Trace operators on bounded subanalytic manifolds

Abstract

We prove that if $M\subset {\mathbb {R}}^n$ is a bounded subanalytic submanifold of ${\mathbb {R}}^n$ such that ${\textbf{B}}(x_0,\varepsilon )\cap M$ is connected for every $x_0\in {{\overline{M}}}$ and $\varepsilon >0$ small, then, for $p\in [1,\infty )$ sufficiently large, the space ${\mathscr {C}}^\infty ( {{\overline{M}}})$ is dense in the Sobolev space $W^{1,p}(M)$. We also show that for p large, if $A\subset {{\overline{M}}}\setminus M$ is subanalytic then the restriction mapping ${\mathscr {C}}^\infty ( {{\overline{M}}})\ni u\mapsto u_{|A}\in L^p(A)$ is continuous (if A is endowed with the Hausdorff measure), which makes it possible to define a trace operator, and then prove that compactly supported functions are dense in the kernel of this operator. We finally generalize these results to the case where our assumption of connectedness at singular points of ${{\overline{M}}}$ is dropped.

1 Introduction

In [21, 22], we proved a Poincaré type inequalities for the functions of Sobolev spaces of bounded subanalytic open subsets of ${\mathbb {R}}^n$. The novelty was that the boundary of the domain was not assumed to be Lipschitz regular. In the present article, we continue our study of Sobolev spaces of subanalytic domains and investigate the trace operator on $W^{1,p}(M)$, where M is a bounded subanalytic submanifold of ${\mathbb {R}}^n$, in the case where p is large. This manifold M may of course admit singularities in its closure which are not metrically conical.

The trace operator plays a crucial role in the theory of partial differential equations, as it helps to find weak formulations of the problems. This theory, which is very satisfying on domains that have Lipschitz regular boundary, is much more challenging when singularities arise [15]. Our ultimate aim is therefore to develop all the material necessary to find weak formulations of basic problems of PDE on a subanalytic or semi-algebraic open subset of ${\mathbb {R}}^n$, such as for instance elliptic differential equations with Dirichlet boundary conditions. Our method, which relies on constructions and techniques that emanate from real algebraic geometry from which many algorithms were implemented, is highly likely to provide effective algorithms for computing approximations of solutions of partial differential equations on semi-algebraic domains.

We first show that if a subanalytic bounded submanifold $M\subset {\mathbb {R}}^n$ is normal (Definition 2.6), i.e., if M is connected at every point of its frontier, then, for every sufficiently large p (not infinite), the set ${\mathscr {C}}^\infty ( {{\overline{M}}})$ is dense in $W^{1,p}(M)$ (Theorem 2.7 (i)). The condition of being normal is proved to be necessary (Corollary 3.10). All the results that we establish are no longer true if we drop the assumption “p large enough”, and the value from which the theorem holds heavily depends on the Lipschitz geometry of the manifold near the singularities of the boundary. This result, which generalizes the famous theorem which is known on manifolds with Lipschitz boundary, is useful to define the trace, and we prove that, if A is a subanalytic subset of ${{\overline{M}}}\setminus M$ then the operator ${\mathscr {C}}^\infty ( {{\overline{M}}})\ni u\mapsto u_{|A}\in L^p(A)$ is bounded, if A is endowed with the Hausdorff measure (for p large enough, Theorem 2.7 (ii)). We also show (Theorem 2.7 (iii), still assuming the manifold to be normal, see also Remark 2.8 (2)) that, when A has pure dimension k, every function that belongs to the kernel of this operator can be approximated by functions of ${\mathscr {C}}^\infty ( {{\overline{M}}})$ that vanish in the vicinity of A. In particular, compactly supported functions are dense among the functions that vanish on the boundary (Corollary 3.9). The situation is actually a bit tricky here and one has to be careful when the set A is composed by several sets of different dimension (the corresponding Hausdorff measure has then to be considered on the different parts) and this is the reason why the result is stated with a stratification.

The basic idea of our approach relies on the very precise description of the Lipschitz geometry obtained by the second author in [24, 25] (Theorem 1.3), which is the natural generalization to Lipschitz geometry of the topological conic structure, well-known to subanalytic and o-minimal geometers [5, 7]. This theorem makes it possible for us to construct mollifying operators near singularities.

In order to deal with the case of non necessarily normal manifolds, we introduce and construct in the last section the ${\mathscr {C}}^\infty$ normalizations, which are inspired from the ${\mathscr {C}}^0$ normalizations of pseudomanifolds [6, 9]. This enables us to define a multi-valued trace by considering a normalization of the manifold, and very close results about density of smooth functions and trace operators can then be achieved. Finally, let us point out, that since our construction of ${\mathscr {C}}^\infty$ normalizations increase the codimension of the underlying manifold, the case of a submanifold of positive codimension in ${\mathbb {R}}^n$ is useful even if one is only interested in the case of an open subset of ${\mathbb {R}}^n$.

We thank the anonymous referees for their useful remarks which helped to improve the content of the manuscript.

1.1 Some notations

Throughout this article, n, j, and k will stand for integers. By “manifold” we will always mean ${\mathscr {C}}^\infty$ manifold and by “smooth mapping” we mean ${\mathscr {C}}^\infty$ mapping. The letter

M will stand for a bounded subanalytic submanifold of ${\mathbb {R}}^n$ and m for its dimension.

The origin of ${\mathbb {R}}^n$ will be denoted $0_{{\mathbb {R}}^n}$. When the ambient space will be obvious from the context, we will however omit the subscript ${\mathbb {R}}^n$. We write $<,>$ for the euclidean inner product of this space and |.| for the euclidean norm. For $x\in {\mathbb {R}}^n$ and $\varepsilon >0$, we respectively denote by ${\textbf{S}}(x,\varepsilon )$ and ${\textbf{B}}(x,\varepsilon )$ the sphere and the open ball of radius $\varepsilon$ that are centered at x (for the euclidean norm), while $\overline{{\textbf{B}}}(x,\varepsilon )$ will stand for the corresponding closed ball. Given a subset A of ${\mathbb {R}}^n$, we denote the closure of A by $\overline{A}$ and set $\delta A=\overline{A}\setminus A$.

The derivative of a mapping $f:M\rightarrow {\mathbb {R}}^k$ at x will be denoted by $D_xf$ and its norm as a linear map by $|D_xf|$, where the tangent space $T_xM$ is endowed with the euclidean norm. Given a mapping $h:M\rightarrow M$, we write $\text{ jac }\, h(x)$ for the absolute value of the determinant of $D_xh$ (defined at the points where h is differentiable).

We denote by ${\mathcal {H}}^k$ the k-dimensional Hausdorff measure and by $L^p(A,{\mathcal {H}}^k)$ the set of $L^p$ measurable functions on A for the measure ${\mathcal {H}}^k$. 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 ${\mathbb {R}}^n$.

A mapping $h:A\rightarrow B$, $A\subset {\mathbb {R}}^n, B\subset {\mathbb {R}}^k$, is Lipschitz if there is a constant C such that $|h(x)-h(x')|\le C|x-x'|$ for all x and $x'$. 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 A as well as a subset B of A, we write “$\xi \lesssim \zeta$ on B” or “$\xi (x)\lesssim \zeta (x)$ for $x\in B$” when there is a constant C such that $\xi (x) \le C\zeta (x)$ for all $x\in B$.

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

We then let

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

denote the Sobolev space, where $\partial u$ stands for the gradient of u in the sense of distributions.

We will several times make use of the fact that a bi-Lipschitz homeomorphism relating two smooth manifolds $h:M\rightarrow M'$ identifies the respective Sobolev spaces and that we have $\partial (u\circ h)=^{{\textbf{t}}}\hspace{-2mm}Dh (\partial u\circ h)$ almost everywhere, for all $u\in W^{1,p}(M')$ (see for instance [4, proof of Theorem III.2.13]). It is also well-known that this space, equipped with the norm

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

is a Banach space, in which ${\mathscr {C}}^\infty (M)$ is dense for all $p\in [1,\infty )$. We will regard ${\mathscr {C}}^\infty ( {{\overline{M}}})$ as a subset of $W^{1,p}(M)$, and we will write ${\mathscr {C}}_0^\infty (M)$ for the set of elements of ${\mathscr {C}}^\infty (M)$ that are compactly supported.

We write $p'$ for the Hölder conjugate of p, i.e., $p'=\frac{p}{p-1}$. We will refer to the following version of Young’s inequality for convolution [1, 2.25]:

$$\begin{aligned} ||u*v||_{L^p(M)}\le ||u||_{L^p(M)}||v||_{L^1(M)}, \end{aligned}$$

for any $u\in L^p(M)$ and $v\in L^1(M)$, $p\in [1,\infty )$.

2 Subanalytic sets

We refer the reader to [3, 8, 14, 25] for all the basic facts about subanalytic geometry. Actually, [25] 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

$$\begin{aligned} E \cap U = \bigcup _{i=1}^r\bigcap _{j=1} ^{s_i} \{x \in U : g_{ij}(x) > 0 \text{ and } f_{ij}(x) = 0\}. \end{aligned}$$

(1.1)

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 $\pi :{\mathbb {R}}^n\times {\mathbb {R}}^k\rightarrow {\mathbb {R}}^n$ 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 ${\mathcal {V}}_n : {\mathbb {R}}^n \rightarrow (-1,1) ^n$ is the homeomorphism defined by

$$\begin{aligned} {\mathcal {V}}_n(x_1, \dots , x_n) := \left( \frac{x_1}{\sqrt{1+|x|^2}},\dots , \frac{x_n}{\sqrt{1+|x|^2}}\right) . \end{aligned}$$

We say that a mapping $f:A \rightarrow B$ 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 [5, 7]).

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 [5, 7], and consequently admits cell decompositions and stratifications, from which it comes down that definable sets enjoy a large number of finiteness properties (see [5, 25] for more).

Clearly, a bounded subset of ${\mathbb {R}}^n$ is subanalytic if and only if it is globally subanalytic. This is the reason why we generally do not mention “globally” when working with bounded sets. Note that a subanalytic function f on a bounded set A may be unbounded and therefore not globally subanalytic.

Our results about $L^p$ functions will be valid for sufficiently large p, which means that there will be $p_0\in {\mathbb {R}}$ such that the claimed fact will be true for all $p\in (p_0,\infty )$. 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 fundamental work of S. Łojasiewicz [13], 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 (2.4)):

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 $\lim \limits _{t \rightarrow 0} f(\gamma (t))=0$ for every globally subanalytic arc $\gamma :(0,\varepsilon ) \rightarrow A$ satisfying $\lim \limits _{t \rightarrow 0} g(\gamma (t))=0$.

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

$$\begin{aligned} |f(x)|^\nu \le C|g(x)|. \end{aligned}$$

See for instance [19, Proposition 1.1] for a proof. The main ingredient of our approach is the following result achieved by the second author in [24, Theorem 3.1] (see also [25] for a complete expository of the theory) which describes the conic structure of subanalytic sets from the metric point of view. In the theorem below, $x_0* ({\textbf{S}}(x_0,\varepsilon )\cap X)$ stands for the cone over ${\textbf{S}}(x_0,\varepsilon )\cap X$ with vertex at $x_0$.

Theorem 1.3

(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

$$\begin{aligned} H: x_0* ({\textbf{S}}(x_0,\varepsilon )\cap X)\rightarrow \overline{ {\textbf{B}}}(x_0,\varepsilon ) \cap X, \end{aligned}$$

satisfying $H_{| {\textbf{S}}(x_0,\varepsilon )\cap X}=Id$, preserving the distance to $x_0$, and having the following metric properties:

• (i) The natural retraction by deformation onto $x_0$

  $$\begin{aligned} r:[0,1]\times \overline{ {\textbf{B}}}(x_0,\varepsilon )\cap X \rightarrow \overline{ {\textbf{B}}}(x_0,\varepsilon )\cap X, \end{aligned}$$

  defined by

  $$\begin{aligned} r(s,x):=H(sH^{-1}(x)+(1-s)x_0), \end{aligned}$$

  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.

• (ii) For each $\delta >0$, the restriction of $H^{-1}$ to $\{x\in X:\delta \le |x-x_0|\le \varepsilon \}$ is Lipschitz and, for each $s\in (0,1]$, the map $r_s^{-1}:\overline{ {\textbf{B}}}(x_0,s\varepsilon ) \cap X\rightarrow \overline{ {\textbf{B}}}(x_0,\varepsilon ) \cap X$ is Lipschitz.

Remark 1.4

It follows from [24, Remark 3.6] (see also [25, Remark 3.4.2]) that, given finitely many definable set germs $X_1,\dots ,X_k$ at $x_0\in \cap _{i=1}^k X_i$, the respective homeomorphisms of the Lipschitz conic structure of the $X_i$’s provided by Theorem 1.3 can be required to be induced by the same homeomorphism $H:x_0*{\textbf{S}}(x_0,\varepsilon )\rightarrow \overline{ {\textbf{B}}}(x_0,\varepsilon )$.

