6.2) $||u||_{\mathscr{C}^{0,\alpha}(U)}\lesssim||u||_{W^{1,p}(M)}.$

Before starting the first step, let us make some observations that are
consequences of the induction assumption. We already know by induction that
inequality (6.2) holds if replace $U$ with any open subset $V$ of $M$
satisfying $\kappa(V)>k$, for each $u\in W^{1,p}(M)$. Since we can use a
cutoff function, given such $V$, we actually have

(6.3) $||u||_{\mathscr{C}^{0,\alpha}(V)}\lesssim||u||_{W^{1,p}(V^{\prime}\cap
M)},$

for every $u\in W^{1,p}(V^{\prime}\cap M)$ (for $p$ large enough), if
$V^{\prime}$ is any neighborhood of $\overline{V}$ in ${\overline{M}}$. Of
course, the constant involved in this inequality depends on the chosen
neighborhood $V^{\prime}$.

Since $U$ is relatively compact and because we can use a partition of unity,
it is enough to deal with the case of a function $u$ whose support fits in
some prescribed neighborhood $U_{x_{0}}$ of some point $x_{0}\in\overline{U}$
that we can choose arbitrarily small. Let us thus take any
${x_{0}}\in\overline{U}$ and let $S$ denote the element of $\Sigma$ that
contains $x_{0}$. We may assume $S$ to be of dimension $k$ (since, in the case
$\dim S>k$, $x_{0}$ would have a neighborhood $U_{x_{0}}$ such that
$\kappa(U_{x_{0}})>k$, and on the other hand ${x_{0}}\in\overline{U}$ entails
$\dim S\geq\kappa(U)=k$).

We establish in step 1 estimate (6.2) in the case $k=0$ (which is the easiest
case), $x_{0}=0_{\mathbb{R}^{n}}\in\delta M$ and
$U=M^{\frac{\varepsilon}{2}}$, the notations $\varepsilon$, $r_{s}$,
$M^{\eta}$, and $N^{\eta}$ being as introduced in section 1.3. Note that, as
$r_{s}$ is bi-Lipschitz for every $s\in(0,1)$, it follows from Łojasiewicz’s
inequality that there are positive numbers $\gamma$ and $C$ such that for all
$s\in(0,1]$ and all $x,y$ in ${M^{\varepsilon}}$:

(6.4)
$|x-y|\frac{s^{\gamma}}{C}\leq|r_{s}(x)-r_{s}(y)|\leq\frac{C}{s^{\gamma}}|x-y|.$

######  Step 1.

Assuming $(\mathbf{A}_{1})$, we show that there are positive numbers $\alpha$,
$C$ and $p_{0}$, depending only on $\varepsilon$ and $M$, such that for all
$p\geq p_{0}$ and all
$u\in\mathscr{C}^{\infty}(\overline{{M^{\varepsilon}}})$:

(6.5) $||u||_{\mathscr{C}^{0,\alpha}(M^{\frac{\varepsilon}{2}})}\leq
C||u||_{W^{1,p}({M^{\varepsilon}})}.$

###### proof of step 1..

By Theorem 2.6, we have $|u(0)|\lesssim||u||_{W^{1,p}({M^{\varepsilon}})}$.
Hence, since we can subtract $u(0)$ if necessary, we see that it suffices to
prove the claimed estimate for a function $u$ that vanishes at the origin
(which will allow us to apply (3.15), see (6.7) below).

For $\eta\leq\varepsilon$, let $P^{\eta}:=M^{\eta}\setminus
M^{\frac{\eta}{2}}$, and apply $(\mathbf{A}_{1})$ to
$P^{\frac{\varepsilon}{2}}$. This provides constants $\tilde{p}_{0}$ and
${\tilde{\alpha}}$ such that for all $v\in W^{1,p}({M^{\varepsilon}})$ (see
(6.3)), with $p>\tilde{p}_{0}$, we have

(6.6)
$||v||_{\mathscr{C}^{0,{\tilde{\alpha}}}(P^{\frac{\varepsilon}{2}})}\lesssim||v||_{W^{1,p}({M^{\varepsilon}})}.$

As ${M^{\varepsilon}}$ is bounded, this inequality is still valid if we change
${\tilde{\alpha}}$ for a smaller value (for a possibly bigger constant). In
particular, we may assume ${\tilde{\alpha}}<\frac{1}{2\gamma}\,$, where
$\gamma$ is the exponent that appears in (6.4).

