# Uniform extension of definable $C^{m,\omega}$-Whitney jets

Adam Parusiński and Armin Rainer  Adam Parusiński: Université Côte d’Azur,
CNRS, LJAD, UMR 7351, 06108 Nice, France<EMAIL_ADDRESS>Armin Rainer: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-
Platz 1, A-1090 Wien, Austria<EMAIL_ADDRESS>
###### Abstract.

We show that definable Whitney jets of class $C^{m,\omega}$, where $m$ is a
nonnegative integer and $\omega$ is a modulus of continuity, are the
restrictions of definable $C^{m,\omega}$-functions; “definable” refers to an
arbitrary given o-minimal expansion of the real field. This is true in a
uniform way: any definable bounded family of Whitney jets of class
$C^{m,\omega}$ extends to a definable bounded family of
$C^{m,\omega}$-functions. We also discuss a uniform $C^{m}$-version and how
the extension depends on the modulus of continuity.

###### Key words and phrases:

O-minimal structures, Whitney extension theorem, $C^{m,\omega}$-extension of
Whitney jets, uniform boundedness of the extension

###### 2020 Mathematics Subject Classification:

03C64, 14P10, 26B35, 26E25, 32B20, 46E15

Supported by FWF-Project P 32905-N and Oberwolfach Research Fellows (OWRF) ID
2244p.

##  1\. Introduction

Let an o-minimal expansion of the real field be fixed. Throughout the paper, a
set $X\subseteq\mathbb{R}^{n}$ is called _definable_ if it is definable in
this fixed o-minimal structure. A map $\varphi:X\to\mathbb{R}^{m}$ is
definable if its graph $\Gamma(\varphi):=\\{(x,\varphi(x)):x\in X\\}$ is a
definable subset of $\mathbb{R}^{n}\times\mathbb{R}^{m}$. We assume
familiarity with the basics of o-minimal structures; cf. [15] and [14].

Due to Kurdyka and Pawłucki [7, 8] and Thamrongthanyalak [12] we have the
definable $C^{m}$ Whitney extension theorem:

######  Theorem 1.1.

Let $0\leq m\leq p$ be integers. Let $E\subseteq\mathbb{R}^{n}$ be a definable
closed set. Any definable Whitney jet of class $C^{m}$ on $E$ extends to a
definable $C^{m}$-function on $\mathbb{R}^{n}$ which is of class $C^{p}$
outside $E$.

In this paper, we prove a $C^{m,\omega}$-version of this result.

######  Theorem 1.2.

Let $0\leq m\leq p$ be integers. Let $\omega$ be a modulus of continuity. Let
$E\subseteq\mathbb{R}^{n}$ be a definable closed set. Any definable Whitney
jet of class $C^{m,\omega}$ on $E$ extends to a definable
$C^{m,\omega}$-function on $\mathbb{R}^{n}$ which is of class $C^{p}$ outside
$E$.

By a _modulus of continuity_ we always mean a positive, continuous,
increasing, and concave function $\omega:(0,\infty)\to(0,\infty)$ such that
$\omega(t)\to 0$ as $t\to 0$. The class $C^{m,\omega}$ consists of
$C^{m}$-functions that are globally bounded together with its partial
derivatives up to order $m$ and whose partial derivatives of order $m$ satisfy
a global $\omega$-Hölder condition. See Section 3.

We use Theorem 1.2 in our paper [9] to show that a definable function
$f:E\to\mathbb{R}$ on a definable closed set $E\subseteq\mathbb{R}^{n}$ that
has a $C^{1,\omega}$-extension to $\mathbb{R}^{n}$ also has a definable
$C^{1,\omega}$-extension. (In [9] we assume that $\omega$ is definable, but
not in the present paper.) In fact, this application was one of our main
motivations for proving Theorem 1.2.

Furthermore, we will show that the definable extension of Whitney jets of
class $C^{m,\omega}$ can be done in a bounded way:

######  Theorem 1.3.

Let $0\leq m\leq p$ be integers. Let $\omega$ be a modulus of continuity. Let
$(E_{a})_{a\in A}$ be a definable family of closed subsets of
$\mathbb{R}^{n}$. For any definable bounded family $(F_{a})_{a\in A}$ of
Whitney jets of class $C^{m,\omega}$ on $(E_{a})_{a\in A}$ there exists a
definable bounded family $(f_{a})_{a\in A}$ of $C^{m,\omega}$-extensions to
$\mathbb{R}^{n}$ of $(F_{a})_{a\in A}$ such that $f_{a}$ is of class $C^{p}$
outside $E_{a}$ for all $a\in A$.

Clearly, boundedness is understood with respect to the natural norms; see
Section 3 for precise definitions. Note that Theorem 1.2 follows as a special
case from Theorem 1.3. And already the case that $(F_{a})_{a\in A}$ is a
definable bounded family of Whitney jets of class $C^{m,\omega}$ on a _fixed
set_ $E=E_{a}$, for all $a\in A$, is very interesting. However, the method of
proof (by induction on dimension) necessitates to consider the general case
that the families of Whitney jets are defined on definable families of sets
$(E_{a})_{a\in A}$.

The construction of the extension in Theorem 1.3 depends on $\omega$ only in a
weak sense. We may, for instance, let the modulus of continuity $\omega_{a}$
depend as well on $a\in A$ if we impose that there is a constant $C>0$ such
that $C^{-1}\leq\omega_{a}(1)\leq C$ for all $a\in A$. This will be discussed
in detail in Section 5.2 in which we present a more general version of Theorem
1.3. As a consequence, we deduce from Theorem 1.3 a uniform version of the
$C^{m}$-result 1.1 on compact sets:

######  Theorem 1.4.

Let $0\leq m\leq p$ be integers. Let $(E_{a})_{a\in A}$ be a definable family
of compact subsets of $\mathbb{R}^{n}$. For any definable bounded family
$(F_{a})_{a\in A}$ of Whitney jets of class $C^{m}$ on $(E_{a})_{a\in A}$
there exists a definable bounded family $(f_{a})_{a\in A}$ of
$C^{m}$-extensions to $\mathbb{R}^{n}$ of $(F_{a})_{a\in A}$ such that $f_{a}$
is of class $C^{p}$ outside $E_{a}$ for all $a\in A$.

Theorem 1.4 is proved in Section 5.3. Furthermore, we deduce a local version
of Theorem 1.3 in Section 5.1 and apply Theorem 1.3 in Section 5.4 to get a
definable version of a correspondence, due to Shvartsman [11], between Whitney
jets of class $C^{m,\omega}$ and certain Lipschitz maps.

The proof of Theorem 1.3 (which builds upon the one of Theorem 1.1 devised in
[7, 8, 12] and also Pawłucki [10] and is very different from Whitney’s
classical method [16]) rests on two main cornerstones:

  1. (1)

_Two versions of Gromov’s inequality[5];_ one classical, the other
incorporating the modulus of continuity. These are inequalities for the
derivatives of a definable function. Since the constants that appear in them
are universal, it is not difficult to get them uniform for definable families
of functions. See Corollary 2.18 and Proposition 2.19.

  2. (2)

_Uniform $\Lambda_{p}$-stratification of definable families of sets._ Roughly
speaking, definable families of sets admit a stratification into a finite
number of cells that are defined by functions satisfying certain estimates
(for their derivatives up to order $p$). The constants in these estimates and
the number of cells are independent of the parameter of the family. See
Theorem 2.16. This is essential for the uniform extension theorem 1.3. We
think that it is also of independent interest.

It is a natural question if there exists even a continuous and/or linear
extension operator for definable Whitney jets of class $C^{m,\omega}$ (or
$C^{m}$) on a definable closed set $E\subseteq\mathbb{R}^{n}$. This remains an
open problem. The theorem of Bartle and Graves [2] (see also [3, Theorem 1.6])
is not applicable since the normed spaces of definable jets and functions
(defined in Section 3) are not complete.

Note that Azagra, Le Gruyer, and Mudarra [1] give an explicit formula for the
extension of Whitney jets of class $C^{1,1}$ with an optimal control of the
norms; for definable input this explicit formula yields a definable
$C^{1,1}$-extension. See also the discussion in [9, Sections 4.2–4].

Let us point out that Pawłucki [10] presents a continuous linear extension
operator for (not necessarily definable) Whitney jets on definable closed sets
which preserves (up to a multiplicative constant) the modulus of continuity.
This extension operator is a finite composite of operators that preserve
definability on the one hand or are defined by integration with respect to a
parameter (more precisely, convolution) on the other hand; in general, the
latter leads out of the original o-minimal structure.

While [10] was a important source of inspiration for handling the modulus of
continuity, the main difficulty (besides getting everything uniformly bounded)
was to replace the convolution operators by definable operations which at the
same time allow for preserving the modulus of continuity.

The paper is organized as follows. In Section 2 the main geometric tools are
prepared: Gromov’s inequality and uniform $\Lambda_{p}$-stratification. We
present in Section 3 background on definable bounded families of Whitney jets
of class $C^{m,\omega}$, most notably, how they behave under pullback along a
definable family of $\Lambda_{p}$-regular maps. The proof of Theorem 1.3 is
carried out in Section 4. In the final Section 5 we give the mentioned
applications, discuss dependence on the modulus of continuity, and prove
Theorem 1.4.

### Notation

Let $\mathbb{N}=\\{0,1,2,\ldots\\}$ be the set of nonnegative integers. We
denote by $d(x,S):=\inf_{y\in S}|x-y|$ the Euclidean distance in
$\mathbb{R}^{n}$ of a point $x$ to a subset $S$ of $\mathbb{R}^{n}$, with the
convention $d(x,\emptyset):=+\infty$. The open Euclidean ball with center
$x\in\mathbb{R}^{n}$ and radius $r>0$ is denoted by
$B(x,r):=\\{y\in\mathbb{R}^{n}:|x-y|<r\\}$. The closure of a set $S$ is
denoted by $\overline{S}$ and the frontier of $S$ by $\partial
S:=\overline{S}\setminus S$; we will not use the topological boundary
$\overline{S}\cap\overline{\mathbb{R}^{n}\setminus S}$. If $S$ is a subset of
$\mathbb{R}^{k}$, we write $S\times 0$ for the set
$\\{(u,w)\in\mathbb{R}^{k}\times\mathbb{R}^{\ell}:u\in\mathbb{R}^{k},w=0\\}$.
The graph of a map $\varphi$ is denoted by $\Gamma(\varphi)$. For real valued
nonnegative functions $f,g$ we write $f\lesssim g$ if $f\leq Cg$ for some
universal constant $C>0$. In particular, it should be always understood that
$C$ is independent of $a\in A$, i.e., the parameter which we consistently use
in parameterized families of sets and maps. We write $f\approx g$ if
$f\lesssim g$ and $g\lesssim f$. We use standard multi-index notation and in
this context $(i)\in\mathbb{N}^{n}$ is the multi-index
$(0,\ldots,0,1,0,\ldots,0)$ with $1$ in the $i$-th entry.

##  2\. Uniform $\Lambda_{p}$-stratifications

The existence of uniform $\Lambda_{p}$-stratifications (Theorem 2.16) is based
on an inequality of Gromov [5], of which we need two versions, and on uniform
L-regular decomposition due to Kurdyka and Parusiński [6].

###  2.1. Definable families of sets and maps

Let $A$ be a definable subset of $\mathbb{R}^{N}$. A family $(E_{a})_{a\in A}$
of definable sets $E_{a}\subseteq\mathbb{R}^{n}$ is called a _definable
family_ if the associated set

(2.1) $E:=\bigcup_{a\in A}\\{a\\}\times E_{a}$

is a definable subset of $\mathbb{R}^{N}\times\mathbb{R}^{n}$. Conversely, any
definable subset $E\subseteq\mathbb{R}^{N}\times\mathbb{R}^{n}$ defines a
definable family $(E_{a})_{a\in A}$ by setting
$A:=\\{a\in\mathbb{R}^{N}:\exists x\in\mathbb{R}^{n}:(a,x)\in E\\}$ and
$E_{a}:=\\{x\in\mathbb{R}^{n}:(a,x)\in E\\}$, $a\in A$. If we allow
$E_{a}=\emptyset$, we may just take $A=\mathbb{R}^{N}$.

A family $(E_{a}^{\prime})_{a\in A}$ of subsets $E_{a}^{\prime}\subseteq
E_{a}$ is said to be a _definable subfamily_ of $(E_{a})_{a\in A}$ if the
associated set $E^{\prime}$ (defined in analogy to (2.1)) is a definable
subset of $E$.

A family $(\varphi_{a})_{a\in A}$ of definable maps
$\varphi_{a}:E_{a}\to\mathbb{R}^{m}$ is called a _definable family_ if the map
$\varphi:E\to\mathbb{R}^{m}$, where $E$ is the associated set (2.1) and

(2.2) $\varphi(a,u):=\varphi_{a}(u),\quad u\in E_{a},$

is definable. This is consistent with the first paragraph, since

$\displaystyle\Gamma(\varphi)$
$\displaystyle=\\{(a,u,\varphi(a,u))\in\mathbb{R}^{N}\times\mathbb{R}^{n}\times\mathbb{R}^{m}:(a,u)\in
E\\}$ $\displaystyle=\bigcup_{a\in
A}\\{(a,u,\varphi_{a}(u))\in\mathbb{R}^{N}\times\mathbb{R}^{n}\times\mathbb{R}^{m}:u\in
E_{a}\\}$ $\displaystyle=\bigcup_{a\in
A}\\{a\\}\times\\{(u,\varphi_{a}(u))\in\mathbb{R}^{n}\times\mathbb{R}^{m}:u\in
E_{a}\\}$ $\displaystyle=\bigcup_{a\in A}\\{a\\}\times\Gamma(\varphi_{a}).$

###  2.2. Gromov’s inequality

We need two versions of an inequality due to Gromov [5]. We start with a
$C^{m}$-version.

######  Lemma 2.1 ([7, Lemma 2]).

Let $m\geq 1$. Let $f:I\to\mathbb{R}$ be a $C^{m+1}$-function, where
$I=[t_{0}-r,t_{0}+r]\subseteq\mathbb{R}$, $r>0$, is an interval. Suppose that,
for all $j=2,\ldots,m+1$, we have either $f^{(j)}\geq 0$ on $I$ or
$f^{(j)}\leq 0$ on $I$. Then

$|f^{(m)}(t_{0})|\leq 2^{\binom{m+2}{2}-2}\,\frac{\sup_{t\in
I}|f(t)|}{r^{m}}.$

We combine Lemma 2.1 with the following lemma in order to get a
$C^{m,\omega}$-version in Lemma 2.3.

######  Lemma 2.2.

Let $f:I\to\mathbb{R}$ be a $C^{2}$-function, where
$I=[t_{0}-r,t_{0}+r]\subseteq\mathbb{R}$, $r>0$, is an interval, and let
$\omega$ be a modulus of continuity for $f$. Suppose that
$f^{\prime\prime}\geq 0$ on $I$ or $f^{\prime\prime}\leq 0$ on $I$. Then

$|f^{\prime}(t_{0})|\leq\frac{\omega(r)}{r}.$

###### Proof.

We may assume that $t_{0}=0$. Suppose that $f^{\prime\prime}\geq 0$ on $I$.
Then $f$ is convex and, for $0<s<r$,

$\frac{f(s)-f(0)}{s}\leq\frac{f(r)-f(0)}{r}\leq\frac{\omega(r)}{r}.$

Letting $s\to 0$, we find that $f^{\prime}(0)\leq\omega(r)/r$. The same
reasoning applied to $f(-t)$ shows that also $-f^{\prime}(0)\leq\omega(r)/r$
so that the assertion is proved.

The case $f^{\prime\prime}\leq 0$ follows from the previous one by considering
$-f$. ∎

######  Lemma 2.3.

Let $m\geq 1$. Let $f:I\to\mathbb{R}$ be a $C^{m+1}$-function, where
$I=[t_{0}-r,t_{0}+r]\subseteq\mathbb{R}$, $r>0$, is an interval, and let
$\omega$ be a modulus of continuity for $f$. Suppose that, for all
$j=2,\ldots,m+1$, we have either $f^{(j)}\geq 0$ on $I$ or $f^{(j)}\leq 0$ on
$I$. Then

$|f^{(m)}(t_{0})|\leq 2^{\binom{m+1}{2}+m-2}\,\frac{\omega(r)}{r^{m}}.$

###### Proof.

If $m=1$, then the statement is immediate from Lemma 2.2. If $m\geq 2$, then,
by Lemma 2.1 applied to $f^{\prime}$ and in turn Lemma 2.2, we have

$|f^{(m)}(t_{0})|\leq\frac{2^{\binom{m+1}{2}-2}}{(\frac{r}{2})^{m-1}}\sup_{|t-t_{0}|\leq\frac{r}{2}}|f^{\prime}(t)|\leq\frac{2^{\binom{m+1}{2}-2}}{(\frac{r}{2})^{m-1}}\cdot\frac{\omega(\tfrac{r}{2})}{\tfrac{r}{2}}\leq
2^{\binom{m+1}{2}+m-2}\,\frac{\omega(r)}{r^{m}}$

as claimed, since $\omega$ is increasing. ∎

###  2.3. Uniform bounds for definable families of functions

######  Proposition 2.4.

Let $(U_{a})_{a\in A}$ be a definable family of open sets
$U_{a}\subseteq\mathbb{R}^{k}$ and let
$U\subseteq\mathbb{R}^{N}\times\mathbb{R}^{k}$ be the associated definable set
(cf. (2.1)). Let $(\varphi_{a})_{a\in A}$ be a definable family of functions
$\varphi_{a}:U_{a}\to\mathbb{R}$ and let $\varphi:U\to\mathbb{R}$ be the
associated definable function (cf. (2.2)). Let $\alpha\in\mathbb{N}^{k}$ with
$|\alpha|=p$. There exists a definable subset $Z\subseteq U$ such that, for
all $a\in A$, $Z_{a}$ is closed in $U_{a}$, $\dim Z_{a}<k$, $\varphi_{a}$ is
$C^{p}$ on $U_{a}\setminus Z_{a}$, and, for each open ball $B=B(u,r)$, $r>0$,
contained in $U_{a}\setminus Z_{a}$, we have

(2.3) $|\partial^{\alpha}\varphi_{a}(u)|\leq C(k,p)\,\sup_{v\in
B}|\varphi_{a}(v)|\,r^{-|\alpha|}.$

###### Proof.

Consider the definable set

$\displaystyle X$ $\displaystyle:=\\{(b,v)\in U:(a,u)\mapsto\varphi(a,u)\text{
is not }C^{p}\text{ at }(b,v)\text{ in }u\\}$ $\displaystyle=\\{(b,v)\in
U:\varphi_{b}\text{ is not }C^{p}\text{ at }v\\}$

and note that

$X_{a}=\\{u\in U_{a}:\varphi_{a}\text{ is not }C^{p}\text{ at }u\\}$

is closed in $U_{a}$ and, by o-minimality, $\dim X_{a}<k$.

