2.1 Weight sequences and Denjoy–Carleman classes

Let $M=(M_{k})_{k\in \mathbb{N}}$ be a positive sequence of reals. Let $U\subseteq \mathbb{R}^{d}$ be open and let ${\mathcal{C}}^{M}(U,\mathbb{R}^{m})$ be the corresponding Denjoy–Carleman class (of Roumieu type) as defined in § 1.1.

If $N=(N_{k})$ is another positive sequence such that $(M_{k}/N_{k})^{1/k}$ is bounded, then ${\mathcal{C}}^{M}(U)\subseteq {\mathcal{C}}^{N}(U)$ . The converse holds if $k!M_{k}$ is logarithmically convex (log-convex for short). It follows that the class ${\mathcal{C}}^{M}(U)$ is preserved by replacing $M=(M_{k})_{k}$ by $(C^{k}M_{k})_{k}$ for some positive constant  $C$ .

We shall assume that the sequence $M$ is log-convex (which entails log-convexity of $k!M_{k}$ ). We may assume that $M_{0}=1$ and that $M$ is increasing. Indeed, the sequence $N_{k}:=C^{k}M_{k}/M_{0}$ for some constant $C\geqslant M_{0}/M_{1}$ , is log-convex, increasing, satisfies $N_{0}=1$ , and ${\mathcal{C}}^{M}(U)={\mathcal{C}}^{N}(U)$ . This motivates the following definition.

The regularity properties of a weight sequence $M=(M_{k})$ entail stability properties of the class ${\mathcal{C}}^{M}$ ; cf. [Reference Rainer and SchindlRS16]. Of particular interest in this paper is the fact that, for a weight sequence $M$ , the composite of ${\mathcal{C}}^{M}$ mappings is ${\mathcal{C}}^{M}$ . By the celebrated Denjoy–Carleman theorem, the condition

(2.1) $$\begin{eqnarray}\mathop{\sum }_{k}\frac{M_{k}}{(k+1)M_{k+1}}<\infty\end{eqnarray}$$

holds if and only if ${\mathcal{C}}^{M}$ is non-quasianalytic, that is, the Borel mapping which sends germs at some point $a$ of smooth functions to their infinite Taylor expansion at $a$ is not injective on ${\mathcal{C}}^{M}$ -germs. Then there exist non-trivial ${\mathcal{C}}^{M}$ -functions with compact support. Note that (2.1) is equivalent to

(2.2) $$\begin{eqnarray}\mathop{\sum }_{k}(k!\,M_{k})^{-1/k}<\infty .\end{eqnarray}$$

Definition 2.2. Let $M=(M_{k})$ be a weight sequence. We say that $M$ is non-quasianalytic if it satisfies (2.1); otherwise it is said to be quasianalytic. A weight sequence $M$ is called strongly non-quasianalytic if

(2.3) $$\begin{eqnarray}\;\exists C>0\;\forall k\in \mathbb{N}:\mathop{\sum }_{j\geqslant k}\frac{M_{j-1}}{jM_{j}}\leqslant C\frac{M_{k-1}}{M_{k}}.\end{eqnarray}$$

It is said to be of moderate growth if

(2.4) $$\begin{eqnarray}\;\exists C>0\;\forall j,k\in \mathbb{N}:M_{j+k}\leqslant C^{j+k}M_{j}M_{k}.\end{eqnarray}$$

A weight sequence is called strongly regular if it is strongly non-quasianalytic and of moderate growth.

Note that ${\mathcal{C}}^{\unicode[STIX]{x1D714}}(U)\subseteq {\mathcal{C}}^{M}(U)\subseteq {\mathcal{C}}^{\infty }(U)$ for every weight sequence $M$ . In fact, the Denjoy–Carleman classes form an a scale of spaces intermediate between the real analytic and the smooth functions.

2.2 The ${\mathcal{C}}^{M}$ curve lemma revisited

We generalize the ${\mathcal{C}}^{M}$ curve lemma (see [Reference Kriegl, Michor and RainerKMR09, § 3.6] and [Reference Kriegl, Michor and RainerKMR11, § 2.5]) which was inspired by [Reference BomanBom67, Lemma 2].

Lemma 2.4. There are sequences $t_{k}\rightarrow t_{\infty }$ and $s_{k}>0$ in $\mathbb{R}$ with the following property. For any non-quasianalytic weight sequence $M=(M_{k})$ and each $a\in \mathbb{N}_{{\geqslant}2}$ there is a real positive sequence $\unicode[STIX]{x1D706}_{k}\rightarrow 0$ satisfying

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{k}\biggl(\frac{M_{ak}}{M_{k}}\biggr)^{1/(ak+1)}\rightarrow 0\quad \text{as }k\rightarrow \infty\end{eqnarray}$$

such that the following holds. Let $E$ be a Banach space. Let $c_{k}\in C^{\infty }(\mathbb{R},E)$ be a sequence such that

(2.6) $$\begin{eqnarray}\{\unicode[STIX]{x1D706}_{k}^{-1}c_{k}^{(\ell )}(t):t\in I,\ell ,k\in \mathbb{N}\}\end{eqnarray}$$

is bounded in $E$ , for every bounded interval $I\subseteq \mathbb{R}$ . Then there exists a ${\mathcal{C}}^{M}$ -curve $c:\mathbb{R}\rightarrow E$ with compact support and $c(t_{k}+t)=c_{k}(t)$ for $|t|\leqslant s_{k}$ .

Proof. There exists a non-quasianalytic weight sequence $L=(L_{k})$ such that $(M_{k}/L_{k})^{1/k}\rightarrow \infty$ (this follows, for example, from [Reference KomatsuKom79, Lemma 6]). Choose a ${\mathcal{C}}^{L}$ -function $\unicode[STIX]{x1D711}:\mathbb{R}\rightarrow [0,1]$ which is $0$ on $\{t:|t|\geqslant 1/2\}$ and $1$ on $\{t:|t|\leqslant 1/3\}$ .