Remark 1.5

The Lipschitz constant of $r_s^{-1}$ (see (ii)) is bounded away from infinity if s stays bounded away from 0. Indeed, if $s\ge \delta >0$ then $r_\delta =r_{\frac{\delta }{s}}\circ r_s$ entails $r_s^{-1}=r_{\frac{\delta }{s}}\circ r_\delta ^{-1}$, and the Lipschitz constants of both $r_{\frac{\delta }{s}}$ and $r_\delta ^{-1}$ are bounded independently of $s\ge \delta$.

2.1 Stratifications

Definition 1.6

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 $\Sigma '$ compatible with the strata of $\Sigma$.

A stratification $\Sigma$ of a set X is locally bi-Lipschitz trivial if for every $S\in \Sigma$, there are an open neighborhood $V_S$ of S in X and a smooth retraction $\pi _S:V_S\rightarrow S$ such that every $x_0\in S$ has an open neighborhood W in S for which there is a bi-Lipschitz homeomorphism

$$\begin{aligned} \Lambda :\pi _S^{-1}(W)\rightarrow \pi _S^{-1}(x_0) \times W, \end{aligned}$$

satisfying:

• (i) $\pi _S(\Lambda ^{-1}(x,y))= y$, for all $(x,y)\in \pi _S^{-1}(x_0)\times W$.

• (ii) $\Sigma _{x_0}:=\{ \pi _S^{-1}(x_0)\cap Y:Y\in \Sigma \}$ is a stratification of $\pi _S^{-1}(x_0)$, and $\Lambda (\pi _S^{-1}(W)\cap Y)=(\pi _S^{-1}(x_0)\cap Y)\times W$, for all $Y\in \Sigma$.

T. Mostowski [16, Proposition 1.2] proved that every complex analytic set admits a locally bi-Lipschitz trivial stratification. This result was extended to the subanalytic category by A. Parusiński [18], to polynomially bounded o-minimal structures expanding ${\mathbb {R}}$ in [17], and to polynomially bounded o-minimal structures expanding an arbitrary real closed field in [12]. The second author gives in [26] a construction of locally bi-Lipschitz trivial stratifications for definable sets in polynomially bounded o-minimal structures (expanding ${\mathbb {R}}$) for which the local trivializations are in addition definable. The stratifications constructed in [17, 18, 26] can be required to be compatible with finitely many definable sets, and therefore, to refine a given stratification. Hence, the theorem below follows from [18, Theorem 1.4], [17, Theorem 2.6], or [26, Corollary 1.6.8].

Theorem 1.7

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

Remark 1.8

• (1) All the results of this article are actually still valid with no modification in the proofs in the larger framework provided by polynomially bounded o-minimal structures admitting ${\mathscr {C}}^\infty$ cell decompositions, since existence of locally bi-Lipschitz trivial stratifications goes over this framework [12, 17]. With minor modifications in the proofs, one could actually also prove that the trace is an $L^p$ bounded operator without assuming existence of ${\mathscr {C}}^\infty$ cell decompositions (see Remark 3.5).

• (2) A stratification is said to be Whitney (a) regular if for every sequence of points $y_i$ in a stratum Y tending to some point z in another stratum Z in such a way that $T_{y_i}Y$ converges to a vector space $\tau$ (in the Grassmannian), we have $T_z Z\subset \tau$. It is easy to see that if $\Sigma$ is a Whitney (a) regular stratification of ${{\overline{M}}}$ of which M is a stratum, S is a stratum included in $\delta M$, and $\pi :U\rightarrow S$ is a smooth retraction, then $\pi ^{-1}(x)\cap M$ is a smooth manifold for every $x\in S$, after shrinking U to a smaller neighborhood of S if necessary. In the proofs, we will assume that our stratifications refine such a stratification and therefore that $\pi ^{-1}(x)\cap M$ is a manifold (the existence of such a stratification $\Sigma$ is well-known [25]).

• (3) Mostowski’s Lipschitz stratifications provide a trivialization to any given smooth retraction $\pi _S:V_S\rightarrow S$, if S is a stratum. Corollary 1.6.8 of [26] provides a stratification such that a given smooth definable retraction, that can be for instance the closest point retraction (which is definable [25, Proposition 2.4.1]), can be trivialized. Hence, the stratification as provided by the above theorem admits definable smooth retractions onto strata that are bi-Lipschitz trivial.

3 The case of normal manifolds

The definition of the trace will appear in Sect. 2.5 (Theorem 2.7). It requires some preliminary local computations that are presented in Sects. 2.2,  2.3, and 2.4. For this purpose, we assume that $0_{{\mathbb {R}}^n}\in \overline{M}$, and then set for $\eta >0$:

$$\begin{aligned} M^{\eta }:={\textbf{B}}(0_{{\mathbb {R}}^n},\eta )\cap M\;\; \quad \text{ and } \quad \;\; N^{\eta }:={\textbf{S}}(0_{{\mathbb {R}}^n},\eta )\cap M. \end{aligned}$$

Apply now Theorem 1.3 to $X=M\cup \{0_{{\mathbb {R}}^n}\}$ 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 smooth manifold, and let r denote the mapping provided by this theorem. Observe that since r is subanalytic, it is smooth almost everywhere.

We first establish in Sect. 2.1 some estimates that rely on the information we have about the derivative of r. They shed light on the role that Łojasiewicz’s inequality will play.

3.1 Some key facts

There is a positive constant C such that:

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

  $$\begin{aligned} \left| \frac{\partial r}{\partial s}(s,x)\right| \le C|x|. \end{aligned}$$

  (2.2)

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

  $$\begin{aligned} \left( \int _0 ^\eta ||v||_{L^p (N^\zeta )}^p d\zeta \right) ^{1/p} \le ||v||_{L^p (M^\eta )} \le C \left( \int _0 ^\eta ||v||_{L^p (N^\zeta )}^p d\zeta \right) ^{1/p}. \end{aligned}$$

  (2.3)

• (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 $s\in (0,1)$:

  $$\begin{aligned} ||v\circ r_s||_{L^p(N^\eta )}\le Cs^{-\nu /p}||v||_{L^p(N^{s\eta })} , \end{aligned}$$

  (2.4)

The constant C is independent of v.

Proof

Let H be the Lipschitz subanalytic homeomorphism provided by Theorem 1.3. By definition of r, we have $\frac{\partial r}{\partial s}(s,x)=D_{sH^{-1}(x)} H(H^{-1}(x))$ (recall that $x_0=0$), and hence (as $H^{-1}$ preserves the distance to the origin):

$$\begin{aligned} \left| \frac{\partial r}{\partial s}(s,x)\right| \le |D_{sH^{-1}(x)} H|\cdot |H^{-1}(x)|= |D_{sH^{-1}(x)} H|\cdot |x|. \end{aligned}$$

Since H is Lipschitz, $|D_{sH^{-1}(x)}H|$ is bounded by some positive constant $C_1$ independently of x and s, which proves (2.2).

To prove (2.3), let $\rho : M\rightarrow {\mathbb {R}}$ be defined by $\rho (x):=|x|$, which is Lipschitz and note that we have (see for instance [10, Theorem 5.3.9] and the remark just before the cited theorem):

$$\begin{aligned} \int _0 ^\eta \int _{N^\zeta } |v(x)|^p dx d\zeta = \int _{M^\eta } |v(x)|^p \left| \partial \rho (x)\right| dx. \end{aligned}$$

(2.5)

As $|\partial \rho (x)|\le 1$, we immediately get the first inequality. To show the second one, let ${\tilde{\rho }}:=\rho \circ H$ and observe that since H preserves the distance to the origin and $H^{-1}(M^\varepsilon )$ is a cone, we have $\partial {\tilde{\rho }}(y)=\frac{y}{|y|}$, for all y. Consequently, $\partial \rho (x)=\frac{^\textbf{t}D_xH^{-1} (H^{-1}(x))}{|H^{-1}(x)|}$, for almost all x. As H is Lipschitz, there exists some positive constant A such that for all x we have $\inf _{|z|=1} |{^\textbf{t}D_x H^{-1}(z)}|>A$, from which we can conclude that there exists $c>0$ such that $|\partial \rho (x)|>c$ for all x, and therefore

$$\begin{aligned} c^{1/p}||v||_{L^p(M^\eta )}\le \left( \int _{M^\eta } |v(x)|^p \left| \partial \rho (x)\right| dx\right) ^{1/p}\overset{(|xm{2.5})}{=}\left( \int _0 ^\eta ||v||_{L^p (N^\zeta )}^p d\zeta \right) ^{1/p}, \end{aligned}$$

which ends the proof of (2.3).

To show (2.4), observe that since $r_s$ is bi-Lipschitz for every $s>0$, $\text{ jac }\, r_s$ can only tend to zero if s is itself going to zero (see Remark 1.5). Hence, by Łojasiewicz’s inequality (see Proposition 1.2), there are an integer $\nu$ and a constant $C_2$ such that for all $s\in (0,1)$ we have for almost all $x\in M^\varepsilon$:

$$\begin{aligned} \text{ jac }\,r_s(x) \ge \frac{s^{\nu }}{C_2}\, , \end{aligned}$$

(2.6)

and therefore

$$\begin{aligned} ||v\circ r_s||_{L^p (N^\eta )}= \left( \int _{N^\eta }|v\circ r_s|^p\right) ^{1/p}\overset{(2.6)}{\le }C_2^{1/p} \left( \int _{N^\eta } | v(r_s(x))|^p\, \frac{\text{ jac }\, r_s (x)}{s^\nu }\ dx \right) ^{1/p}. \end{aligned}$$

As $r_s(N^\eta )=N^{s\eta }$ we get (2.4). Finally, we put $C:=\max \{C_1,C_2^{1/p}, c^{-1/p}\}$, which does not depend on v. $\square$

Remark 2.1

• (i) Fact (3) is still valid if we replace N with M, i.e., there is $\nu \in {\mathbb {N}}$ such that:

  $$\begin{aligned} ||v\circ r_s||_{L^p(M^\eta )}\lesssim s^{-\nu /p}||v||_{L^p(M^{s\eta })} , \end{aligned}$$

  (2.7)

  for $v\in L^p(M^\varepsilon )$, $p\in [1,\infty )$, $s\in (0,1)$, and $\eta {\le } \varepsilon$. This can be proved directly (like above for $N^\eta$) or deduced from (2.4) and (2.3).

• (ii) In Fact (2), we suppose that v is $L^p$. Actually, (2.5) yields that as soon as $\eta \mapsto ||v||_{L^p(N^\eta )}^p$ is $L^1$ then v is $L^p$ (since $|\partial \rho |$ is bounded below away from zero) and (2.3) holds.

3.2 The operator $\Theta ^M$

Let us set for $u\in W^{1,p}(M^\varepsilon )$ and $x\in M^\varepsilon$:

$$\begin{aligned} \Theta ^M u(x):= \int _0^1\frac{\partial (u\circ r)}{\partial s}(s,x)\ ds= \int _0^1<\partial u(r_s(x)),\frac{\partial r}{\partial s}(s,x)>\, ds. \end{aligned}$$

(2.8)

This definition makes sense (for almost every x) as soon as the considered function is sufficiently integrable, as established by the lemma below.

