# Quantitative tame properties of differentiable functions with controlled
derivatives

Armin Rainer  Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-
Platz 1, A-1090 Wien, Austria<EMAIL_ADDRESS>
###### Abstract.

We show that differentiable functions, defined on a convex body
$K\subseteq\mathbb{R}^{d}$, whose derivatives are controlled by a suitable
given sequence of positive real numbers share many properties with
polynomials. The role of the degree of a polynomial is hereby played by an
integer associated with the given sequence of reals, the diameter of $K$, and
a real parameter linked to the $C^{0}$-norm of the function. We give
quantitative information on the size of the zero set, show that it admits a
local parameterization by Sobolev functions, and prove an inequality of Remez-
type. From the latter, we deduce several consequences, for instance, a bound
on the volume of sublevel sets and a comparison of $L^{p}$-norms reversing
Hölder’s inequality. The validity of many of the results only depends on the
derivatives up to some finite order; the order can be specified in terms of
the given data.

###### Key words and phrases:

Differentiable functions, controlled derivatives, quasianalyticity, zero sets,
level sets, sublevel sets, Remez inequality, reverse Hölder inequality, Markov
inequality, Bernstein inequality

###### 2020 Mathematics Subject Classification:

26D07, 26E10, 41A17, 41A63, 58C25

Supported by FWF-Project P 32905-N

##  1\. Introduction

Yomdin showed in [24] that the zero set $Z_{f}$ of a non-zero
$\mathcal{C}^{\infty}$-function $f:B\to\mathbb{R}$ defined on the unit ball
$B\subseteq\mathbb{R}^{d}$ behaves in many respects as the zero set of a
polynomial, if the partial derivatives of some order $j\geq 2$ are
sufficiently small on $B$. To be precise, let
$\|f\|_{j,B}:=\sum_{|\alpha|=j}\frac{j!}{\alpha!}\|f^{(\alpha)}\|_{B}$ and
$\|f\|_{B}:=\|f\|_{0,B}=\sup_{x\in B}|f(x)|$. If, for some $j\geq 2$,

(1) $\|f\|_{j,B}\leq\frac{1}{2^{j+1}}\|f\|_{0,B},$

then $Z_{f}$ has the following properties:

  1. (a)

There is some ball $B^{\prime}\subseteq B$ such that for each affine line
$\ell$ in $\mathbb{R}^{d}$ that meets $B^{\prime}$ the restriction $f|_{\ell}$
has at most $j-1$ zeros (counted with multiplicities).

  2. (b)

$Z_{f}$ is contained in a countable union of compact
$\mathcal{C}^{\infty}$-hypersurfaces.

  3. (c)

The $(d-1)$-dimensional Hausdorff measure of $Z_{f}$ satisfies

$\mathcal{H}^{d-1}(Z_{f})\leq C(d,j),$

where $C(d,j)$ is a positive constant depending only on $d$ and $j$.

An important fact (which is needed to get (c)) is that the radius of
$B^{\prime}$ depends only on $j$.

This result is in drastic contrast to the classical result that any closed
subset of $\mathbb{R}^{d}$ is the zero set of some
$\mathcal{C}^{\infty}$-function. In fact, by (a), a non-zero function
satisfying (1) cannot have points of infinite flatness.

In this paper, we will show that the zero sets of smooth functions with a
different type of constraints have similar quantitative tame properties.
Instead of assuming that the Fréchet derivative of _some_ order is “small”, as
in (1), we ask that the growth of the sequence of derivatives of _all_ orders
is controlled.

In a spirit similar to [24], Yomdin proved in [25] a Remez-type inequality for
smooth functions involving a “remainder term” expressible through bounds on
the derivatives. We will prove a Remez-type inequality (without “remainder
term”) and deduce several consequences for smooth functions with controlled
derivatives.

###  1.1. Functions with controlled derivatives

Let a positive increasing sequence $(\mu_{j})_{j\geq 1}$ and a positive real
number $M_{0}>0$ be given and set $M_{j}:=M_{0}\mu_{1}\mu_{2}\cdots\mu_{j}$
for all $j\geq 1$. Let $K\subseteq\mathbb{R}^{d}$ be a $d$-dimensional convex
body (i.e., compact with non-empty interior $K^{\circ}$). We will study
$\mathcal{C}^{\infty}$-functions $f:K\to\mathbb{R}$ such that

(2) $\|f\|_{j,K}\leq M_{j},\quad j\in\mathbb{N}.$

Now it is well-known that the qualitative behavior of functions satisfying (2)
— let us call them _$(M_{j})_{j}$ -smooth functions_ — fundamentally depends
on the convergence or divergence of the series $\sum_{j}\frac{1}{\mu_{j}}$
(the _quasianalyticity threshold_). The series $\sum_{j}\frac{1}{\mu_{j}}$ and
its partial sums play a central role in our quantitative analysis. In fact,
this analysis is based on a quantity
$\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(b)$ which depends on the sequence
$(\delta_{K}\mu_{j})_{j}$ and an additional parameter $b>0$. Here
$\delta_{K}:=\operatorname{diam}(K)$ is the diameter of $K$. Roughly speaking,
$\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(b)$ is the greatest integer $n$
(possibly infinite) such that

$\sum_{j=j_{0}(b)+1}^{n}\frac{1}{\mu_{j}}<\delta_{K}e,$

where the integer $j_{0}(b)$ is explicitly computed from $b$; see Section 2
for precise definitions. We call $\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(b)$
the _$(\delta_{K}\mu_{j})_{j}$ -degree_, since it has similar properties as
the degree of a polynomial. Note that it is closely related to the _Bang
degree_ introduced by Nazarov, Sodin, and Volberg [15].

The $(\delta_{K}\mu_{j})_{j}$-degree
$\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(b)$ is finite if $\sum_{j\geq
j_{0}(b)+1}\frac{1}{\mu_{j}}$ exceeds $\delta_{K}e$ which is always the case
provided that the series $\sum_{j}\frac{1}{\mu_{j}}$ diverges. But also if
$\sum_{j}\frac{1}{\mu_{j}}$ converges to a large enough sum,
$\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(b)$ is finite and contains useful
information on functions satisfying (2). In fact, this may occur if
$\delta_{K}$ is relatively small.

###  1.2. The zero set of functions with controlled derivatives

We will see that the zero set $Z_{f}$ of $(M_{j})_{j}$-smooth functions
$f:K\to\mathbb{R}$ has properties similar to (a), (b), and (c) whenever the
$(\delta_{K}\mu_{j})_{j}$-degree
$\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(\tfrac{b}{2M_{0}})$ is finite. The
parameter $b>0$ hereby acts as a lower bound on the $C^{0}$-norm of $f$, that
is, we require that $\|f\|_{K}\geq b$.

Let us summarize our results in the following statements (A)–(D):

  1. (A)

There is a ball $B$ contained in the interior $K^{\circ}$ of $K$ whose radius
depends only on $K$ and the ratio $\frac{b}{M_{1}}$ such that for each affine
line $\ell$ in $\mathbb{R}^{d}$ that meets $B$ the restriction $f|_{\ell}$ has
at most $2\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(\tfrac{b}{2M_{0}})$ zeros
(counted with multiplicities). (Theorem 4.2)

The key to this result is a bound for the number of zeros of
$(M_{j})_{j}$-smooth univariate functions which goes back to Bang [2]; see
also [15] for an exposition of Bang’s ideas. We revisit Bang’s result in some
detail in Section 3.

As a consequence of Malgrange’s preparation theorem and Yomdin’s observation
[24, Lemma 6] we obtain:

  1. (B)

$Z_{f}$ is contained in a countable union of compact
$\mathcal{C}^{\infty}$-hypersurfaces. (Theorem 4.6)

Then, using a Crofton-type argument, (A) and (B) allow to conclude:

  1. (C)

The $(d-1)$-dimensional Hausdorff measure of $Z_{f}$ satisfies

$\mathcal{H}^{d-1}(Z_{f})\leq
C\,\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(\tfrac{b}{2M_{0}})\delta_{K}^{d-1},$

where the constant $C>0$ depends only on $d$ and the ratio
$\frac{\delta_{B}}{\delta_{K}}$; $B$ is the ball from (A). (Theorem 4.6)

Combining Malgrange’s preparation theorem with results of Parusiński and
Rainer [17], we find that $Z_{f}$ locally admits a Sobolev parameterization:

  1. (D)

There is a finite cover of $K$ by rectangular boxes $U$ with corresponding
orthogonal coordinates $(x_{1},x_{2},\ldots,x_{d})=(x^{\prime},x_{d})$ such
that $Z_{f}\cap U$ is contained in the graphs
$\\{x_{d}=\xi_{i}(x^{\prime})\\}$ of at most
$2\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(\tfrac{b}{2M_{0}})$ continuous
functions $\xi_{i}$ of Sobolev class $W^{1,p}$, for all $1\leq p<p_{0}$, where

$p_{0}:=\frac{2\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(\tfrac{b}{2M_{0}})}{2\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(\tfrac{b}{2M_{0}})-1}.$

(Theorem 4.8)

That means that the partial derivatives $\partial_{k}\xi_{i}$ of first order
of $\xi_{i}$ exist almost everywhere, agree with the weak partial derivatives,
and $\xi_{i}$ as well as $\partial_{k}\xi_{i}$ are in the Lebesgue space
$L^{p}$.

As a by-product, our reasoning shows that (D) remains true for
$\mathcal{C}^{\infty}$-functions satisfying (1) for some $j\geq 2$ with the
role of $2\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(\tfrac{b}{2M_{0}})$ replaced
by $j-1$.

We discuss uniformity of these results at the end of Section 4.

###  1.3. A Remez-type inequality for functions with controlled derivatives

The classical Remez inequality [21] states that for a Lebesgue measurable
subset $E$ of $[0,1]$ with Lebesgue measure $|E|>0$ we have

$\|p\|_{[0,1]}\leq T_{n}\Big{(}\frac{2-|E|}{|E|}\Big{)}\,\|p\|_{E}$

for all polynomials $p\in\mathbb{R}[x]$ with degree $\deg p\leq n$. Here
$T_{n}(x)=\cos(n\arccos x)$ is the $n$-th Chebyshev polynomial. In higher
dimensions, there is the generalization due to Yu. Brudnyi and Ganzburg [8]:
for a convex body $K\subseteq\mathbb{R}^{d}$ and a measurable subset
$E\subseteq K$ with $|E|>0$ we have the sharp inequality

$\|p\|_{K}\leq
T_{n}\Big{(}\frac{1+(1-\frac{|E|}{|K|})^{1/d}}{1-(1-\frac{|E|}{|K|})^{1/d}}\Big{)}\,\|p\|_{E}$

for all polynomials $p\in\mathbb{R}[x_{1},\ldots,x_{d}]$ with degree $\deg
p\leq n$. It implies

$\|p\|_{K}\leq\Big{(}\frac{4d\,|K|}{|E|}\Big{)}^{n}\,\|p\|_{E}.$

A version of the latter inequality for analytic functions was obtained by A.
Brudnyi [7], where the role of $n$ is played by the _analytic degree_ ; cf.
Section 6.2. As already mentioned there is a version for smooth function due
to [25] which however contains a “remainder term”.

In this paper, we are interested in Remez-type inequalities for
$(M_{j})_{j}$-smooth functions $f:K\to\mathbb{R}$ (without “remainder term”).
On the unit interval $K=[0,1]$, such an inequality was obtained in [15,
Theorem B]. We recall and slightly adjust this result in Theorem 5.1. The
crucial difference is that, instead of the Bang degree, we work with the
$(\delta_{K}\mu_{j})_{j}$-degree
$\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(\tfrac{b}{M_{0}})$ which has the
advantage to be, to a certain extent, independent of $f$; we require again
that $\|f\|_{K}\geq b$. (In contrast to above, there is no factor $2$ in the
denominator.)

In this way, we are able to apply the strategy of restriction to
$1$-dimensional sections of Brudnyi and Ganzburg and obtain a Remez-type
inequality for $(M_{j})_{j}$-smooth functions in several variables (Theorem
5.4). Actually, the Remez-type inequality is invariant under the
$\mathbb{R}^{*}$-action on $f$ by multiplication with non-zero constants. This
observation leads to a useful version (Theorem 5.5) without _a priori_
conditions on $\|f\|_{K}$ and with $(\delta_{K}\mu_{j})_{j}$-degree
$\mathfrak{d}_{(\delta_{K}\mu_{j})_{j}}(1)$.

From this version, we deduce (in a standard way, see for instance [6, 7] and
[8]) a number of consequences for functions with controlled derivatives:

  1. (1)

A bound for the volume of sublevel sets. (Corollary 5.6)

  2. (2)

A comparison of $L^{p}$-norms (reversing Hölder’s inequality). (Corollary 5.8)

  3. (3)

A bound for the mean oscillation of $\log|f|$. (Corollary 5.10 and Corollary
5.11)

###  1.4. Complementary results

In Section 6.1 and Section 6.2, we attempt to clarify the relation between the
$(\mu_{j})_{j}$-degree on the one hand and polynomial and analytic degree on
the other hand. The $(\mu_{j})_{j}$-degree is a more general object, but in a
polynomial or analytic setting it is generally larger than the polynomial or
analytic degree, respectively. This is related to the general assumption that
$(\mu_{j})_{j}$ is increasing, which is crucial for many of the results.

As already mentioned, the $(\mu_{j})_{j}$-degree can be finite also in the
_non-quasianalytic_ setting (i.e., when $\sum_{j}\frac{1}{\mu_{j}}$
converges). In fact, we obtain in Corollary 6.3 a quantitative necessary
condition for the existence of $\mathcal{C}^{\infty}$-functions with uniform
bounds, like (2), and compactly supported in a given ball, in terms of the
size of the ball and the $C^{0}$-norm of the function. For completeness, we
sketch a proof of the fact that, if $\sum_{j}\frac{1}{\mu_{j}}<\infty$, each
non-empty closed subset of $\mathbb{R}^{d}$ is the zero set of a
$\mathcal{C}^{\infty}$-function $f$ such that $\|f\|_{j,\mathbb{R}^{d}}\leq
A^{j+1}M_{j}$ for all $j\in\mathbb{N}$ and some $A>0$ (Remark 6.4).