Let $T\in (0,1]$ and $R>0$ . Assume that $\unicode[STIX]{x1D6FE}\in C^{\infty }(\mathbb{R},E)$ is such that

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FE}^{(\ell )}(t)\Vert \leqslant R\quad \text{for all }|t|\leqslant 1/2,\ell \in \mathbb{N}.\end{eqnarray}$$

Then there exist $C,\unicode[STIX]{x1D70C}\geqslant 1$ such that for the curve $c(t):=\unicode[STIX]{x1D711}(t/T)\unicode[STIX]{x1D6FE}(t)$ we have

(2.7) $$\begin{eqnarray}\displaystyle \Vert c^{(\ell )}(t)\Vert & = & \displaystyle \biggl\|\mathop{\sum }_{j=0}^{\ell }\binom{\ell }{j}T^{-j}\unicode[STIX]{x1D711}^{(j)}\biggl(\frac{t}{T}\biggr)\unicode[STIX]{x1D6FE}^{(\ell -j)}(t)\biggr\|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle R\mathop{\sum }_{j=0}^{\ell }\binom{\ell }{j}T^{-j}C\unicode[STIX]{x1D70C}^{j}j!L_{j}\leqslant CR\biggl(1+\frac{\unicode[STIX]{x1D70C}}{T}\biggr)^{\ell }\ell !L_{\ell }\leqslant CR\biggl(\frac{2\unicode[STIX]{x1D70C}}{T}\biggr)^{\ell }\ell !L_{\ell }.\end{eqnarray}$$

Choose a sequence

(2.8) $$\begin{eqnarray}T_{j}\in (0,1]\text{ with }\mathop{\sum }_{j}T_{j}<\infty ,\text{ and let }t_{k}:=2\mathop{\sum }_{j<k}T_{j}+T_{k}.\end{eqnarray}$$

Now choose $\unicode[STIX]{x1D706}_{j}$ such that the following conditions are fulfilled:

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle 0<\frac{\unicode[STIX]{x1D706}_{j}}{T_{j}^{k}}\leqslant \frac{M_{k}}{L_{k}}\quad \text{for all }j,k, & \displaystyle\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D706}_{j}}{T_{j}^{k}}\rightarrow 0\quad \text{as }j\rightarrow \infty \text{for all }k. & \displaystyle\end{eqnarray}$$

It suffices to take $\unicode[STIX]{x1D706}_{j}\leqslant \inf _{k}T_{j}^{k+1}M_{k}/L_{k}$ . Clearly, we may in addition require that $\unicode[STIX]{x1D706}_{j}$ tends to zero fast enough so that (2.5) holds.

By (2.6), there is $R>0$ such that

$$\begin{eqnarray}\Vert c_{k}^{(\ell )}(t)\Vert \leqslant R\unicode[STIX]{x1D706}_{k}\quad \text{for all }|t|\leqslant 1/2,~\ell ,k\in \mathbb{N}.\end{eqnarray}$$

Define

$$\begin{eqnarray}c(t):=\mathop{\sum }_{j}\unicode[STIX]{x1D711}\biggl(\frac{t-t_{j}}{T_{j}}\biggr)c_{j}(t-t_{j}).\end{eqnarray}$$

The summands have disjoint supports (the support of the $j$ th summand is contained in $[t_{j}-T_{j}/2,t_{j}+T_{j}/2]$ ). Thus $c$ is ${\mathcal{C}}^{\infty }$ on $\mathbb{R}\setminus \{t_{\infty }\}$ . By (2.7),

$$\begin{eqnarray}\Vert c^{(\ell )}(t)\Vert \leqslant CR\unicode[STIX]{x1D706}_{j}\biggl(\frac{2\unicode[STIX]{x1D70C}}{T_{j}}\biggr)^{\ell }\ell !L_{\ell }\quad \text{for }|t-t_{j}|\leqslant \frac{T_{j}}{2}.\end{eqnarray}$$

Consequently, by (2.9),

$$\begin{eqnarray}\Vert c^{(\ell )}(t)\Vert \leqslant CR(2\unicode[STIX]{x1D70C})^{\ell }\ell !M_{\ell }\quad \text{for }t\neq t_{\infty }.\end{eqnarray}$$

It follows that $c:\mathbb{R}\rightarrow E$ has compact support and is ${\mathcal{C}}^{M}$ ; cf. [Reference Kriegl and MichorKM97, Lemma 2.9] and [Reference Kriegl, Michor and RainerKMR09, Lemma 3.7].◻

The next lemma is a variant of [Reference Kriegl and MichorKM97, Lemma 2.8]. Recall that, given some sequence $\unicode[STIX]{x1D707}_{k}\rightarrow \infty$ , a sequence $x_{k}$ in $E$ is called $\unicode[STIX]{x1D707}$ -convergent to $x$ if $\unicode[STIX]{x1D707}_{k}(x_{k}-x)$ is bounded.

Proof. Let $L=(L_{k})$ be a non-quasianalytic weight sequence with $(M_{k}/L_{k})^{1/k}\rightarrow \infty$ . Set $T_{j}:=1/(j(j+1))$ and choose $\unicode[STIX]{x1D706}_{j}$ such that the conditions (2.9) and (2.10) are satisfied. Let $\unicode[STIX]{x1D711}:\mathbb{R}\rightarrow [0,1]$ be a ${\mathcal{C}}^{L}$ -function which vanishes on $(-\infty ,0]$ and is $1$ on $[1,\infty )$ . Let $t_{n}:=1/n$ and define

$$\begin{eqnarray}c(t):=\left\{\begin{array}{@{}ll@{}}\displaystyle x\quad & \displaystyle \text{if }t\leqslant 0,\\ \displaystyle x_{n+1}+\unicode[STIX]{x1D711}\biggl(\frac{t-t_{n+1}}{t_{n}-t_{n+1}}\biggr)(x_{n}-x_{n+1})\quad & \displaystyle \text{if }t_{n+1}\leqslant t\leqslant t_{n},\\ x_{1}\quad & \text{if }t\geqslant 1.\end{array}\right.\end{eqnarray}$$

Clearly, $c$ is ${\mathcal{C}}^{\infty }$ on $\mathbb{R}\setminus \{0\}$ . For $t_{n+1}\leqslant t\leqslant t_{n}$ we have

$$\begin{eqnarray}\displaystyle c^{(k)}(t) & = & \displaystyle \unicode[STIX]{x1D711}^{(k)}\biggl(\frac{t-t_{n+1}}{t_{n}-t_{n+1}}\biggr)(n(n+1))^{k}(x_{n}-x_{n+1})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D711}^{(k)}\biggl(\frac{t-t_{n+1}}{t_{n}-t_{n+1}}\biggr)\cdot \frac{\unicode[STIX]{x1D706}_{n}}{T_{n}^{k}}\cdot \frac{x_{n}-x_{n+1}}{\unicode[STIX]{x1D706}_{n}}.\nonumber\end{eqnarray}$$

Condition (2.10) guarantees that $c^{(k)}(t)\rightarrow 0$ as $t\rightarrow 0$ for all $k$ , and hence $c$ is ${\mathcal{C}}^{\infty }$ on $\mathbb{R}$ . That $c$ is of class ${\mathcal{C}}^{M}$ follows from (2.9).◻

3.1 Uniform cusp property and Hölder sets

We denote by $B(x,\unicode[STIX]{x1D716}):=\{y\in \mathbb{R}^{d}:|x-y|<\unicode[STIX]{x1D716}\}$ the open ball with center $x$ and radius $\unicode[STIX]{x1D716}$ in $\mathbb{R}^{d}$ .