Lemma 2.2

For p sufficiently large, we have for $u \in W^{1,p}(M^\varepsilon )$ and $t\in [0,1]$

$$\begin{aligned} \int _0^t||\frac{\partial (u\circ r_s)}{\partial s}||_{L^p(M^\varepsilon )} ds \lesssim ||\partial u||_{L^p (M^{t\varepsilon })}. \end{aligned}$$

(2.9)

In particular, the function $[0,1] \ni s \mapsto ||\frac{\partial (u\circ r_s)}{\partial s}||_{L^p(M^\varepsilon )}$ belongs to $L^1([0,1])$ and $\Theta ^M u$ is well-defined for $u \in W^{1,p}(M^\varepsilon )$. Moreover, for $u\in W^{1,p}(M^\varepsilon )$ and $\eta <\varepsilon$, we then have:

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

(2.10)

Proof

For $u \in W^{1,p}(M^\varepsilon )$ and $t\in [0,1]$, we have:

$$\begin{aligned} \int _0^t||\frac{\partial (u\circ r_s)}{\partial s}||_{L^p(M^\varepsilon )} ds= & {} \int _0^t\left( \int _{M^\varepsilon }\left| \frac{\partial (u\circ r)}{\partial s}(s,x)\right| ^p\ dx \right) ^{1/p} ds \\\lesssim & {} \int _0^t\left( \int _{M^\varepsilon } |\partial u(r_s(x))|^p\ dx \right) ^{1/p} ds \quad \text{(since } r \text{ is } \text{ Lipschitz) } \\\overset{(2.7)}{\lesssim } & {} \int _0^t s^\frac{-\nu }{p}||\partial u||_{L^p (M^{s\varepsilon })} ds \\\le & {} ||\partial u||_{L^p (M^{t\varepsilon })}\int _0^t|s|^{-\nu /p}ds ,\\\lesssim & {} ||\partial u||_{L^p (M^{t\varepsilon })} \quad \text{(for } p >\nu \text{) }. \end{aligned}$$

That $\Theta ^M u$ is well-defined now follows from Minkowski’s integral inequality. This proves the first part of the Lemma. For the second part, applying again Minkowski’s integral inequality, we can write:

$$\begin{aligned} ||\Theta ^M u||_{L^p(N^\eta )}\le & {} \int _0^1\left( \int _{N^\eta } |\partial u(r_s(x))|^p\left| \frac{\partial r}{\partial s}(s,x)\right| ^p\ dx \right) ^{1/p} ds \\\overset{(2.2)}{\lesssim } & {} \eta \int _0^1\left( \int _{N^\eta } |\partial u(r_s(x))|^p\ dx \right) ^{1/p} ds \\\overset{(2.4)}{\lesssim } & {} \eta \int _0^1|s|^{-\nu /p} ||\partial u||_{L^p(N^{s\eta })}ds\\\le & {} \eta \left( \int _0^1|s|^{-\nu p'/p}ds\right) ^{1/p'}\left( \int _0^1 ||\partial u||_{L^p(N^{s\eta })}^pds\right) ^{1/p}\\\lesssim & {} \eta \left( \int _0^1 ||\partial u||_{L^p(N^{s\eta })}^pds\right) ^{1/p}\;\; \text{(for } \; p>\nu p'\text{) }\\= & {} \eta ^{1-1/p} \left( \int _0^\eta ||\partial u||^p_{L^p(N^t)}dt\right) ^{1/p}\quad (\hbox { setting}\ t:=s\eta )\\\overset{(2.3)}{\lesssim } & {} \eta ^{1-1/p} ||\partial u||_{L^p(M^\eta )}. \end{aligned}$$

$\square$

Remark 2.3

Observe that (2.10) and (2.3) yield that for p large enough, u and $\eta$ as in (2.10), we have:

$$\begin{aligned} ||\Theta ^M u||_{L^p(M^\eta )}\lesssim \eta ||u||_{W^{1,p}(M^\eta )}. \end{aligned}$$

(2.11)

Indeed,

$$\begin{aligned} ||\Theta ^M u||_{L^p(M^\eta )} \overset{(2.3)}{\lesssim }\left( \int _0^\eta ||\Theta ^M u||_{L^p(N^\xi )}^p\,d\xi \right) ^{1/p} \overset{(2.10)}{\lesssim }\left( \int _0^\eta \xi ^{p-1} ||\partial u||_{L^p(M^\xi )}^p d\xi \right) ^{1/p}\\ \le ||\partial u||_{L^p(M^\eta )}\left( \int _0^\eta \xi ^{p-1}d\xi \right) ^{1/p} = ||\partial u||_{L^p(M^\eta )}\eta p^{-1/p}\le \eta p^{-1/p}||u||_{W^{1,p}(M^\eta )}. \end{aligned}$$

In particular, if C is the constant of (2.3) and $C'$ the constant of (2.10) then $CC' p^{-1/p}$ is the constant of (2.11), which means that it is independent of both u and $\eta$.

3.3 The operator ${\mathscr {R}}^M$

Set now for $u \in W^{1,p}(M^\varepsilon )$ (and p sufficiently large for $\Theta ^M$ to be defined):

$$\begin{aligned} {\mathscr {R}}^M u:=u-\Theta ^M u. \end{aligned}$$

(2.12)

Lemma 2.4

For p sufficiently large, ${\mathscr {R}}^M u$ is constant on every connected component of $M^\varepsilon$, for all $u\in W^{1,p}(M^\varepsilon )$. Moreover, $\Theta ^M$ and ${\mathscr {R}}^M$ are then continuous projections and, in the case where u extends to a continuous function on $\overline{M^\varepsilon }$, we have ${\mathscr {R}}^M u\equiv u(0)$.

Proof

Take p sufficiently large for Lemma 2.2 to hold. For $u\in W^{1,p}(M^\varepsilon )\cap {\mathscr {C}}^\infty (M^\varepsilon )$, $t\in (0,1)$, as well as $x\in M^\varepsilon$, set

$$\begin{aligned} \Theta ^M_t u(x):= \int _t^1\frac{\partial \left( u\circ r\right) }{\partial s}(s,x)\ ds=u(x)-u(r_t(x)). \end{aligned}$$

By Minkowski’s integral inequality, we have for p sufficiently large for such u and t:

$$\begin{aligned} ||\Theta ^M u -\Theta ^M_t u||_{L^p(M^\varepsilon )} \le \int _0 ^t ||\frac{\partial (u\circ r_s)}{\partial s} ||_{L^p(M^\varepsilon )}ds \overset{(2.9)}{\lesssim }||\partial u||_{L^p(M^{t\varepsilon })}. \end{aligned}$$

(2.13)

which must tend to 0 as t goes to 0. Observe also that since $u-\Theta ^M_t u=u\circ r_t$, we have for $t\in (0,1)$:

$$\begin{aligned} |\partial (u-\Theta ^M_t u)|=|^\textbf{t}Dr_t \left( \partial u\circ r_t\right) |\le C t |\partial u\circ r_t|, \end{aligned}$$

(2.14)

where the last inequality follows from Theorem 1.3 (i). By (2.7), the $L^p$-norm of the right-hand-side goes to 0 as t goes to 0, for all p large. But since (2.13) yields that $(u-\Theta ^M_t u)$ converges to $(u-\Theta ^M u)$ in the $L^p$-norm, we now can write:

$$\begin{aligned} \partial ({\mathscr {R}}^Mu)=\partial (u-\Theta ^Mu)=\lim _{t\rightarrow 0}\partial (u-\Theta ^M_t u) \overset{(2.14)}{=}0, \end{aligned}$$

since the $L^p$ convergence yields the convergence as distribution. In particular, ${\mathscr {R}}^Mu$ and $\Theta ^Mu$ belong to $W^{1,p}(M^\varepsilon )$ (for p large). It also means that $\partial \Theta ^Mu=\partial u$. Thanks to (2.11) and the density of smooth functions in Sobolev spaces, we derive that $\Theta ^M$ uniquely extends to a continuous operator on $W^{1,p}(M^\varepsilon )$. Remark then that $\Theta ^M {\mathscr {R}}^M u=0$ (since ${\mathscr {R}}^M u$ is locally constant), so that, applying $\Theta ^M$ to (2.12), we see that $\Theta ^M$ is a projection.

Finally, in the case where u extends continuously on $\overline{M^\varepsilon }$, $u\circ r_t$ tends uniformly to the constant function u(0). Thanks to the definition of $\Theta ^M_t$ and (2.13), we therefore see that $\Theta ^M u=u-u(0)$. $\square$

3.4 Mollifying with parameters

One ingredient of the smoothing process of the proof of Theorem 1.3 will be a “mollifying operator along the strata”. We give some facts needed for this purpose in this section.

Let $F\subset {\mathbb {R}}^n$ be a subanalytic manifold and $p\in [1,\infty )$. For $v\in L^p(F\times {\mathbb {R}}^k)$, we define two families of functions on F and ${\mathbb {R}}^k$ respectively, parameterized by ${\mathbb {R}}^k$ and F respectively, by setting for almost every $(x,y)\in F\times {\mathbb {R}}^k$

$$\begin{aligned} u^y(x)=u_x(y):=u(x,y). \end{aligned}$$

(2.15)

For $u\in L^p(F\times {\mathbb {R}}^k)$ and $\psi \in L^1({\mathbb {R}}^k)$ we put for almost every $(x,y)\in F\times {\mathbb {R}}^k$

$$\begin{aligned} u*_k\psi (x,y)=\int _{{\mathbb {R}}^k}u(x,z)\psi (y-z)dz=u_x*\psi (y). \end{aligned}$$

(2.16)

Let $\varphi : {\mathbb {R}}^k\rightarrow {\mathbb {R}}$ be a nonnegative smooth compactly supported function satisfying $\int \varphi =1$, and set $\varphi _\sigma (z)=\frac{1}{\sigma ^k}\varphi (\frac{z}{\sigma })$. We then define a mollifying operator $\Phi _\sigma$, by setting for $\sigma >0$ and $u\in L^p(F\times {\mathbb {R}}^k)$:

$$\begin{aligned} \Phi _\sigma u(x,y)=u*_k \varphi _\sigma (x,y)=\int _{{\mathbb {R}}^k} u_x(y-z)\varphi _\sigma (z)\ dz. \end{aligned}$$

(2.17)

Lemma 2.5

• (i) For $u\in L^p(F\times {\mathbb {R}}^k)$ and $\psi \in L^1({\mathbb {R}}^k)$ we have

  $$\begin{aligned} ||u*_k\psi ||_{L^p(F\times {\mathbb {R}}^k)}\le ||u||_{L^p(F\times {\mathbb {R}}^k)} ||\psi ||_{L^1( {\mathbb {R}}^k)}. \end{aligned}$$

  (2.18)

• (ii) If $u\in W^{1,p}(F\times {\mathbb {R}}^k)$ then $\Phi _\sigma u(x,y)$ belongs to $W^{1,p}(F\times {\mathbb {R}}^k)$ and is ${\mathscr {C}}^\infty$ with respect to y. Moreover,

  $$\begin{aligned} ||\Phi _\sigma u||_{W^{1,p}(F\times {\mathbb {R}}^k)}\le ||u||_{W^{1,p}(F\times {\mathbb {R}}^k)}\quad \text{ and }\quad \lim _{\sigma \rightarrow 0}||\Phi _\sigma u-u||_{W^{1,p}(F\times {\mathbb {R}}^k)}=0. \end{aligned}$$

  (2.19)

Proof

By Fubini’s Theorem and Young’s inequality, we have

$$\begin{aligned} ||u*_k\psi ||_{L^p(F\times {\mathbb {R}}^k)}=\left( \int _F||\left( u*_k\psi \right) _x||^p_{L^p({\mathbb {R}}^k)} dx\right) ^{1/p} \overset{(2.16)}{=}\left( \int _F|| u_x * \psi ||^p_{L^p({\mathbb {R}}^k)} dx\right) ^{1/p} \end{aligned}$$

$$\begin{aligned} \le \left( \int _F|| u_x ||^p_{L^p({\mathbb {R}}^k)} dx\right) ^{1/p}\cdot ||\psi ||_{L^1( {\mathbb {R}}^k)}=||u||_{L^p(F\times {\mathbb {R}}^k)} \cdot ||\psi ||_{L^1( {\mathbb {R}}^k)} , \end{aligned}$$

(2.20)

which shows (2.18). To show (ii) observe first that thanks to the properties of the convolution product

$$\begin{aligned} \frac{\partial }{\partial y_i}(u*_k\varphi _\sigma ) (x,y)\overset{(2.16)}{=}\frac{\partial }{\partial y_i} (u_x*\varphi _\sigma ) (x,y) =u_x*\frac{\partial \varphi _\sigma }{\partial y_i} (x,y), \end{aligned}$$

(2.21)

as distribution. Hence, $u*_k\varphi _\sigma$ must be smooth with respect to y. Note that (2.18) shows that $||\Phi _\sigma u||_{L^p(F\times {\mathbb {R}}^k)} \le ||u||_{L^p(F\times {\mathbb {R}}^k)}$ if $u\in L^p(F\times {\mathbb {R}}^k)$. To prove the first claim of (2.19), it thus suffices to show $||\partial \Phi _\sigma u||_{L^p(F\times {\mathbb {R}}^k)} \lesssim ||\partial u||_{L^p(F\times {\mathbb {R}}^k)}$, for $u\in W^{1,p}(F\times {\mathbb {R}}^k)$. For such u, let $\partial ^F u$ be the partial derivative along F, i.e., $\partial ^F u(x,y):=\partial (u^y)(x)\in T_x F$, so that we have $\partial u =(\partial ^F u, \frac{\partial u}{\partial y_1}, \dots , \frac{\partial u}{\partial y_k})$. By the same argument as in (2.21)

$$\begin{aligned} \frac{\partial }{\partial y_i}(u*_k\varphi _\sigma ) (x,y)=\frac{\partial u}{\partial y_i} *_k\varphi _\sigma (x,y), \end{aligned}$$

and since $\varphi _\sigma$ is independent of x, we clearly have $\partial ^F (u*_k\varphi _\sigma ) (x,y)=(\partial ^F u)*_k\varphi _\sigma (x,y)$ (defining the convolution of a vector by the convolution of each of its components). It means that applying (2.18) to both $\partial ^F u$ and $\frac{\partial u}{\partial y_i}$ (with $\psi =\varphi _\sigma$) yields the first claim of (2.19). It also means that the second claim reduces to show that $u*_k \varphi _\sigma$ tends to u in the $L^p$ norm as $\sigma$ tends to zero for all $u\in L^p(F\times {\mathbb {R}}^k)$. By (2.20) and the density of ${\mathscr {C}}^\infty _0$ in the $L^p$ space, we can assume $u\in {\mathscr {C}}^\infty _0(F\times {\mathbb {R}}^k)$. For such u, as $(u_x*\varphi _\sigma )(y)$ tends to $u_x(y)$ uniformly in x and y (see [1], p. 37), the needed fact is clear. $\square$

3.5 The trace on a normal manifold

Definition 2.6

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

Our constructions will be inductive on the dimension of the strata that meet the support of the considered function, which requires to regard the support as a subset of $\overline{M}$. We start by introducing the necessary notations on this issue.