Bang’s univariate result, i.e. Proposition 3.2, not only contains useful
information on the numbers of zeros but also on the locus of points, where
derivatives of higher order vanish. For instance, we get a uniform bound for
the number of critical points on certain affine lines (Proposition 6.5).

###  1.5. Finite determinacy

So far we only considered $\mathcal{C}^{\infty}$-functions with controlled
derivatives of all orders. Actually, many of our results depend only on the
derivatives up to some finite order and remain true even for functions that
are only differentiable up to said finite order. We will make this precise
below, in particular, we will specify the finite order. To account for this
fact, we introduce a customized terminology in Section 2, Section 4.1, and
Section 5.4.

###  1.6. Notation

We often write $|E|$ for the Lebesgue measure $\mathcal{L}^{d}(E)$ of a
measurable set $E\subseteq\mathbb{R}^{d}$. If not stated otherwise,
“measurable” always means “Lebesgue measurable”. The $k$-dimensional Hausdorff
measure of $E$ is denoted by $\mathcal{H}^{k}(E)$.

Let $B_{r}(a):=\\{x\in\mathbb{R}^{d}:|x-a|<r\\}$ and
$\overline{B}_{r}(a):=\\{x\in\mathbb{R}^{d}:|x-a|\leq r\\}$ denote the open
and closed Euclidean ball in $\mathbb{R}^{d}$ centered at $a$ with radius
$r>0$, respectively. If $a$ is the origin, we simply write $B_{r}:=B_{r}(0)$
and $\overline{B}_{r}:=\overline{B}_{r}(0)$.

Throughout the paper, we write $\delta_{K}:=\operatorname{diam}(K)$ for the
diameter of a set $K\subseteq\mathbb{R}^{d}$. We denote by $\overline{K}$,
$K^{\circ}$, and $\partial K$ the closure, the interior, and the boundary of
$K$ in $\mathbb{R}^{d}$, respectively.

The integral part of a real number $x$ is denoted by $\lfloor
x\rfloor:=\max\\{n\in\mathbb{Z}:n\leq x\\}$. Similarly, $\lceil
x\rceil:=\min\\{n\in\mathbb{Z}:n\geq x\\}$, but most of the time we will use
$\lceil x\rceil_{\mathbb{N}}:=\min\\{n\in\mathbb{N}:n\geq x\\}$, where
$\mathbb{N}:=\mathbb{Z}_{\geq 0}:=\\{n\in\mathbb{Z}:n\geq 0\\}$. Similarly, we
use $\mathbb{N}_{\geq m}:=\\{n\in\mathbb{N}:n\geq m\\}$,
$\mathbb{R}_{>a}:=\\{x\in\mathbb{R}:x>a\\}$, and variations thereof.

##  2\. $M^{|N}$-smooth functions and $\mu^{|N}$-degree

###  2.1. Admissible weights

Let $(\mu_{j})_{j\geq 1}$ be an infinite increasing sequence of elements in
$\mathbb{R}_{>0}\cup\\{\infty\\}$, i.e., $0<\mu_{1}\leq\mu_{2}\leq\cdots$. We
allow that sequence elements attain the value $\infty$. By the requirement
that the sequence is increasing, if $\mu_{j_{0}}=\infty$, then
$\mu_{j}=\infty$ for all $j\geq j_{0}$. We say that such a sequence
$(\mu_{k})_{k\geq 1}$ is an _admissible weight_. It will be convenient to keep
track of the index (if any), where the sequence ceases to be finite. So we
write

$\displaystyle\mu^{|N}$ $\displaystyle:=(\mu_{j})_{j\geq
1}=(\mu_{1},\ldots,\mu_{N},\infty,\infty,\ldots)\quad\text{ if
}\mu_{N}<\infty=\mu_{N+1}\text{ for }N\geq 1,$ $\displaystyle\mu^{|0}$
$\displaystyle:=(\mu_{j})_{j\geq 1}=(\infty,\infty,\ldots)\quad\text{ if
}\mu_{j}=\infty\text{ for all }j\geq 1,$ $\displaystyle\mu^{|\infty}$
$\displaystyle:=(\mu_{j})_{j\geq 1}\quad\text{ if }\mu_{j}<\infty\text{ for
all }j\geq 1.$

From now on, we let $N$ be an element of $\mathbb{N}_{\geq 1}\cup\\{\infty\\}$
so that the notation $\mu^{|N}$ also includes the case $\mu^{|\infty}$. (We
exclude the case $N=0$, where the weight is worthless.) Let us adopt the usual
conventions for the arithmetic with $\infty$: $\infty\pm 1=\infty$,
$\frac{1}{\infty}=0$, and $r\cdot\infty=\infty$ if
$r\in\mathbb{R}_{>0}\cup\\{\infty\\}$.

If $\mu^{|N}$ is an admissible weight and $M_{0}>0$ is any positive real
number, then we call the pair $(\mu^{|N},M_{0})$ a _full admissible weight_.
With the full admissible weight $(\mu^{|N},M_{0})$ we associate a sequence
$M^{|N}=(M_{j})_{j\geq 0}$ by setting

$M_{j}:=M_{0}\mu_{1}\mu_{2}\cdots\mu_{j},\quad j\geq 1.$

Then $M_{j}=\infty$ if $N<\infty$ and $j>N$. That $\mu^{|N}$ is increasing
amounts to the property $M^{2}_{j}\leq M_{j-1}M_{j+1}$ for $j\geq 1$. Given a
sequence $M^{|N}=(M_{j})_{j\geq 0}$ with this property, we may recover the
pair $(\mu^{|N},M_{0})$ in a unique way by

$\mu_{j}:=\frac{M_{j}}{M_{j-1}}\quad\text{ for }1\leq j\leq N\quad\text{ and
}\quad\mu_{j}:=\infty\quad\text{ for }j>N,$

if $N<\infty$, and simply by $\mu_{j}:=\frac{M_{j}}{M_{j-1}}$ for all $j\geq
1$ otherwise. Thus there is a one-to-one correspondence between
$(\mu^{|N},M_{0})$ and $M^{|N}$. We also call $M^{|N}$ a _full admissible
weight_.

Notice that, given a full admissible weight $(\mu^{|N},M_{0})$ and positive
constants $r,C>0$, the pair $(r\mu^{|N},CM_{0})=((r\mu_{j})_{j\geq 1},CM_{0})$
is again a full admissible weight; it corresponds to $(Cr^{j}M_{j})_{j\geq
0}$.

###  2.2. $M^{|N}$-smooth functions

Let $M^{|N}$ be a full admissible weight. Let $K\subseteq\mathbb{R}^{d}$ be a
convex body and $f:K\to\mathbb{R}$ a function. We say that $f$ is _$M^{|N}$
-smooth_ (or _$(\mu^{|N},M_{0})$ -smooth_) if $f$ is of class
$\mathcal{C}^{N}$ (in an open neighborhood of $K$) and

(3) $\|f\|_{j,K}\leq M_{j},\quad 0\leq j<N+1,$

where we set

$\|f\|_{j,K}:=\sum_{|\alpha|=j}\frac{j!}{\alpha!}\|f^{(\alpha)}\|_{K}$

and $\|f\|_{K}=\|f\|_{0,K}=\sup_{x\in K}|f(x)|$ denotes the sup-norm. If
$N=\infty$, this means that $f$ is of class $\mathcal{C}^{\infty}$ and
$\|f\|_{j,K}\leq M_{j}$ for all $j\in\mathbb{N}$.

###  2.3. The $\mu^{|N}$-degree

Let $\mu^{|N}$ be an admissible weight. We consider the function
$\Sigma_{\mu^{|N}}:\mathbb{N}_{\geq 1}\times\mathbb{N}_{\geq 1}\to[0,\infty)$
defined by

$\Sigma_{\mu^{|N}}(m,n):=\sum_{j=m}^{n}\frac{1}{\mu_{j}}$

and extend the definition to $(\mathbb{N}_{\geq
1}\cup\\{\infty\\})\times(\mathbb{N}\cup\\{\infty\\})$ by setting

$\displaystyle\Sigma_{\mu^{|N}}(m,n):=\begin{cases}0&\text{ if }1\leq
m\leq\infty,~{}n=0,\\\ \sum_{j=m}^{\infty}\frac{1}{\mu_{j}}&\text{ if }1\leq
m<\infty,~{}n=\infty,\\\ 0&\text{ if }m=\infty,~{}1\leq
n\leq\infty.\end{cases}$

Note that, if $m>n$, then $\Sigma_{\mu^{|N}}(m,n)$ is an empty sum and hence
has the value $0$. For $n\geq N$, we have
$\Sigma_{\mu^{|N}}(m,n)=\Sigma_{\mu^{|N}}(m,N)$ (since $\frac{1}{\infty}=0$).
The map $(m,n)\mapsto\Sigma_{\mu^{|N}}(m,n)$ is increasing in $n$ and
decreasing in $m$ (even strictly in the range, where $\mu_{k}$ is finite).

For $b>0$ let $j_{0}(b)$ be the smallest integer $j\in\mathbb{N}$ with
$j\geq\log b^{-1}$, i.e.,

$j_{0}(b):=\lceil\log b^{-1}\rceil_{\mathbb{N}}.$

We define the _$\mu^{|N}$ -degree_
$\mathfrak{d}_{\mu^{|N}}(b)\in\mathbb{N}\cup\\{\infty\\}$ by setting

$\displaystyle\mathfrak{d}_{\mu^{|N}}(b):=\sup\big{\\{}n\in\mathbb{N}:\Sigma_{\mu^{|N}}(j_{0}(b)+1,n)<e\big{\\}}.$

We remark that $\mathfrak{d}_{\mu^{|N}}(b)\geq j_{0}(b)$ and
$\mathfrak{d}_{\mu^{|N}}(b)=j_{0}(b)$ occurs precisely if
$\mu_{j_{0}(b)+1}\leq 1/e$. Clearly, $\mathfrak{d}_{\mu^{|N}}(b)=\infty$ if
$\Sigma_{\mu^{|N}}(j_{0}(b)+1,n)<e$ for all $n$. The map
$\mathbb{R}_{>0}\times\mathbb{R}_{>0}\ni(a,b)\mapsto\mathfrak{d}_{a\mu^{|N}}(b)$
is increasing in $a$ and decreasing in $b$.

We could assign a $\mu^{|N}$-degree to a $(\mu^{|N},M_{0})$-smooth function
$f:[0,1]\to\mathbb{R}$ by putting

$\mu^{|N}\text{-degree of
}f:=\mathfrak{d}_{\mu^{|N}}(\tfrac{\|f\|_{[0,1]}}{M_{0}});$

like the _Bang degree_ in [15]. But the unspecified argument $b$ allows for
more flexibility; it will however always be related to the ratio of the
$\mathcal{C}^{0}$-norm of $f$ by $M_{0}$.

If the domain $K$ of $f$ is a $d$-dimensional convex body, then the diameter
$\delta_{K}$ of the domain $K$ (not the dimension!) will affect the
$\mu^{|N}$-degree as a multiplicative factor of the sequence $\mu^{|N}$, i.e.,

$\mathfrak{d}_{\delta_{K}\mu^{|N}}(b).$

We will see below that this follows easily from a scaling argument.

##  3\. Zeros of univariate functions

In this section, we revisit some results of Bang [2] (see also [15]). Since we
have to rephrase them in our terminology, we give detailed proofs.

###  3.1. Bang’s metric theory revisited

Let $M^{|N}=(M_{j})_{j\geq 0}$ be a full admissible weight. Let
$I\subseteq\mathbb{R}$ be a non-trivial compact interval and
$f:I\to\mathbb{R}$ an $M^{|N}$-smooth function.

Following Bang [2], we associate with $f$ and $M^{|N}$ a sequence
$(b_{n})_{n\geq 0}$ of functions defined by

(4) $b_{n}(t)=b_{f,M^{|N},n}(t):=\sup_{j\geq
n}\frac{|f^{(j)}(t)|}{e^{j}M_{j}},\quad t\in I,$

with the interpretation that $\frac{|f^{(j)}(t)|}{e^{j}M_{j}}=0$ if
$M_{j}=\infty$ (even if $f^{(j)}(t)$ is not defined). In other words, if $N$
is finite, then $b_{n}(t)=\sup_{n\leq j\leq N}\frac{|f^{(j)}(t)|}{e^{j}M_{j}}$
for $n\leq N$ and $b_{n}\equiv 0$ for $n>N$.

######  Lemma 3.1.

The sequence $(b_{n})_{n\geq 0}$ has the following properties.

  1. (i)

$e^{-n}\geq b_{n}$ for all $n\geq 0$.

  2. (ii)

$b_{n-1}\geq b_{n}$ and $f^{(n-1)}(t_{0})=0$ implies
$b_{n-1}(t_{0})=b_{n}(t_{0})$ for all $n\geq 1$.

  3. (iii)

For all $k>n$ and all distinct $t,s\in I$,

(5) $b_{n}(s)<\max\\{b_{n}(t),e^{-k}\\}\,e^{e|t-s|\mu_{k}},$

where the right-hand side is interpreted as $\infty$ if $\mu_{k}=\infty$. In
particular, all $b_{n}$ are continuous.

###### Proof.

(i) and (ii) are obvious (by (3)).

Now (iii) is trivial if $N$ is finite and $k>N$ which means that
$\mu_{k}=\infty$. So we may assume that $k<N+1$ and that $\mu_{k}$ and $M_{k}$
are finite. Let $n\leq j<k$ and $t,s\in I$. Then, by Taylor’s formula, for
some $\xi$ between $t$ and $s$,

$\displaystyle\frac{|f^{(j)}(s)|}{e^{j}M_{j}}$
$\displaystyle\leq\sum_{i=0}^{k-j-1}\frac{|f^{(j+i)}(t)|\,|t-s|^{i}}{e^{j}M_{j}\,i!}+\frac{|f^{(k)}(\xi)|\,|t-s|^{k-j}}{e^{j}M_{j}\,(k-j)!}$
$\displaystyle=\sum_{i=0}^{k-j-1}\frac{M_{j+i}}{M_{j}}\frac{|f^{(j+i)}(t)|}{e^{j+i}M_{j+i}}\frac{(e|t-s|)^{i}}{i!}+e^{-k}\frac{M_{k}}{M_{j}}\frac{|f^{(k)}(\xi)|}{M_{k}}\frac{(e|t-s|)^{k-j}}{(k-j)!}$
$\displaystyle\leq
b_{n}(t)\sum_{i=0}^{k-j-1}\mu_{k}^{i}\frac{(e|t-s|)^{i}}{i!}+e^{-k}\mu_{k}^{k-j}\frac{(e|t-s|)^{k-j}}{(k-j)!}$
$\displaystyle<\max\\{b_{n}(t),e^{-k}\\}\,e^{e|t-s|\mu_{k}},$

where we used that $(\mu_{j})_{j}$ is increasing. If $j\geq k$, then trivially

$\displaystyle\frac{|f^{(j)}(s)|}{e^{j}M_{j}}\leq
e^{-j}<\max\\{b_{n}(t),e^{-k}\\}\,e^{e|t-s|\mu_{k}}.$

This implies (5). The continuity of $b_{n}$ follows from (5) if $n<N$.
Moreover, for finite $N$, $b_{N}(t)=\frac{|f^{(N)}(t)|}{e^{N}M_{N}}$ and
$b_{n}\equiv 0$, for $n>N$, are visibly continuous. ∎

The following proposition is due to [2]; we implement several modifications.