The operator $\partial^{\alpha}$ is a linear combination of directional
derivatives $d_{v}^{p}$ for a finite collection $V$ of suitably chosen unit
directions $v$ in $\mathbb{R}^{k}$. Let $\varphi_{1},\ldots,\varphi_{s}$ be an
enumeration of all functions $d^{j}_{v}\varphi:U\setminus X\to\mathbb{R}$,
$j=2,\ldots,p+1$, $v\in V$ (where $d_{v}^{j}$ acts only in the $u$-variable:
$d_{v}^{j}\varphi(a,u)=\partial_{t}^{j}|_{t=0}\,\varphi(a,u+tv)$). For
$i=1,\ldots,s$, set

$\displaystyle Y_{i}$ $\displaystyle:=\bigcup_{a\in A}\\{(a,u)\in U\setminus
X:\exists\epsilon>0\,\forall v\in B(u,\epsilon):\varphi_{i}(a,v)=0\\},$
$\displaystyle Z_{i}$ $\displaystyle:=\\{(a,u)\in U\setminus
X:\varphi_{i}(a,u)=0\\}\setminus Y_{i},$ and $\displaystyle Z$
$\displaystyle:=X\cup\bigcup_{i=1}^{s}Z_{i}.$

Then $Z$ is a definable subset of $U$ and, for all $a\in A$, $Z_{a}$ is closed
in $U_{a}$ and $\dim Z_{a}<k$.

Now (2.3) follows easily by applying Lemma 2.1 to $t\mapsto\varphi(a,u+tv)$. ∎

######  Corollary 2.5.

Let $(U_{a})_{a\in A}$ be a definable family of open sets
$U_{a}\subseteq\mathbb{R}^{k}$ and let $(\varphi_{a})_{a\in A}$ be a definable
family of $C^{1}$-functions $\varphi_{a}:U_{a}\to\mathbb{R}$. Suppose that
there is a constant $M>0$ such that

$|\partial_{j}\varphi_{a}(u)|\leq M,\quad a\in A,\,u\in U_{a},\,j=1,\ldots,k.$

Let $p$ be a positive integer. There exists a definable family $(Z_{a})_{a\in
A}$ of closed definable sets $Z_{a}\subseteq U_{a}$ of dimension $\dim
Z_{a}<k$ such that, for all $a\in A$, $\varphi_{a}$ is of class $C^{p}$ on
$U_{a}\setminus Z_{a}$ and

$|\partial^{\gamma}\varphi_{a}(u)|\leq C(k,p)\,M\,d(u,Z_{a}\cup\partial
U_{a})^{1-|\gamma|},\quad u\in U_{a}\setminus Z_{a},\,1\leq|\gamma|\leq p.$

###### Proof.

Apply Proposition 2.4 to $\partial_{j}\varphi_{a}$. ∎

######  Remark 2.6.

We may assume that $Z_{a}$ is not empty so that $d(u,Z_{a}\cup\partial U_{a})$
is always finite. We will tacitly make this assumption in all subsequent
results of this type.

######  Proposition 2.7.

Let $(U_{a})_{a\in A}$ be a definable family of open sets
$U_{a}\subseteq\mathbb{R}^{k}$ and let $(\varphi_{a})_{a\in A}$ be a definable
family of continuous functions $\varphi_{a}:U_{a}\to\mathbb{R}$. Let $p$ be a
positive integer. Then there exists a definable family $(Z_{a})_{a\in A}$ of
closed subsets $Z_{a}\subseteq U_{a}$ of dimension $\dim Z_{a}<k$ such that,
for all $a\in A$, $\varphi_{a}$ is $C^{p}$ on $U_{a}\setminus Z_{a}$ and

(2.4) $|\partial^{\gamma}\varphi_{a}(x)|\leq
C(k,p)\,\frac{\omega(d(x,Z_{a}\cup\partial U_{a}))}{d(x,Z_{a}\cup\partial
U_{a})^{|\gamma|}},\quad x\in U_{a}\setminus Z_{a},\,1\leq|\gamma|\leq p,$

where $\omega$ is a modulus of continuity for $\varphi_{a}$.

###### Proof.

Follow the proof of Proposition 2.4 and use Lemma 2.3. ∎

######  Remark 2.8.

We want to emphasize that the construction of $(Z_{a})_{a\in A}$ is
independent of $\omega$.

###  2.4. $\Lambda_{p}$-regular mappings

Let $U\subseteq\mathbb{R}^{k}$ be an open set. Let $p$ be a positive integer.
A $C^{p}$-mapping $\varphi:U\to\mathbb{R}^{n}$ is said to be _$\Lambda_{p}$
-regular_ if there exists a constant $C>0$ such that

(2.5) $|\partial^{\gamma}\varphi(u)|\leq C\,d(u,\partial U)^{1-|\gamma|},\quad
u\in U,\,1\leq|\gamma|\leq p.$

$\Lambda_{p}$-regular maps behave nicely on quasiconvex sets. Let us first
recall the definition of quasiconvexity.

######  Definition 2.9 (Quasiconvex sets).

A set $E\subseteq\mathbb{R}^{n}$ is called _quasiconvex_ if there is a
constant $C>0$ such any two points $x,y\in E$ can be joined in $E$ by a
rectifiable path of length at most $C\,|x-y|$.

Let $\varphi:U\to\mathbb{R}^{n}$ by $\Lambda_{p}$-regular. If $E\subseteq U$
is a quasiconvex subset, then $\varphi|_{E}$ is Lipschitz on $E$ and extends
continuously to a map $\overline{\varphi}$ on $\overline{E}$.

###  2.5. $\Lambda_{p}$-cells

Let $p$ be a positive integer. We define recursively $\Lambda_{p}$-cells in
$\mathbb{R}^{n}$:

A definable subset $S\subseteq\mathbb{R}^{n}$ is an _open $\Lambda_{p}$-cell
in $\mathbb{R}^{n}$_ if,

in case $n=1$:

$S$ is an open interval in $\mathbb{R}$,

in case $n>1$:

$S$ is of the form

$S=(\psi_{1},\psi_{2},T):=\\{(x^{\prime},x_{n}):x^{\prime}\in
T,\,\psi_{1}(x^{\prime})<x_{n}<\psi_{2}(x^{\prime})\\},$

where $T$ is an open $\Lambda_{p}$-cell in $\mathbb{R}^{n-1}$ and each
$\psi_{i}$, $i=1,2$, is either a $\Lambda_{p}$-regular definable function
$T\to\mathbb{R}$ or identically $-\infty$ or $+\infty$, and
$\psi_{1}<\psi_{2}$ on $T$. (Here $x^{\prime}=(x_{1},\ldots,x_{n-1})$.)

Note that $S$ is quasiconvex. Moreover, if $\psi_{i}$ is finite, then it is
Lipschitz on $T$ and has a continuous extension $\overline{\psi}_{i}$ to
$\overline{T}$.

A definable subset $S$ of $\mathbb{R}^{n}$ is a _$k$ -dimensional
$\Lambda_{p}$-cell in $\mathbb{R}^{n}$_, where $k=0,\ldots,n-1$, if

$S=\\{(u,w):u\in T,\,w=\varphi(u)\\}=\Gamma(\varphi),$

where $u=(x_{1},\ldots,x_{k})$, $w=(x_{k+1},\ldots,x_{n})$, $T$ is an open
$\Lambda_{p}$-cell in $\mathbb{R}^{k}$, and $\varphi:T\to\mathbb{R}^{n-k}$ is
a $\Lambda_{p}$-regular definable map.

######  Definition 2.10 ($\Lambda_{p}$-cell with constant $C$).

A _$\Lambda_{p}$ -cell $S$ in $\mathbb{R}^{n}$_ is an open or a
$k$-dimensional $\Lambda_{p}$-cell in $\mathbb{R}^{n}$. We say that $S$ is a
$\Lambda_{p}$-cell in $\mathbb{R}^{n}$ _with constant $C$_ if all the
$\Lambda_{p}$-regular maps involved in the recursive definition of $S$ satisfy
(2.5) with the same constant $C$.

###  2.6. Associated functions

We associate with any open $\Lambda_{p}$-cell $S$ in $\mathbb{R}^{n}$ a
sequence of $2n+1$ definable functions $\rho_{j}:\overline{S}\to[0,\infty]$,
$j=0,1,2,\ldots,2n$. We put $\rho_{0}\equiv 1$ and define $\rho_{j}$ for
$j\geq 1$ as follows:

Case $n=1$:

If $S=(a_{1},a_{2})$ we set

$\displaystyle\rho_{1}(x)$ $\displaystyle:=\begin{cases}x-a_{1}&\text{ if
}a_{1}\in\mathbb{R},\\\ +\infty&\text{ if }a_{1}=-\infty,\end{cases}$
$\displaystyle\rho_{2}(x)$ $\displaystyle:=\begin{cases}a_{2}-x&\text{ if
}a_{2}\in\mathbb{R},\\\ +\infty&\text{ if }a_{2}=+\infty.\end{cases}$

Case $n>1$:

If $S=(\psi_{1},\psi_{2},T)$ and $\sigma_{j}$, $j=1,\ldots,2n-2$, are the
functions associated with $T$, we set $\rho_{j}(x)=\sigma_{j}(x^{\prime})$,
for $j=1,\ldots,2n-2$, and

$\displaystyle\rho_{2n-1}(x)$
$\displaystyle:=\begin{cases}x_{n}-\overline{\psi}_{1}(x^{\prime})&\text{ if
}\psi_{1}\text{ is finite},\\\ +\infty&\text{ if
}\psi_{1}\equiv-\infty,\end{cases}$ $\displaystyle\rho_{2n}(x)$
$\displaystyle:=\begin{cases}\overline{\psi}_{2}(x^{\prime})-x_{n}&\text{ if
}\psi_{2}\text{ is finite},\\\ +\infty&\text{ if
}\psi_{2}\equiv+\infty.\end{cases}$

######  Remark 2.11.

We add the function $\rho_{0}$ (which is not present in [7, 8, 12]) in order
to handle the extension from unbounded $\Lambda_{p}$-cells (cf. proof of
Theorem 1.3).

Each of the functions $\rho_{j}$, that is finite, is $\Lambda_{p}$-regular on
$S$ and Lipschitz on $\overline{S}$, cf. [7, Lemma 4]. There is a positive
constant $C>0$ such that

(2.6) $\frac{1}{C}\min_{j\geq 1}\rho_{j}(x)\leq d(x,\partial S)\leq\min_{j\geq
1}\rho_{j}(x),\quad x\in\overline{S},$

where $d(x,\emptyset)=+\infty$ by convention, since the faces of $S$ are
graphs of Lipschitz maps; cf. [7, Lemma 3]. Consequently,

(2.7) $\frac{1}{C}\min_{j\geq 0}\rho_{j}(x)\leq\min\\{1,d(x,\partial
S)\\}\leq\min_{j\geq 0}\rho_{j}(x),\quad x\in\overline{S}.$

Furthermore, if $\rho_{j}$ for $j\geq 1$ is finite, then there exists a
positive constant $C>0$ such that

(2.8) $|\partial^{\gamma}(1/\rho_{j})(x)|\leq C\,d(x,\partial
S)^{-|\gamma|-1},\quad x\in S,\,|\gamma|\leq p,$

cf. [7, Lemma 5] and (2.10). It follows that for all finite $\rho_{j}$, $j\geq
0$, we have

(2.9) $|\partial^{\gamma}(1/\rho_{j})(x)|\leq C\,\min\\{1,d(x,\partial
S)\\}^{-|\gamma|-1},\quad x\in S,\,|\gamma|\leq p.$

######  Remark 2.12.

The constants $C$ in (2.6)–(2.9) only depend on the constants of the
$\Lambda_{p}$-regular maps involved in the definition of $S$.

For later reference, we recall that for a non-vanishing $C^{p}$-function $r$
we have, for $1\leq|\gamma|\leq p$.

(2.10)
$\partial^{\gamma}(1/r)=\sum_{j=1}^{|\gamma|}\Big{(}\sum_{\begin{subarray}{c}\delta_{1}+\cdots+\delta_{j}=\gamma\\\
\delta_{1}\neq 0,\ldots,\delta_{j}\neq
0\end{subarray}}a^{\gamma}_{\delta_{1}\ldots\delta_{j}}\partial^{\delta_{1}}r\cdots\partial^{\delta_{j}}r\Big{)}r^{-j-1},$

where $a^{\gamma}_{\delta_{1}\ldots\delta_{j}}$ are integers that only depend
on $\gamma$ and $\delta_{1},\ldots,\delta_{j}$.

###  2.7. $\Lambda_{p}$-stratification of definable sets

Recall that a definable _$C^{p}$ -stratification_ of a definable set
$E\subseteq\mathbb{R}^{n}$ is a finite decomposition $\mathscr{S}$ of $E$ into
definable $C^{p}$-submanifolds of $\mathbb{R}^{n}$, called strata, such that,
for each stratum $S\in\mathscr{S}$, the frontier $(\overline{S}\setminus
S)\cap E$ in $E$ is the union of some strata of dimension $<\dim S$. A
stratification is called _compatible_ with a collection of finitely many
definable subsets of $E$ if each subset is a union of strata.

A definable $C^{p}$-stratification $\mathscr{S}$ of $E$ is called a
_$\Lambda_{p}$ -stratification_ if each stratum $S\in\mathscr{S}$ is a
$\Lambda_{p}$-cell in $\mathbb{R}^{n}$ in some linear coordinate system.

######  Theorem 2.13 ([7, Proposition 4] and [8, Theorem 3]).

Let $E\subseteq\mathbb{R}^{n}$ be a definable set and let
$E_{1},\ldots,E_{\ell}$ be definable subsets of $E$. Then there exists a
$\Lambda_{p}$-stratification $\mathscr{S}$ of $E$ that is compatible with
$E_{1},\ldots,E_{\ell}$.

###  2.8. Uniform $\Lambda_{p}$-stratifications of definable families of sets

We prove a uniform version of Theorem 2.13. Let us first recall a result on
uniform $L$-regular decompositions.

######  Theorem 2.14 ([6, Proposition 1.4]).

Let $E^{k}\subseteq\mathbb{R}^{N}\times\mathbb{R}^{n}$, $k\in K$, be a finite
collection of definable sets. Then there exist finitely many disjoint
definable sets $B^{i}\subseteq\mathbb{R}^{N}\times\mathbb{R}^{n}$, $i\in I$,
and linear orthogonal mappings $\varphi^{i}:\mathbb{R}^{n}\to\mathbb{R}^{n}$,
$i\in I$, such that:

  1. (1)

For every $a\in\mathbb{R}^{N}$, each $\varphi^{i}(B^{i}_{a})$ is a standard
L-regular cell in $\mathbb{R}^{n}$ with constant $C=C(n)$.

  2. (2)

For every $a\in\mathbb{R}^{N}$, the family $B^{i}_{a}$, $i\in I$, is a
stratification of $\mathbb{R}^{n}$.

  3. (3)

For any $k\in K$, there exists $I_{k}\subseteq I$ such that
$E^{k}_{a}=\bigcup_{i\in I_{k}}B^{i}_{a}$ for every $a\in\mathbb{R}^{N}$.

Here a _standard $L$-regular cell in $\mathbb{R}^{n}$ with constant $C=C(n)$_
(which is terminology used in [6]) is by definition nothing else than a
$\Lambda_{1}$-cell with constant $C=C(n)$.

######  Definition 2.15 (Uniform $\Lambda_{p}$-stratification).

Let $(E_{a})_{a\in A}$ be a definable family of sets
$E_{a}\subseteq\mathbb{R}^{n}$ and let
$E\subseteq\mathbb{R}^{N}\times\mathbb{R}^{n}$ be the associated definable set
(cf. (2.1)). Let $p$ be a positive integer.

A finite collection $\mathscr{S}=\\{S^{j}\\}_{j\in J}$ of disjoint definable
sets $S^{j}\subseteq\mathbb{R}^{N}\times\mathbb{R}^{n}$ is called a _uniform
$\Lambda_{p}$-stratification of $(E_{a})_{a\in A}$_ if

  1. (1)

There exist linear orthogonal maps
$\varphi^{j}:\mathbb{R}^{n}\to\mathbb{R}^{n}$, $j\in J$, such that, for each
$a\in A$ and each $j\in J$, $\varphi^{j}(S^{j}_{a})$ is a $\Lambda_{p}$-cell
in $\mathbb{R}^{n}$ with constant $C$ independent of $a\in A$.

  2. (2)

For each $a\in A$, the family $S^{j}_{a}$, $j\in J$, is a stratification of
$E_{a}$.

For all $a\in A$, let us set $\mathscr{S}_{a}:=\\{S^{j}_{a}\\}_{j\in J}$.
Abusing notation, we will also say that _$(\mathscr{S}_{a})_{a\in A}$ is a
uniform $\Lambda_{p}$-stratification of $(E_{a})_{a\in A}$_.

Let $(E^{i}_{a})_{a\in A}$, $i\in I$, be a finite collection of definable
subfamilies of $(E_{a})_{a\in A}$. A uniform $\Lambda_{p}$-stratification
$(\mathscr{S}_{a})_{a\in A}$ of $(E_{a})_{a\in A}$ is said to be _compatible
with $(E^{i}_{a})_{a\in A}$, $i\in I$,_ if additionally

  1. (3)

For each $i\in I$, there exists a subset $J_{i}\subseteq J$ such that
$E^{i}_{a}=\bigcup_{j\in J_{i}}S^{j}_{a}$ for each $a\in A$.