Definition 3.1 (Truncated open cusp).

Let us consider $\mathbb{R}^{d}=\mathbb{R}^{d-1}\times \mathbb{R}$ with the Euclidean coordinates $x=(x_{1},\ldots ,x_{d})=(x^{\prime },x_{d})$ . For $0<\unicode[STIX]{x1D6FC}\leqslant 1$ and $r,h>0$ , consider the truncated open cusp

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\unicode[STIX]{x1D6FC}}(r,h):=\{(x^{\prime },x_{d})\in \mathbb{R}^{d-1}\times \mathbb{R}:|x^{\prime }|<r,h(|x^{\prime }|/r)^{\unicode[STIX]{x1D6FC}}<x_{d}<h\}.\end{eqnarray}$$

For $\unicode[STIX]{x1D6FC}=1$ this is a truncated open cone.

We denote by $\mathscr{H}^{\unicode[STIX]{x1D6FC}}(\mathbb{R}^{d})$ the collection of all $\unicode[STIX]{x1D6FC}$ -sets in $\mathbb{R}^{d}$ and by

$$\begin{eqnarray}\mathscr{H}(\mathbb{R}^{d}):=\mathop{\bigcup }_{0<\unicode[STIX]{x1D6FC}\leqslant 1}\mathscr{H}^{\unicode[STIX]{x1D6FC}}(\mathbb{R}^{d})\end{eqnarray}$$

the collection of all Hölder sets in $\mathbb{R}^{d}$ . Note that $\mathscr{H}^{\unicode[STIX]{x1D6FC}}(\mathbb{R}^{d})\subseteq \mathscr{H}^{\unicode[STIX]{x1D6FD}}(\mathbb{R}^{d})$ if $\unicode[STIX]{x1D6FC}\geqslant \unicode[STIX]{x1D6FD}$ .

The boundary of an $\unicode[STIX]{x1D6FC}$ -set with $\unicode[STIX]{x1D6FC}<1$ can be quite irregular. It may have Hausdorff dimension strictly larger than $d-1$ and hence its Hausdorff measure ${\mathcal{H}}^{d-1}$ may be locally infinite. See [Reference Delfour and ZolésioDZ11, Theorem 6.10, p. 116].

3.2 $c^{\infty }$ -topology on Hölder sets

The $c^{\infty }$ -topology on a locally convex space $E$ is the final topology with respect to all smooth curves $c:\mathbb{R}\rightarrow E$ . The $c^{\infty }$ -topology on $\mathbb{R}^{d}$ coincides with the usual topology; cf. [Reference Kriegl and MichorKM97, Theorem 4.11]. The $c^{\infty }$ -topology on a subset $X\subseteq E$ is the final topology with respect to all smooth curves $c:\mathbb{R}\rightarrow E$ satisfying $c(\mathbb{R})\subseteq X$ .

Proof. Let $A\subseteq X$ be $c^{\infty }$ -closed in $X$ . Let $\overline{A}$ be the closure of $A$ in $\mathbb{R}^{d}$ . We have to show that $\overline{A}\,\cap \,X=\overline{A}\subseteq A$ . The converse implication is obvious.

Let $x\in \overline{A}$ . Then there is a sequence $x_{n}\in A$ which tends to $x$ . It suffices to find a smooth curve $c\in {\mathcal{C}}^{\infty }(\mathbb{R},X)$ passing through a subsequence of $x_{n}$ and through $x$ . Since $A$ is $c^{\infty }$ -closed in $X$ , this shows $x\in A$ .

Since $X$ is an $\unicode[STIX]{x1D6FC}$ -set, for some $0<\unicode[STIX]{x1D6FC}\leqslant 1$ , we may assume that there exist a neighborhood $U$ of $x$ in $X$ and a cusp $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{d}^{\unicode[STIX]{x1D6FC}}(r,h)$ such that for all $y\in U$ we have $y+\unicode[STIX]{x1D6E4}\subseteq \operatorname{int}(X)$ . By rescaling, we may assume that $r=h=1$ .

Consider $C(y,r):=y+\unicode[STIX]{x1D6E4}_{d}^{\unicode[STIX]{x1D6FC}}(r,r^{\unicode[STIX]{x1D6FC}})$ for $0<r\leqslant 1$ . It is easy to see that there is a universal constant $c>0$ such that $C(y_{1},r_{1})\,\cap \,C(y_{2},r_{2})\neq \emptyset$ provided that $|y_{1}-y_{2}|\leqslant c\min \{r_{1},r_{2}\}$ .

Choose a decreasing sequence $\unicode[STIX]{x1D707}_{n}$ which tends to $0$ faster than any polynomial. By passing to a subsequence of $x_{n}$ (again denoted by $x_{n}$ ), we may assume that $|x-x_{n}|\leqslant c\unicode[STIX]{x1D707}_{n+1}/2$ for all $n$ . Then, for all $n$ ,

$$\begin{eqnarray}|x_{n}-x_{n+1}|\leqslant |x-x_{n}|+|x-x_{n+1}|\leqslant c\unicode[STIX]{x1D707}_{n+1}.\end{eqnarray}$$

Setting $C_{n}:=C(x_{n},\unicode[STIX]{x1D707}_{n})$ , this guarantees the existence of a sequence $u_{n}$ such that $u_{n+1}\in C_{n}\,\cap \,C_{n+1}$ for all $n$ . By construction, $x_{n}$ and $u_{n}$ tend to $x$ faster than any polynomial.

Figure 1. The polygon $P_{n}$ .

For $u\in C_{n}$ define $\unicode[STIX]{x1D70B}_{n}(u):=x_{n}+u_{d}e_{d}$ (where $\{e_{i}\}$ is the standard basis in $\mathbb{R}^{d}$ ). Consider the polygon $P_{n}$ through the points $u_{n}$ , $\unicode[STIX]{x1D70B}_{n}(u_{n})$ , $x_{n}$ , $\unicode[STIX]{x1D70B}_{n}(u_{n+1})$ , $u_{n+1}$ (see Figure 1). It is contained in $C_{n}$ . The infinite polygon consisting of the concatenation of all $P_{n}$ satisfies the assumptions of [Reference Kriegl and MichorKM97, Lemma 2.8] and can hence by parameterized by a smooth curve $c$ which is contained in $X$ and satisfies $c(0)=x$ .◻

3.3 Further properties of Hölder sets

The following proposition is well known. We include a proof for the convenience of the reader.