Applying (6.6) to $v=u\circ r_{s}$, $s\in(0,1)$,
$u\in\mathscr{C}^{\infty}(\overline{{M^{\varepsilon}}})$ satisfying $u(0)=0$,
$p$ large enough, by (3.15), we immediately see that

(6.7) $||u\circ
r_{s}||_{\mathscr{C}^{0,{\tilde{\alpha}}}(P^{\frac{\varepsilon}{2}})}\lesssim
s^{1-\frac{\nu}{p}}||u||_{W^{1,p}(M^{s\varepsilon})},$

which implies

(6.8) $||u||_{L^{\infty}(P^{s\frac{\varepsilon}{2}})}=||u\circ
r_{s}||_{L^{\infty}(P^{\frac{\varepsilon}{2}})}\lesssim
s^{1-\frac{\nu}{p}}||u||_{W^{1,p}(M^{s\varepsilon})}.$

As the constant involved in this estimate is independent of $s$, we already
see that for $p$ large enough, we have for
$u\in\mathscr{C}^{\infty}(\overline{{M^{\varepsilon}}})$ vanishing at the
origin

$||u||_{L^{\infty}(M^{\frac{\varepsilon}{2}})}\lesssim||u||_{W^{1,p}({M^{\varepsilon}})}.$

It therefore only remains to show that there are positive constants $C$,
$p_{0}$, and $\alpha$ such that for all $p\geq p_{0}$,
$u\in\mathscr{C}^{\infty}(\overline{{M^{\varepsilon}}})$ vanishing at the
origin, as well as $x$ and $y$ in $M^{\frac{\varepsilon}{2}}$, we have:

(6.9) $|u(x)-u(y)|\leq
C||u||_{W^{1,p}({M^{\varepsilon}})}\cdot|x-y|^{\alpha}.$

To see this, observe that (6.7) also implies that for such $u$ we have for all
$x$ and $y$ in $P^{\frac{\varepsilon}{2}}$:

(6.10) $|u(r_{s}(x))-u(r_{s}(y))|\lesssim
s^{1-\frac{\nu}{p}}||u||_{W^{1,p}({M^{\varepsilon}})}\cdot|x-y|^{\tilde{\alpha}},$

which, thanks to (6.4), entails

(6.11) $|u(r_{s}(x))-u(r_{s}(y))|\lesssim
s^{1-\frac{\nu}{p}}||u||_{W^{1,p}({M^{\varepsilon}})}\cdot|r_{s}(x)-r_{s}(y)|^{\tilde{\alpha}}\cdot
s^{-{\tilde{\alpha}}\gamma}.$

As ${\tilde{\alpha}}\gamma<\frac{1}{2}$, we see that if $p$ is sufficiently
large then we have for $s\in(0,1)$ and
$u\in\mathscr{C}^{\infty}(\overline{{M^{\varepsilon}}})$ vanishing at the
origin:

(6.12)
$|u|_{\mathscr{C}^{0,{\tilde{\alpha}}}(P^{s\frac{\varepsilon}{2}})}\lesssim||u||_{W^{1,p}({M^{\varepsilon}})}.$

To show that this implies (6.9), take $x$ and $y$ in
$M^{\frac{\varepsilon}{2}}$, and set $s:=\frac{3|x|}{\varepsilon}$, in order
to have $x\in P^{s\frac{\varepsilon}{2}}$. It is no loss of generality to
assume $|x|\geq|y|$. If $y\in P^{s\frac{\varepsilon}{2}}$, by (6.12), the
desired inequality holds (with $\alpha={\tilde{\alpha}}$ in this case) for the
couple $(x,y)$. Otherwise, $y\in
M^{\frac{s\varepsilon}{4}}=M^{\frac{3|x|}{4}}$, which entails
$|x-y|>\frac{|x|}{4}=\frac{s\varepsilon}{12}$, which implies in turn for $p$
sufficiently large

$|u(x)-u(y)|\leq|u(x)|+|u(y)|\overset{(\ref{eq_linfty_0})}{\lesssim}s^{\frac{1}{2}}\cdot||u||_{W^{1,p}({M^{\varepsilon}})}\lesssim|x-y|^{\frac{1}{2}}\cdot||u||_{W^{1,p}({M^{\varepsilon}})},$

yielding the claimed fact, completing the proof of step $1$. ∎

By definition of bi-Lipschitz triviality, the point ${x_{0}}$ that we have
fixed has a neighborhood $U_{x_{0}}$ in ${\overline{M}}$, for which there is a
bi-Lipschitz homeomorphism
$\Lambda:U_{x_{0}}\to(\pi^{-1}(0_{\mathbb{R}^{n}})\cap\overline{M})\times
W_{x_{0}}$, where $\pi:U_{x_{0}}\to S$ is a $\mathscr{C}^{\infty}$ retraction
and $W_{x_{0}}$ is a neighborhood of $x_{0}$ in $S$.