######  Proposition 3.2.

Let $M^{|N}=(M_{j})_{j\geq 0}$ be a full admissible weight. Let
$I\subseteq\mathbb{R}$ be a non-trivial compact interval and
$f:I\to\mathbb{R}$ an $M^{|N}$-smooth function. Let $m\in\mathbb{N}$ be such
that $m+1\leq N$. Assume that for all $0\leq j\leq m$ there is $x_{j}\in I$
such that $f^{(j)}(x_{j})=0$. Let $x_{-1}$ be an arbitrary point in $I$. Then,

(6)
$\sum_{j=0}^{m}|x_{j-1}-x_{j}|\geq\frac{1}{e}\Sigma_{\mu^{|N}}(j_{0}+1,m+1),$

where

(7) $j_{0}=j_{0}(b_{f,M^{|N},0}(x_{-1}))=\lceil\log
b_{f,M^{|N},0}(x_{-1})^{-1}\rceil_{\mathbb{N}}.$

If $j_{0}\leq m$, then the inequality (6) is strict.

######  Remark 3.3.

A few remarks are in order.

(1) Note that $j_{0}<\infty$ if and only if there exists $0\leq j\leq N$ with
$f^{(j)}(x_{-1})\neq 0$.

(2) The right-hand side of (6) is zero if $j_{0}>m$. In that case, the
statement is trivial. For instance, if $N=\infty$, we do here not exclude the
case that $f$ has points of infinite flatness. If $x_{-1}$ is such a point,
then $j_{0}=\infty$ and hence (6) is trivially true. In that case, the
assumptions of the proposition are satisfied for the choice $x_{j}:=x_{-1}$,
$j=0,\ldots,m$, entailing that also the left-hand side of (6) is zero.

###### Proof of Proposition 3.2.

Let $(b_{n})_{n\geq 0}$ be the decreasing sequence of continuous functions
(4); cf. Lemma 3.1. We will construct a new continuous function $\beta$ by
tracing through the graphs of the $b_{n}$ (for $0\leq n\leq m$) and switching
from $b_{n}$ to $b_{n+1}$ at $x_{n}$. For $0\leq k\leq m$, set
$\tau_{k}:=\sum_{j=0}^{k}|x_{j-1}-x_{j}|$ and $\tau_{-1}:=0$. For
$t\in[\tau_{n-1},\tau_{n}]$, where $0\leq n\leq m$, define

$\beta_{n}(t):=\begin{cases}b_{n}(x_{n-1}+\tau_{n-1}-t)&\text{if
}x_{n}<x_{n-1},\\\ b_{n}(x_{n-1}-\tau_{n-1}+t)&\text{if }x_{n}\geq
x_{n-1}.\end{cases}$

Then each $\beta_{n}$ is continuous and
$\beta_{n}(\tau_{n})=b_{n}(x_{n})=b_{n+1}(x_{n})=\beta_{n+1}(\tau_{n})$; by
Lemma 3.1. Thus,

$\beta(t):=\beta_{n}(t)\quad\text{ if }t\in[\tau_{n-1},\tau_{n}],~{}0\leq
n\leq m,$

defines a continuous function on $[0,\tau_{m}]$. By Lemma 3.1, we have
$\beta(t)\leq e^{-n}$ for all $t\geq\tau_{n-1}$ as well as

(8) $\beta(\tau_{m})=\beta_{m}(\tau_{m})=b_{m}(x_{m})=b_{m+1}(x_{m})\leq
e^{-m-1}.$

On the other hand, with $j_{0}$ as defined in (7),

(9) $\beta(0)=\beta_{0}(\tau_{-1})=b_{0}(x_{-1})\geq e^{-j_{0}}.$

It might be that $e^{-j_{0}}\leq e^{-m-1}$ (including the case
$j_{0}=\infty$). Then the right-hand side of (6) is zero so that (6) is
trivially true. Thus we may assume that $j_{0}\leq m$. In view of (8) and (9),
the range of $\beta$ then contains all numbers $e^{-j}$ for $j_{0}\leq j\leq
m+1$. So we find a strictly increasing sequence $t_{j}$, for $j_{0}\leq j\leq
m+1$, such that $\beta(t_{j})=e^{-j}$ and $\beta(t)>e^{-j}$ if $t<t_{j}$
(starting in the point $(0,\beta(0))$ let $t_{j}$ be the first time that the
graph of $\beta$ meets the horizontal line with ordinate $e^{-j}$). Then

(10) $\beta(t_{j-1})<\beta(t_{j})\,e^{e(t_{j}-t_{j-1})\mu_{j}},\quad
j_{0}+1\leq j\leq m+1.$

To see this, we apply (iii) of Lemma 3.1 to each interval in the subdivision
of $(t_{j-1},t_{j})$ induced by the points $\tau_{n}$ between $t_{j-1}$ and
$t_{j}$, and notice that, since $t_{j}\leq\tau_{j-1}$ (as $\beta(t)\leq
e^{-j}$ if $t\geq\tau_{j-1}$), we have $n<j$ for all such $n$ and
$\max\\{b_{n}(t),e^{-j}\\}=b_{n}(t)$ for all $t\in(t_{j-1},t_{j})$.

In view of $\beta(t_{j})=e^{-j}$, (10) amounts to

$t_{j}-t_{j-1}>\frac{1}{e\mu_{j}},\quad j_{0}+1\leq j\leq m+1.$

Summing over $j$, we find

$t_{m+1}\geq
t_{m+1}-t_{j_{0}}>\frac{1}{e}\sum_{k=j_{0}+1}^{m+1}\frac{1}{\mu_{j}}.$

Since $\tau_{m}\geq t_{m+1}$, this yields (6). It also shows that the
inequality is strict provided that $j_{0}\leq m$. ∎

###  3.2. Bounds for the number of zeros

Now it is easy to deduce a lower bound for the length of the interval and an
upper bound for the number of zeros.

######  Corollary 3.4.

Let $M^{|N}=(M_{j})_{j\geq 0}$ be a full admissible weight. Let
$I\subseteq\mathbb{R}$ be a non-trivial compact interval and
$f:I\to\mathbb{R}$ an $M^{|N}$-smooth function. Let $z_{1}\leq
z_{2}\leq\cdots\leq z_{m}$ be an increasing enumeration of some of the zeros
of $f$, where $m\leq N$, and let $x_{-1}\in I\setminus(z_{1},z_{m})$ be
arbitrary. Then we have a lower bound for the length $|I|$ of $I$,

(11) $|I|>\frac{1}{e}\Sigma_{\mu^{|N}}(j_{0}+1,m),\qquad(j_{0}=\lceil\log
b_{f,M^{|N},0}(x_{-1})^{-1}\rceil_{\mathbb{N}}),$

and an upper bound for the number of zeros,

(12) $m\leq\mathfrak{d}_{|I|\mu^{|N}}(b_{f,M^{|N},0}(x_{-1})).$

###### Proof.

Suppose that $x_{-1}\leq z_{1}$; if $x_{-1}\geq z_{m}$ the proof is similar.
By Rolle’s theorem, there is a sequence of points $x_{-1}\leq x_{0}=z_{1}\leq
x_{1}\leq\cdots\leq x_{m-1}$ in $I$ such that $f^{(j)}(x_{j})=0$ for all
$0\leq j\leq m-1$. By (6),

$|I|\geq\sum_{j=0}^{m-1}(x_{j}-x_{j-1})\geq\frac{1}{e}\Sigma_{\mu^{|N}}(j_{0}+1,m),$

with strict inequality if $j_{0}\leq m-1$. If $j_{0}\geq m$, then
$\Sigma_{\mu^{|N}}(j_{0}+1,m)=0$ and also in that case the inequality is
strict, since $I$ is assumed to be non-trivial. So we proved (11). By the
definition of $\mathfrak{d}_{\mu^{|N}}$, also (12) follows. ∎

######  Remark 3.5.

Variations of the argument yield further useful information. For instance, if
$f(x_{-1})\neq 0$ and $x_{0}$ is an $m$-fold zero of $f$, then

$|x_{-1}-x_{0}|\geq\frac{1}{e}\Sigma_{\mu^{|N}}(j_{0}+1,m)\geq\frac{1}{e}\Sigma_{\mu^{|N}}(\lceil\log\tfrac{M_{0}}{|f(x_{-1})|}\rceil_{\mathbb{N}}+1,m),$

since in this case $\sum_{j=0}^{m-1}|x_{j}-x_{j-1}|=|x_{-1}-x_{0}|$ and
$b_{f,M^{|N},0}(x_{-1})\geq\tfrac{|f(x_{-1})|}{M_{0}}$.

In Corollary 3.4, the list of zeros $z_{j}$ may not comprise all zeros of $f$.
If $z_{j}$ is in the list and $z_{j}$ is a multiple zero of $f$, we do not
even require that all multiplicities of $z_{j}$ appear in the list.

Under an additional assumption, we find that the total number of zeros is
finite and get an upper bound for it.

######  Corollary 3.6.

Let $M^{|\infty}=(M_{j})_{j\geq 0}$ be a full admissible weight. Let
$I\subseteq\mathbb{R}$ be a non-trivial compact interval and
$f:I\to\mathbb{R}$ an $M^{|\infty}$-smooth function. Let $x_{-1}\in I$. If

(13) $\Sigma_{\mu^{|\infty}}(j_{0}+1,\infty)>|I|e,\qquad(j_{0}=\lceil\log
b_{f,M^{|\infty},0}(x_{-1})^{-1}\rceil_{\mathbb{N}}),$

then the total number $m$ of zeros of $f$ in $I$ counted with multiplicities
is finite. In that case,
$\mathfrak{d}_{|I|\mu^{|\infty}}(b_{f,M^{|\infty},0}(x_{-1}))<\infty$ and

(14) $m\leq 2\mathfrak{d}_{|I|\mu^{|\infty}}(b_{f,M^{|\infty},0}(x_{-1})).$

For each $x_{-1}\in I$ satisfying (13) and not lying strictly between the
smallest and the largest zero of $f$, we even have

(15) $m\leq\mathfrak{d}_{|I|\mu^{|\infty}}(b_{f,M^{|\infty},0}(x_{-1})).$

If $\mathfrak{d}_{|I|\mu^{|\infty}}(b_{f,M^{|\infty},0}(x_{-1}))=0$, then $f$
has no zeros in $I$.

###### Proof.

By (13), it is obvious that
$\mathfrak{d}_{|I|\mu^{|\infty}}(b_{f,M^{|\infty},0}(x_{-1}))<\infty$. Suppose
for contradiction that there are infinitely many zeros of $f$. Then we find an
increasing infinite sequence of zeros on the right of $x_{-1}$ or a decreasing
infinite sequence of zeros on the left of $x_{-1}$ (or on both sides). But
that contradicts Corollary 3.4. The upper bounds for positive $m$ follow again
from Corollary 3.4; if there are $m_{l}$ zeros left and $m_{r}$ zeros right of
$x_{-1}$, then $m=m_{l}+m_{r}\leq
2\mathfrak{d}_{|I|\mu^{|\infty}}(b_{f,M^{|\infty},0}(x_{-1}))$. In particular,
if $f$ has zeros in $I$, then
$\mathfrak{d}_{|I|\mu^{|\infty}}(b_{f,M^{|\infty},0}(x_{-1}))\neq 0$. ∎

######  Remark 3.7.

If in Corollary 3.6 we assume that $f$ is an $M^{|N}$-smooth function with
finite $N$, then in general $f$ could have more zeros than $N$ so that
Corollary 3.4 is not applicable.

On the other hand, in the setting of Corollary 3.6 we conclude that the number
of zeros of $f$ left and right of $x_{-1}$ is bounded by

(16) $N:=\mathfrak{d}_{|I|\mu^{|\infty}}(b_{f,M^{|\infty},0}(x_{-1})).$

So _a posteriori_ we obtain that total number of zeros $m$ of $f$ in $I$
satisfies

(17) $m\leq 2\mathfrak{d}_{|I|\mu^{|N}}(b_{f,M^{|N},0}(x_{-1})),$

where $N$ is given by (16) and $\mu^{|N}$, $M^{|N}$ are obtained from
$\mu^{|\infty}$, $M^{|\infty}$ simply by setting all elements with index $\geq
N+1$ equal to $\infty$. In particular, the bound on the number of zeros in
(17) depends only on the derivatives up order $N$ of $f$.

###  3.3. Quasianalyticity

Let $\mu^{|\infty}=(\mu_{j})_{j\geq 0}$ be an admissible weight. We see that,
if

(18) $\sum_{j}\frac{1}{\mu_{j}}=\infty$

and $f$ is not identically zero, then condition (13) is always satisfied
(provided that $j_{0}$ is finite) and Corollary 3.6 applies. An admissible
weight $\mu^{|\infty}$ satisfying (18) is said to be _quasianalytic_ ; if (18)
is not fulfilled we say that $\mu^{|\infty}$ is _non-quasianalytic_
(analogously for full admissible weights $M^{|\infty}$). If we speak of
(non-)quasianalytic admissible weights, we always presuppose that $N=\infty$.