Proposition 3.8. Let $X\in \mathscr{H}^{\unicode[STIX]{x1D6FC}}(\mathbb{R}^{d})$ . Then for each $x\in X$ there exist a compact neighborhood $K$ of $x$ in $X$ and a constant $D>0$ such that any two points $y_{1},y_{2}\in K$ can be joined by a polygon $\unicode[STIX]{x1D6FE}$ contained in $K$ with $\unicode[STIX]{x2202}X\,\cap \,\unicode[STIX]{x1D6FE}\subseteq \{y_{1},y_{2}\}$ of length

$$\begin{eqnarray}\ell (\unicode[STIX]{x1D6FE})\leqslant D|y_{1}-y_{2}|^{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

Proof. Clearly each $x\in \operatorname{int}(X)$ has this property. Let $x\in \unicode[STIX]{x2202}X$ . We may assume that in a compact neighborhood $K$ of $x$ the set $X$ is the epigraph $\{x_{d}\geqslant f(x^{\prime })\}$ of a $\unicode[STIX]{x1D6FC}$ -Hölder function $f$ with respect to an orthogonal system of coordinates $(x^{\prime },x_{d})=(x_{1},\ldots ,x_{d})$ . For two points $y_{1},y_{2}\in K$ consider the segments $S:=[y_{1},y_{2}]$ and $S^{\prime }:=[y_{1}^{\prime },y_{2}^{\prime }]$ . If $(y_{1},y_{2})\subseteq K\,\cap \,\operatorname{int}(X)$ there is nothing to prove. Otherwise let $z^{\prime }\in S^{\prime }$ be such that $f(z^{\prime })=\max _{y^{\prime }\in S^{\prime }}f(y^{\prime })$ and let $z=(z^{\prime },z_{d})$ with $z_{d}:=f(z^{\prime })+\unicode[STIX]{x1D716}|y_{1}-y_{2}|$ for some small $\unicode[STIX]{x1D716}>0$ such that $z\in K\,\cap \,\operatorname{int}(X)$ . It is possible to choose $\unicode[STIX]{x1D716}$ such that it only depends on $K$ , not on $y_{1},y_{2}$ . We have $(y_{i})_{d}\leqslant f(z^{\prime })$ and thus $|(y_{i})_{d}-f(z^{\prime })|\leqslant |f(y_{i}^{\prime })-f(z^{\prime })|$ for at least one $i\in \{1,2\}$ , say for $i=1$ . If $(y_{2})_{d}\leqslant f(z^{\prime })$ , then the polygon with vertices $y_{1}$ , $(y_{1}^{\prime },z_{d})$ , $(y_{2}^{\prime },z_{d})$ , $y_{2}$ is contained in $K$ , meets $\unicode[STIX]{x2202}X$ at most at one of the points $y_{i}$ , and has length

$$\begin{eqnarray}\displaystyle & & \displaystyle |(y_{1})_{d}-z_{d}|+|(y_{2})_{d}-z_{d}|+|y_{1}^{\prime }-y_{2}^{\prime }|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \,|f(y_{1}^{\prime })-f(z^{\prime })|+|f(y_{2}^{\prime })-f(z^{\prime })|+2\unicode[STIX]{x1D716}|y_{1}-y_{2}|+|y_{1}^{\prime }-y_{2}^{\prime }|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \,C|y_{1}^{\prime }-z^{\prime }|^{\unicode[STIX]{x1D6FC}}+C|y_{2}^{\prime }-z^{\prime }|^{\unicode[STIX]{x1D6FC}}+(1+2\unicode[STIX]{x1D716})|y_{1}-y_{2}|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \,D|y_{1}-y_{2}|^{\unicode[STIX]{x1D6FC}},\nonumber\end{eqnarray}$$

for constants only depending on $K$ . If $(y_{2})_{d}>f(z^{\prime })$ , then the segment joining $z$ and $y_{2}$ is contained in $K\,\cap \,\operatorname{int}(X)$ , and thus the polygon with vertices $y_{1}$ , $(y_{1}^{\prime },z_{d})$ , $z$ , $y_{2}$ is contained in $K$ , meets $\unicode[STIX]{x2202}X$ at most at one of the points $y_{i}$ , and has length

$$\begin{eqnarray}\displaystyle & & \displaystyle |(y_{1})_{d}-z_{d}|+|y_{2}-z|+|y_{1}^{\prime }-z^{\prime }|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \,|f(y_{1}^{\prime })-f(z^{\prime })|+(1+\unicode[STIX]{x1D716})|y_{1}-y_{2}|+|y_{1}^{\prime }-y_{2}^{\prime }|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \,D|y_{1}-y_{2}|^{\unicode[STIX]{x1D6FC}}.\nonumber\end{eqnarray}$$

This finishes the proof. ◻

5.1 Subanalytic sets

Let $M$ be a real analytic manifold. A subset $X$ of $M$ is called subanalytic if for each $x\in M$ there is an open neighborhood $U$ of $x$ in $M$ such that $X\,\cap \,U$ is the projection of a relatively compact semianalytic subset of $M\times N$ , where $N$ is a real analytic manifold. Recall that a subset $X$ of a real analytic manifold $M$ is semianalytic if for each $x\in M$ there exist an open neighborhood $U$ of $x$ in $M$ and finitely many real analytic functions $f_{ij},g_{ij}$ on $U$ such that

$$\begin{eqnarray}X\,\cap \,U=\mathop{\bigcup }_{i}\mathop{\bigcap }_{j}\{f_{ij}=0,g_{ij}>0\}.\end{eqnarray}$$

If $\dim M\leqslant 2$ , then the family of subanalytic sets in $M$ coincides with the family of semianalytic sets. In higher dimensions the family of subanalytic sets is essentially larger.

Henceforth we restrict to the case $M=\mathbb{R}^{d}$ .

A quadrant in $\mathbb{R}^{d}$ is a set

$$\begin{eqnarray}Q(I_{0},I_{-},I_{+})=\{x\in \mathbb{R}^{d}:x_{i}=0\text{ if }i\in I_{0},\,x_{i}\leqslant 0\text{ if }i\in I_{-},\,x_{i}\geqslant 0\text{ if }i\in I_{+}\},\end{eqnarray}$$

where $I_{0}$ , $I_{-}$ , $I_{+}$ is any partition of $\{1,2,\ldots ,d\}$ .

5.2 Bounded fat subanalytic sets are uniformly polynomially cuspidal

This is due to Pawłucki and Pleśniak [Reference Pawłucki and PleśniakPP86]. We recall some steps of the proof which will be needed later.

Theorem 5.4 [Reference Pawłucki and PleśniakPP86, Proposition 6.3].