The next following steps are devoted to the proof of the induction step in the
case $k>0$. The idea is to split the proof of (6.2) for $k>0$ into two steps:
we first establish the Hölder condition for the restriction of $u$ to the
stratum $S$ (step 2 below) and then prove it for a couple of points $(x,y)$
for which $\pi(x)$ and $\pi(y)$ are close to each other (step 3). Step 4 will
then derive (6.2) from these two steps. We therefore from now assume $k>0$.
Before proceeding, we introduce some notations.

We can assume that $F:=\pi^{-1}(0_{\mathbb{R}^{n}})\cap M$ is a
$\mathscr{C}^{\infty}$ manifold (see Remark $1.6$ of [27]). Set for simplicity
for $\eta\leq\varepsilon$:

$F^{\eta}:=\pi^{-1}(0_{\mathbb{R}^{n}})\cap M^{\eta}\quad\mbox{and}\quad
U_{0}^{\eta}:=F^{\eta}\times\mathbf{B}(0_{\mathbb{R}^{k}},\eta).$

Since we can work up to a bi-Lipschitz homeomorphism, we will identify
$U_{x_{0}}$ with $U_{0}^{\varepsilon}$, ${x_{0}}$ with the origin, and $S\cap
U_{x_{0}}$ with
$W_{0}:=\\{0_{\mathbb{R}^{n-k}}\\}\times\mathbf{B}(0_{\mathbb{R}^{k}},\varepsilon)$.

Observe that $\pi$ is then induced by the canonical projection onto
$\\{0_{\mathbb{R}^{n-k}}\\}\times\mathbb{R}^{k}$. We shall sometimes regard
$W_{0}$ as a subset of $\mathbb{R}^{k}$, identifying
$\mathbf{B}(0_{\mathbb{R}^{k}},\varepsilon)$ with
$\\{0_{\mathbb{R}^{n-k}}\\}\times\mathbf{B}(0_{\mathbb{R}^{k}},\varepsilon)$
(and $x\in\mathbf{B}(0_{\mathbb{R}^{k}},\varepsilon)$ with
$(0_{\mathbb{R}^{n-k}},x)\in W_{0}$).

Apply now Theorem 1.6 to the germ of $F$ at the origin. This provides a
Lipschitz definable mapping $r:[0,1]\times F^{\varepsilon}\to F^{\varepsilon}$
such that for every $s\in(0,1)$, $r_{s}(F^{\varepsilon})=F^{s\varepsilon}$,
that enables us define a retraction by deformation of $U_{0}^{\varepsilon}$
onto the origin:

(6.13) $\tilde{r}:[0,1]\times U_{0}^{\varepsilon}\to
U_{0}^{\varepsilon},\qquad(s,z)\mapsto\tilde{r}_{s}(z):=\big{(}r_{s}(z-\pi(z)),s\pi(z)\big{)}.$

More generally, given $x$ in the set

(6.14)
$W_{0}^{\prime}:=\\{0_{\mathbb{R}^{n-k}}\\}\times\mathbf{B}(0_{\mathbb{R}^{k}},\frac{\varepsilon}{4})$

and $z\in U_{0}^{\varepsilon}$ such that $(z-x)\in U_{0}^{\varepsilon}$, we
set

$\displaystyle\tilde{r}_{x,s}(z):=\tilde{r}_{s}(z-x)+x,$

in order to get a retraction by deformation onto $\\{x\\}$.

Observe that the second component of this mapping is just a homothetic
transformation of $\mathbb{R}^{k}$ of ratio $s$ centered at $x$. Hence, by
(1.5), we see that there is a constant $C$ such that for all $s\in(0,1)$ we
have for almost all $z\in\mathbf{B}(x,\frac{\varepsilon}{2})\cap
U_{0}^{\varepsilon}$:

(6.15) $\left|\frac{\partial\tilde{r}_{x}}{\partial s}(s,z)\right|\leq
C|z-x|.$

As $\tilde{r}_{x,s}$ is merely $\tilde{r}_{s}$ up to a translation, we can
require the constant involved in the above estimate to be independent of $x\in
W_{0}^{\prime}$. Moreover, thanks to Łojasiewicz’s inequality, there are
positive real numbers $c$ and $\nu$ which are independent of $(x,s)\in
W_{0}^{\prime}\times(0,1)$ and such that on
$\mathbf{B}(x,\frac{\varepsilon}{2})\cap U_{0}^{\varepsilon}$ we have:

(6.16) $\mbox{jac}\,\tilde{r}_{x,s}\geq cs^{\nu}.$

######  Step 2.

There exist positive real numbers $C$ and $\mu$, depending only on
$U_{0}^{\varepsilon}$, such that for all $p$ sufficiently large, we have for
all $u\in\mathscr{C}^{0,1}(\overline{U_{0}^{\varepsilon}})$:

(6.17) $|u|_{\mathscr{C}^{0,1-\frac{\mu}{p}}(W^{\prime}_{0})}\leq
C||u||_{W^{1,p}(U_{0}^{\varepsilon})}.$

###### proof of step 2..

Since strata are smooth, we will here follow an argument used to prove
Morrey’s inequality on smoothly bounded domains of $\mathbb{R}^{n}$. For $x$
and $y$ in
$W^{\prime}_{0}=\\{0_{\mathbb{R}^{n-k}}\\}\times\mathbf{B}(0_{\mathbb{R}^{k}},\frac{\varepsilon}{4})$,
let

$A_{x,y}:=M\cap\mathbf{B}(x,\eta)\cap\mathbf{B}(y,\eta),\;\;\mbox{where
}\;\;\;\eta:=|x-y|.$

We first prove that there are positive real numbers $C$ and $\mu$ such that
for all $x$ and $y$ in $W^{\prime}_{0}$ we have:

(6.18) $|x-y|^{\mu}\leq C\,\mathcal{H}^{m}(A_{x,y}).$

To prove this fact, observe that as obviously

$F^{\frac{\eta}{4}}\times\left(W_{0}\cap\mathbf{B}\big{(}\frac{x+y}{2},\frac{\eta}{4}\big{)}\right)\subset
A_{x,y}$

(where as above $\eta=|x-y|$), there must be constants $C$ and $l$ such that
for all $x$ and $y$ in $W_{0}^{\prime}$ we have

(6.19) $\eta^{k}\cdot\mathcal{H}^{m-k}(F^{\frac{\eta}{4}})\leq
C\,\mathcal{H}^{m}(A_{x,y}).$

It is well-known that (see [14] or [28, section $5.2$]), as $F$ is definable,
there is a constant (also denoted $C$) and a real number $l$ such that
$\zeta^{l}\leq C\mathcal{H}^{m-k}(F^{\zeta})$, for all $\zeta>0$, which means
that (6.19) yields (6.18).

To show (6.17), observe now that if $x$ and $y$ belong to $W_{0}^{\prime}$,
then, by the fundamental theorem of calculus (it is enough to establish (6.17)
for a function $u\in\mathscr{C}^{\infty}(\overline{U_{0}^{\varepsilon}})$), we
have:

(6.20) $\displaystyle\int_{A_{x,y}}|u(z)-u(x)|\,dz$ $\displaystyle\leq$
$\displaystyle\int_{A_{x,y}}\int_{0}^{1}\left|\partial
u(\tilde{r}_{x,s}(z))\cdot\frac{\partial\tilde{r}_{x}}{\partial
s}(s,z)\right|ds\,dz$
$\displaystyle\overset{(\ref{eq_der_r_s_tilda})}{\lesssim}$
$\displaystyle\int_{A_{x,y}}\int_{0}^{1}|\partial
u(\tilde{r}_{x,s}(z))|\cdot|z-x|\,ds\,dz$ $\displaystyle\leq$
$\displaystyle|x-y|\int_{A_{x,y}}\int_{0}^{1}|\partial
u(\tilde{r}_{x,s}(z))|\,ds\,dz,$

since $|z-x|<|x-y|$, for all $z\in A_{x,y}$. Remark now that if we write

$|u(x)-u(y)|\leq|u(z)-u(x)|+|u(z)-u(y)|,$

and then integrate this inequality with respect to $z\in A_{x,y}$, we get

$\displaystyle\mathcal{H}^{m}(A_{x,y})\,|u(x)-u(y)|=\int_{A_{x,y}}|u(z)-u(x)|dz\;+\;\int_{A_{x,y}}|u(z)-u(y)|dz$
(6.21)
$\displaystyle\overset{(\ref{eq_uxz})}{\lesssim}|x-y|\int_{A_{x,y}}\int_{0}^{1}|\partial
u(\tilde{r}_{x,s}(z))|dsdz+|x-y|\int_{A_{x,y}}\int_{0}^{1}|\partial
u(\tilde{r}_{y,s}(z))|dsdz.$

We now can estimate the integrals that appear in (6) as follows:

$\displaystyle\int_{A_{x,y}}\int_{0}^{1}|\partial
u(\tilde{r}_{x,s}(z))|ds\,dz\overset{(\ref{eq_jac_tir})}{\lesssim}\int_{A_{x,y}}\int_{0}^{1}|\partial
u(\tilde{r}_{x,s}(z))|\cdot\mbox{jac}\,\tilde{r}_{x,s}(z)^{1/p}\cdot
s^{-\nu/p}ds\,dz$ $\displaystyle\leq$
$\displaystyle\left(\int_{A_{x,y}}\int_{0}^{1}|\partial
u(\tilde{r}_{x,s}(z))|^{p}\cdot\mbox{jac}\,\tilde{r}_{x,s}(z)ds\,dz\right)^{1/p}\cdot\left(\int_{A_{x,y}}\int_{0}^{1}s^{-\nu
p^{\prime}/p}ds\,dz\right)^{1/p^{\prime}}$ $\displaystyle=$
$\displaystyle\left(\int_{0}^{1}\int_{\tilde{r}_{x,s}(A_{x,y})}|\partial
u(z)|^{p}dz\,ds\right)^{1/p}\cdot\left(\int_{A_{x,y}}\int_{0}^{1}s^{-\nu/p-1}dsdz\right)^{1-1/p}$
$\displaystyle\leq$ $\displaystyle\left(\int_{0}^{1}||\partial
u||_{L^{p}(U_{0}^{\varepsilon})}^{p}ds\right)^{1/p}\cdot\left(\int_{A_{x,y}}\int_{0}^{1}s^{-\nu/p-1}ds\,dz\right)^{1-1/p}$
$\displaystyle\lesssim$ $\displaystyle||\partial
u||_{L^{p}(U_{0}^{\varepsilon})}\cdot\mathcal{H}^{m}(A_{x,y})^{1-1/p}\quad\mbox{(for
$p\geq\nu+2$).}$

By the symmetry of our assumptions, this computation is valid to estimate the
integral of $|\partial u(\tilde{r}_{y,s}(z))|$ as well, so that, plugging the
two inequalities resulting from this computation into (6), we get

$\displaystyle|u(x)-u(y)|\lesssim|x-y|\cdot\mathcal{H}^{m}(A_{x,y})^{-1/p}\cdot||\partial
u||_{L^{p}(U_{0}^{\varepsilon})}\overset{(\ref{eq_vol_A_x})}{\lesssim}|x-y|^{1-\frac{\mu}{p}}\cdot||\partial
u||_{L^{p}(U_{0}^{\varepsilon})},$

yielding the claimed estimate and completing step 2.∎

We are still using the notations introduced before step 2. Step 3 will provide
an estimate of the norm of the restriction of a function
$u\in\mathscr{C}^{0,1}(\overline{U_{0}^{\varepsilon}})$ to a neighborhood of a
fiber $\pi^{-1}(a)$, $a\in W^{\prime}_{0}$ (with a constant independent of
$a$), proceeding in a similar way as in the case $k=0$ (step 1). Its proof
will require a generalization of (3.15) to $\tilde{r}_{s}$ (see (6.13) for
$\tilde{r}_{s}$) that we present now (inequality (6.23) below).

For $u\in L^{p}(U_{0}^{\frac{\varepsilon}{2}})$ and $s\in(0,1)$, we have:

$||u\circ\tilde{r}_{s}||_{L^{p}(U_{0}^{\frac{\varepsilon}{4}})}\overset{(\ref{eq_jac_tir})}{\lesssim}s^{-\frac{\nu}{p}}\cdot||u||_{L^{p}(\tilde{r}_{s}(U_{0}^{\frac{\varepsilon}{4}}))}\leq
s^{-\frac{\nu}{p}}\cdot||u||_{L^{p}(M^{s\frac{\varepsilon}{2}})},$

since $U_{0}^{\frac{\varepsilon}{4}}\subset M^{\frac{\varepsilon}{2}}$. In
particular, in the case where $u$ belongs to
$\mathscr{C}^{0,1}\big{(}\;\overline{U_{0}^{\varepsilon/2}}\;\big{)}$ and
satisfies $u(0)=0$, by (3.14), we get for $s\in(0,1)$ (for $p$ large):

(6.22) $||u\circ\tilde{r}_{s}||_{L^{p}(U_{0}^{\frac{\varepsilon}{4}})}\lesssim
s^{1-\frac{\nu}{p}}\cdot||u||_{W^{1,p}(M^{\frac{\varepsilon}{2}})}.$

Furthermore, for such $u$ and $s$

$||\partial(u\circ\tilde{r}_{s})||_{L^{p}(U_{0}^{\frac{\varepsilon}{4}})}=||^{\mathbf{t}}D\tilde{r}_{s}\left(\partial
u\circ\tilde{r}_{s}\right)||_{L^{p}(U_{0}^{\frac{\varepsilon}{4}})}\lesssim
s||\partial
u\circ\tilde{r}_{s}||_{L^{p}(U_{0}^{\frac{\varepsilon}{4}})}\overset{(\ref{eq_jac_tir})}{\lesssim}s^{1-\frac{\nu}{p}}||\partial
u||_{L^{p}(M^{\frac{\varepsilon}{2}})}$