By Theorem 2.14, there always exist uniform $\Lambda_{1}$-stratifications. We
shall see that there exist uniform $\Lambda_{p}$-stratifications for all
$p\geq 1$.

######  Theorem 2.16.

Let $(E_{a})_{a\in A}$ be a definable family of sets
$E_{a}\subseteq\mathbb{R}^{n}$ and $(E^{i}_{a})_{a\in A}$, $i\in I$, a finite
collection of definable subfamilies of $(E_{a})_{a\in A}$. Let $p$ be a
positive integer. Then there exists a uniform $\Lambda_{p}$-stratification
$(\mathscr{S}_{a})_{a\in A}$ of $(E_{a})_{a\in A}$ compatible with
$(E^{i}_{a})_{a\in A}$, $i\in I$.

###### Proof.

Let $k=\max_{a\in A}\dim E_{a}$. We proceed by induction on $k$. If $k=0$,
then all $E_{a}$ are finite and the number of elements of $E_{a}$ is bounded
by a constant independent of $a$. In that case, the assertion is trivially
true.

Suppose that $k>0$. We claim that there exist a finite collection of disjoint
definable sets $T^{j}\subseteq\mathbb{R}^{N}\times\mathbb{R}^{n}$, $j\in J$,
and linear orthogonal maps $\varphi^{j}:\mathbb{R}^{n}\to\mathbb{R}^{n}$,
$j\in J$, such that, for each $a\in A$ and each $j\in J$,

  * •

$T^{j}_{a}$ is either empty or open in $E_{a}$ and compatible with
$E^{i}_{a}$, $i\in I$,

  * •

if $T^{j}_{a}\neq\emptyset$ then $\varphi^{j}(T^{j}_{a})$ is a $k$-dimensional
$\Lambda_{p}$-cell in $\mathbb{R}^{n}$ with constant $C$ independent of $a\in
A$, and

  * •

$\dim E_{a}\setminus\bigcup_{j\in J}T^{j}_{a}<k$.

We allow $T^{j}_{a}=\emptyset$ to account for the case $\dim E_{a}<k$.

Then we can use the induction hypothesis for the definable family
$(E^{\prime}_{a})_{a\in A}$, where
$E^{\prime}_{a}:=E_{a}\setminus\bigcup_{j\in J}T^{j}_{a}$, and the definable
subfamilies $(E^{i}_{a}\cap E^{\prime}_{a})_{a\in A}$, $i\in I$, and
$((\overline{T}^{j}_{a}\setminus T^{j}_{a})\cap E^{\prime}_{a})_{a\in A}$,
$j\in J$. The statement follows.

Let us prove the claim. Theorem 2.14 implies that the claim holds for $p=1$:
let $T^{j}$, $j\in J$, be the corresponding sets with all the properties as
listed in the claim. Now we apply Corollary 2.5 and induction on the
dimension. In fact, for each fixed $j\in J$, $(T^{j}_{a})_{a\in A}$ is a
definable family of $k$-dimensional $\Lambda_{1}$-cells $T^{j}_{a}$ in
$\mathbb{R}^{n}$ that are open in $E_{a}$. After the change of coordinates
$\varphi^{j}$, we may assume that $T^{j}_{a}$ is a $\Lambda_{1}$-cell with
constant $C$ independent of $a\in A$. By Corollary 2.5, there is a definable
family $(Z^{j}_{a})_{a\in A}$ of closed definable sets $Z^{j}_{a}\subseteq
T^{j}_{a}$, $\dim Z^{j}_{a}<k$, such that the functions defining the cell
$T^{j}_{a}$ are $\Lambda_{p}$-regular with uniform constants independent of
$a\in A$ in the complement of $Z^{j}_{a}$. Thus there exists a definable
family $(S^{j}_{a})_{a\in A}$ of subsets $S^{j}_{a}\subseteq T^{j}_{a}$ such
that, for all $a\in A$, $S^{j}_{a}$ is a finite disjoint union of
$k$-dimensional definable $\Lambda_{p}$-cells $S^{j,\ell}_{a}$ that are open
in $E_{a}$ with constant $C$ independent of $a\in A$ and $\dim
T^{j}_{a}\setminus S^{j}_{a}<k$. Thus the number of connected components
$S^{j,\ell}_{a}$ of $S^{j}_{a}$ is uniformly bounded by a constant independent
of $a\in A$. This implies the claim. ∎

###  2.9. Consequences

We may use Theorem 2.16 in order to refine Proposition 2.4, Corollary 2.5, and
Proposition 2.7.

######  Corollary 2.17.

Let $(U_{a})_{a\in A}$ be a definable family of open sets
$U_{a}\subseteq\mathbb{R}^{k}$. Let $(\varphi_{a})_{a\in A}$ be a definable
family of functions $\varphi_{a}:U_{a}\to\mathbb{R}$. Let $p$ be a nonnegative
integer. There exists a uniform $\Lambda_{p}$-stratification
$(\mathscr{S}_{a})_{a\in A}$ of $(U_{a})_{a\in A}$ such that, for all $a\in A$
and each open stratum $S_{a}\in\mathscr{S}_{a}$, $\varphi_{a}$ is $C^{p}$ on
$S_{a}$ and

$|\partial^{\gamma}\varphi_{a}(u)|\leq
C(k,p)\,\frac{\sup\\{|\varphi_{a}(v)|:v\in S_{a},\,|v-u|<d(u,\partial
S_{a})\\}}{d(u,\partial S_{a})^{|\gamma|}},\quad u\in S_{a},\,|\gamma|\leq p.$

###### Proof.

This follows from Theorem 2.16 and Proposition 2.4. ∎

######  Corollary 2.18.

Let $(U_{a})_{a\in A}$ be a definable family of open sets
$U_{a}\subseteq\mathbb{R}^{k}$. Let $(\varphi_{a})_{a\in A}$ be a definable
family of $C^{1}$-functions $\varphi_{a}:U_{a}\to\mathbb{R}$. Suppose that
there is a constant $M>0$ such that

$|\partial_{j}\varphi_{a}(u)|\leq M,\quad a\in A,\,u\in U_{a},\,j=1,\ldots,k.$

Let $p$ be a positive integer. There exists a uniform
$\Lambda_{p}$-stratification $(\mathscr{S}_{a})_{a\in A}$ of $(U_{a})_{a\in
A}$ such that, for all $a\in A$ and each open stratum
$S_{a}\in\mathscr{S}_{a}$, $\varphi_{a}$ is $C^{p}$ on $S_{a}$ and

$|\partial^{\gamma}\varphi_{a}(u)|\leq C(k,p)\,M\,d(u,\partial
S_{a})^{1-|\gamma|},\quad u\in S_{a},\,1\leq|\gamma|\leq p.$

###### Proof.

This follows from Theorem 2.16 and Corollary 2.5. ∎

######  Proposition 2.19.

Let $(U_{a})_{a\in A}$ be a definable family of open sets
$U_{a}\subseteq\mathbb{R}^{k}$. Let $(\varphi_{a})_{a\in A}$ be a definable
family of continuous functions $\varphi_{a}:U_{a}\to\mathbb{R}$. Let $p$ be a
positive integer. There exists a uniform $\Lambda_{p}$-stratification
$(\mathscr{S}_{a})_{a\in A}$ of $(U_{a})_{a\in A}$ such that, for all $a\in A$
and each open stratum $S_{a}\in\mathscr{S}_{a}$, $\varphi_{a}$ is $C^{p}$ on
$S_{a}$ and

$|\partial^{\gamma}\varphi_{a}(u)|\leq C(k,p)\,\frac{\omega(d(u,\partial
S_{a}))}{d(u,\partial S_{a})^{|\gamma|}},\quad u\in S_{a},\,1\leq|\gamma|\leq
p,$

where $\omega$ is a modulus of continuity for $\varphi_{a}$.

###### Proof.

This follows from Theorem 2.16 and Proposition 2.7. ∎

We will need another uniform fact:

######  Proposition 2.20.

Let $(U_{a})_{a\in A}$ be a definable family of open sets
$U_{a}\subseteq\mathbb{R}^{k}$. Let $(\varphi_{a})_{a\in A}$ a definable
family of functions $\varphi_{a}:U_{a}\to\mathbb{R}$. Let $p$ be a positive
integer. There exists a uniform $\Lambda_{p}$-stratification
$(\mathscr{S}_{a})_{a\in A}$ of $(U_{a})_{a\in A}$ such that, for all $a\in A$
and each open stratum $S_{a}\in\mathscr{S}_{a}$, $\varphi_{a}$ is $C^{p}$ on
$S_{a}$ and, for all $j=1,\ldots,k$,

(2.11) $\text{ either }\quad|\partial_{j}\varphi_{a}|\geq 1\text{ on
}S_{a}\quad\text{ or }\quad|\partial_{j}\varphi_{a}|<1\text{ on }S_{a}.$

###### Proof.

Let $U\subseteq\mathbb{R}^{N}\times\mathbb{R}^{n}$ and
$\varphi:U\to\mathbb{R}$ be the associated definable set and function (cf.
(2.1) and (2.2)). Let $X\subseteq U$ be the set defined in the proof of
Proposition 2.4.

For $j=1,\ldots,k$, let $\partial_{j}\varphi(a,u):=\partial_{j}\varphi_{a}(u)$
and set

$\displaystyle Y_{j}$ $\displaystyle:=\bigcup_{a\in A}\\{(a,u)\in U\setminus
X:\exists\epsilon>0\,\forall v\in
B(u,\epsilon):\partial_{j}\varphi(a,v)=1\\},$ $\displaystyle Z_{j}$
$\displaystyle:=\\{(a,u)\in U\setminus
X:\partial_{j}\varphi(a,u)=1\\}\setminus Y_{j},$ and $\displaystyle Z$
$\displaystyle:=X\cup\bigcup_{j=1}^{k}Z_{j}.$

Then $Z$ is a definable subset of $U$ and, for all $a\in A$, $Z_{a}$ is closed
in $U_{a}$ and $\dim Z_{a}<k$. Now the statement follows from Theorem 2.16. ∎

##  3\. Bounded definable families of Whitney jets

Recall that a modulus of continuity $\omega$ is by definition a positive,
continuous, increasing, and concave function $\omega:(0,\infty)\to(0,\infty)$
such that $\omega(t)\to 0$ as $t\to 0$.

###  3.1. $C^{m,\omega}$-functions

Let $\omega$ be a modulus of continuity. Let $U\subseteq\mathbb{R}^{n}$ be an
open set. Let $C^{0,\omega}(U)$ be the set of all continuous bounded functions
$f:U\to\mathbb{R}$ such that

$|f|_{C^{0,\omega}(U)}:=\inf\\{C>0:|f(x)-f(y)|\leq C\,\omega(|x-y|)\text{ for
all }x,y\in U\\}<\infty.$

For a nonnegative integer $m$, the set $C^{m,\omega}(U)$ consists of all
$C^{m}$-functions such that $\partial^{\alpha}f$ is globally bounded for all
$|\alpha|\leq m$ and $\partial^{\alpha}f\in C^{0,\omega}(U)$ for all
$|\alpha|=m$. Then $C^{m,\omega}(U)$ is a Banach space with the norm

$\|f\|_{C^{m,\omega}(U)}:=\sup_{x\in U}\sup_{|\alpha|\leq
m}|\partial^{\alpha}f(x)|+\sup_{|\alpha|=m}|\partial^{\alpha}f|_{C^{0,\omega}(U)}.$

We say that $f\in C^{m,\omega}(U)$ is _$m$ -flat_ outside an open set
$V\subseteq U$ if all $\partial^{\alpha}f$, $|\alpha|\leq m$, vanish on
$U\setminus V$.

Assume that the open set $U\subseteq\mathbb{R}^{n}$ is definable. We denote by
$C^{m,\omega}_{\operatorname{def}}(U)$ the subspace of $C^{m,\omega}(U)$
consisting of the definable functions in the latter space. Note that the
normed space $C^{m,\omega}_{\operatorname{def}}(U)$ is not complete.

######  Definition 3.1 (Bounded families of $C^{m,\omega}$-functions).

Let $m\in\mathbb{N}$ and $\omega$ a modulus of continuity. A family
$(f_{a})_{a\in A}$ of $C^{m,\omega}$-functions $f_{a}:U_{a}\to\mathbb{R}$,
where $U_{a}\subseteq\mathbb{R}^{n}$ is open, is said to be a _bounded family
of $C^{m,\omega}$-functions_ if

$\sup_{a\in A}\|f_{a}\|_{C^{m,\omega}(U_{a})}<\infty.$

We say that $(f_{a})_{a\in A}$ is a _definable bounded family of
$C^{m,\omega}$-functions_ if it is a bounded family of
$C^{m,\omega}$-functions and, additionally, the families $(U_{a})_{a\in A}$
and $(f_{a})_{a\in A}$ are definable. Moreover, $(f_{a})_{a\in A}$ is called
_$m$ -flat outside $(V_{a})_{a\in A}$_ if, for each $a\in A$, $V_{a}\subseteq
U_{a}$ is open and $f_{a}$ is $m$-flat on $U_{a}\setminus V_{a}$. We will say
that $(f_{a})_{a\in A}$ is _$C^{p}$ outside $(E_{a})_{a\in A}$_ if, for each
$a\in A$, $E_{a}\subseteq U_{a}$ is closed and $f_{a}$ is $C^{p}$ on
$U_{a}\setminus E_{a}$.

###  3.2. Whitney jets of class $C^{m,\omega}$

Let $E$ be a locally closed subset of $\mathbb{R}^{n}$. An _$m$ -jet on $E$_
is a collection $F=(F^{\alpha})_{|\alpha|\leq m}$ of continuous functions
$F^{\alpha}:E\to\mathbb{R}$. An $m$-jet $F=(F^{\alpha})_{|\alpha|\leq m}$ on
$E$ is said to be _flat_ on a subset $E^{\prime}\subseteq E$ if all functions
$F^{\alpha}$, $|\alpha|\leq m$, vanish on $E^{\prime}$.

For $a\in E$ we denote by $T^{m}_{a}F$ the Taylor polynomial

$T^{m}_{a}F(x)=\sum_{|\alpha|\leq
m}\frac{1}{\alpha!}F^{\alpha}(a)(x-a)^{\alpha},\quad x\in\mathbb{R}^{n},$

and define the $m$-jet

$R^{m}_{a}F:=F-J_{E}^{m}(T^{m}_{a}F),$

where $J_{E}^{m}(f):=(\partial^{\alpha}f|_{E})_{|\alpha|\leq m}$ for $f\in
C^{m}(\mathbb{R}^{n})$.

A $C^{m,\omega}$ (or $C^{m}$) function $f:\mathbb{R}^{n}\to\mathbb{R}$ is an
_extension to $\mathbb{R}^{n}$ of $F$_ if $J_{E}^{m}(f)=F$. A necessary and
sufficient condition for an $m$-jet to have a $C^{m,\omega}$-extension to
$\mathbb{R}^{n}$ is to be a Whitney jet of class $C^{m,\omega}$ ([16], [4]):

An $m$-jet $F=(F^{\alpha})_{|\alpha|\leq m}$ on $E$ is a _Whitney jet of class
$C^{m,\omega}$ on $E$_ if there exists $C>0$ such that

(3.1) $\sup_{x\in E}\sup_{|\alpha|\leq m}|F^{\alpha}(x)|\leq C$

and, for all $x,y\in E$ and $|\alpha|\leq m$,

(3.2) $|(R^{m}_{x}F)^{\alpha}(y)|\leq C\,\omega(|x-y|)|x-y|^{m-|\alpha|}.$

######  Remark 3.2.

A condition equivalent to (3.2) is

(3.3) $|T^{m}_{x}F(z)-T^{m}_{y}F(z)|\leq
C^{\prime}\,\omega(|x-y|)(|z-x|^{m}+|z-y|^{m})$

for all $x,y\in E$ and $z\in\mathbb{R}^{n}$; see [13, Proposition IV.1.5].
Moreover, (3.2) holds if and only if

(3.4) $\displaystyle|F^{0}(x)-T^{m}_{y}F(x)|\leq
C\,\omega(|x-y|)|x-y|^{m}\quad\text{ for all $x,y\in E$,}$ and, if $m\geq 1$,
(3.5) $\displaystyle\partial_{i}F:=(F^{\alpha+(i)})_{|\alpha|\leq m-1}\text{
is a Whitney jet of class }C^{m-1,\omega}\text{ for all }i=1,\ldots,n.$

If $E$ is quasiconvex with constant $C^{\prime}$ (see Definition 2.9), then
(3.2) follows from

$|F^{\alpha}(x)-F^{\alpha}(y)|\leq C^{\prime\prime}\,\omega(|x-y|),\quad
x,y\in E,\,|\alpha|=m,$

cf. [13, IV (2.5.1)]; then the constant $C$ in (3.2) depends only on $n$, $m$,
$C^{\prime}$, and $C^{\prime\prime}$.

It is not hard to see that the set of all Whitney jets of class $C^{m,\omega}$
on $E$ with the natural addition and the multiplication
$FG:=J_{E}^{m}(T^{m}F\cdot T^{m}G)$ is an $\mathbb{R}$-algebra.

Let $F$ be an $m$-jet on $E\subseteq\mathbb{R}^{n}$. Let $G_{1},\ldots,G_{n}$
be $m$-jets on $A\subseteq\mathbb{R}^{k}$ such that
$(G_{1}^{0},\ldots,G_{n}^{0})(A)\subseteq E$. The _composite_ of $F$ and $G$
is the $m$-jet $F\circ G=F\circ(G_{1},\ldots,G_{n})$ on $A$ defined by

$(F\circ G)(x):=J_{A}^{m}(T^{m}_{G^{0}(x)}F\circ T^{m}_{x}G)(x).$

Note that

$T^{m}_{y}(F\circ
G)(x)=\pi_{m}\big{(}T^{m}_{G^{0}(y)}F(T^{m}_{y}G(x))\big{)},$

where $\pi_{m}$ is the natural truncation operator (which truncates monomials
of order $>m$). One can show (using Remark 3.2) that, for $m\geq 1$, the
composite $F\circ G$ is a Whitney jet of class $C^{m,\omega}$ if $F$ and $G$
are Whitney jets of class $C^{m,\omega}$. We will not use this fact, but the
pullback of Whitney jets of class $C^{m,\omega}$ along a $\Lambda_{m}$-regular
map will be crucial; see Proposition 3.5.