Let $X$ be a bounded open subanalytic subset of $\mathbb{R}^{d}$ . Then there is a map $h:\overline{X}\times \mathbb{R}\rightarrow \mathbb{R}^{d}$ such that $h(x,t)$ is a polynomial in $t$ with degree $n$ independent of $x\in \overline{X}$ with $h(x,0)=x$ for all $x\in \overline{X}$ , $h(\overline{X}\times (0,1])\subseteq X$ , and there exist positive constants $M,m$ such that

$$\begin{eqnarray}\operatorname{dist}(h(x,t),\mathbb{R}^{d}\setminus X)\geqslant Mt^{m}\quad \text{for all }x\in \overline{X},t\in [0,1].\end{eqnarray}$$

We give a sketch of the proof in order to explicate the uniformity of $h_{x}$ which will be of importance later.

The following is a corollary of the rectilinearization theorem.

Proposition 5.5 [Reference Pawłucki and PleśniakPP86, Proposition 6.3].

Let $X$ be a relatively compact subanalytic subset of $\mathbb{R}^{d}$ of pure dimension $d$ . Then there are a finite number of real analytic maps $\unicode[STIX]{x1D711}_{j}:\mathbb{R}^{d}\times \mathbb{R}\rightarrow \mathbb{R}^{d}$ such that, for $I^{d}:=[-1,1]^{d}$ ,

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D711}_{j}(I^{d}\times (0,1])\subseteq X\quad \text{for all }j, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathop{\bigcup }_{j}\unicode[STIX]{x1D711}_{j}(I^{d}\times \{0\})=\overline{X}. & \displaystyle \nonumber\end{eqnarray}$$

Let $X$ be a bounded open subanalytic subset of $\mathbb{R}^{d}$ . Let $\unicode[STIX]{x1D711}_{j}$ be the maps provided by Proposition 5.5. Then, for each $j$ , the function

$$\begin{eqnarray}I^{d}\times [0,1]\ni (y,t)\mapsto \operatorname{dist}(\unicode[STIX]{x1D711}_{j}(y,t),\mathbb{R}^{d}\setminus X)\end{eqnarray}$$

is subanalytic (cf. [Reference Bierstone and MilmanBM88, Remark 3.11]). By the Łojasiewicz inequality (cf. [Reference Bierstone and MilmanBM88, Theorem 6.4]), there exist positive constants $L,m$ such that

$$\begin{eqnarray}\operatorname{dist}(\unicode[STIX]{x1D711}_{j}(y,t),\mathbb{R}^{d}\setminus X)\geqslant Lt^{m},\quad (y,t)\in I^{d}\times [0,1].\end{eqnarray}$$

The constants $L$ , $m$ may be assumed independent of $j$ , by taking the minimum and maximum, respectively. Choose an integer $n\geqslant m$ and write

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{j}(y,t)=T_{j}(y,t)+t^{n+1}Q_{j}(y,t),\quad (y,t)\in \mathbb{R}^{d}\times \mathbb{R},\end{eqnarray}$$

where $T_{j}(y,\cdot )$ is the Taylor polynomial at $0$ of degree $n$ of $\unicode[STIX]{x1D711}_{j}(y,\cdot )$ and $Q_{j}:\mathbb{R}^{d}\times \mathbb{R}\rightarrow \mathbb{R}^{d}$ is real analytic. If we choose $\unicode[STIX]{x1D6FF}\in (0,1]$ such that $|tQ_{j}(y,t)|\leqslant L/2$ for all $j$ , $y\in I^{d}$ , and $t\in [0,\unicode[STIX]{x1D6FF}]$ , then

$$\begin{eqnarray}\operatorname{dist}(T_{j}(y,t),\mathbb{R}^{d}\setminus X)\geqslant Lt^{m}-\frac{L}{2}t^{n}\geqslant \frac{L}{2}t^{m},\quad (y,t)\in I^{d}\times [0,\unicode[STIX]{x1D6FF}].\end{eqnarray}$$

Replacing $t$ by $\unicode[STIX]{x1D6FF}t$ , we obtain

$$\begin{eqnarray}\operatorname{dist}(T_{j}(y,\unicode[STIX]{x1D6FF}t),\mathbb{R}^{d}\setminus X)\geqslant Mt^{m},\quad (y,t)\in I^{d}\times [0,1],\end{eqnarray}$$

where $M:=L\unicode[STIX]{x1D6FF}^{m}/2$ . Clearly, $\bigcup _{j}T_{j}(I^{d}\times \{0\})=\bigcup _{j}\unicode[STIX]{x1D711}_{j}(I^{d}\times \{0\})=\overline{X}$ . Theorem 5.4 follows.

5.3 Fat closed subanalytic sets are $m$ -regular

Another property of fat closed subanalytic sets we need is the fact that they are $m$ -regular in the following sense.

Theorem 5.6 ([Reference BierstoneBie80, Theorem 6.17], [Reference HardtHar83], [Reference Bierstone and MilmanBM88, Theorem 6.10]).

Let $X\subseteq \mathbb{R}^{d}$ be a fat closed subanalytic set. For each $a\in X$ there exist a compact neighborhood $K$ in $X$ , a constant $C>0$ , and a positive integer $m$ such that any two points $x,y\in K$ can be joined by a semianalytic path $\unicode[STIX]{x1D6FE}$ in $X$ which intersects $\unicode[STIX]{x2202}X$ in at most finitely many points and satisfies

$$\begin{eqnarray}\ell (\unicode[STIX]{x1D6FE})\leqslant C|x-y|^{1/m}.\end{eqnarray}$$

5.4 L-regular decomposition

Let us recall the L-regular decomposition of subanalytic sets.

First we introduce sets which are L-regular with respect to a given system of coordinates. Let $X\subseteq \mathbb{R}^{d}$ be a subanalytic set of dimension $d$ . If $d=1$ , then $X$ is called L-regular if $X$ is a non-empty compact interval. If $d>1$ , then $X$ is L-regular if it is of the form

(5.1) $$\begin{eqnarray}X=\{(x^{\prime },x_{d})\in \mathbb{R}^{d}:f(x^{\prime })\leqslant x_{d}\leqslant g(x^{\prime }),x\in X^{\prime }\},\end{eqnarray}$$

where $X^{\prime }\subseteq \mathbb{R}^{d-1}$ is L-regular and $f$ , $g$ are continuous subanalytic functions on $X^{\prime }$ , analytic and satisfying $f<g$ on $\operatorname{int}(X^{\prime })$ with bounded partial derivatives of first order. If $\dim X=k<d$ , then $X$ is L-regular if

(5.2) $$\begin{eqnarray}X=\{(y,z)\in \mathbb{R}^{k}\times \mathbb{R}^{d-k}:z=h(y),y\in X^{\prime }\},\end{eqnarray}$$

where $X^{\prime }\subseteq \mathbb{R}^{k}$ is L-regular, $\dim X^{\prime }=k$ , and $h$ is continuous subanalytic on $X^{\prime }$ , analytic on $\operatorname{int}(X^{\prime })$ with bounded partial derivatives of first order.