In fact, let $I\subseteq\mathbb{R}$ be a non-trivial compact interval and let
$\mathcal{C}^{M^{|\infty}}(I)$ denote the set of all functions
$f:I\to\mathbb{R}$ such that there exists a constant $\rho>0$ such that
$\|f^{(j)}\|_{I}\leq\rho^{j+1}M_{j}$ for all $j\in\mathbb{N}$. Then
$\mathcal{C}^{M^{|\infty}}(I)$ is called _quasianalytic_ if, for all
$f\in\mathcal{C}^{M^{|\infty}}(I)$ and $x_{0}\in I$, triviality of the Taylor
series $\widehat{f}_{x_{0}}$ of $f$ at $x_{0}$, i.e., $\widehat{f}_{x_{0}}=0$,
implies that $f$ is identically zero. By the Denjoy–Carleman theorem (see e.g.
[19]), $\mathcal{C}^{M^{|\infty}}(I)$ is quasianalytic if and only if
$M^{|\infty}$ is quasianalytic (that is, (18) holds).

Note that Corollary 3.6 implies one direction of this equivalence. For, assume
that $f\in\mathcal{C}^{M^{|\infty}}(I)$, $x_{0}\in I$, and
$\widehat{f}_{x_{0}}=0$. Then there exists $\rho>0$ such that $f$ is
$(\rho\mu^{|\infty},\rho M_{0})$-smooth. If $f$ is not identically zero, we
find $x_{-1}\in I$ with $f(x_{-1})\neq 0$ and thus $j_{0}=\lceil\log
b_{f,(\rho^{j+1}M_{j})_{j},0}(x_{-1})^{-1}\rceil_{\mathbb{N}}$ is finite.
Since $\widehat{f}_{x_{0}}=0$ and thus the number of zeros of $f$ is infinite,
Corollary 3.6 implies that (13) must be violated. Since $j_{0}$ is finite, we
may infer that (18) is violated, that is, $M^{|\infty}$ is non-quasianalytic.

##  4\. The zero set of multivariate functions

Let $K\subseteq\mathbb{R}^{d}$ be a convex body. We will derive quantitative
information on the zero set of $M^{|\infty}$-smooth functions
$f:K\to\mathbb{R}$ in terms of the $\delta_{K}\mu^{|\infty}$-degree; recall
that $\delta_{K}$ is the diameter of $K$. It turns out that these results
actually depend only on a finite number of derivatives. To account for this we
introduce the following bit of notation.

###  4.1. Finite determinacy

Let $M^{|\infty}=(M_{j})_{j\geq 0}$ be a full admissible weight,
$K\subseteq\mathbb{R}^{d}$ a convex body, and $b>0$. Assume that

(19)
$\Sigma_{\mu^{|\infty}}(j_{0}+1,\infty)>\delta_{K}e,\qquad(j_{0}=\lceil\log(\tfrac{b}{2M_{0}})^{-1}\rceil_{\mathbb{N}}).$

(Note that (19) is in any case satisfied if $M^{|\infty}$ is a quasianalytic
full admissible weight. The factor $2$ in the definition of $j_{0}$ could be
replaced by any real number $>1$ without changing the validity of the
results.) Then

(20) $N:=\mathfrak{d}_{\delta_{K}\mu^{|\infty}}(\tfrac{b}{2M_{0}})+1$

is a positive (finite) integer. Let $\mu^{|N}$ (resp. $M^{|N}$) be the (resp.
full) admissible weight obtained from $\mu^{|\infty}$ (resp. $M^{|\infty}$) by
setting all elements with index $\geq N+1$ equal to $\infty$. Then
$\Sigma_{\mu^{|\infty}}(j_{0}+1,n)=\Sigma_{\mu^{|N}}(j_{0}+1,n)$ for $n\leq N$
so that

(21)
$\mathfrak{d}_{\delta_{K}\mu^{|\infty}}(\tfrac{b}{2M_{0}})=\mathfrak{d}_{\delta_{K}\mu^{|N}}(\tfrac{b}{2M_{0}})=N-1.$

Furthermore,

(22)
$\Sigma_{\mu^{|\infty}}(j_{0}+1,\infty)>\Sigma_{\mu^{|N}}(j_{0}+1,\infty)=\Sigma_{\mu^{|N}}(j_{0}+1,N)\geq\delta_{K}e$

which follows easily from the definitions.

######  Definition 4.1 (Admissible data).

We call the triple $(M^{|\infty},K,b)$, where $M^{|\infty}=(M_{j})_{j\geq 0}$
is a full admissible weight, $K\subseteq\mathbb{R}^{d}$ a convex body, and
$b>0$, _admissible data_ if (19) holds. In that case, $N$ defined by (20) is
called the _integer associated with the data $(M^{|\infty},K,b)$_. In this
setting, $\mu^{|N}$ (resp. $M^{|N}$) will always denote the (resp. full)
admissible weight resulting from $\mu^{|\infty}$ (resp. $M^{|\infty}$) by
setting all elements with index $\geq N+1$ equal to $\infty$.

###  4.2. Number of zeros on affine lines

Recall that $K^{\circ}$ denotes the interior of $K$.

######  Theorem 4.2.

Let $(M^{|\infty},K,b)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $M^{|N}$-smooth function such that $\|f\|_{K}\geq
b$. Then there is a ball $B\subseteq K^{\circ}$, whose radius only depends on
$K$ and the ratio $\frac{b}{M_{1}}$, such that for each affine line $\ell$ in
$\mathbb{R}^{d}$ that meets $B$ the restriction $f|_{\ell}$ has at most
$2(N-1)$ zeros.

###### Proof.

Consider $U:=\\{x\in K:|f(x)|>\frac{b}{2}\\}$. Since $\|f\|_{K}\geq b$, there
is $a\in U$ with $|f(a)|\geq b$. For all $x\in B_{s}(a)\cap K$ with
$s:=\frac{b}{3M_{1}}$, we have

$|f(x)-f(a)|\leq\|f\|_{1,K}\,|x-a|\leq M_{1}s=\tfrac{b}{3},$

so that

$|f(x)|\geq|f(a)|-|f(x)-f(a)|\geq\tfrac{2b}{3}.$

It follows that $a\in B_{s}(a)\cap K\subseteq U$. But the intersection
$B_{s}(a)\cap K^{\circ}$ contains an open ball $B$, whose radius depends only
on $s$ and on the “thickness” of $K$ near $a$.

Let $\ell\subseteq\mathbb{R}^{d}$ be any affine line that meets the ball $B$.
Let $x_{0}$ and $x_{1}$ be the intersection points of $\ell$ with the boundary
of $K$. Then $g:[0,1]\to\mathbb{R}$ given by $g(t):=f(x_{0}+t(x_{1}-x_{0}))$
defines a $\mathcal{C}^{N}$-function satisfying

$\|g^{(j)}\|_{[0,1]}\leq|x_{1}-x_{0}|^{j}\|f\|_{j,K}\leq\delta_{K}^{j}M_{j},\quad
0\leq j\leq N.$

Thus, $g:[0,1]\to\mathbb{R}$ is $(\delta_{K}\mu^{|N},M_{0})$-smooth. For all
$t\in[0,1]$ with $x_{0}+t(x_{1}-x_{0})\in B$, we find

$b_{g,(\delta_{K}^{j}M_{j})_{j},0}(t)=\sup_{j\geq
0}\frac{|g^{(j)}(t)|}{(\delta_{K}e)^{j}M_{j}}\geq\frac{|g(t)|}{M_{0}}>\frac{b}{2M_{0}},$

whence
$\mathfrak{d}_{\delta_{K}\mu^{|N}}(b_{g,(\delta_{K}^{j}M_{j})_{j},0}(t))\leq\mathfrak{d}_{\delta_{K}\mu^{|N}}(\tfrac{b}{2M_{0}})=N-1$
(see (21)). Fix such a $t$. Suppose that $g$ has (at least) $N$ zeros left or
right of $t$. Then Corollary 3.4 implies

$N\leq\mathfrak{d}_{\delta_{K}\mu^{|N}}(b_{g,(\delta_{K}^{j}M_{j})_{j},0}(t))\leq
N-1,$

a contradiction. Thus the total number of zeros of $g$, and hence of
$f|_{\ell}$, is finite and bounded by $2(N-1)$. ∎

######  Corollary 4.3.

In the setting of Theorem 4.2, if $K=\overline{B}_{r}$, then $\delta_{K}=2r$
and the radius of $B$ is at least $\frac{1}{2}\min\\{\frac{b}{3M_{1}},r\\}$.

###### Proof.

The intersection $B_{s}(a)\cap B_{r}$ contains an open ball $B$ of radius at
least $\frac{1}{2}\min\\{s,r\\}$. ∎

######  Remark 4.4.

In analogy to Remark 3.5, we can give at each point in $K$ a lower bound for
the distance to the closest zero of $f$ of a specific multiplicity; often the
bound is trivial, but not always:

Let $M^{|N}$ be a full admissible weight and $K\subseteq\mathbb{R}^{d}$ a
convex body. Let $f:K\to\mathbb{R}$ be an $M^{|N}$-smooth function and
$x_{-1}\in K\setminus Z_{f}$. Then $f$ has no zeros of multiplicity at least
$m$, where $m\leq N$, in the ball $B_{\epsilon_{m}}(x_{-1})$ with center
$x_{-1}$ and radius

$\epsilon_{m}:=\frac{1}{e}\Sigma_{\delta_{K}\mu^{|N}}(j_{0}+1,m),\qquad(j_{0}=\lceil\log\tfrac{M_{0}}{|f(x_{-1})|}\rceil_{\mathbb{N}}).$

This follows from Remark 3.5 applied to all affine lines through $x_{-1}$.

In particular, $f$ has no zeros in $B_{\epsilon_{1}}(x_{-1})$. But
$\epsilon_{1}\neq 0$ only if $j_{0}=0$ which is equivalent to
$|f(x_{-1})|=M_{0}$; in that case $\epsilon_{1}=\frac{1}{\delta_{K}e\mu_{1}}$.
Furthermore, $f$ has no zeros that are also critical points in
$B_{\epsilon_{2}}(x_{-1})$, where

$\epsilon_{2}=\begin{cases}\frac{1}{\delta_{K}e}(\frac{1}{\mu_{1}}+\frac{1}{\mu_{2}}),&\text{
if }|f(x_{-1})|=M_{0},\\\ \frac{1}{\delta_{K}e\mu_{2}},&\text{ if
}\frac{M_{0}}{e}\leq|f(x_{-1})|<M_{0},\\\ 0,&\text{ otherwise}.\end{cases}$

For later use, we remark that the zero set of a function $f$ as in Theorem 4.2
has zero Lebesgue measure. This will be a trivial consequence of Theorem 4.6,
if $f$ is additionally assumed to be of class $\mathcal{C}^{\infty}$, but we
will need it without this assumption.

######  Corollary 4.5.

In the setting of Theorem 4.2, $\mathcal{L}^{d}(Z_{f})=0$.

###### Proof.

First of all, $f$ is continuous so that $Z_{f}$ is closed, hence measurable.
The union of all affine lines of a fixed direction meeting $B$ form an open
cylinder $U$. By compactness, $K$ is covered by finitely many such cylinders
$U$. Now $Z_{f}\cap U$ is measurable, whence we can apply Fubini’s theorem to
its characteristic function. By Theorem 4.2, this gives
$\mathcal{L}^{d}(Z_{f}\cap U)=0$. ∎

###  4.3. Hausdorff measure of the zero set

######  Theorem 4.6.

Let $(M^{|\infty},K,b)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $M^{|N}$-smooth $\mathcal{C}^{\infty}$-function such
that $\|f\|_{K}\geq b$. Then the zero set $Z_{f}$ is contained in a countable
union of compact $\mathcal{C}^{\infty}$-hypersurfaces and its
$(d-1)$-dimensional Hausdorff measure satisfies

(23) $\mathcal{H}^{d-1}(Z_{f})\leq C\,(N-1)\,\delta_{K}^{d-1},$

where the constant $C>0$ depends only on $d$ and on the ratio
$\tfrac{\delta_{B}}{\delta_{K}}$ and where $B$ is the ball from Theorem 4.2.
In the case $K=\overline{B}_{r}$, the constant $C$ depends only on $d$ and
$\min\\{\frac{b}{6rM_{1}},\frac{1}{2}\\}$; it blows up as $b\to 0$ or
$r\to\infty$.

###### Proof.

The first statement follows from Malgrange’s preparation theorem and [24,
Lemma 6]. Thus, we may apply the Crofton-type result [24, Lemma 7] to conclude
(23) from Theorem 4.2.

If $K=\overline{B}_{r}$, then we may take
$\delta_{B}=\min\\{\frac{b}{3M_{1}},r\\}$, by Corollary 4.3, so that
$\tfrac{\delta_{B}}{\delta_{K}}=\min\\{\frac{b}{6rM_{1}},\frac{1}{2}\\}$. This
also yields the asymptotics of the constant $C$. ∎

######  Remark 4.7.

In Theorem 4.6 (and in the next result), we assume that $f$ is of class
$\mathcal{C}^{\infty}$ in order to apply Malgrange’s preparation theorem. That
$f:K\to\mathbb{R}$ is an $M^{|N}$-smooth $\mathcal{C}^{\infty}$-function means
that $f$ is of class $\mathcal{C}^{\infty}$ and satisfies $\|f\|_{j,K}\leq
M_{j}$ for all $0\leq j\leq N$.

###  4.4. Sobolev parameterization of the zero set

Let $(M^{|\infty},K,b)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $M^{|N}$-smooth $\mathcal{C}^{\infty}$-function such
that $\|f\|_{K}\geq b$.