######  Definition 3.3 (Bounded families of Whitney jets of class
$C^{m,\omega}$).

A family $(F_{a})_{a\in A}$ of Whitney jets $F_{a}$ of class $C^{m,\omega}$ on
$E_{a}\subseteq\mathbb{R}^{n}$ is said to be a _bounded family of Whitney jets
of class $C^{m,\omega}$_ if the constant $C>0$ in (3.1) and (3.2) can be
chosen independent of $a\in A$, that is,

(3.6) $\displaystyle\sup_{a\in A}\sup_{x\in E_{a}}\sup_{|\gamma|\leq
m}|F^{\gamma}_{a}(x)|<\infty$ and (3.7) $\displaystyle\sup_{a\in A}\sup_{x\neq
y\in E_{a}}\sup_{|\gamma|\leq
m}\frac{|(R^{m}_{x}F_{a})^{\gamma}(y)|}{\omega(|x-y|)|x-y|^{m-|\gamma|}}<\infty.$

We say that $(F_{a})_{a\in A}$ is a _definable bounded family of Whitney jets
of class $C^{m,\omega}$_ if it is a bounded family of Whitney jets of class
$C^{m,\omega}$ and, additionally, the families $(E_{a})_{a\in A}$ and
$(F^{\gamma}_{a})_{a\in A}$, $|\gamma|\leq m$, are definable. We say that it
is _flat_ on a subfamily $(E^{\prime}_{a})_{a\in A}$ of $(E_{a})_{a\in A}$ if
$F_{a}$ is flat on $E^{\prime}_{a}$ for all $a\in A$.

A (definable bounded) family $(f_{a})_{a\in A}$ of $C^{m,\omega}$-functions
$f_{a}:\mathbb{R}^{n}\to\mathbb{R}$ is called a _(definable bounded) family of
$C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of $(F_{a})_{a\in A}$_ if
$f_{a}$ is a $C^{m,\omega}$-extension of $F_{a}$ to $\mathbb{R}^{n}$, for each
$a\in A$.

###  3.3. Separation

Let $X,Y,Z$ be subsets of $\mathbb{R}^{n}$. Recall that $X$ and $Y$ are said
to be _$Z$ -separated_ if there exists $C>0$ such that

(3.8) $d(x,Y)\geq C\,d(x,Z),\quad x\in X,$

or equivalently, if there is $C^{\prime}>0$ such that

$d(x,X)+d(x,Y)\geq C^{\prime}\,d(x,Z),\quad x\in\mathbb{R}^{n}.$

If $X$ and $Y$ are $X\cap Y$-separated, then we will simply say that $X$ and
$Y$ are _separated_.

######  Definition 3.4 (Uniformly separated families of sets).

Let $(X_{a})_{a\in A}$, $(Y_{a})_{a\in A}$, and $(Z_{a})_{a\in A}$ be
definable families of subsets of $\mathbb{R}^{n}$. Then $(X_{a})_{a\in A}$ and
$(Y_{a})_{a\in A}$ are said to be _uniformly $(Z_{a})_{a\in A}$-separated_ if,
for all $a\in A$, $X_{a}$ and $Y_{a}$ are $Z_{a}$-separated with a constant
$C>0$ (in (3.8)) independent of $a\in A$. We will say that $(X_{a})_{a\in A}$
and $(Y_{a})_{a\in A}$ are _uniformly separated_ if they are uniformly
$(X_{a}\cap Y_{a})_{a\in A}$-separated.

###  3.4. Pullback along a definable family of $\Lambda_{p}$-regular maps

Let $\varphi:U\to\mathbb{R}^{\ell}$ be a $\Lambda_{m+1}$-regular map, where
$U\subseteq\mathbb{R}^{k}$ is open and quasiconvex. Let
$\overline{\varphi}:\overline{U}\to\mathbb{R}^{\ell}$ the continuous extension
of $\varphi$; cf. Section 2.4. Consider

$\displaystyle\varphi_{+}$ $\displaystyle:U\times\mathbb{R}^{\ell}\to
U\times\mathbb{R}^{\ell},\quad(u,w)\mapsto(u,w+\varphi(u)),$ and
$\displaystyle\overline{\varphi}_{+}$
$\displaystyle:\overline{U}\times\mathbb{R}^{\ell}\to\overline{U}\times\mathbb{R}^{\ell},\quad(u,w)\mapsto(u,w+\overline{\varphi}(u)).$

Let $M$ be a closed subset of $U\times\mathbb{R}^{\ell}$ and let $F$ be an
$m$-jet on $M$. The _pullback of $F$ along $\varphi_{+}$_ is the $m$-jet

$\varphi_{+}^{*}F:=F\circ J^{m}_{N}(\varphi_{+})$

on $N:=\varphi_{+}^{-1}(M)=\\{(u,w)\in
U\times\mathbb{R}^{\ell}:(u,w+\varphi(u))\in M\\}$.

We shall need the following result on the pullback of a definable bounded
family of Whitney jets of class $C^{m,\omega}$ along a definable family
$(\varphi_{a})_{a\in A}$ of $\Lambda_{m+1}$-regular maps. For each $a\in A$,
let $\varphi_{a,+}$ and $\overline{\varphi}_{a,+}$ be defined in analogy to
$\varphi_{+}$ and $\overline{\varphi}_{+}$.

######  Proposition 3.5 ([10, Proposition 4.3]).

Let $(U_{a})_{a\in A}$ be a definable family of open quasiconvex sets
$U_{a}\subseteq\mathbb{R}^{k}$ with constant (in Definition 2.9) independent
of $a\in A$. Let $(\varphi_{a})_{a\in A}$ be a definable family of
$\Lambda_{m+1}$-regular maps $\varphi_{a}:U_{a}\to\mathbb{R}^{\ell}$ with
constant (in (2.5)) independent of $a\in A$. Let $(M_{a})_{a\in A}$ be a
definable family of closed quasiconvex subsets $M_{a}$ of
$U_{a}\times\mathbb{R}^{\ell}$ such that $(\overline{M}_{a})_{a\in A}$ and
$(\partial U_{a}\times\mathbb{R}^{\ell})_{a\in A}$ are uniformly separated.

If $(F_{a})_{a\in A}$ is a definable bounded family of Whitney jets of class
$C^{m,\omega}$ on $(\overline{M}_{a})_{a\in A}$ which is flat on $(\partial
M_{a})_{a\in A}$, then $(G_{a})_{a\in A}$ is a definable bounded family of
Whitney jets of class $C^{m,\omega}$ on $(N_{a})_{a\in A}$, where
$G_{a}:=\varphi_{a,+}^{*}F_{a}$ and $N_{a}:=\varphi_{a,+}^{-1}(M_{a})$.
Moreover, if, for each $a\in A$, $t_{a}:U_{a}\to(0,\infty)$ is a function
satisfying $t_{a}(u)\leq d(u,\partial U_{a})$ for $u\in U_{a}$ and

$|F_{a}^{\kappa}(u,w)|\leq
C_{1}\,\omega(t_{a}(u))t_{a}(u)^{m-|\kappa|},\quad(u,w)\in
M_{a},\,|\kappa|\leq m,$

with $C_{1}>0$ independent of $a\in A$, then, for each $a\in A$,

$|G_{a}^{\kappa}(u,w)|\leq
C_{2}\,\omega(t_{a}(u))t_{a}(u)^{m-|\kappa|},\quad(u,w)\in
N_{a},\,|\kappa|\leq m,$

with $C_{2}>0$ independent of $a\in A$.

###### Proof.

Follows from the proof of [10, Proposition 4.3]. ∎

We will be interested in the case that $M_{a}=\Gamma(\varphi_{a})$, $a\in A$.
Then $(G_{a})_{a\in A}$ extends to a definable bounded family of Whitney jets
of class $C^{m,\omega}$ on $(\overline{N}_{a}=\overline{U}_{a}\times 0)_{a\in
A}$ which is flat on $(\partial N_{a}=\partial U_{a}\times 0)_{a\in A}$. This
follows from the following lemma and Hestenes’ lemma (e.g. [12, Theorem
1.10]); cf. [10, Remark 4.2].

######  Lemma 3.6.

Let $(E_{a})_{a\in A}$ be a family of locally closed, quasiconvex sets
$E_{a}\subseteq\mathbb{R}^{n}$ with constant (in Definition 2.9) independent
of $a\in A$. Suppose that $(F_{a})_{a\in A}$ is a bounded family of Whitney
jets of class $C^{m,\omega}$ on $(E_{a})_{a\in A}$ such that all
$F_{a}^{\alpha}$, $a\in A$, $|\alpha|\leq m$, have a continuous extension
$\overline{F}_{a}^{\alpha}$ to $\overline{E}_{a}$. Then
$(\overline{F}_{a})_{a\in A}$ is a bounded family of Whitney jets of class
$C^{m,\omega}$ on $(\overline{E}_{a})_{a\in A}$.

###### Proof.

Let $K\subseteq\mathbb{R}^{n}$ be compact and connected by rectifiable paths.
Let $\delta$ be the inner metric on $K$. Let $F=(F^{\alpha})_{|\alpha|\leq m}$
be a Whitney jet of class $C^{m}$ on $K$. Then, by [13, IV (2.5.1)], for all
$x,y\in K$ and $|\alpha|\leq m$,

$|(R^{m}_{x}F)^{\alpha}(y)|\leq
n^{\frac{m-|\alpha|}{2}}\delta(x,y)^{m-|\alpha|}\sup_{\xi\in
K}\sup_{|\beta|=m}|F^{\beta}(\xi)-F^{\beta}(x)|.$

Let $x,y\in\overline{E}_{a}$. There exist sequences $(x_{k}),(y_{k})\subseteq
E_{a}$ such that $x_{k}\to x$ and $y_{k}\to y$. By assumption, there exist a
constant $C_{1}>0$, independent of $a\in A$ and of $k$, and a rectifiable path
$\sigma_{k}$ joining $x_{k}$ and $y_{k}$ in $E_{a}$ such that for the length
of $\sigma_{k}$ we have

$\ell(\sigma_{k})\leq C_{1}\,|x_{k}-y_{k}|.$

Applying the above inequality (to $K=\sigma_{k}$), we get

$|(R^{m}_{x_{k}}F_{a})^{\alpha}(y_{k})|\leq
n^{\frac{m-|\alpha|}{2}}C_{1}^{m-|\alpha|}|x_{k}-y_{k}|^{m-|\alpha|}\sup_{\xi\in\sigma_{k}}\sup_{|\beta|=m}|F_{a}^{\beta}(\xi)-F_{a}^{\beta}(x_{k})|.$

We may assume that $\sigma_{k}$ is parameterized by $t\in[0,1]$ with
$\sigma_{k}(0)=x_{k}$ and $\sigma_{k}(1)=y_{k}$. By (3.7), for $t\in[0,1]$,

$\displaystyle\sup_{|\beta|=m}|F^{\beta}(\sigma_{k}(t))-F^{\beta}(x_{k})|$
$\displaystyle\leq C_{2}\,\omega(|\sigma_{k}(t)-x_{k}|)\leq
C_{2}\,\omega(\ell(\sigma_{k}|_{[0,t]}))$ $\displaystyle\leq
C_{2}\,\omega(\ell(\sigma_{k}))\leq C_{2}\,\omega(C_{1}|x_{k}-y_{k}|)\leq
C_{3}\,\omega(|x_{k}-y_{k}|),$

for constants $C_{i}>0$ independent of $a\in A$. Thus

$|(R^{m}_{x_{k}}F_{a})^{\alpha}(y_{k})|\leq
n^{\frac{m-|\alpha|}{2}}C_{1}^{m-|\alpha|}|x_{k}-y_{k}|^{m-|\alpha|}\,C_{3}\,\omega(|x_{k}-y_{k}|)$

and letting $k\to\infty$ shows that (3.7) is satisfied for
$(\overline{F}_{a})_{a\in A}$. That also (3.6) is satisfied is clear. ∎

###  3.5. Cutoff

We finish this section with a technical cutoff result which will be used in
the proof of Theorem 1.3.

######  Proposition 3.7 ([10, Proposition 3.9]).

Assume that there are constants $C_{i}$, $i=0,\ldots,3$, such that the
following holds. Let $(U_{a})_{a\in A}$ be a definable family of open
quasiconvex sets $U_{a}\subseteq\mathbb{R}^{k}$ with constant (in Definition
2.9) independent of $a\in A$. Let $(h_{a})_{a\in A}$ be a definable family of
$C^{m}$-functions $h_{a}:U_{a}\times\mathbb{R}^{\ell}\to\mathbb{R}$ and
$(\rho_{a})_{a\in A}$ a definable family of $C^{m+1}$-functions
$\rho_{a}:U_{a}\to\mathbb{R}$. Let $(t_{a})_{a\in A}$ be a definable family of
positive $C_{0}$-Lipschitz functions $t_{a}:U_{a}\to(0,\infty)$ such that
$t_{a}(u)\leq d(u,\partial U_{a})$ for all $a\in A$, $u\in U_{a}$. For
$\epsilon>0$ consider the definable family $(\Delta^{\epsilon}_{a})_{a\in A}$,
where

$\Delta^{\epsilon}_{a}:=\\{(u,w)\in
U_{a}\times\mathbb{R}^{\ell}:|w|<\epsilon\,t_{a}(u)\\}.$

Assume that, for all $a\in A$,

(3.9) $|\partial^{\alpha}(1/\rho_{a})(u)|\leq
C_{1}\,t_{a}(u)^{-|\alpha|-1},\quad u\in U_{a},\,|\alpha|\leq m+1.$

Let $\xi:\mathbb{R}\to\mathbb{R}$ be a definable $C^{m}$-function with compact
support, fix $1\leq i\leq\ell$, and set, for all $a\in A$,

$f_{a}(u,w):=\xi\Big{(}\frac{w_{i}}{\rho_{a}(u)}\Big{)}h_{a}(u,w),\quad(u,w)\in
U_{a}\times\mathbb{R}^{\ell}.$

If $(h_{a})_{a\in A}$ is a definable bounded family of
$C^{m,\omega}$-functions on $(\Delta^{\epsilon}_{a})_{a\in A}$ such that, for
each $a\in A$,

$|\partial^{\gamma}h_{a}(u,w)|\leq
C_{2}\,\omega(t_{a}(u))t_{a}(u)^{m-|\gamma|},\quad(u,w)\in\Delta^{\epsilon}_{a},\,|\gamma|\leq
m,$

then $(f_{a})_{a\in A}$ is a definable bounded family of
$C^{m,\omega}$-functions on $(\Delta^{\epsilon}_{a})_{a\in A}$ such that, for
each $a\in A$,

$|\partial^{\gamma}f_{a}(u,w)|\leq
C_{3}\,\omega(t_{a}(u))t_{a}(u)^{m-|\gamma|},\quad(u,w)\in\Delta^{\epsilon}_{a},\,|\gamma|\leq
m.$

###### Proof.

It suffices to repeat the proof of Proposition 3.9 in [10] (as well as Lemma
3.5 and Proposition 3.6 which are used in the proof). ∎

######  Remark 3.8.

Proposition 3.5 and Proposition 3.7 remain true if we remove everywhere the
attribute “definable”.

##  4\. Bounded definable extension of Whitney jets

This section is devoted to the proof of Theorem 1.3. Let us recall the setup.

_Let $0\leq m\leq p$ be integers and $\omega$ a modulus of continuity. Let
$(E_{a})_{a\in A}$ be a definable family of closed subsets of
$\mathbb{R}^{n}$. Let $(F_{a})_{a\in A}$ be a definable bounded family of
Whitney jets of class $C^{m,\omega}$ on $(E_{a})_{a\in A}$. We will show that
there exists a definable bounded family $(f_{a})_{a\in A}$ of
$C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of $(F_{a})_{a\in A}$ that is
$C^{p}$ outside $(E_{a})_{a\in A}$._

For each $a\in A$, let $\operatorname{supp}F_{a}$ denote the closure of
$\bigcup_{|\kappa|\leq m}\\{x\in E_{a}:F^{\kappa}_{a}(x)\neq 0\\}$ and let
$(E^{\prime}_{a})_{a\in A}$ be a definable subfamily of $(E_{a})_{a\in A}$
consisting of closed subsets $E^{\prime}_{a}$ of $E_{a}$ such that
$\operatorname{supp}F_{a}\subseteq E^{\prime}_{a}$.

Let $A^{\prime}:=\\{a\in A:\operatorname{supp}F_{a}=\emptyset\\}$. The family
$(F_{a})_{a\in A^{\prime}}$ can be extended by $(0)_{a\in A^{\prime}}$ to
$\mathbb{R}^{n}$. So we may assume that, for all $a\in A$,
$\operatorname{supp}F_{a}\neq\emptyset$ and thus
$E^{\prime}_{a}\neq\emptyset$.

We proceed by induction on $k:=\max_{a\in A}\dim E^{\prime}_{a}$ and show:

($\mathbf{I}_{k}$) Let $(E_{a})_{a\in A}$ be a definable family of closed
subsets $E_{a}$ of $\mathbb{R}^{n}$ and $(F_{a})_{a\in A}$ a definable bounded
family of Whitney jets of class $C^{m,\omega}$ on $(E_{a})_{a\in A}$. Let
$(E^{\prime}_{a})_{a\in A}$ be a definable subfamily of $(E_{a})_{a\in A}$ of
closed subsets $E^{\prime}_{a}$ of $E_{a}$ such that
$\operatorname{supp}F_{a}\subseteq E^{\prime}_{a}$ and $\dim
E^{\prime}_{a}\leq k$, for all $a\in A$. Then there exists a definable bounded
family $(f_{a})_{a\in A}$ of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of
$(F_{a})_{a\in A}$ that is $C^{p}$ outside $(E^{\prime}_{a})_{a\in A}$.

Let us fix an integer $p\geq m+1$.