In general a subanalytic set $X$ in $\mathbb{R}^{d}$ is said to be L-regular if it is L-regular with respect to some linear (or equivalently orthogonal) system of coordinates. It is called an L-regular cell if it is the relative interior of an L-regular set, that is, it is $\operatorname{int}(X)$ in case (5.1) and the graph of $h$ restricted to $\operatorname{int}(X^{\prime })$ in case (5.2). By definition, every point is a zero-dimensional L-regular cell.

It is well known that L-regular sets and L-regular cells are quasiconvex (cf. [Reference KurdykaKur92], [Reference ParusińskiPar94a, Lemma 2.2], or [Reference Kurdyka and ParusińskiKP06]): there is a constant $C>0$ such that any two points $x,y$ in the set can be joined in the set by a subanalytic path of length at most $C|x-y|$ .

For the proof of Theorem 1.14 we need the following preparatory results.

Lemma 5.9. Let $X\subseteq \mathbb{R}^{d}$ be a fat closed subanalytic set. Let $x\in \unicode[STIX]{x2202}X$ and suppose there is a basis of neighborhoods $\mathscr{U}$ of $x$ such that $U\,\cap \,\operatorname{int}(X)$ is connected for all $U\in \mathscr{U}$ . Then there exist $U_{0}\in \mathscr{U}$ and a positive constant $C$ such that the following holds. For all $U\in \mathscr{U}_{0}:=\{U\in \mathscr{U}:U\subseteq U_{0}\}$ and for any two points $y,z\in U\,\cap \,\operatorname{int}(X)$ , there exists a rectifiable path $\unicode[STIX]{x1D6FE}$ in $\operatorname{int}(X)$ which connects $y$ and $z$ and satisfies

$$\begin{eqnarray}\ell (\unicode[STIX]{x1D6FE})\leqslant C\operatorname{diam}(U).\end{eqnarray}$$

Proof. We may assume that $X$ is bounded, by intersecting with a ball centered at $x$ . Let $U_{0}$ be any member of $\mathscr{U}$ which is contained in this ball. By Theorem 5.7, $\operatorname{int}(X)$ is a finite disjoint union of L-regular cells $\{A_{1},\ldots ,A_{k}\}$ .

Fix $U\in \mathscr{U}_{0}$ and let $y,z\in U\,\cap \,\operatorname{int}(X)$ . Since $U\,\cap \,\operatorname{int}(X)$ is connected, there is a path $\unicode[STIX]{x1D70E}:[0,1]\rightarrow U\,\cap \,\operatorname{int}(X)$ with $\unicode[STIX]{x1D70E}(0)=y$ and $\unicode[STIX]{x1D70E}(1)=z$ . Then we have a finite disjoint union $[0,1]=\bigcup _{i=1}^{k}E_{i}$ , where $E_{i}:=\unicode[STIX]{x1D70E}^{-1}(A_{i})$ .

Let $E_{i}^{\prime }$ be the set of limit points of $E_{i}$ . Then $[0,1]=\bigcup _{i=1}^{k}E_{i}^{\prime }$ . Let $i_{1}\in \{1,\ldots ,k\}$ be such that $t_{0}:=0\in E_{i_{1}}^{\prime }$ . If $t_{1}<1$ , then there exists $i_{2}\in \{1,\ldots ,k\}\setminus \{i_{1}\}$ such that $t_{1}\in E_{i_{2}}^{\prime }$ and $t_{2}:=\sup E_{i_{2}}^{\prime }>t_{1}$ , by Lemma 5.8. Moreover, $[t_{1},b]=\bigcup _{j\neq i_{1}}E_{j}^{\prime }\,\cap \,[t_{1}\,\cap \,b]$ . If $t_{2}<b$ we may apply Lemma 5.8 again and find $i_{3}\in \{1,\ldots ,k\}\setminus \{i_{1},i_{2}\}$ such that $t_{2}\in E_{i_{3}}^{\prime }$ and $t_{3}:=\sup E_{i_{3}}^{\prime }>t_{2}$ . This procedure ends after finitely many steps and gives a finite partition $0=t_{0}<t_{1}<\cdots <t_{h-1}<t_{h}=1$ of $[0,1]$ . The points $y=z_{0},z_{1},\ldots ,z_{h}=z$ , where $z_{j}=\unicode[STIX]{x1D70E}(t_{j})$ , all lie in $U\,\cap \,\operatorname{int}(X)$ . Let $\unicode[STIX]{x1D716}>0$ be sufficiently small such that the balls $B_{j}:=B(z_{j},\unicode[STIX]{x1D716})$ are all contained in $U\,\cap \,\operatorname{int}(X)$ . For all $j=1,2,\ldots ,h-1$ , there exist $z_{j}^{-}\in B_{j}\,\cap \,A_{i_{j}}$ and $z_{j}^{+}\in B_{j}\,\cap \,A_{i_{j+1}}$ , by construction. Additionally, there exist $z_{0}^{+}\in B_{0}\,\cap \,A_{i_{1}}$ and $z_{h}^{-}\in B_{h}\,\cap \,A_{i_{h}}$ .

Since the cells are quasiconvex, for all $j=1,2,\ldots ,h$ , there exist rectifiable paths $\unicode[STIX]{x1D6FE}_{j}\in A_{i_{j}}$ joining $z_{j-1}^{+}$ and $z_{j}^{-}$ such that

$$\begin{eqnarray}\ell (\unicode[STIX]{x1D6FE}_{j})\leqslant C_{j}|z_{j-1}^{+}-z_{j}^{-}|,\end{eqnarray}$$

where the constant $C_{j}$ depends only on $A_{i_{j}}$ . Joining the paths $\unicode[STIX]{x1D6FE}_{j}$ with the line segments $[z_{0},z_{0}^{+}]$ , $[z_{j}^{-},z_{j}^{+}]$ , for $j=1,\ldots ,h-1$ , and $[z_{h}^{-},z_{h}]$ , we obtain a rectifiable path $\unicode[STIX]{x1D6FE}$ in $\operatorname{int}(X)$ which connects $y$ and $z$ and has length

$$\begin{eqnarray}\ell (\unicode[STIX]{x1D6FE})\leqslant C\operatorname{diam}(U),\end{eqnarray}$$

for a constant $C$ which depends only on the $C_{j}$ and the number of cells $k$ , since all points $z_{j}$ , $z_{j}^{-}$ , $z_{j}^{+}$ lie in $U$ .◻

5.5 Proof of Theorem 1.14

The inclusion ${\mathcal{C}}^{\infty }(X)\subseteq {\mathcal{A}}^{\infty }(X)$ is clear.

Let $f\in {\mathcal{A}}^{\infty }(X)$ . Then $f$ is smooth in $\operatorname{int}(X)$ , by Result 1.1. We must show that $f\in {\mathcal{C}}^{\infty }(X)$ . This is a local problem, so we may assume without loss of generality that $X$ is compact (by intersecting with a suitable ball). By Lemma 1.10 and Theorem 5.6, it suffices to show that $f$ satisfies Lemma 1.10(3).