Let U be an open subset of M and let V be an open subset of ${{\overline{M}}}$ satisfying $V\cap M=U$. For a distribution u on U, we define $\text{ supp}_V u$ (the support of u in V) as the closure in V of the support of u, denoted $\text{ supp }\, u$. When $\text{ supp}_V u$ is compact, we will say that u has compact support in V.

If B is an open subset of ${{\overline{M}}}$ containing M, regarding the elements of ${\mathscr {C}}^\infty ( {{\overline{M}}})$ as functions on M, we then can set

$$\begin{aligned} {\mathscr {C}}^\infty _{B}( {{\overline{M}}}):=\{u\in {\mathscr {C}}^\infty ({\overline{M}}): \text{ supp}_B u \; \text{ is } \text{ compact } \}. \end{aligned}$$

Observe that, as M is bounded, given an $L^p$ function $u:M\rightarrow {\mathbb {R}}$, $\text{ supp}_ {{\overline{M}}}u\cap \delta M=\emptyset$ if and only if u is compactly supported (in M).

Theorem 2.7

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

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

• (ii) The linear operator

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

  is bounded for $||\cdot ||_{W^{1,p}(M)}$ and thus extends to a mapping $\textbf{tr}_A:W^{1,p}(M)\rightarrow L^p(A,{\mathcal {H}}^k)$.

• (iii) If ${\mathcal {S}}$ is a stratification of A, then ${\mathscr {C}}^\infty _{\overline{M}\setminus \overline{A}}( {{\overline{M}}})$ is a dense subspace of $\bigcap \limits _{Y\in {\mathcal {S}}}\ker \textbf{tr}_Y$.

Proof of (i)

Fix a locally bi-Lipschitz trivial stratification $\Sigma$ of $\overline{M}$ compatible with $\delta M$. The idea is that, using some “cut-off functions” (see (2.23)) together with the mollifying operators constructed in Sect. 2.4, we can reduce our approximation problem to approximating a function of $W^{1,p}(M)$ that vanishes near the strata of dimension zero, and then to approximating a function of $W^{1,p}(M)$ that vanishes near the strata of dimension less than or equal to one, and so on. Eventually, it remains to approximate a function that vanishes near $\delta M$, which is easy since such a function is compactly supported. More concretely, we will argue by induction on the dimension of the smallest stratum that meets the closure of the support of the function that we wish to approximate by smooth functions, the first step of the induction being the case of compactly supported functions. We will consider functions defined on $V\cap M$, with V open subset of ${{\overline{M}}}$, for we will work up to a partition of unity subordinated to a finite covering of ${{\overline{M}}}$. We thus set, for $u\in W^{1,p}(V\cap M)$, with V an open subset of $\overline{M}$,

$$\begin{aligned} \kappa _u: =\min \{\dim S\,:S\in \Sigma , \, S\subset \delta M,\, S\cap \text{ supp}_{V} u\ne \emptyset \}, \end{aligned}$$

(2.22)

with the convention that $\kappa _u=m$ whenever this set is empty, and we will show by decreasing induction on k:

$({\textbf{A}}_k).$ Let V be as above and $V'$ be an open subset satisfying $\overline{V'}\subset V$. Given $u\in W^{1,p}(V\cap M)$ satisfying $\kappa _u\ge k$, there is a family of functions $v_\mu \in W^{1,p}(V\cap M)\cap {\mathscr {C}}^\infty (V)$, $\mu >0$, such that $||u-v_\mu ||_{W^{1,p}(V'\cap M)}$ tends to zero as $\mu$ goes to zero.

The desired fact follows from $({\textbf{A}}_0)$ for $V=V'= {{\overline{M}}}$. It is well-known that functions that are compactly supported in M can be approximated by smooth compactly supported functions, which already means that $({\textbf{A}}_m)$ holds true. We thus fix $k< m$, assume that $({\textbf{A}}_j)$ holds for all $j>k$, and take a function $u\in W^{1,p}(V\cap M)$, $V\subset \overline{M}$ open, with $\kappa _u\ge k$.

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 $\overline{V'}$ is compact, we can extract a finite covering of $\overline{V'}$ by such open sets. Notice that if $(\phi _i)_{i\in I}$ is a smooth partition of unity subordinated to a finite covering $(U_i)_{i\in I}$ of $\overline{V'}$ then, as $u=\sum _{i \in I} \phi _i u$, if $u_{i,j}:U_i\cap {{\overline{M}}}\rightarrow {\mathbb {R}}$ is a sequence of smooth functions tending to $\phi _i u$ in $W^{1,p}(U'_i\cap M)$ as $j\rightarrow \infty$ (like provided by $({\textbf{A}}_k)$ for $\phi _i u$), with $U'_i$ neighborhood of $\text{ supp }\, \phi _i$ satisfying $\overline{U_i'}\subset U_i$, and $\lambda _i$ is a smooth compactly supported function on $U_i'$ that is equal to 1 on $\text{ supp }\, \phi _i$, then $\lambda _i u_{i,j}$ tends to $\phi _i u$. In other words, we may work up to such a partition of unity. We will thus assume that $\text{ supp}_{V} u$ 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 $U_{x_0}$. Moreover, in the case where $\dim S>k$, the result follows by our decreasing induction on $\kappa _u$ (since $\text{ supp}_{V}u\subset U_{x_0}$ which cannot meet any stratum of dimension smaller than $\dim S$), which means that we can assume that $\dim S\le k$. As $k\le \kappa _u$, we will thus assume that $\dim S=\kappa _u =k$ and, without loss of generality, we will do the proof for $x_0=0_{{\mathbb {R}}^n}\in \delta M$.

We start with the case $k=0$, i.e., we assume for the moment that $S=\{0_{{\mathbb {R}}^n}\}$. Since we argue by decreasing induction on k, this corresponds to the last step of the induction but we begin with this case because it is much easier and will help the reader to understand the case $k>0$ which is a natural generalization. We will also suppose that $\text{ supp}_{V} u\subset {\textbf{B}}(0_{{\mathbb {R}}^n},\varepsilon )$, with $\varepsilon$ sufficiently small for the operators $\Theta ^M$ and ${\mathscr {R}}^M$ to be defined on $M^\varepsilon$ (refining our partition of unity if necessary).

As M is connected at $0_{{\mathbb {R}}^n}$, ${\mathscr {R}}^Mu$ is constant on $M^\varepsilon$ (see Lemma 2.4), which implies that it is a smooth function. It thus suffices to approximate $\Theta ^M u$.

Let $\psi :{\mathbb {R}}\rightarrow [0,1]$ 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 $\mu$ positive and $x\in M$, $\psi _\mu (x):=\psi \left( \frac{|x|^2}{\mu ^2}\right)$. As $\partial \psi _\mu (x)=2\psi '\left( \frac{|x|^2}{\mu ^2}\right) \cdot \frac{P_x(x)}{\mu ^2}$, where $P_x$ denotes the orthogonal projection onto $T_xM^\varepsilon$, and since $\psi '\left( \frac{|x|^2}{\mu ^2}\right) =0$ for $|x|\ge \mu$ we have

$$\begin{aligned} \sup _{x\in M} |\partial \psi _\mu (x)|\lesssim 1/\mu \quad \text{ and } \quad \text{ supp}_{M} |\partial \psi _\mu |\subset M^{\mu }. \end{aligned}$$

(2.23)

We claim that this entails that $v_\mu :=\psi _\mu \cdot \Theta ^M u$ tends to $\Theta ^M u$ in $W^{1,p}(M^\varepsilon )$ for all p large enough as $\mu$ goes to 0. Indeed, $\psi _\mu \cdot u$ clearly tends to u in the $L^p$ norm. Moreover, as $\partial \Theta ^Mu=\partial u$ (since ${\mathscr {R}}^M u$ is constant), by the product rule we have $\partial (\psi _\mu \Theta ^M u)= \Theta ^M u\cdot \partial \psi _\mu +\psi _\mu \partial u$ so that

$$\begin{aligned} || \partial (\psi _\mu \Theta ^M u)-\partial u||_{L^p(M^\varepsilon )}\le || \Theta ^M u\cdot \partial \psi _\mu ||_{L^p(M^\varepsilon )}+||( \psi _\mu -1) \partial u||_{L^p(M^\varepsilon )} \end{aligned}$$

(2.24)

$$\begin{aligned} \overset{(2.3)}{\lesssim }\frac{1}{\mu } || \Theta ^M u ||_{L^p(M^\mu )} +||( \psi _\mu -1) \partial u||_{L^p(M^\varepsilon )}. \end{aligned}$$

which tends to 0 by (2.11) (since $\psi _\mu$ is bounded and tends to 1 pointwise). As $\kappa _{v_\mu } >0$, this completes the induction step in the case $k=0$.

We now deal with the case $k>0$ that we shall address with a similar argument. The situation is however more complicated and we shall need to apply a mollifier along the stratum S.

By definition of bi-Lipschitz triviality, there is a bi-Lipschitz homeomorphism $\Lambda : U_0 \rightarrow (\pi ^{-1}(0_{{\mathbb {R}}^n})\cap \overline{M})\times W_0$ (here $U_0=U_{0_{{\mathbb {R}}^n}}$), where $\pi :U_0 \rightarrow S$ is a definable (see Remark 1.8 (3)) smooth retraction and $W_0$ is a neighborhood of the origin in S that, up to a coordinate system of S, we can identify with ${\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )$, $\alpha >0$. Notice that the sets

$$\begin{aligned} F:=\pi ^{-1}(0_{{\mathbb {R}}^n}) \cap M \quad \text{ and } \quad U':=F\times {\textbf{B}}(0_{{\mathbb {R}}^k},\alpha ) \end{aligned}$$

are smooth manifolds (see Remark 1.8 (2)). As in (2.15), if v is a function on $U'$, we put for $(x,y)\in U'$, $v_x(y)=v^y(x):=v(x,y).$ Let us also set for $v\in W^{1,p}(U')$

$$\begin{aligned} {\tilde{\Theta }}v(x,y)=\Theta ^Fv^y(x), \end{aligned}$$

(2.25)

where $\Theta ^F$ is the operator defined in Sect. 2.2, for the manifold F (we assume here that $U_0$ is sufficiently small for $\Theta ^F$ to be defined).

In order to generalize the argument we used in the case $k=0$, we first check that ${\tilde{\Theta }}$ satisfies a similar inequality as $\Theta ^F$ (see (2.11)). Indeed, if we set $F^\eta :=F\cap {\textbf{B}}(0_{{\mathbb {R}}^n},\eta )$, then, by (2.11) for $\Theta ^F$ (and $u:=v^y$), we see that there is a constant C (independent of y) such that for $v\in W^{1,p}(F\times {\textbf{B}}(0_{{\mathbb {R}}^k},\alpha ))$ (for all p large enough)

$$\begin{aligned} ||({\tilde{\Theta }} v)^y||^p_{L^p(F^\eta )}\le C \eta ^{p}|| v^y||^p_{W^{1,p}(F^\eta )}. \end{aligned}$$

Integrating with respect to y, we deduce that

$$\begin{aligned} ||{\tilde{\Theta }} v||_{L^p(E^\eta )} \lesssim \eta || v||_{W^{1,p}(E^\eta )}, \end{aligned}$$

(2.26)

where $E^\eta =F^\eta \times {\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )$. In particular, ${\tilde{\Theta }} v$ is $L^p$ if p is sufficiently large. Let $\Phi _\sigma$ be the operator defined in (2.17). By Lemma 2.5, if p is sufficiently large then $\Phi _\sigma v \in W^{1,p}(E^\eta )$. By Fubini’s Theorem, we therefore have for such p:

$$\begin{aligned} {\tilde{\Theta }}\Phi _\sigma =\Phi _\sigma {\tilde{\Theta }}. \end{aligned}$$

(2.27)

We now claim that this entails that ${\tilde{\Theta }} \Phi _\sigma v$ belongs to $W^{1,p}(U')$ for every v in this space. Indeed, thanks to Lemma 2.5 (ii) and (2.26), we see that ${\tilde{\Theta }}\Phi _\sigma v$ is $L^p$. Moreover, by (2.27)

$$\begin{aligned} \frac{\partial }{\partial y_i} {\tilde{\Theta }} \Phi _\sigma v(x,y)=\frac{\partial }{\partial y_i} \Phi _\sigma {\tilde{\Theta }}v(x,y)= ({\tilde{\Theta }}v)_x * \frac{\partial \varphi _\sigma }{\partial y_i}(y){\overset{(2.16)}{=}({\tilde{\Theta }}v) *_k \frac{\partial \varphi _\sigma }{\partial y_i}(x,y)}, \end{aligned}$$

which, by (2.18), belongs to $L^p(U')$. Furthermore, as ${\tilde{\Theta }}\Phi _\sigma v(x,y)=\Theta ^F (\Phi _\sigma v)^y(x)$, Lemma 2.4 entails that the partial derivative of ${\tilde{\Theta }}\Phi _\sigma v$ with respect to x is also $L^p$, yielding the claimed fact.