### Induction basis ($\mathbf{I}_{0}$)

If $k=0$, then each $E^{\prime}_{a}$ is a finite set (but $E_{a}$ might be
infinite) and there is a constant which bounds the number $|E^{\prime}_{a}|$
of elements of $E^{\prime}_{a}$ independently of $a\in A$.

Let us make induction on $s:=\max_{a\in A}|E^{\prime}_{a}|$. The base case
$s=1$ is treated in the following lemma.

######  Lemma 4.1.

Let $(E_{a})_{a\in A}$ be a definable family of closed subsets $E_{a}$ of
$\mathbb{R}^{n}$ and $(E^{\prime}_{a})_{a\in A}$ a definable subfamily of
$(E_{a})_{a\in A}$ such that, for each $a\in A$, $E^{\prime}_{a}=\\{x_{a}\\}$.
Let $(F_{a})_{a\in A}$ be a definable bounded family of Whitney jets of class
$C^{m,\omega}$ on $(E_{a})_{a\in A}$ such that
$\operatorname{supp}F_{a}\subseteq\\{x_{a}\\}$, for all $a\in A$. Then there
exists a definable bounded family $(f_{a})_{a\in A}$ of
$C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of $(F_{a})_{a\in A}$ that is
$C^{p}$ outside $(E^{\prime}_{a})_{a\in A}$.

###### Proof.

Note that, for each $a\in A$, $x_{a}$ is an isolated point of $E_{a}$, by
continuity of $F_{a}$.

Let $\chi:\mathbb{R}^{n}\to\mathbb{R}$ be a definable $C^{p}$-function that
equals $1$ in a neighborhood of $0$ and has support contained in the unit
ball. For each $a\in A$, set

$d_{a}:=\begin{cases}\min\\{1,d(x_{a},E_{a}\setminus\\{x_{a}\\})\\}&\text{if
}E_{a}\setminus\\{x_{a}\\}\neq\emptyset,\\\ 1&\text{otherwise.}\end{cases}$

Define, for each $a\in A$,

$f_{a}(x):=\chi\Big{(}\frac{x-x_{a}}{d_{a}}\Big{)}\cdot
T^{m}_{x_{a}}F_{a}(x),\quad x\in\mathbb{R}^{n}.$

Then $(f_{a})_{a\in A}$ is a definable family of $C^{p}$-functions
$f_{a}:\mathbb{R}^{n}\to\mathbb{R}$ such that each $f_{a}$ has support
contained in the ball $B_{a}:=B(x_{a},d_{a})$ with radius $d_{a}$ around
$x_{a}$ and extends the jet $F_{a}$. We will prove that the family
$(f_{a})_{a\in A}$ is bounded in $C^{m,\omega}(\mathbb{R}^{n})$.

Let $\gamma\in\mathbb{N}^{n}$, $|\gamma|\leq m$. Then

$\displaystyle\partial^{\gamma}f_{a}(x)=\sum_{\alpha+\beta=\gamma}\binom{\gamma}{\alpha}d_{a}^{-|\alpha|}\,\partial^{\alpha}\chi\Big{(}\frac{x-x_{a}}{d_{a}}\Big{)}\partial^{\beta}T^{m}_{x_{a}}F_{a}(x).$

By (3.7) (for $y\in E_{a}\setminus\\{x_{a}\\}$ with $d_{a}=|x_{a}-y|$ if
$d_{a}<1$) and (3.6) (if $d_{a}=1$),

$|F_{a}^{\beta}(x_{a})|\leq C\,\omega(d_{a})d_{a}^{m-|\beta|},\quad|\beta|\leq
m,$

for a constant $C>0$ independent of $a\in A$. For the rest of the proof, $C$
will denote a constant independent of $a\in A$; its actual value may change.
Thus, for $x\in B_{a}$,

$\displaystyle|\partial^{\beta}T^{m}_{x_{a}}F_{a}(x)|$
$\displaystyle=\Big{|}\sum_{|\kappa|\leq
m-|\beta|}\frac{1}{\kappa!}F^{\kappa+\beta}_{a}(x_{a})(x-x_{a})^{\kappa}\Big{|}$
$\displaystyle\leq C\,\omega(d_{a})d_{a}^{m-|\beta|},\quad|\beta|\leq m.$

It follows that, for all $x\in\mathbb{R}^{n}$,

(4.1) $\displaystyle|\partial^{\gamma}f_{a}(x)|\leq
C\,\omega(d_{a})d_{a}^{m-|\gamma|}\leq C\,\omega(1),\quad|\gamma|\leq m.$

Now assume that $|\gamma|=m$. To see that
$|\partial^{\gamma}f_{a}|_{C^{0,\omega}(\mathbb{R}^{n})}$ is bounded by a
constant independent of $a\in A$, it suffices to estimate, for
$\alpha+\beta=\gamma$,

$\displaystyle
D(x,y):=\Big{|}d_{a}^{-|\alpha|}\,\partial^{\alpha}\chi\Big{(}\frac{x-x_{a}}{d_{a}}\Big{)}\partial^{\beta}T^{m}_{x_{a}}F_{a}(x)-d_{a}^{-|\alpha|}\,\partial^{\alpha}\chi\Big{(}\frac{y-x_{a}}{d_{a}}\Big{)}\partial^{\beta}T^{m}_{x_{a}}F_{a}(y)\Big{|}.$

Let us first assume that $x,y\in\overline{B}_{a}$. Then

$\displaystyle
d_{a}^{-|\alpha|}\,\Big{|}\partial^{\alpha}\chi\Big{(}\frac{x-x_{a}}{d_{a}}\Big{)}-\partial^{\alpha}\chi\Big{(}\frac{y-x_{a}}{d_{a}}\Big{)}\Big{|}|\partial^{\beta}T^{m}_{x_{a}}F_{a}(x)|$
$\displaystyle\leq C\,\frac{\omega(d_{a})}{d_{a}}|x-y|\leq 2C\,\omega(|x-y|),$

since $\omega$ is concave and $|x-y|\leq 2d_{a}$. On the other hand,

$\displaystyle
d_{a}^{-|\alpha|}\,\Big{|}\partial^{\alpha}\chi\Big{(}\frac{y-x_{a}}{d_{a}}\Big{)}\Big{|}|\partial^{\beta}T^{m}_{x_{a}}F_{a}(x)-\partial^{\beta}T^{m}_{x_{a}}F_{a}(y)|$
$\displaystyle\leq C\,d_{a}^{-|\alpha|}\,\sum_{|\kappa|\leq
m-|\beta|}\frac{1}{\kappa!}|F^{\kappa+\beta}_{a}(x_{a})||(x-x_{a})^{\kappa}-(y-x_{a})^{\kappa}|$
$\displaystyle\leq C^{\prime}\,\frac{\omega(d_{a})}{d_{a}}\,|x-y|\leq
2C^{\prime}\,\omega(|x-y|).$

So $D(x,y)\leq C\,\omega(|x-y|)$ for a constant $C>0$ independent of $a\in A$.

If $x$ and $y$ lie outside of $B_{a}$, then $D(x,y)=0$. If $x\in B_{a}$ and
$y\not\in\overline{B}_{a}$ and $z$ is the point, where the line segment
$[x,y]$ meets $\partial B_{a}$, then

$D(x,y)=D(x,z)\leq C\,\omega(|x-z|)\leq C\,\omega(|x-y|).$

This ends the proof. ∎

Now assume that $s>1$. For each $a\in A$, choose a numbering of the elements
of $E^{\prime}_{a}=\\{x_{a,1},\ldots,x_{a,s_{a}}\\}$, where $s_{a}\leq s$. By
the induction hypothesis,
$(F_{a}|_{E_{a}\setminus\\{x_{a,2},\ldots,x_{a,s_{a}}\\}})_{a\in A}$ admits a
definable bounded family $(f^{1}_{a})_{a\in A}$ of $C^{m,\omega}$-extensions
to $\mathbb{R}^{n}$ that is $C^{p}$ outside $(\\{x_{a,1}\\})_{a\in A}$. Then
$(F_{a}-J^{m}_{E_{a}}(f^{1}_{a}))_{a\in A}$ is a definable bounded family of
Whitney jets of class $C^{m,\omega}$ on $(E_{a})_{a\in A}$ which is flat on
$(E_{a}\setminus\\{x_{a,2},\ldots,x_{a,s_{a}}\\})_{a\in A}$ and has a
definable bounded family $(f^{2}_{a})_{a\in A}$ of $C^{m,\omega}$-extensions
to $\mathbb{R}^{n}$ that is $C^{p}$ outside
$(\\{x_{a,2},\ldots,x_{a,s_{a}}\\})_{a\in A}$, again by the induction
hypothesis. Thus, $(f^{1}_{a}+f^{2}_{a})_{a\in A}$ is the desired definable
bounded family of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of
$(F_{a})_{a\in A}$.

This ends the induction on $s$ and the base case ($\mathbf{I}_{0}$) of the
induction on $k$.

### Setup for the induction step

Let $k>0$ and suppose that ($\mathbf{I}_{k-1}$) holds. We will prove
($\mathbf{I}_{k}$). This will be accomplished by showing Proposition 4.2
below, but first we make a few preparatory reductions.

By ($\mathbf{I}_{k-1}$), we may assume that $\dim E^{\prime}_{a}=k$ for all
$a\in A$.

By Theorem 2.16, there is a uniform $\Lambda_{p}$-stratification
$(\mathscr{S}_{a})_{a\in A}$ of $(E_{a})_{a\in A}$ compatible with
$(E^{\prime}_{a})_{a\in A}$ such that, for each $a\in A$ and each
$|\kappa|\leq m$, $F^{\kappa}_{a}$ is of class $C^{p}$ on the strata in
$\mathscr{S}_{a}$.

By ($\mathbf{I}_{k-1}$), there is a definable bounded $C^{m,\omega}$-extension
$(f^{0}_{a})_{a\in A}$ to $\mathbb{R}^{n}$ of the restriction of
$(F_{a})_{a\in A}$ to $(E_{a}\setminus P_{a})_{a\in A}$, where

$P_{a}=\bigcup\\{S_{a}\in\mathscr{S}_{a}:S_{a}\subseteq E^{\prime}_{a},\,\dim
S_{a}=k\\}.$

Replacing $F_{a}$ by $F_{a}-J^{m}_{E_{a}}(f^{0}_{a})$, for each $a\in A$, we
may assume that $F_{a}$ is flat on all strata $S_{a}\in\mathscr{S}_{a}$,
$S_{a}\subseteq E^{\prime}_{a}$, with $\dim S_{a}<k$ and also on
$E_{a}\setminus E^{\prime}_{a}$.

Let us now see that we may furthermore reduce to the case that, for each $a\in
A$, $E^{\prime}_{a}$ is the closure of just one $k$-dimensional stratum
$S_{a}$ and that $F_{a}$ is flat on its frontier. Indeed, the number $s_{a}$
of $k$-dimensional strata of $E^{\prime}_{a}$ is uniformly bounded by a
constant not depending on $a\in A$. We may use induction on $s:=\max_{a\in
A}s_{a}$ of which the above statement is the base case that we take for
granted for the moment. The induction step works just as for finite sets
$E^{\prime}_{a}$: for each $a\in A$, let $S_{a,1},\ldots,S_{a,s_{a}}$ be a
numbering of the $k$-dimensional strata of $E^{\prime}_{a}$. By the induction
hypothesis, $(F_{a}|_{E_{a}\setminus R_{a}})_{a\in A}$, where
$R_{a}:=\bigcup_{i\geq 2}S_{a,i}$, admits a definable bounded family
$(f^{1}_{a})_{a\in A}$ of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ that
is $C^{p}$ outside $(E^{\prime}_{a}\setminus R_{a})_{a\in A}$. Then
$(F_{a}-J^{m}_{E_{a}}(f^{1}_{a}))_{a\in A}$ is a definable bounded family of
Whitney jets of class $C^{m,\omega}$ on $(E_{a})_{a\in A}$ which is flat on
$(E_{a}\setminus R_{a})_{a\in A}$ and has a definable bounded family
$(f^{2}_{a})_{a\in A}$ of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ that
is $C^{p}$ outside $(R_{a})_{a\in A}$, again by the induction hypothesis.
Thus, $(f^{1}_{a}+f^{2}_{a})_{a\in A}$ is the desired definable bounded family
of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of $(F_{a})_{a\in A}$.

In the case that $k=n$, $S_{a}$ is open in $\mathbb{R}^{n}$ and extending
$F_{a}$ by $0$ outside $S_{a}$, for all $a\in A$, yields a definable bounded
family $(F_{a})_{a\in A}$ of Whitney jets of class $C^{m,\omega}$ on
$(\mathbb{R}^{n})_{a\in A}$ so that $(F^{0}_{a})_{a\in A}$ is the desired
family of $C^{m,\omega}$-extensions. This follows from Hestenes’ lemma (e.g.
[12, Theorem 1.10]); indeed, if $x\in S_{a}$ and $y\not\in\overline{S}_{a}$
and $z$ is the point, where the line segment $[x,y]$ meets $\partial S_{a}$,
then, by (3.7) and Remark 3.2, for any $u\in\mathbb{R}^{n}$,

$\displaystyle|T^{m}_{x}F_{a}(u)-T^{m}_{y}F_{a}(u)|$
$\displaystyle=|T^{m}_{x}F_{a}(u)|=|T^{m}_{x}F_{a}(u)-T^{m}_{z}F_{a}(u)|$
$\displaystyle\leq C\,\omega(|x-z|)(|u-x|^{m}+|u-z|^{m})$ $\displaystyle\leq
2C\,\omega(|x-y|)(|u-x|^{m}+|u-y|^{m}),$

for a constant $C>0$ independent of $a\in A$, since
$|u-z|\leq\max\\{|u-x|,|u-y|\\}$.

Consequently, we may assume that $\ell:=n-k>0$.

We reduced the proof to showing the following. (We may assume that $S_{a}$ is
a $\Lambda_{p}$-cell in a fixed orthogonal system of coordinates of
$\mathbb{R}^{n}$, which is independent of $a\in A$, thanks to Theorem 2.16.)

######  Proposition 4.2.

Let $(E_{a})_{a\in A}$ be a definable family of closed sets $E_{a}$ in
$\mathbb{R}^{n}$. Let $(E^{\prime}_{a})_{a\in A}$ be a definable subfamily of
$(E_{a})_{a\in A}$ of closed subsets $E^{\prime}_{a}$ of $E_{a}$ with $\dim
E^{\prime}_{a}=k$ such that $E^{\prime}_{a}=\overline{S}_{a}$, where

$S_{a}=\\{(u,\varphi_{a}(u))\in\mathbb{R}^{k}\times\mathbb{R}^{\ell}:u\in
T_{a}\\}=\Gamma(\varphi_{a}),$

and $(\varphi_{a})_{a\in A}$ is a definable family of $\Lambda_{p}$-regular
maps $\varphi_{a}:T_{a}\to\mathbb{R}^{\ell}$, $T_{a}$ an open
$\Lambda_{p}$-cell in $\mathbb{R}^{k}$, and all constants in the definition of
$T_{a}$ and $\varphi_{a}$ are independent of $a\in A$. Then any definable
bounded family $(F_{a})_{a\in A}$ of Whitney jets of class $C^{m,\omega}$ on
$(E_{a})_{a\in A}$ such that, for all $a\in A$,
$\operatorname{supp}F_{a}\subseteq E^{\prime}_{a}$, $F_{a}$ is flat on
$\partial S_{a}$, and $F_{a}^{\kappa}$, $|\kappa|\leq m$, is $C^{p}$ on
$S_{a}$, admits a definable bounded family $(f_{a})_{a\in A}$ of
$C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ that is $C^{p}$ outside
$(E^{\prime}_{a})_{a\in A}$.

The proposition is proved in three gradually more general steps.

Step 1:

$\varphi_{a}\equiv 0$ and $E^{\prime}_{a}=E_{a}$ for all $a\in A$.

Step 2:

$\varphi_{a}\equiv 0$ for all $a\in A$.

Step 3:

The general case.

### Step 1

For all $a\in A$, $E_{a}=E^{\prime}_{a}=\overline{T}_{a}\times 0$, where
$T_{a}\subseteq\mathbb{R}^{k}$ is an open $\Lambda_{p}$-cell with constant $C$
independent of $a\in A$. We will prove Proposition 4.2 in this special case
with the additional property that $(f_{a})_{a\in A}$ is $m$-flat outside
$(\Delta(T_{a}\times 0))_{a\in A}$, where

(4.2) $\Delta(T_{a}\times 0):=\\{(u,w)\in
T_{a}\times\mathbb{R}^{\ell}:|w|<\min\\{1,d(u,\partial T_{a})\\}\\}.$

For each $a\in A$, we write

$F_{a}=(F_{a}^{(\alpha,\beta)})_{(\alpha,\beta)\in\mathbb{N}^{k}\times\mathbb{N}^{\ell},|\alpha|+|\beta|\leq
m}.$

Fix $\beta\in\mathbb{N}^{\ell}$ with $|\beta|\leq m$. Let $F_{a,\beta}$ be the
$m$-jet which results from $F_{a}$ by setting all
$F_{a}^{(\alpha,\beta^{\prime})}$ equal to $0$ whenever
$\beta^{\prime}\neq\beta$. Then $(F_{a,\beta})_{a\in A}$ is a definable
bounded family of Whitney jets of class $C^{m,\omega}$ on
$(\overline{T}_{a}\times 0)_{a\in A}$. Indeed, for each $a\in A$, the
definable Whitney jet $F_{a}$ of class $C^{m,\omega}$ on
$\overline{T}_{a}\times 0$ can be identified with a collection
$\tilde{F}_{a,\beta}$, $|\beta|\leq m$, where $\tilde{F}_{a,\beta}$ is a
definable $C^{m-|\beta|,\omega}$-function on $\overline{T}_{a}$ such that
$\partial^{\alpha}\tilde{F}_{a,\beta}(u)=F_{a}^{(\alpha,\beta)}(u,0)$ for all
$u\in\overline{T}_{a}$ and $\alpha\in\mathbb{N}^{k}$, $|\alpha|\leq
m-|\beta|$; see [8, Remark 5] and [4, pp. 87-88]. It suffices to prove that,
for each $\beta$, $(F_{a,\beta})_{a\in A}$ admits a definable bounded family
of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ that is $m$-flat outside
$(\Delta(T_{a}\times 0))_{a\in A}$ and $C^{p}$ outside
$(\overline{T}_{a}\times 0)_{a\in A}$. Thus, we may suppose that, for each
$a\in A$, $F_{a}^{(\alpha,\beta^{\prime})}=0$ whenever
$\beta^{\prime}\neq\beta$. By assumption, $F_{a}^{(\alpha,\beta)}$ is $C^{p}$
on $T_{a}\times 0$.