Fix $x\in \unicode[STIX]{x2202}X$ . By Theorem 5.4, there is a polynomial curve $h_{x}:\mathbb{R}\rightarrow \mathbb{R}^{d}$ of degree at most $n$ with the properties:

1. (i) $h_{x}((0,1])\subseteq \operatorname{int}(X)$ and $h_{x}(0)=x$ ,

2. (ii) $\operatorname{dist}(h_{x}(t),\mathbb{R}^{d}\setminus X)\geqslant Mt^{m}$ for all $t\in (0,1]$ ,

where $n,M,m$ are independent of $x$ and $t$ . Then there is a positive integer $k=k(x)$ such that $h_{x}(t)-x=t^{k}\tilde{h}_{x}(t)$ , where $\tilde{h}_{x}(0)\neq 0$ . Set $v_{1}:=\tilde{h}_{x}(0)/|\tilde{h}_{x}(0)|\in S^{d-1}$ . Choose $d-1$ directions $v_{2},\ldots ,v_{d}\in S^{d-1}$ such that $v_{1},v_{2},\ldots ,v_{d}$ are linearly independent and define

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{x,v}(t_{1},t_{2},\ldots ,t_{d}):=h_{x}(t_{1})+t_{2}v_{2}+\cdots +t_{d}v_{d}\end{eqnarray}$$

for $t=(t_{1},\ldots ,t_{d})$ in the set

$$\begin{eqnarray}Y:=\{(t_{1},\ldots ,t_{d})\in \mathbb{R}^{d}:t_{1}\in (0,\unicode[STIX]{x1D6FF}),|t_{j}|<Ct_{1}^{m}\text{ for }2\leqslant j\leqslant d\}.\end{eqnarray}$$

If $C:=M/(2(d-1))$ and $\unicode[STIX]{x1D6FF}>0$ is chosen small enough, then $\unicode[STIX]{x1D6F9}_{x,v}$ is a diffeomorphism of $Y$ onto the open subset $H_{x,v}:=\unicode[STIX]{x1D6F9}_{x,v}(Y)$ of $\operatorname{int}(X)$ and it extends to a homeomorphism between $Y\cup \{0\}$ and $H_{x,v}\cup \{x\}$ ; indeed, by (2),

$$\begin{eqnarray}\displaystyle \operatorname{dist}(\unicode[STIX]{x1D6F9}_{x,v}(t),\mathbb{R}^{d}\setminus X) & {\geqslant} & \displaystyle \operatorname{dist}(h_{x}(t_{1}),\mathbb{R}^{d}\setminus X)-|t_{2}|-\cdots -|t_{d}|\nonumber\\ \displaystyle & {>} & \displaystyle Mt_{1}^{m}-(d-1)Ct_{1}^{m}=\frac{M}{2}t_{1}^{m}>0,\nonumber\end{eqnarray}$$

for $t\in Y$ . Since $f$ is smooth in $\operatorname{int}(X)$ , we have

(5.3) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t_{2}}^{\unicode[STIX]{x1D6FC}_{2}}\cdots \unicode[STIX]{x2202}_{t_{d}}^{\unicode[STIX]{x1D6FC}_{d}}(f\,\circ \,\unicode[STIX]{x1D6F9}_{x,v})(t)=d_{v_{2}}^{\unicode[STIX]{x1D6FC}_{2}}\cdots d_{v_{d}}^{\unicode[STIX]{x1D6FC}_{d}}f(\unicode[STIX]{x1D6F9}_{x,v}(t))\quad \text{for all }t\in Y,\unicode[STIX]{x1D6FC}_{j}\geqslant 0.\end{eqnarray}$$

The left-hand side of (5.3) extends continuously to $t=0$ , since $f\,\circ \,\unicode[STIX]{x1D6F9}_{x,v}\in {\mathcal{A}}^{\infty }(\overline{Y})$ and ${\mathcal{A}}^{\infty }(\overline{Y})={\mathcal{C}}^{\infty }(\overline{Y})$ , by Theorem 1.13, as $\overline{Y}$ is a Hölder set. Since $\unicode[STIX]{x1D6F9}_{x,v}$ is a homeomorphism $Y\cup \{0\}\rightarrow H_{x,v}\cup \{x\}$ , we may conclude that the directional derivatives $d_{v_{2}}^{\unicode[STIX]{x1D6FC}_{2}}\cdots d_{v_{d}}^{\unicode[STIX]{x1D6FC}_{d}}f$ , $\unicode[STIX]{x1D6FC}_{j}\geqslant 0$ , extend continuously from $H_{x,v}$ to $x$ .

If we perturb the directions $v_{2},\ldots ,v_{d}$ a little such that $v_{1},v_{2},\ldots ,v_{d}$ remain linearly independent and take the intersection $H_{x}$ of the corresponding sets $H_{x,v}$ , then $H_{x}$ still is an open subset of $\operatorname{int}(X)$ with $h_{x}(t)\in H_{x}$ for small $t>0$ and $x\in \overline{H}_{x}$ . Then $d_{w_{2}}^{\unicode[STIX]{x1D6FC}_{2}}\cdots d_{w_{d}}^{\unicode[STIX]{x1D6FC}_{d}}f$ , $\unicode[STIX]{x1D6FC}_{j}\geqslant 0$ , extend continuously from $H_{x}$ to $x$ for all $w_{2},\ldots ,w_{d}$ near $v_{2},\ldots ,v_{d}$ . By Lemma 1.10, we infer that the Fréchet derivatives $f^{(p)}$ of all orders of $f$ extend continuously from $H_{x}$ to $x$ .

Thus for all $x\in \unicode[STIX]{x2202}X$ and $p\in \mathbb{N}$ , we have a candidate for the Fréchet derivative $f^{(p)}(x)$ of $f$ at $x$ and an open set $H_{x}\subseteq \operatorname{int}(X)$ on which $f^{(p)}(y)$ tends to this candidate as $y\rightarrow x$ . It remains to prove that the extension of $f^{(p)}$ to $X$ thus defined is continuous on $X$ . First we show that it is bounded.

Let $p\in \mathbb{N}$ be fixed. It suffices to show that $f^{(p)}$ is bounded on $\operatorname{int}(X)$ (since $X$ is fat). For contradiction suppose that there is a sequence $(x_{\ell })$ in $\operatorname{int}(X)$ such that $\Vert f^{(p)}(x_{\ell })\Vert _{L_{p}}\rightarrow \infty$ . Since $X$ is compact, we may assume that $x_{\ell }\rightarrow x$ . Then $x\in \unicode[STIX]{x2202}X$ , since we already know that $f$ is smooth on $\operatorname{int}(X)$ .