(again, since $U_{0}^{\frac{\varepsilon}{4}}\subset
M^{\frac{\varepsilon}{2}}$). Combining this estimate with (6.22), we conclude
that for $u\in\mathscr{C}^{0,1}(\;\overline{U_{0}^{\varepsilon/2}}\;)$
vanishing at the origin, we have:

(6.23)
$||u\circ\tilde{r}_{s}||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{4}})}\lesssim
s^{1-\frac{\nu}{p}}\cdot||u||_{W^{1,p}(M^{\frac{\varepsilon}{2}})}.$

The statement of the next step requires to introduce the following open
neighborhoods of $\pi^{-1}(a)\cap M^{\eta}$ in $U_{0}^{\varepsilon}$, for
$a\in\\{0_{\mathbb{R}^{n-k}}\\}\times\mathbf{B}(0_{\mathbb{R}^{k}},\frac{\varepsilon}{4})$
and $\eta\in(0,\varepsilon)$:

$V^{\eta}_{a}:=\\{x\in U_{0}^{\varepsilon}:2|\pi(x)-a|<|x-\pi(x)|<\eta\\}.$

######  Step 3.

There are positive constants $C$, $p_{0}$, and $\kappa$ such that for all
$p>p_{0}$, $u\in\mathscr{C}^{0,1}(\overline{U_{0}^{\varepsilon}})$, and all
$a\in\\{0_{\mathbb{R}^{n-k}}\\}\times\mathbf{B}(0_{\mathbb{R}^{k}},\frac{\varepsilon}{4})$,
we have

(6.24) $||u||_{\mathscr{C}^{0,\kappa}(V_{a}^{\frac{\varepsilon}{8}})}\leq
C||u||_{W^{1,p}(U_{0}^{\varepsilon})}.$

###### proof of step 3..

We first focus on the case $a=0_{\mathbb{R}^{n}}$. For
$\eta\in(0,\varepsilon)$, set
$Z^{\eta}:=V^{\eta}_{0}\setminus\overline{V_{0}^{\eta/2}}$ and apply
$(\mathbf{A}_{k+1})$ to $Z^{\frac{\varepsilon}{8}}$. This provides positive
constants $C$, $p_{0}$, and $\kappa$ such that for all $p>p_{0}$, we have for
all $v\in\mathscr{C}^{0,1}(\overline{U^{\varepsilon}_{0}})$ (see (6.3)):

(6.25) $||v||_{\mathscr{C}^{0,\kappa}(Z^{\frac{\varepsilon}{8}})}\leq
C||v||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{4}})}.$

Note that we can change $\kappa$ for a smaller constant. In particular, we can
assume that $\kappa<\frac{1}{2\gamma}$, where $\gamma$ is the exponent for
which $\tilde{r}_{s}$ satisfies (6.4).

Remark first that we have for $p$ sufficiently large, $s\in(0,1)$, and
$u\in\mathscr{C}^{0,1}(\overline{U_{0}^{\varepsilon}})$ vanishing at the
origin:

$||u||_{L^{\infty}(Z^{\frac{s\varepsilon}{8}}_{0})}=||u\circ\tilde{r}_{s}||_{L^{\infty}(Z^{\frac{\varepsilon}{8}}_{0})}\overset{(\ref{eq_hrk})}{\lesssim}||u\circ\tilde{r}_{s}||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{4}})}\overset{(\ref{eq_utir_wp})}{\lesssim}s^{1-\frac{\nu}{p}}\cdot||u||_{W^{1,p}(M^{\frac{\varepsilon}{2}})}.$

As the involved constants are independent of $s$ and since
$M^{\frac{\varepsilon}{2}}\subset U_{0}^{\frac{\varepsilon}{2}}$, this implies
that for such $u$ we have for $p$ large enough:

(6.26) $\displaystyle||u||_{L^{\infty}(V^{\frac{\varepsilon}{8}}_{0})}\lesssim
s^{\frac{1}{2}}||u||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{2}})}.$

Given now any function
$u\in\mathscr{C}^{0,1}(\overline{U_{0}^{\varepsilon}})$, by Theorem 2.6, we
have $|u(0)|\lesssim||u||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{4}})}.$
Consequently, since it is possible to subtract $u(0)$ if necessary, we see
that (6.26) is still valid for a function which does not vanish at the origin.