By Theorem 2.16, Corollary 2.17, and Proposition 2.19, there is a uniform
$\Lambda_{p}$-stratification $(\mathscr{D}_{a})_{a\in A}$ of
$(\overline{T}_{a})_{a\in A}$ such that, for all $a\in A$, each open
$k$-dimensional $D_{a}\in\mathscr{D}_{a}$, and all $\alpha,\beta$,
$F^{(\alpha,\beta)}_{a}$ is $C^{p}$ on $D_{a}\times 0$, and, for all $u\in
D_{a}$ and $\gamma\in\mathbb{N}^{k}$ with $|\gamma|\leq p$, we have

(4.3) $|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u,0)|\leq
L\,\frac{\sup\\{|F_{a}^{(\alpha,\beta)}(v,0)|:v\in D_{a},\,|v-u|<d(u,\partial
D_{a})\\}}{d(u,\partial D_{a})^{|\gamma|}},$

and, if $1\leq|\gamma|\leq p$,

(4.4) $|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u,0)|\leq
L\,\frac{\omega(d(u,\partial D_{a}))}{d(u,\partial D_{a})^{|\gamma|}},$

where $L>0$ is a constant independent of $a\in A$.

For each $a\in A$, let $Z_{a}:=\bigcup\\{D_{a}\in\mathscr{D}_{a}:\dim
D_{a}<k\\}$. Setting

$G_{a}(x):=\begin{cases}F_{a}(x)&\text{if }x\in Z_{a}\times 0,\\\ 0&\text{if
}x\in\mathbb{R}^{n}\setminus\Delta(T_{a}\times 0),\end{cases}$

defines a definable bounded family $(G_{a})_{a\in A}$ of Whitney jets of class
$C^{m,\omega}$ on $((Z_{a}\times
0)\cup(\mathbb{R}^{n}\setminus\Delta(T_{a}\times 0)))_{a\in A}$. This follows
from Hestenes’ lemma (e.g. [12, Theorem 1.10]) and the following reasoning.
Clearly, $(G_{a})_{a\in A}$ satisfies (3.6). To see (3.7) it suffices to
consider the case that $x\in Z_{a}\times 0$ and
$y\in\mathbb{R}^{n}\setminus\Delta(T_{a}\times 0)$ and to show that

(4.5) $|F_{a}^{\kappa}(x)|\leq
C\,\omega(|x-y|)|x-y|^{m-|\kappa|},\quad|\kappa|\leq m,$

for a constant $C>$ independent of $a\in A$. We have

$|F_{a}^{\kappa}(x)|\leq C\,\omega(d(u,\partial T_{a}))d(u,\partial
T_{a})^{m-|\kappa|},\quad|\kappa|\leq m,$

by (3.7), since $(F_{a})_{a\in A}$ is flat on $(\partial T_{a}\times 0)_{a\in
A}$, and

(4.6) $|F_{a}^{\kappa}(x)|\leq
C=\frac{C}{\omega(1)}\cdot\omega(1)1^{m-|\kappa|},\quad|\kappa|\leq m,$

by (3.6). Then (4.5) follows, since we have $|x-y|\geq
c\,\min\\{1,d(u,\partial T_{a})\\}$ for a universal constant $c>0$, by (4.2).

By ($\mathbf{I}_{k-1}$), there exists a definable bounded family
$(g_{a})_{a\in A}$ of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of
$(G_{a})_{a\in A}$ that is $C^{p}$ outside $(Z_{a}\times 0)_{a\in A}$. So,
instead of $(F_{a})_{a\in A}$, it is enough to consider
$(F_{a}-J^{m}_{E_{a}}(g_{a}))_{a\in A}$.

If $D_{a}$ and $D^{\prime}_{a}$ are distinct open $k$-dimensional strata in
$\mathscr{D}_{a}$, then $\Delta(D_{a}\times 0)\subseteq\Delta(T_{a}\times 0)$
and $\overline{\Delta(D_{a}\times 0)}\cap\overline{\Delta(D^{\prime}_{a}\times
0)}\subseteq Z_{a}\times 0$. Thus it suffices to find, separately for each
$(D_{a})_{a\in A}$, a definable bounded family $(f_{a})_{a\in A}$ of
$C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of
$((F_{a}-J^{m}_{E_{a}}(g_{a}))|_{\overline{D}_{a}\times 0})_{a\in A}$ that is
$m$-flat outside $(\Delta(D_{a}\times 0))_{a\in A}$ and $C^{p}$ outside
$(\overline{D}_{a}\times 0)_{a\in A}$.

For each $a\in A$, set

$h_{a}(u,w):=\frac{1}{\beta!}F_{a}^{(0,\beta)}(u,0)w^{\beta}-g_{a}(u,w),$

and define $f_{a}:\mathbb{R}^{n}\to\mathbb{R}$ by

$f_{a}(u,w):=\begin{cases}r_{a}(u,w)h_{a}(u,w)&\text{ if }u\in D_{a},\\\
0&\text{ otherwise, }\end{cases}$

where

(4.7)
$r_{a}(u,w):=\prod_{i=1}^{\ell}\prod_{j=0}^{2k}\xi\Big{(}C\,\sqrt{\ell}\frac{w_{i}}{\rho_{a,j}(u)}\Big{)}$

with $\xi:\mathbb{R}\to\mathbb{R}$ a semialgebraic $C^{p}$-function which is
$1$ near $0$ and vanishes outside $(-1,1)$,
$\rho_{a,0},\rho_{a,1},\ldots,\rho_{a,2k}$ the functions associated with the
open $\Lambda_{p}$-cell $D_{a}$ (cf. Section 2.6), and $C$ is the constant
from (2.7) which may be taken independent from $a\in A$, since it is
determined by the constants in the definition of the $\Lambda_{p}$-cells
$D_{a}$, $a\in A$; see Remark 2.12. Note that the $m$-jet of $h_{a}$ at
$(u,0)$ coincides with $(F_{a}-J^{m}_{E_{a}}(g_{a}))(u,0)$ for all $u\in
D_{a}$.

By construction, $(f_{a})_{a\in A}$ is a definable family. We will see that it
is a bounded family of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of
$((F_{a}-J^{m}_{E_{a}}(g_{a}))|_{\overline{D}_{a}\times 0})_{a\in A}$. It is
$m$-flat outside $(\Delta(D_{a}\times 0))_{a\in A}$, thanks to the properties
of $r_{a}$, and it is $C^{p}$ outside $(\overline{D}_{a}\times 0)_{a\in A}$.
Indeed, if $(u,w)\in(D_{a}\times\mathbb{R}^{\ell})\setminus\Delta(D_{a}\times
0)$, then, by (2.7),

$\displaystyle\sqrt{\ell}\max_{1\leq
i\leq\ell}|w_{i}|\geq|w|\geq\min\\{1,d(u,\partial
D_{a}\\}\geq\frac{1}{C}\min_{0\leq j\leq 2k}\rho_{a,j}(u)$

so that $r_{a}$ is identically zero on
$(D_{a}\times\mathbb{R}^{\ell})\setminus\Delta(D_{a}\times 0)$. It remains to
check that the family $(f_{a})_{a\in A}$ is contained and bounded in
$C^{m,\omega}(\mathbb{R}^{n})$. To this end, we need two lemmas.

######  Lemma 4.3.

For each $a\in A$, $h_{a}$ is of class $C^{m,\omega}$ on $\Delta(D_{a}\times
0)$ and the $C^{m,\omega}$-norm of $h_{a}$ on $\Delta(D_{a}\times 0)$ is
bounded by a constant independent of $a\in A$.

###### Proof.

By construction, each $h_{a}$ is of class $C^{m}$. Since $(g_{a})_{a\in A}$ is
a bounded family of $C^{m,\omega}$-functions on $\mathbb{R}^{n}$, it suffices
it consider $(u,w)\mapsto F_{a}^{(0,\beta)}(u,0)w^{\beta}$. We have to check
that there is a constant $C>0$ such that, for all $a\in A$, all
$\kappa=(\sigma,\tau)\in\mathbb{N}^{k}\times\mathbb{N}^{\ell}$, $|\kappa|\leq
m$, and all $(u,w)\in\Delta(D_{a}\times 0)$,

(4.8) $|\partial^{\kappa}\big{(}F_{a}^{(0,\beta)}(u,0)w^{\beta}\big{)}|\leq
C,$

and, if $|\kappa|=m$, for all $x_{i}=(u_{i},w_{i})\in\Delta(D_{a}\times 0)$,
$i=1,2$,

(4.9)
$|\partial^{\kappa}\big{(}F_{a}^{(0,\beta)}(u_{1},0)w_{1}^{\beta}\big{)}-\partial^{\kappa}\big{(}F_{a}^{(0,\beta)}(u_{2},0)w_{2}^{\beta}\big{)}|\leq
C\,\omega(|x_{1}-x_{2}|).$

We may assume that $\tau\leq\beta$. Let $\alpha,\gamma\in\mathbb{N}^{k}$ be
such that $\alpha+\gamma=\sigma$ and $|\alpha|+|\beta|\leq m$ is maximal. To
see (4.8), observe that, by (4.3), (3.6), and $|w|<\min\\{1,d(u,\partial
D_{a})\\}$,

$\displaystyle|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u,0)w^{\beta-\tau}|$
$\displaystyle\leq L\,\frac{\sup\\{|F_{a}^{(\alpha,\beta)}(v,0)|:v\in
D_{a},\,|v-u|<d(u,\partial D_{a})\\}}{d(u,\partial
D_{a})^{|\gamma|}}|w|^{|\beta-\tau|}$ $\displaystyle\leq
CL\,|w|^{|\beta-\tau|-|\gamma|}\leq CL,$

where $C>0$ is the supremum in (3.6); indeed, if $\gamma\neq 0$ then
$|\alpha|+|\beta|=m$ and thus $|\beta-\tau|\geq|\gamma|$.

Let us prove (4.9). Now $|\kappa|=m$ and $|\alpha|+|\beta|=m$, whence
$|\beta-\tau|=|\gamma|$. Then it is enough to show

(4.10)
$|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{1},0)w_{1}^{\beta-\tau}-\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{2},0)w_{2}^{\beta-\tau}|\leq
C\,\omega(|x_{1}-x_{2}|).$

If $\gamma=0$, this follows from (3.7). So let us assume that $|\gamma|\geq
1$.

Set $t_{a}(u):=\tfrac{1}{2}d(u,\partial D_{a})$. Then

(4.11) $|t_{a}(u_{1})-t_{a}(u_{2})|\leq\tfrac{1}{2}|u_{1}-u_{2}|.$

Note that, for $i=1,2$,

(4.12) $|w_{i}|<d(u_{i},\partial D_{a})=2t_{a}(u_{i}).$

We consider two cases.

#### Case 1

Suppose that $t_{a}(u_{i})\leq|x_{1}-x_{2}|$ for $i=1,2$. Then, by (4.4) and
(4.12),

$\displaystyle|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{i},0)w_{i}^{\beta-\tau}|\leq
L\,\omega(2t_{a}(u_{i}))\leq 2L\,\omega(|x_{1}-x_{2}|),$

since $\omega$ is concave and increasing.

#### Case 2

Assume (without loss of generality) that $t_{a}(u_{1})>|x_{1}-x_{2}|$. Then
$|u_{1}-u_{2}|\leq|x_{1}-x_{2}|<t_{a}(u_{1})=\tfrac{1}{2}d(u_{1},\partial
D_{a})$ so that the line segment $[x_{1},x_{2}]$ is contained in
$D_{a}\times\mathbb{R}^{\ell}$. Furthermore, if $u\in[u_{1},u_{2}]$ then, by
(4.11),

$|t_{a}(u_{1})-t_{a}(u)|\leq\tfrac{1}{2}|u_{1}-u|\leq\tfrac{1}{2}|x_{1}-x_{2}|<\tfrac{1}{2}t_{a}(u_{1}),$

whence

$\tfrac{1}{2}t_{a}(u_{1})<t_{a}(u)<\tfrac{3}{2}t_{a}(u_{1}),\quad
u\in[u_{1},u_{2}].$

The left-hand side of (4.10) is bounded by

$\displaystyle|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{1},0)-\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{2},0)||w_{1}|^{|\beta-\tau|}+|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{2},0)||w_{1}^{\beta-\tau}-w_{2}^{\beta-\tau}|.$

By (4.4) and (4.12),

$\displaystyle|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{1},0)-\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{2},0)||w_{1}|^{|\beta-\tau|}$
$\displaystyle\lesssim\sup_{u\in[u_{1},u_{2}]}\sum_{j=1}^{k}|\partial^{\gamma+(j)}F_{a}^{(\alpha,\beta)}(u,0)||u_{1}-u_{2}|\,t_{a}(u_{1})^{|\gamma|}$
$\displaystyle\lesssim\sup_{u\in[u_{1},u_{2}]}\frac{\omega(2t_{a}(u))}{t_{a}(u)^{|\gamma|+1}}|u_{1}-u_{2}|\,t_{a}(u_{1})^{|\gamma|}$
$\displaystyle\lesssim\frac{\omega(t_{a}(u_{1}))}{t_{a}(u_{1})}|x_{1}-x_{2}|$
$\displaystyle\leq\omega(|x_{1}-x_{2}|),$

since $\omega$ is concave. Again, by (4.4) and (4.12),

$\displaystyle|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u_{2},0)||w_{1}^{\beta-\tau}-w_{2}^{\beta-\tau}|$
$\displaystyle\lesssim\frac{\omega(2t_{a}(u_{2}))}{t_{a}(u_{2})^{|\gamma|}}|w_{1}-w_{2}|\,t_{a}(u_{1})^{|\gamma|-1}\lesssim\frac{\omega(t_{a}(u_{1}))}{t_{a}(u_{1})}|x_{1}-x_{2}|\leq\omega(|x_{1}-x_{2}|).$

The proof is complete. ∎

The proof shows that (4.9) actually holds on the larger set $\\{(u,w)\in
D_{a}\times\mathbb{R}^{\ell}:|w|<d(u,\partial D_{a})\\}$.

######  Lemma 4.4.

For each $a\in A$,

(4.13) $|\partial^{\kappa}h_{a}(u,w)|\leq C\,\omega(d(u,\partial
D_{a}))d(u,\partial D_{a})^{m-|\kappa|},$

for all $(u,w)\in\Delta(D_{a}\times 0)$, all $\kappa\in\mathbb{N}^{n}$,
$|\kappa|\leq m$, and a constant $C>0$ independent of $a\in A$.

###### Proof.

Fix $x=(u,w)\in\Delta(D_{a}\times 0)$. If $d(u,\partial D_{a})<d(u,\partial
T_{a})$, then let $u^{\prime}\in\partial D_{a}$ be such that
$|u-u^{\prime}|=d(u,\partial D_{a})$ and set $x^{\prime}=(u^{\prime},0)$. The
open line segment $(x,x^{\prime})$ is contained in $\Delta(D_{a}\times 0)$.
Since $u^{\prime}\in T_{a}$, where $F_{a}^{(0,\beta)}$ is of class $C^{p}$,
and $h_{a}$ is of class $C^{m,\omega}$ on $\Delta(D_{a}\times 0)$ with
$C^{m,\omega}$-norm bounded by a constant independent of $a\in A$, by Lemma
4.3, we may conclude the assertion from Taylor’s theorem.

So we assume that $d(u,\partial D_{a})=d(u,\partial T_{a})$. Let
$u^{\prime}\in\partial T_{a}$ such that $|u-u^{\prime}|=d(u,\partial T_{a})$.
Let us first assume that
$\kappa=(\sigma,\tau)\in\mathbb{N}^{k}\times\mathbb{N}^{\ell}$ with
$|\kappa|=m$. By construction, $\partial^{\kappa}g_{a}(u^{\prime},0)=0$ so
that

$|\partial^{\kappa}g_{a}(u,w)|=|\partial^{\kappa}g_{a}(u,w)-\partial^{\kappa}g_{a}(u^{\prime},0)|\lesssim\omega(|u-u^{\prime}|)=\omega(d(u,\partial
D_{a})),$

where we used $|w|<d(u,\partial D_{a})=|u-u^{\prime}|$. Hence it suffices to
consider $\partial^{\kappa}\big{(}F_{a}^{(0,\beta)}(u,0)w^{\beta}\big{)}$ or
equivalently $\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u,0)w^{\beta-\tau}$,
where $\alpha,\gamma\in\mathbb{N}^{k}$ are such that $\alpha+\gamma=\sigma$,
$|\alpha|+|\beta|=m$, and $\tau\leq\beta$. Thus $|\beta-\tau|=|\gamma|$. If
$|\gamma|\geq 1$, (4.4) implies

$\displaystyle|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u,0)w^{\beta-\tau}|\leq
L\,\frac{\omega(d(u,\partial D_{a}))}{d(u,\partial
D_{a})^{|\gamma|}}|w|^{|\gamma|}\leq L\,\omega(d(u,\partial D_{a})),$

and, if $\gamma=0$, (3.7) gives

$\displaystyle|\partial^{\gamma}F_{a}^{(\alpha,\beta)}(u,0)w^{\beta-\tau}|=|F_{a}^{(\alpha,\beta)}(u,0)|\lesssim\omega(d(u,\partial
T_{a}))=\omega(d(u,\partial D_{a})),$

since $(F_{a})_{a\in A}$ is flat on $(\partial T_{a}\times 0)_{a\in A}$.