By Proposition 5.5, there are a finite number of real analytic maps $\unicode[STIX]{x1D711}_{j}:\mathbb{R}^{d}\times \mathbb{R}\rightarrow \mathbb{R}^{d}$ such that

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D711}_{j}(I^{d}\times (0,1])\subseteq \operatorname{int}(X)\quad \text{for all }j, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathop{\bigcup }_{j}\unicode[STIX]{x1D711}_{j}(I^{d}\times \{0\})=X, & \displaystyle \nonumber\end{eqnarray}$$

where $I^{d}:=[-1,1]^{d}$ . After passing to a subsequence we may assume that $x_{\ell }\in \unicode[STIX]{x1D711}_{j_{0}}(I^{d}\times \{0\})$ for all $\ell$ and some $j_{0}$ . Choose $y_{\ell }\in I^{d}$ such that $\unicode[STIX]{x1D711}_{j_{0}}(y_{\ell },0)=x_{\ell }$ . Since $I^{d}$ is compact, after passing to a subsequence we may assume that $y_{\ell }\rightarrow y$ and in turn that this convergence is faster than any polynomial. The infinite polygon through the points $y_{\ell }$ and $y$ can be parameterized by a smooth curve $c:\mathbb{R}\rightarrow I^{d}$ such that $c(1/\ell )=y_{\ell }$ and $c(0)=y$ (cf. [Reference Kriegl and MichorKM97, Lemma 2.8]). Then $s\mapsto \unicode[STIX]{x1D711}_{j_{0}}(c(s),0)$ is a smooth curve in $X$ through the points $x_{\ell }$ and $x$ .

Since $\unicode[STIX]{x1D711}_{j_{0}}$ is real analytic, for small $t_{1}$ we have $\unicode[STIX]{x1D711}_{j_{0}}(y,t_{1})=x+t_{1}^{k}\tilde{\unicode[STIX]{x1D711}}_{j_{0}}(y,t_{1})$ for some positive integer $k$ and a real analytic map $\tilde{\unicode[STIX]{x1D711}}_{j_{0}}$ with $\tilde{\unicode[STIX]{x1D711}}_{j_{0}}(y,0)\neq 0$ . Then $\tilde{\unicode[STIX]{x1D711}}_{j_{0}}(z,t_{1})\neq 0$ for $(z,t_{1})$ in a neighborhood of $(y,0)$ . Thus,

$$\begin{eqnarray}v_{1}(z,t_{1}):=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\unicode[STIX]{x2202}_{t_{1}}\unicode[STIX]{x1D711}_{j_{0}}(z,t_{1})}{|\unicode[STIX]{x2202}_{t_{1}}\unicode[STIX]{x1D711}_{j_{0}}(z,t_{1})|}=\frac{k\tilde{\unicode[STIX]{x1D711}}_{j_{0}}(z,t_{1})+t_{1}\tilde{\unicode[STIX]{x1D711}}_{j_{0}}^{\prime }(z,t_{1})}{|k\tilde{\unicode[STIX]{x1D711}}_{j_{0}}(z,t_{1})+t_{1}\tilde{\unicode[STIX]{x1D711}}_{j_{0}}^{\prime }(z,t_{1})|}\quad & \text{if }t_{1}>0,\\ \displaystyle \frac{\tilde{\unicode[STIX]{x1D711}}_{j_{0}}(z,0)}{|\tilde{\unicode[STIX]{x1D711}}_{j_{0}}(z,0)|}\quad & \text{if }t_{1}=0,\end{array}\right.\end{eqnarray}$$

is continuous in $(z,t_{1})$ , where $t_{1}\geqslant 0$ , near $(y,0)$ . It follows that we can find an open set of directions $v\in S^{d-1}$ such that $v_{1}(c(s),0)$ and $v$ are linearly independent for $s$ near $0$ . For such $v$ ,

$$\begin{eqnarray}(s,t_{1},t_{2})\rightarrow f(\unicode[STIX]{x1D711}_{j_{0}}(c(s),t_{1})+t_{2}v)\end{eqnarray}$$

is smooth for small $s\in \mathbb{R}$ , $t_{1}\geqslant 0$ , and $|t_{2}|\leqslant Ct_{1}^{m}$ , by the arguments in § 5.2 and the considerations in the first part of the proof. But this implies that $d_{v}^{p}f(x_{\ell })$ is bounded for all such $v$ , and hence $f^{(p)}(x_{\ell })$ is bounded, a contradiction. Claim 1 is proved.

Let $x\in \unicode[STIX]{x2202}X$ and suppose that $(x_{n})$ and $(y_{n})$ are two sequences in $\operatorname{int}(X)$ both converging to $x$ . By Lemma 5.9, for each $\unicode[STIX]{x1D716}>0$ there exists $n_{0}\in \mathbb{N}$ such that for all $n\geqslant n_{0}$ the points $x_{n}$ and $y_{n}$ can be joined by a rectifiable path $\unicode[STIX]{x1D6FE}_{n}$ in $\operatorname{int}(X)$ with length $\ell (\unicode[STIX]{x1D6FE}_{n})\leqslant \unicode[STIX]{x1D716}$ . Since $f$ is smooth in $\operatorname{int}(X)$ , we may apply the fundamental theorem of calculus and Claim 1 to conclude

$$\begin{eqnarray}\Vert f^{(p)}(x_{n})-f^{(p)}(y_{n})\Vert _{L_{p}}\leqslant \Bigl(\sup _{z\in \unicode[STIX]{x1D6FE}_{n}}\Vert f^{(p+1)}(z)\Vert _{L_{p+1}}\Bigr)\ell (\unicode[STIX]{x1D6FE}_{n})\rightarrow 0\quad \text{as }n\rightarrow \infty .\end{eqnarray}$$

If we assume that the sequence $(x_{n})$ lies in $H_{x}$ , we obtain that $f^{(p)}(y)\rightarrow f^{(p)}(x)$ for all $\operatorname{int}(X)\ni y\rightarrow x$ . Finally, suppose that $\unicode[STIX]{x2202}X\ni x_{n}\rightarrow x$ . Choose $y_{n}\in H_{x_{n}}\,\cap \,B(x_{n},1/n)$ . Then

$$\begin{eqnarray}\Vert f^{(p)}(x)-f^{(p)}(x_{n})\Vert _{L_{p}}\leqslant \Vert f^{(p)}(x)-f^{(p)}(y_{n})\Vert _{L_{p}}+\Vert f^{(p)}(x_{n})-f^{(p)}(y_{n})\Vert _{L_{p}}\rightarrow 0\end{eqnarray}$$

as $n\rightarrow \infty$ . This proves Claim 2 and hence the theorem. ◻