2.1 Bastiani’s differential calculus

In this section, we recall Bastiani’s differential calculus, see also [Reference Glöckner3, Reference Hamilton4, Reference Milnor12–Reference Neeb15]. Let $E,F\in {\textsf {hlcVect}}$ be given. A map

, with $U\subseteq E$ open, is said to be differentiable if

exists for each $x\in U$ and $v\in E$ . The map f is said to be k-times differentiable for $k\geq 1$ if

is defined for all $v_1,\ldots ,v_k\in E$ . Implicitly, this means that f is p-times differentiable for each $1\leq p\leq k$ , and we set

$$ \begin{align*} {\mathrm{d}}^p_xf(v_1,\ldots,v_p)\equiv {\mathrm{d}}^p f(x,v_1,\ldots,v_p):=D_{v_p,\ldots,v_1}f(x)\qquad\quad\forall\: x\in U,\:v_1,\ldots,v_p\in E \end{align*} $$

for $p=1,\ldots ,k$ . We furthermore define ${\mathrm {d}} f:= {\mathrm {d}}^1 f$ , as well as ${\mathrm {d}}_x f:= {\mathrm {d}}^1_x f$ for each $x\in U$ . The map

is said to be

• • of class $C^0$ if it is continuous. In this case, we define ${\mathrm {d}}^0 f:= f$ .

• • of class $C^k$ for $k\geq 1$ if it is k-times differentiable, such that

  
  is continuous for $p=0,\ldots ,k$ . In this case, ${\mathrm {d}}^p_x f$ is symmetric and p-multilinear for each $x\in U$ and $p=1,\ldots ,k$ , see [Reference Glöckner3].

• • of class $C^\infty $ if it is of class $C^k$ for each $k\in {\mathbb {N}}$ .

Remark 2.1 Let $E,F$ be normed spaces. We define $L^0(E,F):=F$ , and let $L^{\ell }(E,F)$ , for $\ell \geq 1$ , denote the space of all continuous $\ell $ -multilinear maps $E^{\ell }\rightarrow F$ equipped with the operator topology.Footnote ² For $k\in {\mathbb {N}}$ and $U\subseteq E$ non-empty open, we denote the set of all k-times Fréchet differentiable maps $U\rightarrow F$ by $\mathcal {F}C^k(U,F)$ . Given $f\in \mathcal {F}C^k(U,F)$ , we denote its $\ell $ th Fréchet differential, for $0\leq \ell \preceq k$ , by . We recall that $C^{k+1}(U,F)\subseteq \mathcal {F}C^k(U,F)\subseteq C^k(U,F)$ holds [Reference Neeb13, Reference Whitney20], with

$$ \begin{align*} D^{(\ell)} f(x)={\mathrm{d}}^{\ell}_x f\qquad\quad\forall\: x\in U,\: 0\leq \ell\preceq k. \end{align*} $$

In particular, we have $C^\infty (U,F)=\mathcal {F}C^\infty (U,F)$ .

We have the following differentiation rules [Reference Glöckner3].

We observe the following.

Corollary 2.3 Let $F,\bar {F},\bar {\bar {F}}\in {\textsf {hlcVect}}$ , $1\leq k\leq \infty $ , as well as

and

be of class $C^k$ . Then, for $1\leq \ell \preceq k$ we have

where

is given as a linear combination of maps of the form

$$ \begin{align*} &U\times F^{\ell}\ni (x,v_1,\ldots,v_{\ell})\\ &\qquad \mapsto {\mathrm{d}}^q \bar{f}(f(x), {\mathrm{d}}^{p_1} f(x,v_1,\ldots,v_{p_1}),\ldots, {\mathrm{d}}^{p_q} f(x,v_{\ell-p_q+1},\ldots,v_{\ell}))\in \bar{\bar{F}}, \end{align*} $$

such that the following conditions are fulfilled:

• • We have $1\leq q< \ell $ , as well as $p_1,\ldots ,p_q\geq 1$ with .