The collection $\mathcal{L}$ of all affine lines in $\mathbb{R}^{d}$ that meet
the open ball $B$ from Theorem 4.2 covers $\mathbb{R}^{d}$, i.e.,
$\mathbb{R}^{d}=\bigcup_{\ell\in\mathcal{L}}\ell$. Fix $z_{0}\in Z_{f}$ and
any line $\ell\in\mathcal{L}$ with $z_{0}\in\ell$. By Theorem 4.2, the
multiplicity $n$ of $z_{0}$ as a zero of $f|_{\ell}$ is at most $2(N-1)$. Let
$x_{1},\ldots,x_{d}$ be an orthogonal coordinate system such that $\ell$
coincides with the $x_{d}$-axis. By Malgrange’s preparation theorem, there is
a rectangular neighborhood $U$ of $z_{0}$ of the form
$U:=\\{x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}:\alpha_{j}<x_{j}<\beta_{j}\text{
for all }1\leq j\leq d\\}$ such that $f(x)=p(x)u(x)$ for $x\in U$, where $u$
does not vanish on $U$ and

(24)
$p(x^{\prime},x_{d})=x_{d}^{n}+a_{1}(x^{\prime})x_{d}^{n-1}+\cdots+a_{d}(x^{\prime})$

is a polynomial in $x_{d}$ with $\mathcal{C}^{\infty}$-coefficients $a_{j}$
which are defined on the projection $U^{\prime}$ of $U$ onto the first $d-1$
coordinates $x^{\prime}:=(x_{1},\ldots,x_{d-1})$. Thus,

$Z_{f}\cap U=Z_{p}\cap U.$

Let $[\zeta_{1}(x^{\prime}),\ldots,\zeta_{n}(x^{\prime})]$ be the unordered
$n$-tuple of complex roots (with multiplicities) of the polynomial
$p(x^{\prime},x_{d})$ in $x_{d}$ for any $x^{\prime}\in U^{\prime}$. Note that
the number of real roots among the $\zeta_{j}(x^{\prime})$ may change with
varying $x^{\prime}$. Let $(\xi_{1}(x^{\prime}),\ldots,\xi_{n}(x^{\prime}))$
be the $n$-tuple consisting of the real parts of the $\zeta_{j}(x^{\prime})$,
ordered increasingly such that
$\xi_{1}(x^{\prime})\leq\xi_{2}(x^{\prime})\leq\cdots\leq\xi_{n}(x^{\prime})$
for all $x^{\prime}$. Since all the real roots among the
$\zeta_{j}(x^{\prime})$ clearly are also among the $\xi_{i}(x^{\prime})$ (with
correct multiplicities), it follows that

$\displaystyle Z_{f}\cap U$ $\displaystyle=Z_{p}\cap U$
$\displaystyle=\\{(x^{\prime},x_{d})\in U:\text{ there is }1\leq j\leq n\text{
with }x_{d}=\zeta_{j}(x^{\prime})\\}$
$\displaystyle\subseteq\\{(x^{\prime},x_{d})\in U:\text{ there is }1\leq i\leq
n\text{ with
}x_{d}=\xi_{i}(x^{\prime})\\}\subseteq\bigcup_{i=1}^{n}\Gamma(\xi_{i}),$

where $\Gamma(\xi_{i}):=\\{(x^{\prime},\xi_{i}(x^{\prime})):x^{\prime}\in
U^{\prime}\\}$ denotes the graph of $\xi_{i}:U^{\prime}\to\mathbb{R}$. The
increasing order of the $\xi_{i}$ implies that each $\xi_{i}$ is a continuous
function $U^{\prime}\to\mathbb{R}$ and so, applying [17, Remark 9] and [18,
Theorem A.1] (see also [16]), we may conclude that each $\xi_{i}$ is of
Sobolev class $\xi_{i}\in W^{1,p}(U^{\prime})$, for all $1\leq
p<\frac{n}{n-1}$, and $\|\xi_{i}\|_{W^{1,p}(U^{\prime})}$ depends uniformly on
$f$; see Remark 4.10. In view of [11, Theorem 1.2], we obtain that each graph
$\Gamma(\xi_{i})$ is countably $\mathcal{H}^{d-1}$-rectifiable and

(25) $\mathcal{H}^{d-1}(\\{(x^{\prime},\xi_{i}(x^{\prime})):x^{\prime}\in
E\\})=\int_{E}\sqrt{1+|\nabla\xi_{i}(x^{\prime})|^{2}}\,dx^{\prime},$

for all measurable subsets $E\subseteq U^{\prime}$. The right-hand side of
(25) depends uniformly on $f$ (see Remark 4.10).

Now it is easy to conclude

######  Theorem 4.8.

Let $(M^{|\infty},K,b)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $M^{|N}$-smooth $\mathcal{C}^{\infty}$-function such
that $\|f\|_{K}\geq b$. There is a finite cover of $K$ by rectangular boxes
$U$ with corresponding orthogonal coordinates $(x^{\prime},x_{d})$ such that
$Z_{f}\cap U$ is contained in the graphs $\\{x_{d}=\xi_{i}(x^{\prime})\\}$ of
at most $2(N-1)$ continuous functions $\xi_{i}$ of Sobolev class $W^{1,p}$,
for all $1\leq p<\frac{2N-2}{2N-3}$, whose $W^{1,p}$-norm depends uniformly on
$f$ (see Remark 4.10). Furthermore, (25) holds.

Note that the Sobolev regularity in the results of [17, 18] used above is
optimal.

######  Remark 4.9.

It evident from the above arguments that the statement of Theorem 4.8 remains
true for $\mathcal{C}^{\infty}$-functions $f:K\to\mathbb{R}$ that satisfy

$\|f\|_{j,K}\leq\frac{1}{2\delta_{K}^{j}}\|f\|_{0,K},\quad\text{ for some
}j\geq 2,$

where $2(N-1)$ is replaced by $j-1$. Use [24, Theorem 3(ii)].

######  Remark 4.10.

In general, the coefficients $a_{j}$ of the polynomial (24) will no longer be
$M^{|N}$-smooth (cf. [1]). But the choice of the functions $a_{j}$ (and $u$)
can be made to depend linearly and continuously (with respect to the Whitney
$\mathcal{C}^{\infty}$ topology) on $f$ (cf. [14]). Furthermore, we have
uniform bounds for the $W^{1,p}$-norm of the functions $\xi_{i}$ in terms of
the $\mathcal{C}^{2(N-1)}$-norm of the $a_{j}$ (cf. [17] and [18]).

###  4.5. Further remarks

Let $(M^{|\infty},K,b)$ be admissible data and $N$ the associated integer.

######  Remark 4.11.

In the above results on $Z_{f}$, the number $N$ is independent of $f$. E.g.,
the bound on $\mathcal{H}^{d-1}(Z_{f})$ in (23) remains unchanged as along as
$f$ satisfies the assumptions of Theorem 4.6.

Let $f:K\to\mathbb{R}$ be an $M^{|N}$-smooth function with $\|f\|_{K}\geq b$.
If we set $V(f):=\sup_{x,y\in K}|f(x)-f(y)|$, then $V(f)\leq 2\|f\|_{K}$ and
hence

$\|f-c\|_{K}\geq\frac{V(f-c)}{2}=\frac{V(f)}{2}\quad\text{ for all
}c\in\mathbb{R}.$

On the other hand,

$\|f-c\|_{K}\leq M_{0}\quad\text{ if }|c|\leq M_{0}-\|f\|_{K}.$

So, if $2b\leq V(f)$, we get the same uniform results for the level sets
$Z_{f-c}=f^{-1}(c)$ for all $|c|\leq M_{0}-\|f\|_{K}$.

######  Remark 4.12.

If we assume that $f:K\to\mathbb{R}$ is
$(\delta_{K}^{-1}\mu^{|\infty},M_{0})$-smooth with $\|f\|_{K}\geq b$ and
accordingly

$\Sigma_{\mu^{|\infty}}(j_{0}+1,\infty)>e,\qquad(j_{0}=\lceil\log(\tfrac{b}{2M_{0}})^{-1}\rceil_{\mathbb{N}}),$

instead of (19), then we get all the above results with both the
$\mu^{|\infty}$-degree $\mathfrak{d}_{\mu^{|\infty}}(\frac{b}{2M_{0}})$ and
the constant $C$ in (23) independent of $\delta_{K}$.

##  5\. Remez inequality for functions with controlled derivatives

In this section, we prove a Remez-type inequality for $M^{|N}$-smooth (and
$\mu^{|N}$-smooth) functions in several variables and derive several
consequences. Our results are based on a univariate version due to [15] which
we recall in slightly modified form in Theorem 5.1.

###  5.1. Definitions and conventions

Let $\mu^{|\infty}=(\mu_{j})_{j\geq 1}$ be an admissible weight. Let us assume
that there is an increasing continuous function
$\tilde{\mu}:[1,\infty)\to(0,\infty)$ that is (piecewise) $\mathcal{C}^{1}$
such that

$\mu_{k}=\tilde{\mu}(k),\quad k\geq 1.$

This is no real restriction, since we may always take $\tilde{\mu}$ piecewise
affine and work consistently with the left derivative at points, where
$\tilde{\mu}$ is not differentiable.

Once we have $\tilde{\mu}$, we define (following [15])

$\gamma_{\tilde{\mu}}(n):=\sup_{1\leq s\leq
n}\frac{s\tilde{\mu}^{\prime}(s)}{\tilde{\mu}(s)}\quad\text{ and
}\quad\Gamma_{\tilde{\mu}}(n):=4e^{4+\gamma_{\tilde{\mu}}(n)},$

for all positive integers $n$. Note that $\gamma_{\tilde{\mu}}$ and
$\Gamma_{\tilde{\mu}}$ depend on the choice of $\tilde{\mu}$ which is not
unique.

In this section, we will again make use of the terminology introduced in
Section 4.1 (see Definition 4.1). But here it is better to replace $b$ by
$2b$. So let _$(M^{|\infty},K,2b)$ be admissible data and $N$ the associated
integer_. Recall that this means the following: $M^{|\infty}$ is a full
admissible weight, $K\subseteq\mathbb{R}^{d}$ a convex body, and $b>0$ such
that

(26)
$\Sigma_{\mu^{|\infty}}(j_{0}+1,\infty)>\delta_{K}e,\qquad(j_{0}=\lceil\log(\tfrac{b}{M_{0}})^{-1}\rceil_{\mathbb{N}}).$

The associated positive integer $N$ is given by

$N:=\mathfrak{d}_{\delta_{K}\mu^{|\infty}}(\tfrac{b}{M_{0}})+1.$

It satisfies (21) and (22) with $b$ replaced by $2b$, in particular,

(27)
$\mathfrak{d}_{\delta_{K}\mu^{|\infty}}(\tfrac{b}{M_{0}})=\mathfrak{d}_{\delta_{K}\mu^{|N}}(\tfrac{b}{M_{0}})=N-1.$

In all occurrences of the Remez inequality for $M^{|N}$-smooth functions, the
constant $\Gamma_{\tilde{\mu}}(2(N-1))^{2(N-1)}$ will appear. If
$\mathfrak{d}_{\delta_{K}\mu^{|\infty}}(\frac{b}{M_{0}})=N-1=0$ it is
undefined. For ease of notation, we set

(28) $C_{N}:=\Gamma_{\tilde{\mu}}(2(N-1))$

with the interpretation $C_{N}^{2(N-1)}=:\infty$ if $N-1=0$.

###  5.2. Remez inequality for univariate functions

We recall and slightly modify [15, Theorem B]. The statement in [15] involves
the so-called Bang degree $\mathfrak{n}_{f}$ (which depends on $f$) and it is
formulated for a quasianalytic full admissible weight $M^{|\infty}$ with
$M_{0}=1$. Here we work with
$\mathfrak{d}_{\mu^{|\infty}}(\frac{b}{M_{0}})=\mathfrak{d}_{\mu^{|N}}(\frac{b}{M_{0}})$
(which is independent of $f$) instead of $\mathfrak{n}_{f}$.

######  Theorem 5.1.

Let $(M^{|\infty},[0,1],2b)$ be admissible data and $N$ the associated
integer. Let $f:[0,1]\to\mathbb{R}$ be any $M^{|2(N-1)}$-smooth function such
that $\|f\|_{[0,1]}\geq b$. Then for any interval $I\subseteq[0,1]$ and any
Lebesgue measurable set $E\subseteq I$ we have

(29) $\|f\|_{I}\leq\Big{(}\frac{C_{N}\,|I|}{|E|}\Big{)}^{2(N-1)}\|f\|_{E}.$

######  Remark 5.2.

If $\mathfrak{d}_{\mu^{|\infty}}(\frac{b}{M_{0}})=N-1=0$, then, by the
interpretation $C_{N}^{2(N-1)}=\infty$, (29) is trivially true.

###### Proof of Theorem 5.1.

First of all, we may assume that $M_{0}=1$, by dividing $f$ by $M_{0}$. In
fact, $\bar{f}:=\frac{1}{M_{0}}f$ is $(\mu^{|2(N-1)},1)$-smooth and
$\|\bar{f}\|_{[0,1]}\geq\frac{b}{M_{0}}=:\bar{b}>0$ so that (26), (29), and
the integer $N$ remain unchanged.

By Remark 5.2, we may assume that $N-1>0$ and follow the proof of [15]; it
involves only derivatives up to order
$2\mathfrak{d}_{\mu^{|\infty}}(b)=2\mathfrak{d}_{\mu^{|N}}(b)=2(N-1)$ (cf.
(27) with $K=[0,1]$ and $M_{0}=1$).

In the only place, where the definition of $\mathfrak{n}_{f}$ actually plays a
role in the argument, we use the following estimate (cf. [15, p. 72]):

(30) $\min_{t\in[0,1]}b_{0}(t)=\min_{t\in[0,1]}\sup_{j\geq
0}\frac{|f^{(j)}(t)|}{e^{j}M_{j}}>e^{-N}.$

Let us justify (30). Suppose it is not true, i.e.,
$\min_{t\in[0,1]}b_{0}(t)\leq e^{-N}$. On the other hand,
$\max_{t\in[0,1]}b_{0}(t)\geq b\geq e^{-j_{0}}$, by the definition of $j_{0}$
(recall that $M_{0}=1$). We have $j_{0}<\mathfrak{d}_{\mu^{|N}}(b)+1=N$ (cf.
Section 2.3), that is $e^{-N}<e^{-j_{0}}$. Since $b_{0}$ is continuous (cf.
Lemma 3.1), there is a monotonic sequence $x_{j}\in[0,1]$ such that
$b_{0}(x_{j})=e^{-j}$ for all $j_{0}\leq j\leq N$ (cf. the proof of
Proposition 3.2). By Lemma 3.1, we have $|x_{j}-x_{j-1}|>\frac{1}{e\mu_{j}}$
for $j_{0}+1\leq j\leq N$ so that

$1\geq\sum_{j=j_{0}+1}^{N}|x_{j}-x_{j-1}|>\frac{1}{e}\sum_{j=j_{0}+1}^{N}\frac{1}{\mu_{j}},$

and thus $N-1=\mathfrak{d}_{\mu^{|\infty}}(\frac{b}{M_{0}})\geq N$, a
contradiction. Hence (30) is proved. ∎

######  Remark 5.3.

As pointed out in [15], if $\overline{\gamma}_{\tilde{\mu}}:=\sup_{s\geq
1}\frac{s\tilde{\mu}^{\prime}(s)}{\tilde{\mu}(s)}<\infty$ (as is the case for
$\tilde{\mu}(s)=s$ or $\tilde{\mu}(s)=s(\log(s+e))^{\delta}$ if $0<\delta\leq
1$), then in (29) the constant $C_{N}$ can be replaced by
$\overline{\Gamma}_{\tilde{\mu}}=4e^{4+\overline{\gamma}_{\tilde{\mu}}}$,
provided that $N-1\neq 0$ (see Remark 5.2).