As $\Lambda$ is bi-Lipschitz, the function $u':=u\circ \Lambda ^{-1}$ belongs to $W^{1,p}(U')$. By (2.27), we thus can write

$$\begin{aligned} \Phi _\sigma u'=\Phi _\sigma (u'-{\tilde{\Theta }}u')+{\tilde{\Theta }}\Phi _\sigma u'. \end{aligned}$$

(2.28)

Let $f_\sigma$ and $g_\sigma$ respectively denote the first and second term composing the right-hand-side.

We first check that $f_\sigma \circ \Lambda$ is smooth. Since M is normal, by Lemma 2.4, $f_\sigma (x,y)$ is constant with respect to x and, as this function is smooth with respect to y (see Lemma 2.5), it induces a smooth function ${\tilde{f}}$ on $W_0$. As a matter of fact, since $\pi (\Lambda ^{-1}(x,y))=y$, we see that $f_\sigma \circ \Lambda (z)= {\tilde{f}}\circ \pi (z)$ is smooth.

But since, by Lemma 2.5, $\Phi _\sigma u'$ tends to $u'$ in $W^{1,p} (U')$ as $\sigma$ goes to zero, thanks to (2.28), it means that we just have to find arbitrarily close smooth approximations of $g_\sigma \circ \Lambda$ in $W^{1,p}(U_0)$ (for each $\sigma >0$ small), which, thanks to our induction hypothesis, reduces to checking that there are arbitrarily close approximations v of $g_\sigma$ (in $W^{1,p}(U')$) satisfying $\kappa _v>k$.

For this purpose, fix $\sigma >0$ and let for $\mu$ positive ${\widetilde{\psi }}_\mu$ denote the “cut-off function” defined by ${\widetilde{\psi }}_\mu (x,y)=\psi (\frac{|x|^2}{\mu ^2})$, for $(x,y)\in U'$, where $\psi :{\mathbb {R}}\rightarrow [0,1]$ is (as in the case $k=0$) a ${\mathscr {C}}^\infty$ function such that $\psi \equiv 0$ on $(-\infty ,\frac{1}{2})$ and $\psi \equiv 1$ near $[1,+\infty )$. We claim that ${\tilde{v}}_\mu :={\widetilde{\psi }}_\mu \cdot g_\sigma$ is the desired sequence of approximations of $g_\sigma$.

Indeed, as ${\tilde{v}}_\mu$ tends to $g_\sigma$ in the $L^p$ norm, it actually suffices to establish the convergence of the first derivative. Remark that since $\text{ supp}_{U'} |\partial {\widetilde{\psi }}_\mu | \subset E^{\mu }$ and $\sup _{U'} |\partial {\tilde{\psi }}_\mu |\lesssim \frac{1}{\mu }$, we have (for all p large enough)

$$\begin{aligned} ||g_\sigma \cdot \partial {\widetilde{\psi }}_\mu ||_{L^p(U')} \lesssim \frac{||g_\sigma ||_{L^p(E^{\mu })} }{\mu } \overset{(2.26)}{\lesssim } ||\Phi _\sigma u' ||_{W^{1,p}(E^{\mu })}, \end{aligned}$$

which tends to zero as $\mu$ goes to zero. Hence,

$$\begin{aligned} ||\partial {\tilde{v}}_\mu -\partial g_\sigma ||_{L^p(U')}\le ||g_\sigma \cdot \partial {\widetilde{\psi }}_\mu ||_{L^p(U')} +||(1 -{\widetilde{\psi }}_\mu ) \cdot \partial g_\sigma ||_{L^p(U')} \end{aligned}$$

(2.29)

also tends to zero, yielding (i).

Proof of (ii)

We argue by induction on m (the case $m=0$ being trivial). Fix a subanalytic set $A\subset \delta M$ of dimension k, and let $\Sigma$ be a stratification of $\overline{M}$ compatible with A and $\delta M$. As the Lipschitz conic structure provided by Theorem 1.3 may be required to be induced by the same homeomorphism for all several set-germs (see Remark 1.4), we will assume the key facts of Sect. 2.1 to hold for the strata and $r_s$ to preserve the strata. Observe that if we show the result for every stratum $Y\subset A$ of dimension k, this will yield the result for A. Notice that if $(\phi _i)_{i\in I}$ is a smooth partition of unity subordinated to a finite covering of ${\overline{M}}$ and $u\in W^{1,p}(M)$ is a function that extends continuously on ${{\overline{M}}}$ then, as $u=\sum _{i \in I} \phi _i u$, it suffices to check the needed fact for each $\phi _i u$. Hence, as ${\overline{M}}$ is compact, we will work in a small neighborhood of a point of $\overline{Y}$, $Y\subset A$ stratum, that we will assume to be the origin, and a function u that is supported in this neighborhood. We set $Y^\eta :={\textbf{B}}(0,\eta )\cap Y$, for $\eta \le \varepsilon$, where $\varepsilon$ is as explained just before Sect. 2.1.

Let $u\in {\mathscr {C}}^0(\overline{M})\cap W^{1,p}(M)$, with $\text{ supp}_{\overline{M}} u\subset {\textbf{B}}(0_{{\mathbb {R}}^n},\varepsilon )$. As ${\mathscr {R}}^M$ is a bounded operator and because ${\mathscr {R}}^M u$ is a constant function, we clearly have (for all p large enough)

$$\begin{aligned} ||{\mathscr {R}}^M u||_{L^p(Y^{\varepsilon })} \lesssim ||u||_{W^{1,p}(M^\varepsilon )}. \end{aligned}$$

We thus simply have to focus on $\Theta ^M u$, which means (since $\Theta ^M$ is a projection) that we can assume that $u=\Theta ^M u$. Notice first that we thus have for all $s\in (0,1)$ (and p large):

$$\begin{aligned} || u\circ r_s||^p_{L^p (N^\varepsilon )}\overset{(2.4)}{\lesssim } s^{-\nu }||u||_{L^p(N^{s\varepsilon }) } ^p \overset{(2.10)}{\lesssim }s^{p-1-\nu } ||u||_{W^{1,p}(M^{s\varepsilon })} ^p. \end{aligned}$$

Integrating with respect to s, we derive:

$$\begin{aligned} \int _0 ^{\frac{\eta }{\varepsilon }}||u\circ r_s||_{L^{p} (N^\varepsilon )}^p ds\lesssim \eta ^{p-\nu } ||u||^p_{W^{1,p}(M^{\eta })}. \end{aligned}$$

(2.30)

Moreover, as $r_s$ is Cs Lipschitz with C independent of s, we have for all $s\in (0,1)$

$$\begin{aligned} ||\partial (u\circ r_s)||_{L^p (N^\varepsilon )}^p= ||^{{\textbf{t}}} Dr_s (\partial u\circ r_s)||_{L^p (N^\varepsilon )}^p\lesssim s^p ||\partial u\circ r_s||_{L^p (N^\varepsilon )}^p \overset{(2.4)}{\lesssim }s^{p-\nu } ||\partial u||^p_{L^p (N^{s\varepsilon })} , \end{aligned}$$

which, after integrating with respect to s, gives:

$$\begin{aligned} \int _0 ^{\frac{\eta }{\varepsilon }}||\partial (u\circ r_s)||_{L^p (N^\varepsilon )}^p ds\lesssim \eta ^{p-\nu } \int _0 ^{\frac{\eta }{\varepsilon }} ||\partial u||^p_{L^p (N^{s\varepsilon })} \overset{(2.3)}{\lesssim }\eta ^{p-\nu } ||\partial u||^p_{L^p (M^{\eta })} , \end{aligned}$$

which, together with (2.30), gives in turn for $\eta \le \varepsilon$ and p sufficiently large

$$\begin{aligned} \int _0 ^{\frac{\eta }{\varepsilon }}||u\circ r_s||_{W^{1,p} (N^\varepsilon )}^p ds\lesssim \eta ^{p-\nu } ||u||^p_{W^{1,p}(M^{\eta })}. \end{aligned}$$

(2.31)

Given $\eta \le \varepsilon$, we set for simplicity $Z^\eta := {\textbf{S}}(0,\eta )\cap Y$. Observe that since $N^\varepsilon$ has dimension $(m-1)$, the induction hypothesis implies that there is a constant C such that for all $v \in W^{1,p}(N^\varepsilon )\cap {\mathscr {C}}^0(\overline{N^\varepsilon })$ we have (for p large enough)

$$\begin{aligned} ||v||_{L^p(Z^\varepsilon )}\le C ||v||_{W^{1,p}(N^\varepsilon )}. \end{aligned}$$

(2.32)

It makes it possible to estimate the $L^p$ norm of the trace on Y as follows:

$$\begin{aligned} ||u||_{L^p(Y^\varepsilon )}^p\overset{(2.3)}{\lesssim } \int _0 ^\varepsilon ||u||^p_{L^p(Z^\zeta )}d\zeta \lesssim \int _0 ^1 ||u\circ r_s||^p_{L^p(Z^{\varepsilon })}ds \overset{(2.31)}{\lesssim }\int _0^1 ||u\circ r_s||^p_{W^{1,p}(N^\varepsilon )} ds, \end{aligned}$$

(here we have applied (2.32) to $u\circ r_s$ for each s) which, by (2.31) (for $\eta =\varepsilon$), gives the desired bound for $||u||_{L^p(Y^\varepsilon )}$.

Remark that (this will be needed to prove (iii)) for $0<\eta <\varepsilon$, integrating on $(0,\frac{\eta }{\varepsilon })$ (instead of (0, 1)), the just above computation actually yields the more general estimate (assuming as above $\Theta ^M u=u$):

$$\begin{aligned} ||u||_{L^p(Y^\eta )}^p \lesssim \int _0 ^\frac{\eta }{\varepsilon } ||u\circ r_s||^p_{L^p(Z^{\varepsilon })}ds \overset{(2.32)}{\lesssim } \int _0^{\frac{\eta }{\varepsilon }} ||u\circ r_s||^p_{W^{1,p}(N^\varepsilon )} ds \overset{(2.31)}{\lesssim } \eta ^{p-\nu } ||u||^p_{W^{1,p}(M^{\eta })}.\nonumber \\ \end{aligned}$$

(2.33)

Proof of (iii)

Fix a stratification ${\mathcal {S}}$ of $A\subset \delta M$. Take then a locally bi-Lipschitz trivial stratification $\Sigma$ of $\overline{M}$, compatible with $\delta M$ as well as with all the strata of ${\mathcal {S}}$. Let $\Sigma _A$ be the stratification of A induced by $\Sigma$ and denote by $\Sigma '_A$ the collection of all the elements of $\Sigma _A$ that are maximal for the partial order relation: $Y\preceq Y'$ if and only if $Y\subset \overline{Y'}$. Note that $\Sigma '_A$ is a stratification of an open dense definable subset of A.

As $\Sigma$ is compatible with the strata of ${\mathcal {S}}$, $\Sigma _A$ is a refinement of ${\mathcal {S}}$. Hence, by definition of the trace, we have $\bigcap _{Y\in {\mathcal {S}}} \ker \textbf{tr}_Y\subset \bigcap _{Y\in \Sigma '_A} \ker \textbf{tr}_Y$. We are going to show that we can approximate every function $u \in \bigcap _{Y\in \Sigma _A'} \ker \textbf{tr}_Y$ by decreasing induction on $\kappa _u$ (see (2.22) for $\kappa _u$). Take for this purpose such a function $u:M\rightarrow {\mathbb {R}}$ and suppose the result true for every $v\in \bigcap _{Y\in \Sigma '_ A} \ker \textbf{tr}_Y$ satisfying $\kappa _v>\kappa _u$ (the case $\kappa _u=m$ is well-known).