• • If $\ell \geq 2$ holds, then we have $p_i\geq 2$ for some $1\leq i\leq q$ .

Proof For $\ell =1$ , the claim is clear from (d) in Proposition 2.2. Moreover, we obtain from the differentiation rules in Proposition 2.2 that

holds, which proves the claim for $\ell =2$ . The rest now follows by induction from Proposition 2.2.▪

Let us finally consider the situation, where

holds for a nonempty open interval $I\subseteq {\mathbb {R}}$ (hence, $E\equiv {\mathbb {R}}$ ). It is then not hard to see that $\gamma $ is of class $C^k$ for

if and only if $\gamma ^{(p)}$ , inductively defined by $\gamma ^{(0)}:=\gamma $ as well asFootnote ³

exists and is continuous for $0\leq p\preceq k$ . If $D\subseteq {\mathbb {R}}$ is an arbitrary interval (connected, nonempty and nonsingleton), we let $C^k(D,F)$ (

) denote the set of all maps

, such that $\gamma ={\widetilde {\gamma }}|_D$ holds for some ${\widetilde {\gamma }}\in C^k(I,F)$ with $I\subseteq {\mathbb {R}}$ an open interval such that $D\subseteq I$ . In this case, we set $\gamma ^{(p)}:={\widetilde {{\gamma }}}^{(p)}|_D$ for each $0\leq p\preceq k$ .

2.2 Locally convex vector spaces

In this section, we collect some elementary statements concerning locally convex vector spaces.

2.2.1 Product spaces and continuous maps

Given $F_1,\ldots ,F_n,F\in {\textsf {hlcVect}}$ , the Tychonoff topology on equals the Hausdorff locally convex topology that is generated by the seminorms

(2.1)

with ${\mathfrak {q}}_p\in {\textsf {Sem}(F_p)}$ for $p=1,\ldots ,n$ . We recall the following statements.

Lemma 2.4 For each ${\mathfrak {q}}\in {\textsf {Sem}(E)}$ , there exist ${\mathfrak {q}}_p\in {\textsf {Sem}(F_p)}$ for $p=1,\ldots ,n$ , with ${\mathfrak {q}}\leq {\textsf {max}}[{\mathfrak {q}}_1,\ldots ,{\mathfrak {q}}_n]$ .

Proof Since the seminorms (2.1) form a fundamental system, the claim is clear from Proposition 22.6 in [Reference Meise and Vogt9], when applied to the identity ${\mathrm {id}}_E$ .Footnote ⁴ ▪

Lemma 2.5 Let X be a topological space, and let

be continuous, such that

is n-multilinear for each $x\in X$ . Then, for each compact ${\mathrm{K}}\subseteq X$ and each ${\mathfrak {p}}\in {\textsf {Sem}(F)}$ , there exist seminorms ${\mathfrak {q}}_p\in {\textsf {Sem}(F_p)}$ for $p=1,\ldots ,n$ , as well as $O\subseteq X$ open with ${\mathrm{K}}\subseteq O$ , such that

Proof See, e.g., Corollary 1 in [Reference Hanusch5].▪

2.2.2 The Riemann integral

Let $\gamma \in C^0([r,r'],F)$ be given. We denote the Riemann integralFootnote ⁵ of $\gamma $ by $\int \gamma (s) \:{\mathrm {d}} s\in {\textsf {comp}({F})}$ , and define

$$ \begin{align*} \textstyle\int_a^b \gamma(s)\:{\mathrm{d}} s:= \int \gamma|_{[a,b]}(s) \:{\mathrm{d}} s\qquad\quad\forall\: r\leq a<b\leq r'.\\[-15pt] \end{align*} $$

The Riemann integral is linear, with

$$ \begin{align*} \textstyle\hat{{\mathfrak{p}}}\big(\int_a^b \gamma(s)\:{\mathrm{d}} s\big)\textstyle\leq\int_a^b {\mathfrak{p}}(\gamma(s))\:{\mathrm{d}} s\qquad\quad\forall\: {\mathfrak{p}}\in {\textsf{Sem}(F)},\: r\leq a<b\leq r'.\\[-15pt] \end{align*} $$