###  5.3. Remez inequality for multivariate functions

Next we prove a Remez-type inequality for multivariate functions with
controlled derivatives. The proof is inspired by the technique of [8].

######  Theorem 5.4.

Let $(M^{|\infty},K,2b)$ be admissible data and $N$ the associated integer.
Let $f:K\to\mathbb{R}$ be any $M^{|2(N-1)}$-smooth function and $L\subseteq K$
any convex body such that $\|f\|_{L}\geq b$. If $E\subseteq L$ is a Lebesgue
measurable subset with $|E|>0$, then

$\|f\|_{L}\leq\Big{(}\frac{C_{N}\,|L|^{1/d}}{|L|^{1/d}-(|L|-|E|)^{1/d}}\Big{)}^{2(N-1)}\|f\|_{E}.$

###### Proof.

Let $\mathcal{F}(E,L)=\mathcal{F}_{M^{|2(N-1)}}(E,L,b)$ denote the set of all
$M^{|2(N-1)}$-smooth functions $f:K\to\mathbb{R}$ satisfying $\|f\|_{L}\geq b$
and $\|f\|_{E}\leq 1$. For $\lambda>0$ consider

$\mathcal{R}(\lambda,L)=\mathcal{R}_{M^{|2(N-1)}}(\lambda,L,b):=\sup_{\begin{subarray}{c}E\subseteq
L\\\ |E|\geq\lambda\end{subarray}}\sup_{f\in\mathcal{F}(E,L)}\|f\|_{L}.$

We claim that

(31)
$\mathcal{R}(\lambda,L)\leq\Big{(}\frac{C_{N}\,|L|^{1/d}}{|L|^{1/d}-(|L|-\lambda)^{1/d}}\Big{)}^{2(N-1)}.$

Fix a measurable subset $E\subseteq L$ with $|E|\geq\lambda$ and
$f\in\mathcal{F}(E,L)$. There is $x_{0}\in L$ with $\|f\|_{L}=|f(x_{0})|$. Let
$\ell$ be any half-line emanating from $x_{0}$ and let $x_{1}$ be the (other)
intersection point of $\ell$ with $\partial L$. Define $g:[0,1]\to\mathbb{R}$
by $g(t):=f(x_{0}+t(x_{1}-x_{0}))$. As seen in the proof of Theorem 4.2,
$g:[0,1]\to\mathbb{R}$ is $(\delta_{K}\mu^{|2(N-1)},M_{0})$-smooth and
$\|g\|_{[0,1]}\geq|g(0)|=|f(x_{0})|\geq b$. Thus (in view of (26) and since
$\|f\|_{E}\leq 1$) Theorem 5.1 implies

$\|f\|_{L}=|f(x_{0})|=|g(0)|\leq\Big{(}\frac{C_{N}\,\mathcal{L}^{1}(L\cap\ell)}{\mathcal{L}^{1}(E\cap\ell)}\Big{)}^{2(N-1)},$

because of the scaling properties of the $1$-dimensional Lebesgue measure
$\mathcal{L}^{1}$ (induced on $\ell$); note that
$\gamma_{\delta_{K}\tilde{\mu}}=\gamma_{\tilde{\mu}}$ and hence
$\Gamma_{\delta_{K}\tilde{\mu}}=\Gamma_{\tilde{\mu}}$. Taking the essential
infimum over all half-lines emanating from $x_{0}$, the supremum over all
$f\in\mathcal{F}(E,L)$, and the supremum over all measurable $E\subseteq L$
with $|E|\geq\lambda$, we find

$\mathcal{R}(\lambda,L)\leq\Big{(}C_{N}\sup_{\begin{subarray}{c}E\subseteq
L\\\
|E|\geq\lambda\end{subarray}}\operatorname*{ess\,inf}_{\ell}\frac{\mathcal{L}^{1}(L\cap\ell)}{\mathcal{L}^{1}(E\cap\ell)}\Big{)}^{2(N-1)},$

and, using [8, Lemma 3 and Remark 2], we conclude (31).

Now we may prove the statement of the theorem. Let $f$, $L$, and $E$ be as in
the assumptions of the theorem. Since $|E|>0$, we have $\|f\|_{E}>0$, by
Corollary 4.5. Then $F:=\frac{f}{\|f\|_{E}}$ is
$\frac{1}{\|f\|_{E}}M^{|2(N-1)}$-smooth and satisfies
$\|F\|_{L}\geq\frac{b}{\|f\|_{E}}$ and $\|F\|_{E}=1$. Thus
$F\in\mathcal{F}_{\frac{1}{\|f\|_{E}}M^{|2(N-1)}}(E,L,\frac{b}{\|f\|_{E}})$;
note that (26), $N$, and $C_{N}$ remain unchanged. Then (31) implies

$\|F\|_{L}\leq\Big{(}\frac{C_{N}\,|L|^{1/d}}{|L|^{1/d}-(|L|-|E|)^{1/d}}\Big{)}^{2(N-1)}.$

Since $\|F\|_{L}=\frac{\|f\|_{L}}{\|f\|_{E}}$, this completes the proof. ∎

###  5.4. An important consequence

The next result is a simple but important consequence of Theorem 5.4.

For its formulation, it is convenient to adapt our terminology. We say that
the pair _$(\mu^{|\infty},K)$ is admissible data with associated integer $N$_
if $\mu^{|\infty}=(\mu_{j})_{j\geq 1}$ is an admissible weight and
$K\subseteq\mathbb{R}^{d}$ a convex body such that

(32) $\Sigma_{\mu^{|\infty}}(1,\infty)>\delta_{K}e$

and

$N:=\mathfrak{d}_{\delta_{K}\mu^{|\infty}}(1)+1.$

As before, $C_{N}:=\Gamma_{\tilde{\mu}}(2(N-1))$ and $C_{N}^{2(N-1)}=:\infty$
if $N-1=0$.

We will specialize Theorem 5.4 to the case $b=M_{0}=\|f\|_{K}$, that is to
$(\mu^{|2(N-1)},\|f\|_{K})$-smooth functions. Recall that, by the definition
in Section 2.2, $f:K\to\mathbb{R}$ is called
$(\mu^{|2(N-1)},\|f\|_{K})$-smooth if $\|f\|_{K}>0$ and

$\|f\|_{j,K}\leq\|f\|_{K}\cdot\mu_{1}\mu_{2}\cdots\mu_{j},\quad 1\leq j\leq
2(N-1).$

######  Theorem 5.5.

Let $(\mu^{|\infty},K)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $(\mu^{|2(N-1)},\|f\|_{K})$-smooth function. If
$E\subseteq K$ is a Lebesgue measurable subset with $|E|>0$, then

(33)
$\|f\|_{K}\leq\Big{(}\frac{C_{N}\,|K|^{1/d}}{|K|^{1/d}-(|K|-|E|)^{1/d}}\Big{)}^{2(N-1)}\|f\|_{E}.$

###### Proof.

Set $L=K$ and $b=M_{0}:=\|f\|_{K}$ in Theorem 5.4. ∎

There is no _a priori_ condition on $\|f\|_{K}>0$ in Theorem 5.5. Visibly,
(33) is invariant under the action of $\mathbb{R}^{*}$ on $f$.

It can happen that $\mathfrak{d}_{\delta_{K}\mu^{|\infty}}(1)=N-1=0$, in which
case Theorem 5.5 contains no information (cf. Remark 5.2). In fact, this
occurs precisely if $\delta_{K}\mu_{1}\leq 1/e$.

###  5.5. Volume of sublevel sets

Theorem 5.5 has several interesting corollaries. We begin with a bound on the
growth of the volume of sublevel sets.

######  Corollary 5.6.

Let $(\mu^{|\infty},K)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $(\mu^{|2(N-1)},\|f\|_{K})$-smooth function. Then
the sublevel set $S_{t}:=\\{x\in K:|f(x)|\leq t\\}$ satisfies

(34) $|S_{t}|\leq
C_{N}d\,|K|\,\Big{(}\frac{t}{\|f\|_{K}}\Big{)}^{\frac{1}{2(N-1)}},\quad t>0,$

with the understanding that the right-hand side is $\infty$ if $N-1=0$.

###### Proof.

We may assume that $N-1\neq 0$. Apply Theorem 5.5 to $E=S_{t}$; note that
$|S_{t}|\neq 0$, for any $t>0$, since $S_{t}$ must contain a non-empty open
set, by continuity. Then, putting $\theta:=\frac{|S_{t}|}{|K|}$,

$\|f\|_{K}\leq\Big{(}\frac{C_{N}}{1-(1-\theta)^{1/d}}\Big{)}^{2(N-1)}t,$

and consequently,

$\frac{\theta}{d}\leq 1-(1-\theta)^{1/d}\leq
C_{N}\,\Big{(}\frac{t}{\|f\|_{K}}\Big{)}^{\frac{1}{2(N-1)}}.$

The statement follows. ∎

Corollary 5.6 implies useful estimates for the distribution function and the
decreasing rearrangement of $f$ (more generally, of $f|_{E}$ for a measurable
subset $E\subset K$). Recall that the distribution function of
$f:K\to\mathbb{R}$ is defined by

$d_{f}(t):=|\\{x\in K:|f(x)|>t\\}|=|K|-|S_{t}|$

and the decreasing rearrangement of $f$ by

$f^{*}(y):=\inf\\{t>0:d_{f}(t)\leq y\\}.$

######  Corollary 5.7.

Let $(\mu^{|\infty},K)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $(\mu^{|2(N-1)},\|f\|_{K})$-smooth function. Let
$E\subseteq K$ be a Lebesgue measurable subset with $|E|>0$. Then

(35)
$(f|_{E})^{*}(|E|\,\lambda)\geq\|f\|_{K}\Big{(}\frac{|E|}{|K|}\cdot\frac{1-\lambda}{C_{N}d}\Big{)}^{2(N-1)},\quad\lambda\in(0,1),$

where the right-hand side is identically zero if $N-1=0$.

###### Proof.

Assume that $N-1\neq 0$. We have

$d_{f|_{E}}(t):=|\\{x\in E:|f(x)|>t\\}|=|E|-|S_{t}\cap E|=|E|\,\lambda_{t},$

where $\lambda_{t}:=1-|S_{t}\cap E|/|E|$. Let $s_{\lambda}$ denote the right-
hand side of (35). By Corollary 5.6,

(36) $d_{f|_{E}}(t)=|E|\,\lambda_{t}>|E|\,\lambda,\quad\text{ if
}t\in(0,s_{\lambda}).$

Indeed, by (34),

$\displaystyle|E|(1-\lambda_{t})=|S_{t}\cap E|\leq|S_{t}|$ $\displaystyle\leq
C_{N}d\,|K|\,\Big{(}\frac{t}{\|f\|_{K}}\Big{)}^{\frac{1}{2(N-1)}}$
$\displaystyle<C_{N}d\,|K|\,\Big{(}\frac{s_{\lambda}}{\|f\|_{K}}\Big{)}^{\frac{1}{2(N-1)}}=|E|(1-\lambda).$

Now (36) implies $(f|_{E})^{*}(|E|\,\lambda)\geq s_{\lambda}$, and (35) is
proved. ∎

###  5.6. Comparison of $L^{p}$-norms

Let us write

$\displaystyle\|f\|_{L^{p}(E)}^{\sharp}$
$\displaystyle:=\Big{(}\frac{1}{|E|}\int_{E}|f(x)|^{p}\,dx\Big{)}^{1/p},\quad
0<p<\infty,$ $\displaystyle\|f\|_{L^{\infty}(E)}^{\sharp}$
$\displaystyle:=\operatorname*{ess\,sup}_{E}|f|,$

for the normalized $L^{p}$-norms of $f$ on a measurable set $E$ with
$0<|E|<\infty$. Then, as a consequence of Hölder’s inequality,

$\|f\|_{L^{q}(E)}^{\sharp}\leq\|f\|_{L^{p}(E)}^{\sharp},\quad\text{ if
}0<q\leq p\leq\infty.$

For functions with controlled derivatives also suitable opposite inequalities
hold.

######  Corollary 5.8.

Let $(\mu^{|\infty},K)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $(\mu^{|2(N-1)},\|f\|_{K})$-smooth function. Let
$E\subseteq K$ be a Lebesgue measurable subset with $|E|>0$. Then, for all
$0<q<p\leq\infty$,

(37)
$\|f\|_{L^{p}(K)}^{\sharp}\leq\Big{(}\frac{C_{N}d\,|K|}{|E|}\Big{)}^{2(N-1)(1-\frac{q}{p})}(2q\,(N-1)+1)^{\frac{1}{q}-\frac{1}{p}}(\|f\|_{L^{q}(K)}^{\sharp})^{\frac{q}{p}}(\|f\|_{L^{q}(E)}^{\sharp})^{1-\frac{q}{p}}.$

In particular,

(38)
$\|f\|_{L^{p}(K)}^{\sharp}\leq(C_{N}d)^{2(N-1)(1-\frac{q}{p})}(2q\,(N-1)+1)^{\frac{1}{q}-\frac{1}{p}}\|f\|_{L^{q}(K)}^{\sharp}$

and

(39)
$\|f\|_{L^{p}(K)}^{\sharp}\leq\Big{(}\frac{C_{N}d\,|K|}{|E|}\Big{)}^{2(N-1)}\,(2q(N-1)+1)^{\frac{1}{q}}\,\|f\|_{L^{q}(E)}^{\sharp}.$

###### Proof.

We may assume that $N-1\neq 0$; otherwise the statements are trivially true.
Note that both (38) and (39) follow from (37): it suffices to specialize to
$E=K$ and to use $\|f\|_{L^{q}(K)}^{\sharp}\leq\|f\|_{L^{p}(K)}^{\sharp}$,
respectively.