We are going to show an analogous estimate for
$|u|_{\mathscr{C}^{0,\alpha}(V^{\frac{\varepsilon}{8}}_{0})}$ (see (6.29)
below). Note first that (6.25) also implies that for
$u\in\mathscr{C}^{0,1}(\overline{U_{0}^{\varepsilon}})$ we have for all $x$
and $y$ in $Z_{0}^{\frac{\varepsilon}{8}}$ and all $s\in(0,1)$:

$|u(\tilde{r}_{s}(x))-u(\tilde{r}_{s}(y))|\lesssim||u\circ\tilde{r}_{s}||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{4}})}\cdot|x-y|^{\kappa},$

which, via (6.23) and (6.4) (for $\tilde{r}_{s}$), entails in the case where
$u$ vanishes at the origin

(6.27) $|u(\tilde{r}_{s}(x))-u(\tilde{r}_{s}(y))|\lesssim
s^{1-\frac{\nu}{p}}||u||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{2}})}\cdot|\tilde{r}_{s}(x)-\tilde{r}_{s}(y)|^{\kappa}\cdot
s^{-\kappa\gamma}.$

As $\kappa\gamma<\frac{1}{2}$ and since
$\tilde{r}_{s}(V_{0}^{\frac{\varepsilon}{8}})=V_{0}^{\frac{s\varepsilon}{8}}$,
we have shown that there are constants $p_{0}$ and $C$ such that if $p>p_{0}$
then we have for all $u\in\mathscr{C}^{0,1}(U_{0}^{\varepsilon})$ vanishing at
the origin, all $s$, and all $x$ and $y$ in $Z_{0}^{\frac{s\varepsilon}{8}}$:

(6.28) $|u(x)-u(y)|\leq
C||u||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{2}})}|x-y|^{\kappa}.$

Again, since it is possible to subtract $u(0)$ if necessary, we see that this
estimate is still valid for a function which does not vanish at the origin.

We claim that this estimate continues to hold if we replace
$Z_{0}^{\frac{s\varepsilon}{8}}$ with $V_{0}^{\frac{\varepsilon}{8}}$, i.e.,
that for $u\in\mathscr{C}^{0,1}(U_{0}^{\varepsilon})$:

(6.29) $|u|_{\mathscr{C}^{0,\kappa}(V_{0}^{\frac{\varepsilon}{8}})}\leq
C||u||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{2}})},$

for some possibly bigger constant $C$.

To show this take any $x$ and $y$ in $V_{0}^{\frac{\varepsilon}{8}}$, with
$|y-\pi(y)|\leq|x-\pi(x)|$, and take any
$s\in(\frac{8|x-\pi(x)|}{\varepsilon},\frac{10|x-\pi(x)|}{\varepsilon})$ (so
that we have $x\in Z_{0}^{s\frac{\varepsilon}{8}}$). If $y$ also belongs to
$Z_{0}^{s\frac{\varepsilon}{8}}$ then the claimed inequality follows from
(6.28). Otherwise, we must have $|y-\pi(y)|<\frac{s\varepsilon}{16}$, which
implies that

$|x-y|\geq|x-\pi(x)|-|y-\pi(y)|\geq\frac{s\varepsilon}{10}-\frac{s\varepsilon}{16}\geq\frac{s\varepsilon}{32},$

from which we can conclude that (for $p$ large)

$|u(x)-u(y)|\leq|u(x)|+|u(y)|\overset{(\ref{eqn_uba_rs_infty_k})}{\lesssim}s^{\frac{1}{2}}\cdot||u||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{2}})}\lesssim|x-y|^{\frac{1}{2}}\cdot||u||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{2}})},$

which yields (6.29).

To prove (6.24), given $u\in W^{1,p}(U_{0}^{\varepsilon})$, $p>p_{0}$, and any
$a\in\\{0_{\mathbb{R}^{n-k}}\\}\times\mathbf{B}(0_{\mathbb{R}^{k}},\frac{\varepsilon}{4})$,
it now suffices to apply (6.29) to $\tilde{u}_{a}(x):=u(x+a)$. As

$||\tilde{u}_{a}||_{W^{1,p}(U_{0}^{\frac{\varepsilon}{2}})}\leq||u||_{W^{1,p}(U_{0}^{\varepsilon})},$

and since $V^{\frac{\varepsilon}{8}}_{a}$ coincides with
$V^{\frac{\varepsilon}{8}}_{0}$ up to a translation, we get the desired
result. ∎

We are now ready to carry out the induction step in the case $k>0$, which will
complete the proof of the theorem:

######  Step 4.

There are positive constants $p_{0}$, $C$, and $\alpha$ such that for all
$p>p_{0}$ and all $u\in\mathscr{C}^{0,1}(\overline{U_{0}^{\varepsilon}})$ we
have

(6.30) $||u||_{\mathscr{C}^{0,\alpha}(U_{0}^{\frac{\varepsilon}{8}})}\leq
C||u||_{W^{1,p}(U_{0}^{\varepsilon})},$

with $\alpha=\min(\kappa,1-\frac{\mu}{p})$, where $\mu$ is provided by step 2
and $\kappa$ by step 3.