As in the proof of (i), we will work near $x_0=0_{{\mathbb {R}}^n}\in \delta M$, assuming $k:=\dim S=\kappa _u<m$, where S is the stratum of $\Sigma$ containing $0_{{\mathbb {R}}^n}$. Again, since we can argue up to a partition of unity, we will also suppose that $\text{ supp}_{ {{\overline{M}}}}u\subset U_{0}$, where $U_{0}$ is a neighborhood of $0_{{\mathbb {R}}^n}$ on which we have a bi-Lipschitz trivialization of $\Sigma$, and that $0_{{\mathbb {R}}^n}\in \overline{A}$ (since otherwise the desired fact follows from assertion (i)). We will identify S with ${\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )$, $\alpha >0$. Note that there is a stratum $Y\in \Sigma _A'$ that contains S in its closure.

We start with the (easier) case $\dim S=0$. We claim that for $p\ge 2\nu$:

$$\begin{aligned} {\mathscr {R}}^M u=0. \end{aligned}$$

(2.34)

Note that Lemma 2.4 yields that ${\mathscr {R}}^M v\equiv v(0)=\textbf{tr}_S v(0)$ for all $v\in {\mathscr {C}}^\infty ( {{\overline{M}}})$, from which we can conclude (by density) that ${\mathscr {R}}^Mu\equiv \textbf{tr}_S u(0)$. Hence, if $S=Y$ then (2.34) is clear since $\textbf{tr}_S u=0$. Otherwise, we have $S\subset \overline{Y}\setminus Y$ and since $\textbf{tr}_Y u=0$, by definition of ${\mathscr {R}}^M$, we see that $\textbf{tr}_Y {\mathscr {R}}^Mu=-\textbf{tr}_Y \Theta ^M u$, so that, setting $Y^\eta :={\textbf{B}}(0,\eta )\cap Y$, we get

$$\begin{aligned} ||\textbf{tr}_Y {\mathscr {R}}^M u||_{L^p(Y^\eta )}^p= ||\textbf{tr}_Y \Theta ^M u||_{L^p(Y^\eta )}^p\overset{(2.33)}{\lesssim } \eta ^{p-\nu } ||\Theta ^M u||^p_{W^{1,p}(M^{\eta })} \end{aligned}$$

(2.35)

(here we apply (2.33) to $v:=\Theta ^M u$, which satisfies $\Theta ^M v=v$ since $\Theta ^M$ is a projection, see Lemma 2.4). Note that for $p\ge 2\nu$ we have for $\eta <\varepsilon$

$$\begin{aligned} \eta ^{p-\nu }\lesssim \eta ^\nu \lesssim {\mathcal {H}}^l(Y^\eta ) \qquad (\text{ by } (2.7) \text{ for } v\equiv 1, \text{ here } l:=\dim Y) . \end{aligned}$$

(2.36)

But if ${\mathscr {R}}^M u\ne 0$ then, as ${\mathscr {R}}^M u$ is constant, we have for $\eta <\varepsilon$

$$\begin{aligned} {\mathcal {H}}^l(Y^\eta )\lesssim ||\textbf{tr}_Y {\mathscr {R}}^M u||_{L^p(Y^\eta )}^p\overset{(2.35)}{\lesssim }\eta ^{p-\nu } ||\Theta ^M u||^p_{W^{1,p}(M^{\eta })}, \end{aligned}$$

(2.37)

in contradiction with the preceding estimate. This yields (2.34).

Consequently, $u=\Theta ^M u$, which means that (see (2.24))

$$\begin{aligned} v_\mu :=\psi _\mu \cdot \Theta ^M u=\psi _\mu \cdot u \end{aligned}$$

tends to u in $W^{1,p}(M)$ as $\mu \in (0,1)$ tends to zero (here $\psi _\mu$ is as in the proof of (i)). As $\kappa _{v_\mu }>0$ and $\textbf{tr}_{Y'} v_\mu =\psi _\mu \cdot \textbf{tr}_{Y'} u = 0$ for all $Y'\in \Sigma '_ A$ and all $\mu >0$, we see that the induction hypothesis ensures that we can find approximations of every $v_\mu$ having the required properties, which completes the induction step in the case $k=0$.

It remains the case $k>0$ that we are going to address with the same argument, using the operator ${\tilde{\Theta }}$ that we constructed in the proof of (i). As $\Sigma$ is a locally bi-Lipschitz trivial stratification, there exists a bi-Lipschitz homeomorphism $\Lambda :U_{0}\rightarrow (\pi ^{-1}(0_{{\mathbb {R}}^n})\cap \overline{M})\times {\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )$, where $\pi$ is a definable (see Remark 1.8 (3)) smooth retraction onto S and $\alpha >0$. To simplify notations, we will identify the strata with their images under $\Lambda$, writing Y instead of $\Lambda (Y\cap U_0)$.

For $v\in W^{1,p}(F\times {\textbf{B}}(0_{{\mathbb {R}}^k},\alpha ))$, set ${\tilde{{\mathscr {R}}}}v(x,y):={\mathscr {R}}^F v^y(x)$, where (as above) $F=\pi ^{-1}(0_{{\mathbb {R}}^n})\cap M$, and remark that, if ${\tilde{\Theta }}$ is as defined in (2.25), we have:

$$\begin{aligned} v={\tilde{\Theta }}v +{\tilde{{\mathscr {R}}}} v. \end{aligned}$$

(2.38)

As shown in the proof of (i), ${\tilde{\Theta }}$ is continuous (and therefore so is ${\tilde{{\mathscr {R}}}}$). We claim that if we set

$$\begin{aligned} u':=u\circ \Lambda ^{-1}\quad \text{ and } \quad w_\sigma := \Phi _\sigma u' \end{aligned}$$

(2.39)

(see (2.17) for $\Phi _\sigma$) then for all $\sigma >0$ small

$$\begin{aligned} {\tilde{{\mathscr {R}}}} w_\sigma =0.\end{aligned}$$

(2.40)

Note for this purpose that Lemma 2.4 yields that ${\mathscr {R}}^F v^y(x)\equiv \textbf{tr}_S v(y)$ for all $v\in {\mathscr {C}}^\infty (\overline{U'})$ (for each $y\in S$), and consequently that $\textbf{tr}_S{\tilde{{\mathscr {R}}}} v=\textbf{tr}_S v$ for such v, from which we can conclude (by density) that $\textbf{tr}_S{\tilde{{\mathscr {R}}}} w_\sigma =\textbf{tr}_S w_\sigma$. Hence, if $S=Y$ then our claim is clear since $\textbf{tr}_S w_\sigma =0$. Otherwise, as ${\tilde{{\mathscr {R}}}}w_{\sigma }(x,y)$ is constant with respect to $x\in F$, so is $\textbf{tr}_Y {\tilde{{\mathscr {R}}}}w_{\sigma }(x,y)$. If we set $X^\eta :=\pi ^{-1}(0_{{\mathbb {R}}^n})\cap {\textbf{B}}(0_{{\mathbb {R}}^n},\eta )\cap Y$ and $Y^\eta := X^\eta \times {\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )$, then, by Fubini’s Theorem for all $\eta <\varepsilon$, we have

$$\begin{aligned} ||\textbf{tr}_Y{\tilde{{\mathscr {R}}}}w_\sigma ||^p_{L^p(Y^\eta )}=\int _{{\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )}\int _{X^\eta }|\textbf{tr}_Y{\tilde{{\mathscr {R}}}}w_\sigma |^p(x,y)\,dx \,dy, \end{aligned}$$

and consequently for each $x\in F$ (since ${\tilde{{\mathscr {R}}}}w_{\sigma }(x,y)$ is constant with respect to x)

$$\begin{aligned} ||\textbf{tr}_Y{\tilde{{\mathscr {R}}}}w_\sigma ||^p_{L^p(Y^\eta )}={\mathcal {H}}^{l-k}(X^\eta )\cdot \int _{{\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )}| {\tilde{{\mathscr {R}}}}w_\sigma |^p(x,y)\ dy, \end{aligned}$$

(2.41)

where $l=\dim Y$. Moreover, it follows from (2.33) (for the manifold F) that for some $\nu$ (depending only on F and $X^\varepsilon$) we have for almost every $y\in {\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )$

$$\begin{aligned} ||\textbf{tr}_{X^\varepsilon }\Theta ^F w_\sigma ^y||_{L^p (X^\eta )}^p \lesssim \eta ^{p-\nu } ||\Theta ^F w_\sigma ^y ||^p_{W^{1,p} (F^\eta )} \end{aligned}$$

(with a constant independent of y), and therefore (integrating with respect to y)

$$\begin{aligned} ||\textbf{tr}_Y{\tilde{\Theta }}w_\sigma ||_{L^p (Y^\eta )}^p \lesssim \eta ^{p-\nu } ||{\tilde{\Theta }}w_\sigma ||^p_{W^{1,p} (E^\eta )} , \end{aligned}$$

(2.42)

where we recall that $E^\eta = (F\cap {\textbf{B}}(0_{{\mathbb {R}}^n},\eta ))\times {\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )$. As $\textbf{tr}_Y w_\sigma =\Phi _\sigma \textbf{tr}_Y u'=0$, (2.38) entails that $\textbf{tr}_Y {\tilde{{\mathscr {R}}}} w_\sigma =-\textbf{tr}_Y{\tilde{\Theta }}w_\sigma$, so that we can deduce from (2.42)

$$\begin{aligned} ||\textbf{tr}_Y {\tilde{{\mathscr {R}}}} w_\sigma ||^p_{L^p(Y^\eta )}\lesssim \eta ^{p-\nu } ||{\tilde{\Theta }}w_\sigma ||^p_{W^{1,p}(E^{\eta })}. \end{aligned}$$

(2.43)

For $p\ge 2\nu$, we have for $\eta >0$ small by (2.7) (applied with $v\equiv 1$ and M equal to the regular locus of $X^\varepsilon$)

$$\begin{aligned} \eta ^{p-\nu }\le \eta ^{\nu } \lesssim {\mathcal {H}}^{l-k}(X^\eta ). \end{aligned}$$

(2.44)

Now if, contrarily to the claim that we are proving (see (2.40)), we assume that for some $\sigma$ we have for some (and hence for any) $x\in F$:

$$\begin{aligned} \int _{{\textbf{B}}(0_{{\mathbb {R}}^k},\alpha )}| {\tilde{{\mathscr {R}}}}w_\sigma |^p(x,y)\ dy\ne 0, \end{aligned}$$

equality (2.41) then yields that (for this $\sigma$)

$$\begin{aligned} {\mathcal {H}}^{l-k}(X^\eta )\lesssim ||\textbf{tr}_Y {\tilde{{\mathscr {R}}}} w_\sigma ||^p_{L^p(Y^\eta )} \overset{(2.43)}{\lesssim }\eta ^{p-\nu } ||{\tilde{\Theta }}w_\sigma ||^p_{W^{1,p}(E^{\eta })}, \end{aligned}$$

which clearly contradicts (2.44) (since $||{\tilde{\Theta }}w_\sigma ||^p_{W^{1,p}(E^{\eta })}$ tends to 0 as $\eta \rightarrow 0$), yielding our claim (see (2.40)).

Hence, $w_\sigma ={\tilde{\Theta }}w_\sigma$, and consequently (see (2.29)), for each $\sigma$,

$$\begin{aligned} {\tilde{v}}_\mu :={\widetilde{\psi }}_\mu \cdot w_\sigma ={\widetilde{\psi }}_\mu \cdot {\tilde{\Theta }}w_\sigma ={\widetilde{\psi }}_\mu \cdot {\tilde{\Theta }}\Phi _\sigma u' \end{aligned}$$

tends to $w_\sigma$ in $W^{1,p}(U')$ as $\mu$ tends to zero. Since $\kappa _{{\tilde{v}}_\mu }> k$ and $\textbf{tr}_{Y'} {\tilde{v}}_\mu ={\widetilde{\psi }}_\mu \cdot \textbf{tr}_{Y'} w_\sigma = 0$ for all $Y'\in \Sigma '_ A$, and as $w_\sigma$ tends to $u'$ as $\sigma$ goes to 0, this completes the induction step. $\square$

Remark 2.8

• (1) The proof of (ii) has actually established that, alike in the case of Lipschitz manifolds with boundary, if u belongs to ${\mathscr {C}}^0(\overline{M})\cap W^{1,p}(M)$ (and not necessarily to ${\mathscr {C}}^\infty (\overline{M})$) then $\textbf{tr}_A u=u_{|A}$.

• (2) For (iii) of the above theorem to be true, it is not really necessary to require the manifold to be normal. We have proved that it suffices that M be connected at every point of $\delta M\setminus \overline{A}$. It is as well worthy of notice that we do not need ${\mathcal {S}}$ to be a stratification of A, as it suffices to have a covering a dense subset of A by some subanalytic sets of pure dimension.