To prove the statement for $|\kappa|<m$, we proceed by induction on
$m-|\kappa|$. Suppose that the assertion is already shown for every
$\lambda\in\mathbb{N}^{n}$ with $|\kappa|<|\lambda|\leq m$. Since the open
line segment $(x,x^{\prime})$ connecting $x=(u,w)$ and
$x^{\prime}=(u^{\prime},0)$ is contained in $\Delta(D_{a}\times 0)$, we have,
by induction hypothesis, where
$x^{\prime\prime}=(u^{\prime\prime},w^{\prime\prime})$,

$\displaystyle|\partial^{\kappa}h_{a}(u,w)|$
$\displaystyle\leq\sup_{x^{\prime\prime}\in(x,x^{\prime})}\sum_{j=1}^{n}|\partial^{\kappa+(j)}h_{a}(u^{\prime\prime},w^{\prime\prime})||x-x^{\prime}|$
$\displaystyle\lesssim\sup_{x^{\prime\prime}\in(x,x^{\prime})}\omega(d(u^{\prime\prime},\partial
D_{a}))d(u^{\prime\prime},\partial D_{a})^{m-|\kappa|-1}\,d(u,\partial D_{a})$
$\displaystyle\lesssim\omega(d(u,\partial D_{a}))d(u,\partial
D_{a})^{m-|\kappa|},$

since $d(u^{\prime\prime},\partial D_{a})\leq d(u,\partial D_{a})$. ∎

It follows from Lemma 4.3 and Lemma 4.4 that, for each $a\in A$,

(4.14) $|\partial^{\kappa}h_{a}(u,w)|\leq C\,\omega(\min\\{1,d(u,\partial
D_{a})\\})\min\\{1,d(u,\partial D_{a})\\}^{m-|\kappa|},$

for all $(u,w)\in\Delta(D_{a}\times 0)$, all $\kappa\in\mathbb{N}^{n}$,
$|\kappa|\leq m$, and a constant $C>0$ independent of $a\in A$. Indeed, by
Lemma 4.3,

$|\partial^{\kappa}h_{a}(u,w)|\leq
C=\frac{C}{\omega(1)}\cdot\omega(1)1^{m-|\kappa|},\quad|\kappa|\leq m,$

for all $(u,w)\in\Delta(D_{a}\times 0)$, which, together with (4.13), gives
(4.14).

Now Proposition 3.7 (see also Remark 3.8) implies that the family
$(f_{a})_{a\in A}$ is bounded in $C^{m,\omega}(\mathbb{R}^{n})$. Indeed, Lemma
4.4, Lemma 4.3, and (4.14) guarantee that the assumptions of Proposition 3.7
are satisfied, where $t_{a}(u)=\min\\{1,d(u,\partial D_{a})\\}$. Condition
(3.9) holds thanks to (2.9) and Remark 2.12. We also get

(4.15) $|\partial^{\kappa}f_{a}(u,w)|\leq C\,\omega(d(u,\partial
D_{a}))d(u,\partial D_{a})^{m-|\kappa|},$

for all $(u,w)\in\Delta(D_{a}\times 0)$, all $\kappa\in\mathbb{N}^{n}$,
$|\kappa|\leq m$, and a constant $C>0$ independent of $a\in A$.

### Step 2

For all $a\in A$, $E^{\prime}_{a}=\overline{S}_{a}=\overline{T}_{a}\times 0$,
but possibly $E^{\prime}_{a}$ is a proper subset of $E_{a}$ for some $a\in A$.
Consider the definable family $(r_{a})_{a\in A}$ of functions
$r_{a}:T_{a}\to(0,\infty)$ given by

$r_{a}(u):=\begin{cases}\inf\\{|w|:(u,w)\in E_{a}\setminus S_{a}\\}&\text{ if
}\\{w:(u,w)\in E_{a}\setminus S_{a}\\}\neq\emptyset,\\\ 1&\text{
otherwise.}\end{cases}$

Since $F_{a}$ is flat on $E_{a}\setminus S_{a}$ we have (by (3.6) and (3.7))

(4.16) $|F^{\kappa}_{a}(u,0)|\leq C\,\omega(r_{a}(u))r_{a}(u)^{m-|\kappa|}$

for all $u\in T_{a}$, all $\kappa\in\mathbb{N}^{n}$, $|\kappa|\leq m$, and a
constant $C>0$ independent of $a\in A$. (In the case that $\\{w:(u,w)\in
E_{a}\setminus S_{a}\\}=\emptyset$, it follows from (3.6) and we have to
replace $C$ by $C/\omega(1)$.)

By Theorem 2.16 and Proposition 2.20, there is a uniform
$\Lambda_{p}$-stratification of $(\overline{T}_{a})_{a\in A}$ such that

$\overline{T}_{a}=Q_{a,1}\cup\cdots\cup Q_{a,s}\cup Z_{a},$

where, for each $a\in A$ and each $i=1,\ldots,s$, $Z_{a}$ is closed with $\dim
Z_{a}<k$, each $Q_{a,i}$ is an open $k$-dimensional $\Lambda_{p}$-cell with
constant independent of $a\in A$, $r_{a}$ is $C^{p}$ on $Q_{a,i}$, and either

  * (i)

$|\partial_{j}r_{a}|\leq 1$, $j=1,\ldots,k$, on $Q_{a,i}$, in which case we
may assume that $|\partial^{\alpha}r_{a}(u)|d(u,\partial
Q_{a,i})^{|\alpha|-1}$, $1\leq|\alpha|\leq p$, is bounded on $Q_{a,i}$ by a
constant independent of $a\in A$, by Corollary 2.18, or

  * (ii)

$|\partial_{j}r_{a}(u)|>1$ for some $j$ on $Q_{a,i}$.

By ($\mathbf{I}_{k-1}$), we may assume that $(F_{a})_{a\in A}$ is flat on
$(Z_{a}\times 0)_{a\in A}$ and hence on $(\partial Q_{a,i}\times 0)_{a\in A}$
for each $i=1,\ldots,s$.

Now it is enough to show that, for every $i=1,\ldots,s$,
$(F_{a}|_{E_{a}\cap(\overline{Q}_{a,i}\times\mathbb{R}^{\ell})})_{a\in A}$
admits a definable bounded family $(f_{a,i})_{a\in A}$ of
$C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ that is $m$-flat outside
$(\Delta(Q_{a,i}\times 0))_{a\in A}$ and $C^{p}$ outside
$(\overline{Q}_{a,i}\times 0)_{a\in A}$. To this end, we fix $i$ and drop it
from the notation.

Step 1 gives a definable bounded family $(g_{a})_{a\in A}$ of
$C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of
$(F_{a}|_{\overline{Q}_{a}\times 0})_{a\in A}$ that is $m$-flat outside
$(\Delta(Q_{a}\times 0)_{a\in A}$ and $C^{p}$ outside $(\overline{Q}_{a}\times
0)_{a\in A}$. By Taylor’s formula and (4.16), for each $a\in A$,

(4.17) $|\partial^{\kappa}g_{a}(u,w)|\leq
C\,\omega(r_{a}(u))\,r_{a}(u)^{m-|\kappa|}$

for all $(u,w)\in Q_{a}\times\mathbb{R}^{\ell}$, $|w|<C^{\prime}\,r_{a}(u)$,
and all $\kappa\in\mathbb{N}^{n}$, $|\kappa|\leq m$, where $C,C^{\prime}>0$
are independent of $a\in A$. Similarly, we have

(4.18) $|\partial^{\kappa}g_{a}(u,w)|\leq C\,\omega(d(u,\partial
Q_{a}))\,d(u,\partial Q_{a})^{m-|\kappa|}$

for all $(u,w)\in Q_{a}\times\mathbb{R}^{\ell}$, $|w|<C^{\prime}\,d(u,\partial
Q_{a})$, and all $\kappa\in\mathbb{N}^{n}$, $|\kappa|\leq m$, where
$C,C^{\prime}>0$ are independent of $a\in A$.

In Case (ii), one can easily see (cf. [8, p. 94]) that, for each $a\in A$,
$r_{a}(u)\geq d(u,\partial Q_{a})$ for $u\in Q_{a}$, so that $E_{a}\setminus
S_{a}\subseteq\mathbb{R}^{n}\setminus\Delta(Q_{a}\times 0)$. That means that
$g_{a}$ is a $C^{m,\omega}$-extension to $\mathbb{R}^{n}$ of
$F_{a}|_{E_{a}\cap(\overline{Q}_{a}\times 0)}$, and we are done.

In Case (i), a modification is necessary: we define, for each $a\in A$,

$f_{a}(u,w):=\begin{cases}\prod_{i=1}^{\ell}\xi\Big{(}\sqrt{\ell}\frac{w_{i}}{r_{a}(u)}\Big{)}\cdot
g_{a}(u,w)&\text{ if }u\in Q_{a},\\\ 0&\text{ otherwise,}\end{cases}$

where $\xi:\mathbb{R}\to\mathbb{R}$ is a semialgebraic $C^{p}$-function that
is $1$ near $0$ and vanishes outside $(-1,1)$. Note that $(f_{a})_{a\in A}$ is
a definable family of functions $f_{a}:\mathbb{R}^{n}\to\mathbb{R}$. Moreover,
we set

$\Delta^{\prime}(Q_{a}\times 0):=\\{(u,w)\in
Q_{a}\times\mathbb{R}^{\ell}:|w|<t_{a}(u)\\}$

with

$t_{a}(u):=\min\\{r_{a}(u),d(u,\partial Q_{a})\\},$

and claim that, for each $a\in A$, $f_{a}$ is of class $C^{m,\omega}$ on
$\Delta^{\prime}(Q_{a}\times 0)$ with $C^{m,\omega}$-norm bounded by a
constant independent of $a\in A$, and

(4.19) $|\partial^{\kappa}f_{a}(u,w)|\leq
C\,\omega(t_{a}(u))\,t_{a}(u)^{m-|\kappa|}$

for $(u,w)\in\Delta^{\prime}(Q_{a}\times 0)$ and all
$\kappa\in\mathbb{N}^{n}$, $|\kappa|\leq m$, where $C>0$ is independent of
$a\in A$.

To see this, let us first assume that $r_{a}(u)<d(u,\partial Q_{a})$ so that
$t_{a}(u)=r_{a}(u)$. Since we are in Case (i), we find that, thanks to (2.10),

$|\partial^{\alpha}(1/r_{a})(u)|\leq C\,r_{a}(u)^{-|\alpha|-1},\quad u\in
Q_{a},\,|\alpha|\leq p,$

for a constant $C>0$ independent of $a\in A$. Thus, the claim follows from
(4.17) and Proposition 3.7.

If $r_{a}(u)\geq d(u,\partial Q_{a})$ (that is $t_{a}(u)=d(u,\partial
Q_{a})$), then similarly

$|\partial^{\alpha}(1/r_{a})(u)|\leq C\,d(u,\partial
Q_{a})^{-|\alpha|-1},\quad u\in Q_{a},\,|\alpha|\leq p,$

Then we infer the claim from (4.18) and Proposition 3.7.

We conclude that $(f_{a})_{a\in A}$ is the required family of definable
bounded $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of
$(F_{a}|_{E_{a}\cap(\overline{Q}_{a}\times\mathbb{R}^{\ell})})_{a\in A}$ that
is $m$-flat outside $(\Delta(Q_{a}\times 0))_{a\in A}$ and $C^{p}$ outside
$(\overline{Q}_{a}\times 0)_{a\in A}$. This ends Step 2.

### Step 3

The general case of Proposition 4.2: for all $a\in A$,
$S_{a}=\Gamma(\varphi_{a})$, $E_{a}^{\prime}=\overline{S}_{a}\subseteq E_{a}$,
where $\varphi_{a}:T_{a}\to\mathbb{R}^{\ell}$ is not necessarily identically
$0$. Consider the definable family $(s_{a})_{a\in A}$ of functions
$s_{a}:S_{a}\to(0,\infty)$ given by

$s_{a}(x):=\min\\{d(x,E_{a}\setminus S_{a}),d(x,\partial S_{a}\\},\quad x\in
S_{a}.$

For each $a\in A$, let
$\overline{\varphi}_{a}:\overline{T}_{a}\to\mathbb{R}^{\ell}$ be the
continuous extension of $\varphi_{a}$; cf. Section 2.4. Furthermore, we
consider the maps

$\displaystyle\varphi_{a,\pm}$ $\displaystyle:T_{a}\times\mathbb{R}^{\ell}\to
T_{a}\times\mathbb{R}^{\ell},\quad(u,w)\mapsto(u,w\pm\varphi_{a}(u))$ and
$\displaystyle\overline{\varphi}_{a,\pm}$
$\displaystyle:\overline{T}_{a}\times\mathbb{R}^{\ell}\to\overline{T}_{a}\times\mathbb{R}^{\ell},\quad(u,w)\mapsto(u,w\pm\overline{\varphi}_{a}(u)).$

Note that $\overline{\varphi}_{a,+}$ is a bi-Lipschitz homeomorphism with
inverse $\overline{\varphi}_{a,-}$ and Lipschitz constants independent of
$a\in A$.

Since $F_{a}$ is flat on $E_{a}\setminus S_{a}$ and on $\partial S_{a}$, we
have (by (3.7))

(4.20) $|F_{a}^{\kappa}(x)|\leq C\,\omega(s_{a}(x))s_{a}(x)^{m-|\kappa|}$

for all $x\in S_{a}$, all $\kappa\in\mathbb{N}^{n}$, $|\kappa|\leq m$, and a
constant $C>0$ independent of $a\in A$. Setting

$t_{a}(u):=s_{a}(u,\varphi_{a}(u)),\quad u\in T_{a},$

(4.20) reads as

(4.21) $|F_{a}^{\kappa}(u,\varphi_{a}(u))|\leq
C\,\omega(t_{a}(u))t_{a}(u)^{m-|\kappa|},\quad u\in T_{a},\,|\kappa|\leq m.$

The uniformity of the constants in the definition of $T_{a}$ and $\varphi_{a}$
implies that $(\overline{S}_{a})_{a\in A}$ and $(\partial
T_{a}\times\mathbb{R}^{\ell})_{a\in A}$ are uniformly separated. Observe that
(by the definition of $s_{a}$) $t_{a}(u)\leq C^{\prime}\,d(u,\partial T_{a})$
for $C^{\prime}>0$ independent of $a\in A$, since $\overline{\varphi}_{a,+}$
is a bi-Lipschitz homeomorphism with Lipschitz constants independent of $a\in
A$.

Thus Proposition 3.5 (and Lemma 3.6) implies that $(G_{a})_{a\in A}$, where
$G_{a}:=\varphi_{a,+}^{*}(F_{a}|_{S_{a}})$, is a definable bounded family of
Whitney jets of class $C^{m,\omega}$ on $(T_{a}\times 0)_{a\in A}$ and extends
to a definable bounded family of Whitney jets of class $C^{m,\omega}$ on
$(\overline{T}_{a}\times 0)_{a\in A}$ which is flat on $(\partial T_{a}\times
0)_{a\in A}$ and such that

(4.22) $|G_{a}^{\kappa}(u,0)|\leq C\,\omega(t_{a}(u))t_{a}(u)^{m-|\kappa|}$

for all $u\in T_{a}$, all $\kappa\in\mathbb{N}^{n}$, $|\kappa|\leq m$, and a
constant $C>0$ independent of $a\in A$. For each $a\in A$, set
$\tilde{E}_{a}:=\overline{\varphi}_{a,-}(E_{a}\cap(\overline{T}_{a}\times\mathbb{R}^{\ell}))$.
Since $\overline{\varphi}_{a,+}$ is a bi-Lipschitz homeomorphism with
constants independent of $a\in A$, we may conclude

(4.23) $|G_{a}^{\kappa}(u,0)|\leq
C\,\omega(d((u,0),\tilde{E}_{a}\setminus(T_{a}\times
0)))d((u,0),\tilde{E}_{a}\setminus(T_{a}\times 0))^{m-|\kappa|}$

for all $u\in T_{a}$, all $\kappa\in\mathbb{N}^{n}$, $|\kappa|\leq m$, and a
constant $C>0$ independent of $a\in A$. Thus $(\tilde{G}_{a})_{a\in A}$, where

$\tilde{G}_{a}(u,w):=\begin{cases}G_{a}(u,0)&\text{ if
}(u,w)\in\overline{T}_{a}\times 0,\\\ 0&\text{ if
}(u,w)\in\tilde{E}_{a}\setminus(\overline{T}_{a}\times 0),\end{cases}$

is a definable bounded family of Whitney jets of class $C^{m,\omega}$ on
$(\tilde{E}_{a})_{a\in E}$ that is flat on
$(\tilde{E}_{a}\setminus(T_{a}\times 0))_{a\in A}$.

By Step 2, there exists a definable bounded family $(\tilde{g}_{a})_{a\in A}$
of $C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of $(\tilde{G}_{a})_{a\in A}$
that is $m$-flat on $(\tilde{E}_{a}\setminus(T_{a}\times 0))_{a\in A}$ as well
as outside $(T_{a}\times\mathbb{R}^{\ell})_{a\in A}$ and $C^{p}$ outside
$(\overline{T}_{a}\times 0)_{a\in A}$.

For each $a\in A$, define $f_{a}:\mathbb{R}^{n}\to\mathbb{R}$ by

$f_{a}(u,w):=\begin{cases}(\tilde{g}_{a}\circ\varphi_{a,-})(u,w)&\text{ if
}(u,w)\in T_{a}\times\mathbb{R}^{\ell},\\\ 0&\text{ otherwise.}\end{cases}$

Then $(f_{a})_{a\in A}$ is a definable bounded family of
$C^{m,\omega}$-extensions to $\mathbb{R}^{n}$ of $(F_{a})_{a\in A}$ that is
$C^{p}$ outside $(\overline{S}_{a})_{a\in A}$, which follows again from
Proposition 3.5 (with $M_{a}=T_{a}\times\mathbb{R}^{\ell}$ and $U_{a}=T_{a}$).

This completes the proof of Proposition 4.2, hence of ($\mathbf{I}_{k}$), and
thus the proof of Theorem 1.3.

##  5\. Further applications

In this section, we present a local version of Theorem 1.3, we discuss the
dependence of the bounded extension on the modulus of continuity which leads
to the proof of Theorem 1.4, and finally we obtain a definable version of a
correspondence between Whitney jets of class $C^{m,\omega}$ and certain
Lipschitz maps, which was originally observed by Shvartsman [11].