Let us prove (37). We begin with the case $p=\infty$. Using (35), we get

$\displaystyle\frac{1}{|E|}\int_{E}|f(x)|^{q}\,dx$
$\displaystyle=\frac{1}{|E|}\int_{0}^{|E|}\big{(}(f|_{E})^{*}(y)\big{)}^{q}\,dy=\int_{0}^{1}\big{(}(f|_{E})^{*}(|E|\,\lambda)\big{)}^{q}\,d\lambda$
$\displaystyle\geq\Big{(}\frac{|E|}{C_{N}d\,|K|}\Big{)}^{2q(N-1)}\int_{0}^{1}(1-\lambda)^{2q(N-1)}\,d\lambda\,\|f\|_{K}^{q}$
$\displaystyle=\Big{(}\frac{|E|}{C_{N}d\,|K|}\Big{)}^{2q(N-1)}\frac{1}{2q(N-1)+1}\,\|f\|_{K}^{q},$

whence

(40)
$\|f\|_{K}\leq\Big{(}\frac{C_{N}d\,|K|}{|E|}\Big{)}^{2(N-1)}(2q\,(N-1)+1)^{\frac{1}{q}}\|f\|_{L^{q}(E)}^{\sharp}$

which is (37) for $p=\infty$.

If $0<q<p<\infty$, then (40) implies

$\displaystyle\frac{1}{|K|}\int_{K}|f(x)|^{p}\,dx\leq\|f\|_{K}^{p-q}\frac{1}{|K|}\int_{K}|f(x)|^{q}\,dx=\|f\|_{K}^{p-q}\,(\|f\|_{L^{q}(K)}^{\sharp})^{q}$
$\displaystyle\leq\Big{(}\Big{(}\frac{C_{N}d\,|K|}{|E|}\Big{)}^{2(N-1)}(2q\,(N-1)+1)^{\frac{1}{q}}\Big{)}^{p-q}\,(\|f\|_{L^{q}(K)}^{\sharp})^{q}(\|f\|_{L^{q}(E)}^{\sharp})^{p-q}$

from which (37) follows easily. The proof is complete. ∎

######  Remark 5.9.

Bourgain proved in [5] the following inequality for polynomials: Let
$K\subseteq\mathbb{R}^{d}$ be a convex body of volume $1$ and
$p\in\mathbb{R}[x_{1},\ldots,x_{d}]$ a polynomial of degree $n$. Then, for
each $q>0$,

$\|p\|_{L^{q}(K)}\leq C(n,q)\,\|p\|_{L^{1}(K)}.$

More precisely,

$\|p\|_{L^{\psi}(K)}\leq C_{1}\,\|p\|_{L^{1}(K)},$

where $L^{\psi}$ it the Orlicz space with Orlicz function
$\psi(t)=\exp(t^{C_{2}/n})-1$, where the constants $C_{1}$ and $C_{2}$ are
absolute.

These inequalities have been generalized by Brudnyi [7] to analytic functions,
where the role of $n$ is played by the _analytic degree_ ; see Section 6.2.

In contrast to (37), (38), and (39), here the constants do not depend on the
dimension $d$. Following Bourgain’s approach, it should be possible to obtain
versions of (37), (38), and (39) that are independent of the dimension: use
Theorem 5.5 for $d=1$ and $K=[0,1]$ to get a replacement for [5, Lemma 3.1].

###  5.7. A bound for the mean oscillation of $\log|f|$

Let $K\subseteq\mathbb{R}^{d}$ be a convex body. Recall that the _mean
oscillation_ of a locally integrable function $g:K\to\mathbb{R}$ over a ball
$B\subseteq K$ is

$\operatorname{mo}_{B}(g):=\frac{1}{|B|}\int_{B}|g(x)-g_{B}|\,dx,$

where

$g_{B}:=\frac{1}{|B|}\int_{B}g(x)\,dx.$

We have the following bound for the mean oscillation of $\log|f|$ if $f$ has
suitably controlled derivatives.

######  Corollary 5.10.

Let $(\mu^{|\infty},K)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $(\mu^{|2(N-1)},\|f\|_{K})$-smooth function. Then,
for each ball $B\subseteq K$,

(41) $\operatorname{mo}_{B}(\log|f|)\leq
4(N-1)\Big{(}\log\Big{(}\frac{C_{N}d\,|K|}{|B|}\Big{)}+1\Big{)}$

where the right-hand side is interpreted as $\infty$ if $N-1=0$.

###### Proof.

Observe that

$\operatorname{mo}_{B}(\log|f|)\leq\frac{2}{|B|}\int_{B}\Big{|}\log\frac{|f(x)|}{\|f\|_{K}}\Big{|}\,dx.$

Indeed,

$\displaystyle\Big{|}\log|f|-\frac{1}{|B|}\int_{B}\log|f|\,dx\Big{|}$
$\displaystyle\leq\Big{|}\log|f|-\log\|f\|_{K}\Big{|}+\Big{|}\log\|f\|_{K}-\frac{1}{|B|}\int_{B}\log|f|\,dx\Big{|}$
$\displaystyle=\Big{|}\log\frac{|f|}{\|f\|_{K}}\Big{|}+\Big{|}\frac{1}{|B|}\int_{B}\log\|f\|_{K}-\log|f|\,dx\Big{|}$
$\displaystyle\leq\Big{|}\log\frac{|f|}{\|f\|_{K}}\Big{|}+\frac{1}{|B|}\int_{B}\Big{|}\log\frac{|f|}{\|f\|_{K}}\Big{|}\,dx.$

We may assume that $N-1\neq 0$. Now, by (35),

$\displaystyle\frac{1}{|B|}\int_{B}\Big{|}\log\frac{|f(x)|}{\|f\|_{K}}\Big{|}\,dx$
$\displaystyle=\frac{1}{|B|}\int_{0}^{|B|}\Big{|}\log\frac{(f|_{B})^{*}(y)}{\|f\|_{K}}\Big{|}\,dy$
$\displaystyle=\int_{0}^{1}\Big{|}\log\frac{(f|_{B})^{*}(|B|\,\lambda)}{\|f\|_{K}}\Big{|}\,d\lambda$
$\displaystyle\leq
2(N-1)\int_{0}^{1}-\log\Big{(}\frac{|B|}{|K|}\cdot\frac{1-\lambda}{C_{N}d}\Big{)}\,d\lambda$
$\displaystyle=2(N-1)\Big{(}\log\Big{(}\frac{C_{N}d\,|K|}{|B|}\Big{)}+1\Big{)},$

and (41) follows. ∎

Corollary 5.10 has similarity with the log-BMO property of analytic functions
[7, Corollary 1.10]. But from (41) it seems not possible to deduce that
$\log|f|$ has bounded mean oscillation, since the right-hand side tends to
infinity if the radius of $B$ tends to zero. Also using (41) in the case $K=B$
does not help, because then $N-1$ becomes eventually zero if $B$ gets small
enough.

This is related to the fact that we do not have good enough control away from
points in $K$, where $\|f\|_{K}$ is attained. Indeed, if $x\in K$ is such that
$|f(x)|=\|f\|_{K}$, then

$\sup_{\begin{subarray}{c}\text{balls }B\subseteq K\\\ x\in
B\end{subarray}}\operatorname{mo}_{B}(\log|f|)\leq 4(N-1)(\log(C_{N}d)+1)$

as follows from the next corollary.

######  Corollary 5.11.

Let $(\mu^{|\infty},K)$ be admissible data and $N$ the associated integer. Let
$f:K\to\mathbb{R}$ be any $(\mu^{|2(N-1)},\|f\|_{K})$-smooth function. Then,
for each convex body $L\subseteq K$ such that $\|f\|_{L}=\|f\|_{K}$,

(42) $\operatorname{mo}_{L}(\log|f|)\leq 4(N-1)(\log(C_{N}d)+1)$

where the right-hand side is interpreted as $\infty$ if $N-1=0$.

###### Proof.

If $L\subseteq K$ is a convex body such that $\|f\|_{L}=\|f\|_{K}$ and $E$ is
a measurable subset of $L$ with $|E|>0$, then we get

$\|f\|_{L}\leq\Big{(}\frac{C_{N}\,|L|^{1/d}}{|L|^{1/d}-(|L|-|E|)^{1/d}}\Big{)}^{2(N-1)}\|f\|_{E}.$

from Theorem 5.4 (in analogy to Theorem 5.5). For $E:=|\\{x\in L:|f(x)|\leq
t\\}|$ we may conclude

$|E|\leq
C_{N}d\,|L|\,\Big{(}\frac{t}{\|f\|_{L}}\Big{)}^{\frac{1}{2(N-1)}},\quad t>0,$

(in analogy to Corollary 5.6) and thus

$(f|_{L})^{*}(|L|\,\lambda)\geq\|f\|_{L}\Big{(}\frac{1-\lambda}{C_{N}d}\Big{)}^{2(N-1)},\quad\lambda\in(0,1),$

(in analogy to Corollary 5.7). Using this estimate in the computations of the
proof of Corollary 5.10, yields (42). ∎

##  6\. Complementary results

In this section, we compare the $\mu^{|\infty}$-degree to the polynomial and
the analytic degree, respectively. Moreover, we discuss the existence of non-
quasianalytic bump functions and show how the technique from Section 3 can be
used to extract information on critical points.

###  6.1. The $\mu^{|\infty}$-degree vs. the polynomial degree

What is the relation between the $\mu^{|\infty}$-degree and the usual degree
of a polynomial?

Since most of our results are based on restriction to one-dimensional
sections, let us assume that $d=1$. Let $p\in\mathbb{R}[t]$ be a univariate
polynomial of degree $n$. We want to find upper and lower bounds in $n$ for
the $2\mu^{|\infty}$-degree $\mathfrak{d}_{2\mu^{|\infty}}(1)$ for any
suitable full admissible weight $(\mu^{|\infty},M_{0})$ such that
$p:[-1,1]\to\mathbb{R}$ is $(\mu^{|\infty},M_{0})$-smooth. (It is natural to
take $N=\infty$, since all derivatives of high order of $p$ are identically
zero anyway. The factor $2$ appears, since $\delta_{[-1,1]}=2$.)

By Markov’s inequality ([12], [13]),

(43)
$\|p^{(k)}\|_{[-1,1]}\leq\frac{n^{2}(n^{2}-1^{2})(n^{2}-2^{2})\cdots(n^{2}-(k-1)^{2})}{1\cdot
3\cdot 5\cdots(2k-1)}\|p\|_{[-1,1]},\quad 1\leq k\leq n.$

Thus a most natural full admissible weight $(\mu^{|\infty},M_{0})$ such that
$p:[-1,1]\to\mathbb{R}$ is $(\mu^{|\infty},M_{0})$-smooth is

(44) $M_{0}:=\|p\|_{[-1,1]},\quad\mu_{j}:=n^{2},\quad j\geq 1.$

In fact, $(\mu_{j})_{j}$ has to be increasing so that any admissible choice
must satisfy $\mu_{j}\geq n^{2}$ for all $j\geq 1$. Making $(\mu_{j})_{j}$
bigger increases also $\mathfrak{d}_{2\mu^{|\infty}}(1)$ and in that way we
could make $\mathfrak{d}_{2\mu^{|\infty}}(1)$ as large as we please. It is
also natural to take the second parameter equal to $1$, since
$\frac{b}{M_{0}}=1$ for the choice $b=\|p\|_{[-1,1]}$ and
$b\mapsto\mathfrak{d}_{2\mu^{|\infty}}(b)$ is decreasing.

For the choice (44) we have

$\Sigma_{\mu^{|\infty}}(1,m)=\sum_{j=1}^{m}\frac{1}{\mu_{j}}=\frac{m}{n^{2}}$

and, consequently,

$\mathfrak{d}_{\mu^{|\infty}}(1)=\lfloor en^{2}\rfloor.$

Note that in this case

(45) $2\mathfrak{d}_{\mu^{|\infty}}(1)\leq\mathfrak{d}_{2\mu^{|\infty}}(1)\leq
2\mathfrak{d}_{\mu^{|\infty}}(1)+1;$

the first inequality is always true, because $(\mu_{j})_{j}$ is increasing.

We get a better result for complex polynomials. Let $p\in\mathbb{C}[z]$ be a
univariate polynomial of degree $n$. By Bernstein’s inequality ([3], [22]),

$\|p^{(k)}\|_{\overline{D}_{1}}\leq\frac{n!}{(n-k)!}\|p\|_{\overline{D}_{1}},\quad
k\geq 1,$

where $\overline{D}_{1}:=\\{z\in\mathbb{C}:|z|\leq 1\\}$ is the closed unit
disk. Thus $p|_{\overline{D}_{1}}$ is $(\mu^{|\infty},M_{0})$-smooth for

$M_{0}:=\|p\|_{\overline{D}_{1}},\quad\mu_{j}=Cn,\quad j\geq 1,$

where the constant $C>0$ accounts for the conversion of the bounds for complex
derivatives to real partial derivatives. In this case, we find

$\mathfrak{d}_{\mu^{|\infty}}(1)=\lfloor en\rfloor,$

and (45) is still valid.

######  Remark 6.1.

Strictly speaking a $(\mu^{|\infty},M_{0})$-smooth function is real valued by
definition (see Section 2.2). Its definition clearly makes sense for complex
valued functions as well, but crucial results of the paper are based on
Rolle’s theorem.

###  6.2. The $\mu^{|\infty}$-degree vs. the analytic degree

The _analytic degree_ $d_{f}(2\epsilon)$ of a holomorphic function $f$ on the
disk $D_{1+2\epsilon}:=\\{z\in\mathbb{C}:|z|<1+2\epsilon\\}$, for some
$\epsilon>0$, is defined in [7] as the best constant $d$ in the inequality

$\|f\|_{I}\leq\Big{(}\frac{4|I|}{|E|}\Big{)}^{d}\|f\|_{E},$

where $I$ is any interval in the intersection of a real affine line in
$\mathbb{C}$ with the unit disk $D_{1}$ and $E\subseteq I$ is any measurable
subset with $|E|>0$.