• (3) Examining the proof of this theorem, one could give a lower bound for p that just depends on $\nu$ (apparently $2\nu$), where $\nu$ is given by (2.4), which is itself given by applying Łojasiewicz’s inequality to the Jacobian of $r_s$ (note that the Lipschitz Conic Structure Theorem was applied to M but also to other manifolds that arose on the way, such as F). Finding $r_s$ is a heavy task. The proof of Theorem 1.3 being constructive it is however theoretically doable. As well-known [2], there are quantifier elimination algorithms that can be implemented and that, given a manifold defined by polynomial inequalities, make it possible to carry out effective computations of the geometric constructions needed to produce such a retract $r_s$.

Example 2.9

Theorem 2.7 no longer holds when M fails to be definable. For instance, whenever $\delta M$ has infinite measure, any constant nonzero function provides an example of function whose trace is not $L^p$. Furthermore, in [15, section 1.7.2], it is explained how to construct an example of open set (not subanalytic) $\Omega$ (choosing in the example presented therein $\varepsilon _i:=2^{-2i^2}$ and $a_i:= 2^{-i}$) with a function $u\in W^{1,p}(\Omega )$, for all $p\in [1,\infty )$, that cannot be approximated by $W^{1,\infty }$ functions. We also wish to emphasize that the trace is not $L^p$ bounded when p is not large, even if M is subanalytic, as shown by the example $\Omega _k:=\{(x,y)\in (0,1)^2: y<x^k\}$, $k\in {\mathbb {N}}\setminus \{0,1\}$, and $u(x,y)=\frac{1}{x}\in W^{1,p}(\Omega _k)$ for all $p\in [1,\frac{k}{2}]$.

4 The general case

4.1 Normalizations

We introduce the (subanalytic) ${\mathscr {C}}^\infty$ normalizations and show their existence and uniqueness. This will be needed to investigate the trace operator on non normal manifolds. This notion of ${\mathscr {C}}^\infty$ normalization is a natural counterpart of topological normalizations of pseudomanifolds [6, 9] in our framework.

Definition 3.1

A ${\mathscr {C}}^\infty$ normalization of M is a definable ${\mathscr {C}}^\infty$ diffeomorphism $h: {\check{M}}\rightarrow M$ satisfying $\sup _{x\in {\check{M}}} |D_x h|<\infty$ and $\sup _{x\in M} |D_x h^{-1}|<\infty$, with ${\check{M}}$ a normal ${\mathscr {C}}^\infty$ submanifold of ${\mathbb {R}}^k$, for some k.

So far, we have regarded the manifold M as endowed with the metric induced by the euclidean metric of ${\mathbb {R}}^n$. We can also regard it as a Riemannian manifold: given x and y in the same connected component of M, let $d_M(x,y)$ denote the length of the shortest ${\mathscr {C}}^1$ arc in M joining x and y.

This metric is sometimes referred as the inner metric of M. Although it is not always equivalent to the euclidean metric, it enjoys the following property that will be useful for our purpose. Let $(A_t)_{t\in {\mathbb {R}}^k}$ be a definable family of manifolds (in the sense that $\bigcup _{t\in {\mathbb {R}}^k} A_t\times \{t\}$ is a definable set and $A_t$ is a smooth manifold for every t). There is a constant C (independent of t) such that for all $t\in {\mathbb {R}}^k$ we have for all x and y in the same connected component of $A_t$:

$$\begin{aligned} d_{A_t}(x,y)\le C \text{ diam }(A_t), \end{aligned}$$

(3.45)

where $\text{ diam }(A_t)$ stands for the euclidean diameter of $A_t$, defined as $\text{ diam }(A_t)=\sup \{ |a-b|:a\in A_t, b\in A_t\}$. See [11, Corollary 1.3] (or [25, Proposition 3.1.24]) for a proof.

Lemma 3.2

Suppose that M is connected at $x_0\in \delta M$ and let $f:M\rightarrow {\mathbb {R}}$ be a definable ${\mathscr {C}}^1$ function. If $\sup _{x\in M} |\partial f(x)|<\infty$ then f extends continuously at $x_0$.

Proof

As f has bounded derivative, by (3.45), it is bounded near $x_0$, and therefore $\lim _{t \rightarrow 0} f(\gamma (t))$ exists for every definable arc $\gamma :(0,1)\rightarrow M$ tending to $x_0$ as t goes to zero. It thus suffices to check that this limit is independent of the arc $\gamma$.

Let for $\varepsilon >0$, $A_\varepsilon := {\textbf{S}}(x_0,\varepsilon )\cap M$ and observe that $A_\varepsilon$ is connected for $\varepsilon >0$ small. By (3.45), there is a constant C such that for any $\varepsilon >0$ small, $d_{A_\varepsilon }(x,y)\le C\varepsilon$, for all x and y in $A_\varepsilon$. As f has bounded derivative, this entails that $|f(x)-f(y)|\le C'\varepsilon$, for all such x and y, where $C'$ is a constant. Consequently, if $\gamma _1$ and $\gamma _2$ are two definable arcs in M tending to $x_0$ (that we can parameterize by their distance to ${x_0}$), $|f(\gamma _1(t))-f(\gamma _2(t))|$ tends to 0 as t goes to 0. $\square$

Observe that if $h:{\check{M}}\rightarrow M$ is a normalization and if $z\in \delta {\check{M}}$ then this lemma entails that h induces a homeomorphism between the germ of ${\check{M}}$ at z and a connected component of the germ of M at $\lim _{x\rightarrow z} h(x)$. It also yields:

Proposition 3.3

Every ${\mathscr {C}}^\infty$ normalization $h: {\check{M}}\rightarrow M$ extends continuously to a mapping from $\overline{{\check{M}}}$ to ${{\overline{M}}}$. Moreover, if $h_1:{\check{M}}_1 \rightarrow M$ and $h_2:{\check{M}}_2 \rightarrow M$ are two ${\mathscr {C}}^\infty$ normalizations of M, then $h_2^{-1} h_1$ extends to a homeomorphism between $\overline{{\check{M}}_1}$ and $\overline{{\check{M}}_2}$.

For $x\in \delta M$, let

$$\begin{aligned} {\mathfrak {c}}_M(x):= \text{ number } \text{ of } \text{ connected } \text{ components } \text{ of } {\textbf{B}}(x,\varepsilon )\cap M, \varepsilon >0 \text{ small. } \end{aligned}$$

If ${\overline{h}}$ stands for the extension of h to $\overline{{\check{M}}}$, then ${\overline{h}}^{-1}(x)$ has exactly ${\mathfrak {c}}_M(x)$ points, for each $x\in \delta M$. In particular, ${\overline{h}}$ must be finite-to-one. Remark also that this proposition somehow establishes uniqueness of ${\mathscr {C}}^\infty$ normalizations, in the sense that the inner Lipschitz geometry of $\overline{{\check{M}}}$ is independent of the chosen normalization ${\check{M}}$. The next result is devoted to their existence.

Given a submanifold $S\subset {\mathbb {R}}^n$ and $x\in {\mathbb {R}}^n$, we denote by $\pi _S(x)$ the point of S at which the euclidean distance to S is reached (this point is unique if x is sufficiently close to S) and by $\rho _S(x)$ the square of this distance. These two mappings, defined on a suitable neighborhood $U_S$ of S, are globally subanalytic and ${\mathscr {C}}^\infty$ if S is globally subanalytic and $U_S$ is small enough (see for instance [25, Proposition 2.4.1]). We then say that $(U_S,\pi _S,\rho _S)$ is a tubular neighborhood of S.

Proposition 3.4

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

Proof

Let $B_M$ be the set of points of $\delta M$ at which ${\mathfrak {c}}_M(x)>1$ and set $k_M:=\dim B_M$. We are going to prove the result by induction on $k_M$ (if $k_M=-1=\dim \emptyset$, we are done).

Take a stratification $\Sigma$ of $\delta M$ such that ${\mathfrak {c}}_M$ is constant on the strata, and let S be a $k_M$-dimensional stratum included in $B_M$. As the construction that we are going to carry out may be realized simultaneously for every such S, we will assume that S is the only element of dimension $k_M$ of $\Sigma$ included in $B_M$.

Let $(U_S,\pi _S,\rho _S)$ be a tubular neighborhood of S and denote by $C_1,\dots ,C_l$ the connected components of $M \cap U_S$. Taking $U_S$ smaller if necessary, we can assume that $l={\mathfrak {c}}_M(x)$, for all $x\in S$.

Let $\alpha$ be a definable ${\mathscr {C}}^\infty$ function on $U_S$ satisfying $|\alpha (x)-d(x,{\mathbb {R}}^n \setminus U_S)|<\frac{d(x,{\mathbb {R}}^n \setminus U_S)}{2}$ (by the density of smooth globally subanalytic functions in the Efroymson topology [20, Theorem 4.8], such a function exists). Remark that if $\gamma :[0,1]\rightarrow U_S$ is a definable arc such that $\lim _{s\rightarrow 0^+}\frac{1}{1+|\partial \alpha (\gamma (s))|}=0$ then $\gamma (s)$ tends to the frontier of $U_S$ (since $\alpha$ is smooth on $U_S$), which entails that $\alpha (\gamma (s))$ tends to 0. Consequently, by Łojasiewicz’s inequality (Proposition 1.2), there is an integer $\kappa _1 \ge 1$ such that $\alpha ^{\kappa _1-1} \lesssim \frac{1}{1+|\partial \alpha |}$ on $U_S$, so that the function $\alpha ^{\kappa _1}$ has bounded gradient on this set. Similarly, there is a definable ${\mathscr {C}}^\infty$ function $\beta$ on ${\mathbb {R}}^n\setminus \delta S$ such that $|\beta (x)-d(x,\delta S)|<\frac{d(x,\delta S)}{2}$ and an integer $\kappa _2$ such that $\beta ^{\kappa _2}$ has bounded gradient. We claim that, choosing $\kappa _2$ bigger if necessary, we can require in addition that there is a constant A such that on the set

$$\begin{aligned} Z:=\{x\in U_S: \rho _S(x) \le \alpha (x) \}, \end{aligned}$$

we have:

$$\begin{aligned} \beta (x)^{\kappa _2} \le A \alpha (x)^{\kappa _1}. \end{aligned}$$

(3.46)

Indeed, thanks to Łojasiewicz’s inequality, it suffices to check that $\lim _{t \rightarrow 0 }\beta (\gamma (t))=0$ for every definable arc $\gamma :[0,1] \rightarrow Z$ such that $\lim _{t\rightarrow 0} \alpha (\gamma (t))=0$. To see this, note that such an arc $\gamma$ must tend to a point of $\delta U_S$ (since $\alpha$ tends to zero) which is a point of $\overline{S}$ (since $\rho _S$ also tends to zero, due to the definition of Z), which entails that $\lim _{t\rightarrow 0} \gamma (t) \in \delta S$ (since $\delta U_S\cap \overline{S}\subset \delta S$), from which we can conclude that $\beta$ tends to zero, as required.

We now are going to construct for every $i\le l$ a definable ${\mathscr {C}}^\infty$ function $\rho _i$ on M satisfying:

• (a) There is a neighborhood $V_S$ of S such that $\rho _i>\frac{\beta ^{\kappa _2}}{2}$ on $C_i\cap V_S$, and $\rho _i<\frac{\beta ^{\kappa _2}}{3}$ on $C_j\cap V_S$ for every $j\ne i$.

• (b) $\rho _i$ has bounded first derivative and tends to zero as we are drawing near $\delta S$.

Choose a ${\mathscr {C}}^1$ definable function $\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ such that $\phi (t)=0$ for $t\ge 1$, $\phi (t)=1$ for $t<\frac{1}{2}$ and let $i\le l$. Define then a ${\mathscr {C}}^1$ function on $C_i$ by setting for $x\in C_i$

$$\begin{aligned} \rho _i(x):= \phi \left( \frac{\rho _S(x)^{\kappa _1}}{\alpha (x)^{\kappa _1}}\right) \cdot \beta (x)^{\kappa _2}, \end{aligned}$$

and extend this function as identically zero on $M\setminus C_i$. We shall establish that $\rho _i$ satisfies properties (a) and (b). This function $\rho _i$ is not ${\mathscr {C}}^\infty$ but just ${\mathscr {C}}^1$. However, as properties (a) and (b) also hold for any sufficiently close approximation of $\rho _i$ (in the ${\mathscr {C}}^1$ topology, see [20]), the existence of a ${\mathscr {C}}^\infty$ function satisfying these properties will then follow from Efroymson’s approximation theorem for globally subanalytic functions [20].

Thanks to (3.46), a straightforward computation of derivative yields that $\rho _i$ has bounded first derivative (note that the support of the function $\phi '\left( \frac{\rho _S(x)^{\kappa _1}}{\alpha (x)^{\kappa _1}}\right)$ is included in the above set Z). Moreover, as $\beta$ tends to zero as x tends to $\delta S$, it is clear that so does $\rho _i$, which yields (b). Fact (a) also holds since $\rho _i$ vanishes on $C_j$ for $j\ne i$, and $\phi \equiv 1$ near S.