###  5.1. Definable $C^{m,\omega}_{\operatorname{loc}}$-extensions

Let $U\subseteq\mathbb{R}^{n}$ be open. We denote by
$C^{m,\omega}_{\operatorname{loc}}(U)$ the space of functions
$f:U\to\mathbb{R}$ such that $f|_{V}\in C^{m,\omega}(V)$, for all relatively
compact open subsets $V\subseteq U$.

Let $E\subseteq\mathbb{R}^{n}$ be a closed set. An $m$-jet $F$ on $E$ is
called a _(definable) Whitney jet of class $C^{m,\omega}_{\operatorname{loc}}$
on $E$_ if $F|_{K}$ is a (definable) Whitney jet of class $C^{m,\omega}$ on
$K$, for all (definable) compact subsets $K\subseteq E$. A
$C^{m,\omega}_{\operatorname{loc}}$-function $f:\mathbb{R}^{n}\to\mathbb{R}$
is a _$C^{m,\omega}_{\operatorname{loc}}$ -extension to $\mathbb{R}^{n}$ of
$F$_ if $J_{E}^{m}(f)=F$.

Let $(E_{a})_{a\in A}$ be a family of closed sets
$E_{a}\subseteq\mathbb{R}^{n}$. A family $(F_{a})_{a\in A}$ of Whitney jets of
class $C^{m,\omega}_{\operatorname{loc}}$ on $E_{a}$ is called a _(definable)
bounded family of Whitney jets of class $C^{m,\omega}_{\operatorname{loc}}$_
if $(F_{a}|_{K_{a}})_{a\in A}$ is a (definable) bounded family of Whitney jets
of class $C^{m,\omega}$ for each (definable) subfamily $(K_{a})_{a\in A}$ of
$(E_{a})_{a\in A}$ consisting of (definable) compact sets $K_{a}\subseteq
E_{a}$.

A family $(f_{a})_{a\in A}$ of $C^{m,\omega}_{\operatorname{loc}}$-functions
$f_{a}:\mathbb{R}^{n}\to\mathbb{R}$ is called a _(definable) bounded family of
$C^{m,\omega}_{\operatorname{loc}}$-extensions to $\mathbb{R}^{n}$ of
$(F_{a})_{a\in A}$_ if $f_{a}$ is a
$C^{m,\omega}_{\operatorname{loc}}$-extension to $\mathbb{R}^{n}$ of $F_{a}$,
for each $a\in A$, and, for each (definable) relatively compact subset
$V\subseteq\mathbb{R}^{n}$, $(f_{a}|_{V})_{a\in A}$ is a (definable) bounded
family of $C^{m,\omega}$-functions.

######  Corollary 5.1.

Let $0\leq m\leq p$ be integers. Let $\omega$ be a modulus of continuity. Let
$(E_{a})_{a\in A}$ be a definable family of closed subsets $E_{a}$ of
$\mathbb{R}^{n}$. For any definable bounded family $(F_{a})_{a\in A}$ of
Whitney jets of class $C^{m,\omega}_{\operatorname{loc}}$ on $(E_{a})_{a\in
A}$ there exists a definable bounded family $(f_{a})_{a\in A}$ of
$C^{m,\omega}_{\operatorname{loc}}$-extensions to $\mathbb{R}^{n}$ of
$(F_{a})_{a\in A}$ that is $C^{p}$ outside $(E_{a})_{a\in A}$.

###### Proof.

This follows by standard arguments from Theorem 1.3 and the existence of
definable partitions of unity of class $C^{p}$. ∎

######  Remark 5.2.

Note that we do not say that $f_{a}$ is definable as a global function
$f_{a}:\mathbb{R}^{n}\to\mathbb{R}$, because the gluing argument (based on the
partition of unity) involves an infinite sum.

Nevertheless, global definability of $f_{a}$ can be achieved by an adaptation
of the whole proof to the $C^{m,\omega}_{\operatorname{loc}}$-setting. We
leave the details to the reader.

###  5.2. Dependence on the modulus of continuity

The main result, Theorem 1.3, only depends in a weak sense on the modulus of
continuity $\omega$, namely, the uniform constant $C$ occasionally must be
multiplied by $\omega(1)$ or by $\omega(1)^{-1}$; see (4.1), (4.6), (4.14),
and (4.16).

Thus, we can allow in Theorem 1.3 that, for each $a\in A$, $F_{a}$ is a
Whitney jet of class $C^{m,\omega_{a}}$ on $E_{a}$, where $\omega_{a}$ is a
modulus of continuity and there is a constant $C>0$ independent of $a\in A$
such that

(5.1) $C^{-1}\leq\omega_{a}(1)\leq C,\quad a\in A.$

Then the statement is the following:

######  Theorem 5.3.

Let $0\leq m\leq p$ be integers. Let $(\omega_{a})_{a\in A}$ be a family of
moduli of continuity satisfying (5.1). Let $(E_{a})_{a\in A}$ be a definable
family of closed subsets $E_{a}$ of $\mathbb{R}^{n}$. For any definable family
$(F_{a})_{a\in A}$ of Whitney jets $F_{a}$ of class $C^{m,\omega_{a}}$ on
$E_{a}$ such that

(5.2) $\displaystyle\sup_{a\in A}\sup_{x\in E_{a}}\sup_{|\gamma|\leq
m}|F_{a}^{\gamma}(x)|<\infty,$ and (5.3) $\displaystyle\sup_{a\in
A}\sup_{x\neq y\in E_{a}}\sup_{|\gamma|\leq
m}\frac{|(R^{m}_{x}F_{a})^{\gamma}(y)|}{\omega_{a}(|x-y|)|x-y|^{m-|\gamma|}}<\infty,$

there exists a definable family $(f_{a})_{a\in A}$ of
$C^{m,\omega_{a}}$-extensions $f_{a}$ to $\mathbb{R}^{n}$ of $F_{a}$ such that
$f_{a}$ is of class $C^{p}$ outside $E_{a}$, for all $a\in A$, and

(5.4) $\sup_{a\in A}\|f_{a}\|_{C^{m,\omega_{a}}(\mathbb{R}^{n})}<\infty.$

###  5.3. Proof of Theorem 1.4

That the definable family $(F_{a})_{a\in A}$ of Whitney jets of class $C^{m}$
on $(E_{a})_{a\in A}$, where $E_{a}\subseteq\mathbb{R}^{n}$ is compact, is
_bounded_ means that

(5.5) $\displaystyle\sup_{a\in A}\sup_{x\in E_{a}}\sup_{|\gamma|\leq
m}|F^{\gamma}_{a}(x)|<\infty$ and (5.6) $\displaystyle\sup_{a\in A}\sup_{x\neq
y\in E_{a}}\sup_{|\gamma|\leq
m}\frac{|(R^{m}_{x}F_{a})^{\gamma}(y)|}{|x-y|^{m-|\gamma|}}<\infty.$

###### Proof of Theorem 1.4.

We modify slightly an argument used in [13, Proposition IV.1.5]. For each
$a\in A$, consider

$\sigma_{a}(t):=\sup_{\begin{subarray}{c}x\neq y\in E_{a}\\\ |x-y|\leq
t\end{subarray}}\sup_{|\gamma|\leq
m}\frac{|(R^{m}_{x}F_{a})^{\gamma}(y)|}{|x-y|^{m-|\gamma|}},\quad
t>0,\quad\sigma_{a}(0):=0.$

Then $\sigma_{a}:[0,\infty)\to[0,\infty)$ is an increasing function that is
continuous at $0$ and

$\sigma_{a}(t)=\sigma_{a}(\operatorname{diam}E_{a}),\quad
t\geq\operatorname{diam}E_{a}.$

Thus also $\tau_{a}:[0,\infty)\to[0,\infty)$ defined by

$\tau_{a}(t):=\begin{cases}\sigma_{a}(t)&\text{ if }t<1,\\\
\max\\{1,\sigma_{a}(t)\\}&\text{ if }t\geq 1,\end{cases}$

is increasing and continuous at $0$ with

(5.7) $\tau_{a}(t)\leq\max\\{1,\sigma_{a}(\operatorname{diam}E_{a})\\},\quad
t\geq 0.$

Let $\omega_{a}$ be the least concave majorant of $\tau_{a}$ which is finite,
thanks to (5.7). Then $\omega_{a}$ is a modulus of continuity and

$\sup_{a\in A}\sup_{x\neq y\in E_{a}}\sup_{|\gamma|\leq
m}\frac{|(R^{m}_{x}F_{a})^{\gamma}(y)|}{\omega_{a}(|x-y|)|x-y|^{m-|\gamma|}}\leq
1.$

Moreover, $\omega_{a}(1)\geq 1$ and, by (5.7),

$\omega_{a}(t)\leq\max\\{1,\sigma_{a}(\operatorname{diam}E_{a})\\}\leq C,\quad
t\geq 0,$

for a constant $C>0$ independent of $a\in A$, thanks to (5.6). In particular,
(5.1) is satisfied.

Thus Theorem 5.3 implies that there is a definable family $(f_{a})_{a\in A}$
such that, for each $a\in A$, $f_{a}$ is a $C^{m,\omega_{a}}$-extension to
$\mathbb{R}^{n}$ of $F_{a}$, $C^{p}$ outside $E_{a}$, and

$\sup_{a\in A}\|f_{a}\|_{C^{m,\omega_{a}}(\mathbb{R}^{n})}<\infty.$

In particular, $(f_{a})_{a\in A}$ is a bounded family of $C^{m}$-functions. ∎

###  5.4. Definable Whitney jets as Lipschitz maps

We end with a few observations on a definable version of a correspondence, due
to Shvartsman [11], between Whitney jets of class $C^{m,\omega}$ and certain
Lipschitz maps. Here the notation follows closely the one of [11].

Let $\omega$ be a modulus of continuity and $m$ a positive integer. For
$\alpha\in\mathbb{N}^{n}$ with $|\alpha|<m$ let $\psi_{\alpha}$ be the inverse
of the (strictly increasing) function $s\mapsto s^{m-|\alpha|}\omega(s)$ and
put $\varphi_{\alpha}:=\omega\circ\psi_{\alpha}$. For $|\alpha|=m$, set
$\varphi_{\alpha}(t):=t$.

Let $\mathcal{P}_{m}$ denote the space of real polynomials of degree at most
$m$ in $n$ variables. For
$T_{i}=(P_{i},x_{i})\in\mathcal{P}_{m}\times\mathbb{R}^{n}$, $i=1,2$, define

$\displaystyle\delta_{\omega}(T_{1},T_{2}):=\max\Big{\\{}\omega(|x_{1}-x_{2}|),\max_{\begin{subarray}{c}|\alpha|\leq
m\\\
i=1,2\end{subarray}}\varphi_{\alpha}(|\partial^{\alpha}(P_{1}-P_{2})(x_{i})|)\Big{\\}}.$

Then we get a metric $d_{\omega}$ on $\mathcal{P}_{m}\times\mathbb{R}^{n}$ by
setting

$d_{\omega}(T,T^{\prime}):=\inf\sum_{j=0}^{k-1}\delta_{\omega}(T_{j},T_{j+1}),$

where the infimum is taken over all finite sequences
$T=T_{1},T_{2},\ldots,T_{k}=T^{\prime}$ in
$\mathcal{P}_{m}\times\mathbb{R}^{n}$. It turns out (cf. [11, Theorem 2.1])
that

$d_{\omega}((P,x),(P^{\prime},x^{\prime}))\leq\delta_{\omega}((P,x),(P^{\prime},x^{\prime}))\leq
d_{\omega}((e^{n}P,x),(e^{n}P^{\prime},x^{\prime})).$

Let $\mathcal{T}_{m,n}$ be the metric space
$(\mathcal{P}_{m}\times\mathbb{R}^{n},d_{\omega})$. For a nonempty subset
$X\subseteq\mathbb{R}^{n}$, we denote by $X_{\omega}$ the metric space
$(X,(x,y)\mapsto\omega(|x-y|))$. Let
$\mathbf{Lip}(X_{\omega},\mathcal{T}_{m,n})$ be the space of Lipschitz maps
$T:x\mapsto(P_{x},z_{x})$ such that $\max_{|\alpha|\leq m}\sup_{x\in
X}|\partial^{\alpha}P_{x}(x)|<\infty$, equipped with the norm

$\displaystyle\|T\|^{*}_{LO(X)}$ $\displaystyle:=\max_{|\alpha|\leq
m}\sup_{x\in X}|\partial^{\alpha}P_{x}(x)|$
$\displaystyle\quad+\inf\\{\lambda>0:d_{\omega}(\lambda^{-1}T(x),\lambda^{-1}T(y))\leq\omega(|x-y|)\text{
for all }x,y\in X\\},$

where $\lambda^{-1}T(x):=(\lambda^{-1}P_{x},z_{x})$. Let $T^{m}_{x}f$ be the
Taylor polynomial of order $m$ at $x$ of a $C^{m}$-function $f$.

Now let us recall a result of [11].

######  Proposition 5.4 ([11, Proposition 1.9 and Proposition 2.8]).

Let $X\subseteq\mathbb{R}^{n}$ be a closed set. Given a family of polynomials
$\\{P_{x}\in\mathcal{P}_{m}:x\in X\\}$, there exists $f\in
C^{m,\omega}(\mathbb{R}^{n})$ such that $T^{m}_{x}f=P_{x}$ for all $x\in X$ if
and only if the map $T:x\mapsto(P_{x},x)$ belongs to
$\mathbf{Lip}(X_{\omega},\mathcal{T}_{m,n})$. We have

$\inf\\{\|f\|_{C^{m,\omega}(\mathbb{R}^{n})}:T^{m}_{x}f=P_{x}\text{ for all
}x\in X\\}\approx\|T\|^{*}_{LO(X)}$

in the sense that either side is bounded by a constant $C(m,n)$ times the
other side. Moreover, if $T:x\mapsto(P_{x},x)$ belongs to
$\mathbf{Lip}(X_{\omega},\mathcal{T}_{m,n})$, then $T$ has an extension
$\widetilde{T}:x\mapsto(\widetilde{P}_{x},x)$ in
$\mathbf{Lip}(\mathbb{R}^{n}_{\omega},\mathcal{T}_{m,n})$ satisfying

$\|\widetilde{T}\|^{*}_{LO(\mathbb{R}^{n})}\leq C(m,n)\,\|T\|^{*}_{LO(X)}.$

These results are based on the classical extension theorem for Whitney jets of
class $C^{m,\omega}$. As a consequence of Theorem 1.2, we may conclude the
following definable version, where, provided that $X$ is definable,
$\mathbf{Lip}_{\operatorname{def}}(X_{\omega},\mathcal{T}_{m,n})$ is the
subspace of definable maps $T:x\mapsto(P_{x},z_{x})$ in
$\mathbf{Lip}(X_{\omega},\mathcal{T}_{m,n})$, which means that $z_{x}$ and the
coefficients of $P_{x}$ are definable maps in $x$. Recall that
$C^{m,\omega}_{\operatorname{def}}(\mathbb{R}^{n})$ is the subspace of
$C^{m,\omega}(\mathbb{R}^{n})$ consisting of all definable functions in
$C^{m,\omega}(\mathbb{R}^{n})$.

######  Proposition 5.5.

Let $X\subseteq\mathbb{R}^{n}$ be a definable closed set. Given a definable
family of polynomials $\\{P_{x}\in\mathcal{P}_{m}:x\in X\\}$, there exists
$f\in C^{m,\omega}_{\operatorname{def}}(\mathbb{R}^{n})$ such that
$T^{m}_{x}f=P_{x}$ for all $x\in X$ if and only if the map
$T:x\mapsto(P_{x},x)$ belongs to
$\mathbf{Lip}_{\operatorname{def}}(X_{\omega},\mathcal{T}_{m,n})$. Moreover,
if $T:x\mapsto(P_{x},x)$ belongs to
$\mathbf{Lip}_{\operatorname{def}}(X_{\omega},\mathcal{T}_{m,n})$, then $T$
has an extension $\widetilde{T}:x\mapsto(\widetilde{P}_{x},x)$ in
$\mathbf{Lip}_{\operatorname{def}}(\mathbb{R}^{n}_{\omega},\mathcal{T}_{m,n})$.

Concerning the existence of uniform bounds for the norms, remarks similar to
the ones in [9, Section 4.4] apply. But Theorem 1.3 implies the following
supplement.

######  Proposition 5.6.

Suppose that in the setting of Proposition 5.5, the family of polynomials
depends definably on additional parameters $a\in A$, i.e., a definable family
of polynomials $\\{P^{a}_{x}\in\mathcal{P}_{m}:x\in X,\,a\in A\\}$ is given.
Then there exists a bounded family $(f^{a})_{a\in A}$ of definable
$C^{m,\omega}$-functions $f^{a}:\mathbb{R}^{n}\to\mathbb{R}$ such that

$T_{x}^{m}f^{a}=P^{a}_{x},\quad\text{ for all }x\in X\text{ and }a\in A,$

if and only if $(T^{a}:x\mapsto(P^{a}_{x},x))_{a\in A}$ forms a bounded subset
of $\mathbf{Lip}_{\operatorname{def}}(X_{\omega},\mathcal{T}_{m,n})$.
Moreover, if $(T^{a}:x\mapsto(P^{a}_{x},x))_{a\in A}$ forms a bounded subset
of $\mathbf{Lip}_{\operatorname{def}}(X_{\omega},\mathcal{T}_{m,n})$, then
there is a family $(\widetilde{T}^{a}:x\mapsto(\widetilde{P}^{a}_{x},x))_{a\in
A}$ of extensions $\widetilde{T}^{a}$ of $T^{a}$ which forms a bounded subset
of
$\mathbf{Lip}_{\operatorname{def}}(\mathbb{R}^{n}_{\omega},\mathcal{T}_{m,n})$.

### Acknowledgements

A large part of the work on this paper has been done at Mathematisches
Forschungsinstitut Oberwolfach (_Oberwolfach Research Fellows (OWRF) ID 2244p,
October 31 - November 19, 2022_). We are grateful for the support and the
excellent working conditions.

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