###### proof of step 4..

As the mapping $u\mapsto u(0)$ is Lipschitz on $W^{1,p}(U_{0}^{\varepsilon})$
(for $p$ large, by Theorem 2.6), it suffices to prove this estimate for the
semi-norm $|u|_{\mathscr{C}^{0,\alpha}(U_{0}^{\frac{\varepsilon}{8}})}$.

Take $x$ and $y$ in $U_{0}^{\frac{\varepsilon}{8}}$. If
$|x-y|<\frac{|x-\pi(x)|}{2}$ then $y\in V_{a}^{\frac{\varepsilon}{8}}$, with
$a:=\pi(x)$, and the needed inequality then directly follows from (6.24).
Otherwise, we have $|x-y|\geq\frac{|x-\pi(x)|}{2}$, and consequently
$|y-\pi(y)|\leq 4|x-y|$, so that writing

$|u(x)-u(y)|\leq|u(x)-u(\pi(x))|+|u(\pi(x))-u(\pi(y))|+|u(y)-u(\pi(y))|,$

we see that the desired inequality comes down from (6.24) (applied to both
couples $(x,\pi(x))$ and $(y,\pi(y))$) and (6.17) (applied to the couple
$(\pi(x),\pi(y))$). ∎

In the case where $M$ is not assumed to be normal, we have:

######  Corollary 6.3.

There is a positive real number $\alpha$ such that for all $p$ sufficiently
large the embedding

$(\mathscr{C}^{\infty}({\overline{M}}),||\cdot||_{W^{1,p}(M)})\hookrightarrow(\check{\mathscr{C}}^{0,\alpha}(M),||\cdot||_{\check{\mathscr{C}}^{0,\alpha}(M)})$

is continuous and therefore extends to a compact embedding
$W^{1,p}(M)\hookrightarrow\check{\mathscr{C}}^{0,\alpha}(M)$.

###### Proof.

It suffices to apply Theorem 6.2 to a normalization of $M$. As
$\mathscr{C}^{\infty}$ normalizations, as well as their inverse mappings, have
bounded first derivative, they are bi-Lipschitz with respect to the inner
metric. ∎

For $u\in W^{1,p}(M)$, we can regard the components of $\partial u$ in the
canonical basis of $\mathbb{R}^{n}$ as the partial derivatives of $u$
(although $M$ may have positive codimension in $\mathbb{R}^{n}$). We then
define the second order partial derivatives as the partial derivatives these
first order partial derivatives, and so on. We have:

######  Corollary 6.4.

Let $u\in L^{1}(M)$. If all the successive partial derivatives of $u$ are
$L^{1}$ then $u$ is Lipschitz with respect to the inner metric. Moreover, in
the case where $M$ is normal, such a function $u$ is Hölder continuous.

###### Proof.

Observe first that by Theorem 5.1 (applied with $p=1$), such a function $u$,
as well as all its subsequent partial derivatives, belong to $L^{1+\mu}(M)$.
Iterating $k$ times the argument, we thus can show that these functions
actually belong to $L^{1+k\mu}(M)$ for all $k\in\mathbb{N}$. By Corollary 6.3,
we deduce that $u$ and $\partial_{1}u,\dots,\partial_{n}u$ all belong to
$\check{\mathscr{C}}^{0,\alpha}(M)$ for some $\alpha>0$, which implies that
these are bounded functions. Hence, $u\in W^{1,\infty}(M)$, which entails that
it coincides almost everywhere with a function that is Lipschitz with respect
to the inner metric. The last sentence is a consequence of Theorem 6.2 (see
also Remark 6.1). ∎

If we define $W^{\infty,1}(M)$ as the space of $L^{1}$ functions that have all
their subsequent partial derivatives $L^{1}$, the just above corollary exactly
shows that

$W^{\infty,1}(M)\subset W^{1,\infty}(M).$

We have actually shown that there is an integer $k$, depending on the
Lipschitz geometry of $M$, such that $W^{k,p}(M)\subset W^{1,\infty}(M)$.

######  Example 6.5.

This corollary is no longer true if the underlying manifold is simply required
to be definable on a non necessarily polynomially bounded o-minimal structure:
if $M$ is the manifold constituted by the interior of the set $Y$ introduced
in Example 1.7 and $u(x,y)=\sqrt{x}$ then $u$ as well as all its subsequent
partial derivatives belong to $L^{1}(M)$ but $u$ is not Lipschitz with respect
to the inner metric.

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