Let now for $x \in M$, $\rho (x):=(\rho _1(x),\dots ,\rho _l(x))\in {\mathbb {R}}^{l},$ and let $\check{M}$ stand for the graph of $\rho$. Denote by $\pi :\check{M}\rightarrow M$ the map induced by the canonical projection onto ${\mathbb {R}}^n$ and by $\overline{\pi }:\overline{\check{M}}\rightarrow {{\overline{M}}}$ its extension. As $\partial \rho _i$ is bounded for all i, the mappings $\pi$ and $\pi ^{-1}$ both have bounded first derivative.

By (a), the sets $\overline{\Gamma _{\rho _{|C_j}}}\cap \overline{\pi }^{-1}(S)$ (here $\Gamma _{\rho _{|C_j}}$ stands for the graph of $\rho _{|C_j}$), $j\le l$, are disjoint from each other. Hence, as for each j, the function $\rho _{|C_j}$ extends continuously on S (in virtue of (b) and Lemma 3.2), $\overline{\Gamma _{\rho }}$ must be connected at every point of $\overline{\pi }^{-1}(S)$. Observe also that, again due to (b) and Lemma 3.2, at a given point $x\in \delta M$, the function $\rho$ cannot have more than ${\mathfrak {c}}_M(x)$ asymptotic values, which means that $\overline{\pi }$ is finite-to-one and that ${\check{M}}$ is connected at every point above ${{\overline{M}}}\setminus B_M$. Hence, $\{x \in {\check{M}}:{\mathfrak {c}}_{{\check{M}}}(x)>1\} \subset \overline{\pi }^{-1}(B_M\setminus S)$ (since ${\mathfrak {c}}_{{\check{M}}}(x)\equiv 1$ on $\overline{\pi }^{-1}(S)\cup \overline{\pi }^{-1}( {{\overline{M}}}\setminus B_M)$), which has dimension less than $k_M$ (since $\overline{\pi }$ is finite-to-one). This completes the induction step. $\square$

Remark 3.5

We have used the subanalytic version of Efroymson’s approximation theorem [20] because we wanted to construct ${\mathscr {C}}^\infty$ normalizations. If one only wishes to construct ${\mathscr {C}}^i$ normalizations $i\in {\mathbb {N}}$ (defined analogously, they seem to be satisfying to define the trace operator), it is easy to spare this theorem which is rather difficult to prove.

4.2 The general case

If the manifold M is not assumed to be normal, it is possible to define a multivalued trace by considering a normalization, and then prove some partial generalizations of the results of Sect. 2. We first point out some facts useful for this purpose.

Fix a normalization $h:{\check{M}}\rightarrow M$. As h and $h^{-1}$ have bounded derivative, the mapping $h_*:W^{1,p}({\check{M}})\rightarrow W^{1,p}(M)$, $u\mapsto u\circ h^{-1}$ is a continuous isomorphism for all p. Theorem 2.7 thus immediately yields that for $p\in [1,\infty )$ sufficiently large, the space

$$\begin{aligned} {\mathscr {C}}^h ( {{\overline{M}}}):=h_* {\mathscr {C}}^\infty (\overline{{\check{M}}})=\{u\circ h^{-1} : u \in {\mathscr {C}}^\infty (\overline{{\check{M}}}) \} \end{aligned}$$

(3.47)

is dense in $W^{1,p}(M)$.

Although the functions of ${\mathscr {C}}^h( {{\overline{M}}})$ may fail to be smooth on ${{\overline{M}}}$, this ring is satisfying for many purposes. A given function v of this ring has the property that for every ${x_0}$ in ${{\overline{M}}}$, the restriction of v to a connected component U of ${\textbf{B}}({x_0},\varepsilon )\cap M$, $\varepsilon >0$ small, extends to a function which is Lipschitz with respect to the inner metric. It actually can be required (see [24, 25]) that the differential of this restriction extends to a stratified 1-form and therefore can be used to perform integrations by parts.

By Proposition 3.3, h extends to a continuous mapping $\overline{h}:\overline{{\check{M}}} \rightarrow \overline{M}$. We also have:

Proposition 3.6

There are stratifications $\check{{\mathcal {S}}}$ and ${\mathcal {S}}$ of $\delta {\check{M}}$ and $\delta M$ respectively such that for each $S\in {\mathcal {S}}$, $\overline{h}^{-1}(S)=\bigcup _{i=1}^j S_i$, where, for each $i\le j$, $S_i$ is a stratum of $\check{{\mathcal {S}}}$ on which $\overline{h}$ induces a diffeomorphism $h_{S_i}:S_i\rightarrow S$ satisfying $\sup _{x\in S_i} |D_x h_{S_i}|<\infty$ and $\sup _{x\in S} |D_x h_{S_i}^{-1}|<\infty$.

Proof

Possibly replacing ${\check{M}}$ with the set $\{(y,x):y=h(x)\}$, we can assume that h is given by a canonical projection. We then take a cell decomposition ${\mathcal {D}}$ compatible with $\overline{{\check{M}}}$ and ${\check{M}}$, which immediately gives rise to a cell decomposition ${\mathcal {E}}$ of ${\mathbb {R}}^n$ (see [25, Remark 1.2.2]). As h is finite-to-one, it maps injectively the cells of ${\mathcal {D}}$ onto the cells of ${\mathcal {E}}$. These two cell decompositions induce stratifications of $\overline{{\check{M}}}$ and $\overline{M}$ respectively. Given a definable mapping that has bounded derivative on an open dense set (definable mappings are smooth on an open dense definable set) there is a stratification such that the restriction of this mapping to every stratum is smooth and has bounded derivative [24, Lemma 3.8] (see also [25, Proposition 2.6.12]). Requiring the cell decomposition ${\mathcal {D}}$ to be compatible with such a stratification, we see that we can demand h and $h^{-1}$ to have bounded first derivative on the strata. $\square$

4.3 Definition of the trace

We first define $\textbf{tr}_S$ when S is a stratum of ${\mathcal {S}}$, where ${\mathcal {S}}$ is provided by Proposition 3.6 above. Let $l:=\sup _{x\in M} {\mathfrak {c}}_M(x)$, fix $S\in {\mathcal {S}}$, and remark that ${\mathfrak {c}}_M$ is constant on S. We thus can define $\textbf{tr}_S : W^{1,p}(M) \rightarrow L^p (S)^l$ by setting for $v\in W^{1,p}(M)$ (and $p\in [1,\infty )$ large):

$$\begin{aligned} \textbf{tr}_S v := \left( (\textbf{tr}_{S_1} v\circ h)\circ h_{S_1}^{-1}, \dots , (\textbf{tr}_{S_j} v\circ h)\circ h_{S_j}^{-1}, 0,\dots , 0\right) , \end{aligned}$$

where $S_1,\dots , S_j$ are provided by the above proposition. Here, we add $(l-j)$ times the zero function because it will be convenient that the trace has the same number of components for all $S \in {\mathcal {S}}$. This mapping of course depends on the way the elements $S_1,\dots , S_j$ are enumerated (see nevertheless on this issue Proposition 3.8 below). Notice that since $h_{S_i}:S_i\rightarrow S$ and its inverse have bounded derivative, Theorem 2.7 ensures that $\textbf{tr}_S$ is a bounded operator.

Take now a subanalytic subset A of $\delta M$ and assume that ${\mathcal {S}}$ is compatible with A. In this situation, we can define $\textbf{tr}_A v$ as the function induced by the mappings $\textbf{tr}_S v$, $S\in {\mathcal {S}}$, $S\subset A$, $\dim S=\dim A$. Since the strata of dimension less than $k:=\dim A$ are ${\mathcal {H}}^k$-negligible this definition is clearly independent of the chosen stratification.

Theorem 2.7 now has the following consequence:

Corollary 3.7

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

$$\begin{aligned} \textbf{tr}_A :W^{1,p}(M) \rightarrow L^p(A,{\mathcal {H}}^k)^l, \end{aligned}$$

is bounded.

We then denote by ${\textbf {tr}}_{A,1},\dots , {\textbf {tr}}_{A,l}$ the components of $\textbf{tr}_A$. As pointed out before Corollary 3.7, the trace depends on the way the strata $S_1,\dots , S_j$ above are enumerated. This choice being made, we however have:

Proposition 3.8

Let $A\subset \delta M$ be subanalytic. If p is sufficiently large then, for every $v\in W^{1,p}(M)$, the set of functions $\{ \textbf{tr}_{A,1}\, v , \dots , \textbf{tr}_{A,l} \,v\}$ does not depend on the chosen ${\mathscr {C}}^\infty$ normalization.

Proof

Let $h_1:{\check{M}}_1 \rightarrow M$ and $h_2:{\check{M}}_2\rightarrow M$ be two ${\mathscr {C}}^\infty$ normalizations of M. By Proposition 3.3, the mapping $H:=h_2^{-1}h_1:{\check{M}}_1 \rightarrow {\check{M}}_2$ is a diffeomorphism which extends continuously on $\delta {\check{M}}_1$, giving rise to a homeomorphism $\overline{H} :\overline{{\check{M}}_1}\rightarrow \overline{{\check{M}}_2}$. The result is therefore clear if $v\circ h^{-1}_1$ extends continuously to $\overline{{\check{M}}_1}$ (see Remark 2.8 (1)), which means that it holds for every $v\in {\mathscr {C}}^{h_1}(M)$ (see (3.47)). But, as this set is dense and $\textbf{tr}_A$ is bounded, it means that the result must hold for each $v\in W^{1,p}(M)$. $\square$

It is worthy of notice that Theorem 2.7 also yields that if $\Sigma$ is any stratification of $A \subset \delta M$, then (for p large) $h_* {\mathscr {C}}^\infty _{B}(\overline{{\check{M}}})$ is dense in $\underset{S\in \Sigma }{\bigcap }\ker \textbf{tr}_S$, where $h:{\check{M}}\rightarrow M$ is any ${\mathscr {C}}^\infty$ normalization of M and $B:=\overline{{\check{M}}} \setminus \overline{\overline{h}^{-1}(A)}$. In particular, in the case where $\overline{h}^{-1}(A)$ is dense in $\delta {\check{M}}$, we get:

Corollary 3.9

If $\Sigma$ is a stratification of a subanalytic subset A of $\delta M$ such that $\overline{h}^{-1}(A)$ is dense in $\delta {\check{M}}$, for some (and hence for any) normalization $h:{\check{M}}\rightarrow M$, then ${\mathscr {C}}^\infty _0(M)$ is dense in $\underset{S\in \Sigma }{\bigcap }\ker \textbf{tr}_S$ for all $p\in [1,\infty )$ sufficiently large.

It is worthy of notice that the above condition “$\overline{h}^{-1}(A)$ is dense in $\delta {\check{M}}$” holds as soon as A is dense in $\delta M$ and contains $\delta M\setminus \partial M$. We also can get a converse of (i) of Theorem 2.7:

Corollary 3.10

${\mathscr {C}}^\infty (\overline{M})$ is dense in $W^{1,p}(M)$ for arbitrarily large values of p if and only if M is normal.

Proof

The “if” part is established by Theorem 2.7. To prove the “only if” part, assume that M fails to be normal and take a point $x_0\in \overline{M}$ at which M is not connected. For $\varepsilon >0$ small enough, let $U_1,\dots , U_l$, $l>1$, be the connected components of $M\cap {\textbf{B}}(x_0,\varepsilon )$. Let then $u:\bigcup _{j=1}^l U_j\rightarrow {\mathbb {R}}$ be the function identically equal to 1 on $U_1$ and 0 on the other $U_j$’s. Multiplying this function by a ${\mathscr {C}}^\infty$ function v which is 1 on some neighborhood of $x_0$ in ${\mathbb {R}}^n$ and 0 outside ${\textbf{B}}(x_0,\varepsilon )$, we get a function ${\tilde{v}}$ which extends continuously to M. Clearly, this extension, still denoted ${\tilde{v}}$, belongs to $W^{1,p}(M)$.

Suppose now that there exists a sequence $(v_i)_{i \in {\mathbb {N}}}$ in ${\mathscr {C}}^\infty (\overline{M})$ tending to ${\tilde{v}}$ in $W^{1,p}(M)$. Notice that we have $\textbf{tr}_{\{x_0\}}v_{i}=(v_{i}(x_0),\dots ,v_i ({x_0}),0,\dots , 0)$, where $v_i(x_0)$ occurs l times, while $\textbf{tr}_{\{x_0\}}{\tilde{v}}= (1,0,\dots ,0)$. Since, due to Corollary 3.7, $\textbf{tr}_{\{x_0\}}v_{i}$ should tend to $\textbf{tr}_{\{x_0\}}{\tilde{v}}$, this contradicts $l>1$. $\square$

Remark 3.11

Of course, if the manifold M is unbounded, the trace is well-defined and $L^p_{loc}$ on the boundary. Moreover, since we can use cutoff functions, the density results remain true in this case.