By the Cauchy estimates,

$\|f^{(k)}\|_{\overline{D}_{1}}\leq\frac{k!\,\|f\|_{D_{1+\epsilon}}}{\epsilon^{k+1}},\quad
k\geq 1.$

So $f$ on $\overline{D}_{1}$ is $(\mu^{|\infty},M_{0})$-smooth for

$M_{0}:=\frac{\|f\|_{D_{1+\epsilon}}}{\epsilon},\quad\mu_{j}:=\frac{Cj}{\epsilon},\quad
j\geq 1,$

where the constant $C>0$ accounts for the conversion of the bounds for complex
derivatives to real partial derivatives. For $\tilde{\mu}(s)=s$ we have
$\overline{\gamma}_{\tilde{\mu}}=1$ and
$\overline{\Gamma}_{\tilde{\mu}}=4e^{5}$; see Remark 5.3. By Theorem 5.1, we
find (as in the proof of Theorem 5.4)

$\|f\|_{I}\leq\Big{(}\frac{4e^{5}|I|}{|E|}\Big{)}^{2\mathfrak{d}_{2\mu^{|\infty}}(\epsilon)}\|f\|_{E}.$

Since $(4e^{5})^{2\mathfrak{d}_{2\mu^{|\infty}}(\epsilon)}\leq
4^{10\,\mathfrak{d}_{2\mu^{|\infty}}(\epsilon)}$, we conclude that

$d_{f}(2\epsilon)\leq 10\,\mathfrak{d}_{2\mu^{|\infty}}(\epsilon).$

We do not know if there is a lower bound for $d_{f}(2\epsilon)$ in terms of
$\mathfrak{d}_{2\mu^{|\infty}}(\epsilon)$.

######  Remark 6.2.

For $\mu^{|\infty}=(j)_{j\geq 1}$, $\Sigma_{\mu^{|\infty}}(1,n)$ is the
partial harmonic series $\sum_{j=1}^{n}\frac{1}{j}=:H_{n}$, and, more
generally, $\Sigma_{\mu^{|\infty}}(m,n)=H_{n}-H_{m-1}$ if $1<m\leq n$. Let
$x\geq 1$ and define the positive integer $n(x)$ by

(46) $H_{n(x)}\leq x<H_{n(x)+1}.$

Comtet [9] (see also Boas and Wrench [4]) showed that, for $x\geq 2$,

$\Big{\lfloor}e^{x-\gamma}-\frac{1}{2}-\frac{3}{2}\frac{1}{e^{x-1}-1}\Big{\rfloor}\leq
n(x)\leq\Big{\lfloor}e^{x-\gamma}-\frac{1}{2}+\frac{1}{12}\frac{1}{e^{x-1}-1}\Big{\rfloor},$

where $\gamma$ is the Euler–Mascheroni constant, which determines one or two
possible values of $n(x)$ for any $x\geq 2$. With this formula it is not
difficult to find explicit estimates of $\mathfrak{d}_{a\mu^{|\infty}}(b)$,
for $a,b>0$. Note that the possible equality $H_{n(x)}=x$ in (46), in contrast
to the strict inequality in the definition of
$\mathfrak{d}_{a\mu^{|\infty}}(1)$, might effect a deviation of at most $1$
between $n(ae)$ and $\mathfrak{d}_{a\mu^{|\infty}}(1)$.

###  6.3. Conditions for non-quasianalytic bump functions

Let $M^{|\infty}$ be a non-quasianalytic full admissible weight. We may infer
from Theorem 4.2 a quantitative necessary condition for the existence of
$M^{|\infty}$-smooth functions $f$ with compact support contained in the
interior $K^{\circ}$ of the convex body $K$. Of course, this is most
informative if $K$ has equal width in all directions, e.g., if $K$ is a ball.

######  Corollary 6.3.

Let $M^{|\infty}$ be a non-quasianalytic full admissible weight,
$K\subseteq\mathbb{R}^{d}$ a convex body, and $b>0$. Suppose that
$f:K\to\mathbb{R}$ is an $M^{|\infty}$-smooth function compactly supported in
$K^{\circ}$ and $\|f\|_{K}\geq b$. Then

$\Sigma_{\mu^{|\infty}}(j_{0}+1,\infty)\leq\delta_{K}e,\qquad(j_{0}=\lceil\log(\tfrac{b}{2M_{0}})^{-1}\rceil_{\mathbb{N}}).$

###### Proof.

Theorem 4.2 shows that (19) must be violated. ∎

######  Remark 6.4.

For completeness, we sketch a proof of the following fact: _Let $Z$ be any
non-empty closed subset of $\mathbb{R}^{d}$ and $M^{|\infty}=(M_{j})_{j\geq
0}$ a non-quasianalytic full admissible weight. There exist
$f\in\mathcal{C}^{\infty}(\mathbb{R}^{d})$ and $A>0$ such that
$\|f^{(\alpha)}\|_{\mathbb{R}^{d}}\leq A^{|\alpha|+1}M_{|\alpha|}$, for all
$\alpha\in\mathbb{N}^{d}$, and $Z_{f}=Z\subseteq Z(f^{(\alpha)})$ for all
$\alpha\neq 0$._

By [19, Lemma 2.3 and Corollary 3.5], there exists a non-quasianalytic full
admissible weight $L^{|\infty}=(L_{k})_{k\geq 0}$ such that

(47) $\Big{(}\frac{M_{j}}{L_{j}}\Big{)}^{1/j}\to\infty.$

By [10, Theorem 1.4.2] or [19, Proposition 3.11], there is a
$\mathcal{C}^{\infty}$-function $0\leq\varphi\leq 1$ with $\varphi(0)=1$ and
support contained in the unit ball $B_{1}$ and $A>0$ such that
$\|\varphi^{(\alpha)}\|_{\mathbb{R}^{d}}\leq A^{|\alpha|+1}L_{|\alpha|}$ for
all $\alpha$. For every $x\in\mathbb{R}^{d}\setminus Z$, let
$d(x):=\frac{1}{2}\inf_{z\in Z}|x-z|$ and
$\varphi_{x}(y):=\varphi(\frac{y-x}{d(x)})$. Then $\varphi_{x}(x)=1$,
$U_{x}:=\\{y:\varphi_{x}(y)\neq 0\\}\subseteq\mathbb{R}^{d}\setminus Z$, and

$\|\varphi_{x}^{(\alpha)}\|_{\mathbb{R}^{d}}=\frac{1}{d(x)^{|\alpha|}}\|\varphi^{(\alpha)}\|_{\mathbb{R}^{d}}\leq\frac{A^{|\alpha|+1}}{d(x)^{|\alpha|}}L_{|\alpha|}.$

The family $\\{U_{x}\\}$ forms an open cover of $\mathbb{R}^{d}\setminus Z$
which admits a countable subcover $\\{U_{n}:=U_{x_{n}}\\}$. Let
$\varphi_{n}:=\varphi_{x_{n}}$, $d_{n}:=1/d(x_{n})$, and choose constants
$s_{n}>0$ such that

$d_{n}^{j}s_{n}\leq\frac{M_{j}}{L_{j}}\quad\text{ for all }n,j\geq 1.$

This is possible by (47), since for each $d_{n}$ we find $j_{n}$ such that
$d_{n}\leq\big{(}\frac{M_{j}}{L_{j}}\big{)}^{1/j}$ for all $j>j_{n}$; so we
may take

$s_{n}:=\min\Big{\\{}\min_{j\leq
j_{n}}\frac{M_{j}}{d_{n}^{j}L_{j}},1\Big{\\}}.$

Then $f:=\sum_{n\geq 1}\frac{s_{n}}{2^{n}}\varphi_{n}$ converges uniformly in
all derivatives,

$\displaystyle\Big{|}\sum_{n\geq
1}\frac{s_{n}}{2^{n}}\varphi_{n}^{(\alpha)}(x)\Big{|}$
$\displaystyle\leq\sum_{n\geq 1}\frac{s_{n}}{2^{n}}\cdot
A^{|\alpha|+1}d_{n}^{|\alpha|}L_{|\alpha|}\leq A^{|\alpha|+1}M_{|\alpha|},$

so that $f$ defines a $\mathcal{C}^{\infty}$-function on $\mathbb{R}^{d}$ with
$f^{(\alpha)}=\sum_{n\geq 1}\frac{s_{n}}{2^{n}}\varphi_{n}^{(\alpha)}$ and
$\|f^{(\alpha)}\|_{\mathbb{R}^{d}}\leq A^{|\alpha|+1}M_{|\alpha|}$ for all
$\alpha$. Since the $U_{n}$ cover $\mathbb{R}^{d}\setminus Z$, $f$ is strictly
positive on $\mathbb{R}^{d}\setminus Z$ and vanishes, together with all
partial derivatives $f^{(\alpha)}$, on $Z$.

###  6.4. Critical points

Let $M^{|\infty}$ be a full admissible weight and $f:K\to\mathbb{R}$ an
$M^{|\infty}$-smooth function, where $K\subseteq\mathbb{R}^{d}$ is a convex
body. In order to get quantitative information on the critical points of $f$,
one can consider the function $g:=|\nabla
f|^{2}=\sum_{j=1}^{d}(\partial_{j}f)^{2}$ and apply the results of Section 4
to $g$. Note that $g$ is $\hat{M}^{|\infty}$-smooth for a full admissible
weight $\hat{M}^{|\infty}$ which can be computed from $M^{|\infty}$ in view of
the Faà di Bruno formula.

But the one-dimensional analysis from Section 3 allows us to extract
information in a more direct way.

######  Proposition 6.5.

Let $M^{|\infty}=(M_{j})_{j\geq 0}$ be a full admissible weight,
$K\subseteq\mathbb{R}^{d}$ a convex body, and $b>0$ such that

(48)
$\Sigma_{\mu^{|\infty}}(j_{0}+1,\infty)>2\delta_{K}e,\qquad(j_{0}=\lceil\log(\tfrac{b}{2M_{0}})^{-1}\rceil_{\mathbb{N}}).$

Then $N:=\mathfrak{d}_{2\delta_{K}\mu^{|\infty}}(\tfrac{b}{2M_{0}})+1$ is a
positive integer. Let $f:K\to\mathbb{R}$ be any $M^{|N}$-smooth function such
that $\|f\|_{K}\geq b$. Let $\ell$ be any affine line that meets $B$ (i.e. the
ball contained in $\\{x\in K:|f(x)|>\frac{b}{2}\\}$ from Theorem 4.2) such
that either $\ell\cap Z_{f}\neq\emptyset$ or $V(f|_{\ell})\geq 2b$. Then $f$
has at most $2(N-1)$ critical points on $\ell$. Thus at all but possibly
$2(N-1)$ points of $\ell$ the level sets of $f$ are
$\mathcal{C}^{N}$-submanifolds.

Notice the factor $2$ in (48) and in the definition of $N$.

###### Proof.

Let $\ell$ be an affine line that meets $B$. If no zero of $f$ lies on $\ell$,
we may assume without loss of generality that $f$ is positive on $\ell$ and
that $c:=\min_{\ell\cap K}f>0$. The assumption $V(f|_{\ell})\geq 2b$
guarantees that we can replace $f$ by $f-c$; clearly the set of critical
points of $f$ and $f-c$ is the same. Indeed, cf. Remark 4.11,

$b\leq\frac{V(f|_{\ell})}{2}\leq\|f-c\|_{\ell\cap K}=\max_{\ell\cap K}f-c\leq
M_{0}-c\leq M_{0},$

and $\|f-c\|_{j,K}=\|f\|_{j,K}$ for all $j\geq 1$. Since $f-c$ has a zero on
$\ell$, we may assume from now on that $f$ has a zero on $\ell$.

Now we restrict $f$ to $\ell$ and work with the
$(\delta_{K}\mu^{|N},M_{0})$-function $g:[0,1]\to\mathbb{R}$ as in the proof
of Theorem 4.2. Since $\ell$ meets $B$, we have $\|g\|_{[0,1]}>\frac{b}{2}$.
And $g$ has at least one zero $s_{0}$ in $[0,1]$, because $f$ vanishes on
$\ell$. As in Theorem 4.2, we see that the number of critical points of $g$
left and right of $s_{0}$ is finite and bounded by $N-1$. Since each critical
point of $f$ on $\ell$ is a critical point of $g$, the proof is complete.

Let us explain in more detail why we need the factor $2$ in the definition of
$N$. To this end, assume that $t_{1}\leq t_{2}\leq\cdots\leq t_{m_{r}}$ is an
increasing enumeration of the critical points of $g$ that lie to the right of
$s_{0}$. By Rolle’s theorem, we find $t_{1}=s_{1}\leq s_{2}\leq\cdots\leq
s_{m_{r}-1}$ such that $g^{(j)}(s_{j})=0$ for $0\leq j\leq m_{r}-1$. Choose
$s_{-1}\in[0,1]$ such that $|g(s_{-1})|>\frac{b}{2}$. If $s_{-1}<s_{0}$, we
find, as in the proof of Corollary 3.4,

$1\geq\sum_{j=0}^{m_{r}-1}|s_{j}-s_{j-1}|>\frac{1}{e}\Sigma_{\delta_{K}\mu^{|\infty}}(j_{0}+1,m_{r}),$

whence $m_{r}\leq\mathfrak{d}_{\delta_{K}\mu^{|\infty}}(\frac{b}{2M_{0}})$. If
$s_{-1}>s_{0}$, then

$\sum_{j=0}^{m_{r}-1}|s_{j}-s_{j-1}|=(s_{-1}-s_{0})+\sum_{j=1}^{m_{r}-1}(s_{j}-s_{j-1})\leq
2$

so that we may conclude that
$m_{r}\leq\mathfrak{d}_{2\delta_{K}\mu^{|\infty}}(\frac{b}{2M_{0}})$. In any
case $m_{r}\leq N-1$. ∎

######  Remark 6.6.

If $M$ has additional regularity properties so that the implicit function
theorem holds in the associated Denjoy–Carleman class, then the submanifolds
are of the respective Denjoy–Carleman class; see e.g. [20].

A comprehensive quantitative study of the critical and near-critical values of
differentiable mappings can be found in [23].

### Acknowledgements

Part of the work on this paper has been done at the Fields Institute in
Toronto, Canada, during the _Thematic Program on Tame Geometry, Transseries
and Applications to Analysis and Geometry (January 1 – June 30, 2022)_. I am
grateful for the kind hospitality and the excellent working conditions. Also I
thank A. Debrouwere and B. Prangoski for pointing out a mistake in an earlier
version of the paper. The author was supported by the Austrian Science Fund
(FWF), Project P 32905-N.

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