It follows that the Riemann integral is $C^0$ -continuous, i.e., continuous w.r.t. the seminorms

$$ \begin{align*} {\mathfrak{p}}_{\infty}(\gamma):=\sup\{{\mathfrak{p}}(\gamma(t))\:|\: t\in [a,b]\}\qquad\quad\forall\: {\mathfrak{p}}\in {\textsf{Sem}(F)},\:\gamma\in C^0([a,b],F),\: a<b.\\[-15pt] \end{align*} $$

For $\gamma \in C^1(I,F)$ ( $I\subseteq {\mathbb {R}}$ an open interval) and $a<b$ with $[a,b]\subseteq I$ , we have by [Reference Glöckner3] that

(2.2) $$ \begin{align} \textstyle\gamma(b)-\gamma(a)\textstyle=\int_a^b \gamma^{(1)}(s)\:{\mathrm{d}} s. \end{align} $$

It is furthermore not hard to see that given $\gamma \in C^0(I,F)$ , then for $a<b$ with $[a,b]\subseteq I$ and , we have

(2.3) $$ \begin{align} \Gamma\in C^1([a,b],{\textsf{comp}({F})})\qquad\:\:\text{with}\qquad\:\: \Gamma^{(1)}=\gamma|_{[a,b]}. \end{align} $$

2.2.3 Harmonic subsets and extensions

Let $\{0\} \neq H\in {\textsf {hlcVect}}$ , $U\subseteq H$ nonempty open, and $\emptyset \neq A\subseteq U$ closed in U w.r.t. the subspace topology on U. Then, A is said to be harmonic if for each $(x,v)\in A\times (H\setminus \{0\})$ , there exists $\delta>0$ as well as continuous at $0$ with $\gamma _{\pm }(0)=0$ , such thatFootnote ⁶

(2.4)

Example 2.6 (Harmonic Subsets)

1. (i) If $A\subseteq U$ is harmonic and $\emptyset \neq B\subseteq A$ closed in U, then $B\subseteq U$ is harmonic.

2. (ii) Each nonempty finite subset of U is harmonic.

   Proof If $\emptyset \neq A\subseteq U$ is finite, then A is closed in U. For $x\in A$ fixed, there exists ${\mathfrak {h}}\in {\mathsf {Sem}(H)}$ with ${\mathrm {B}}_1(x):=\{y\in H\:|\: {\mathfrak {h}}(y-x)<1\}\subseteq U$ , such that holds. For $0\neq v\in H$ fixed, we set $\delta :=\frac {1}{2\max (1,{\mathfrak {h}}(v))}$ and define . Then, we have

   
   for all $\lambda \in (0,\delta )$ , hence .▪

3. (iii) Let $\tilde {H}:= H\times F$ with $F\in {\mathsf {hlcVect}}$ , $\emptyset \neq W\subseteq F$ open, and $\tilde {U}:= U\times W$ . If $A\subseteq U$ is harmonic, then $\tilde {A}:=A\times W\subseteq \tilde {U}$ is harmonic.

   Proof Let $\tilde {x}\equiv (x,z)\in \tilde {A}$ and $\tilde {v}\equiv (v,u)\in \tilde {H}\setminus \{(0,0)\}$ be given.

   • • Let $v\neq 0$ . We choose $\delta>0$ and $\gamma _{\pm }$ as in (2.4). Shrinking $\delta>0$ if necessary, we can assume (as W is open). We set , and obtain

     
     for all $\lambda \in (0,1)$ and $\mu \in (0,\delta )$ , hence .

   • • Let $v=0$ . We fix $0\neq w\in H$ , and choose $\delta>0$ and $\gamma _{\pm }$ as in (2.4) for $v\equiv w$ there. Shrinking $\delta>0$ if necessary, we can assume (as W is open). We set , and obtain (observe and for $\lambda \in (0,1)$ )

     
     for all $\lambda \in (0,1)$ and $\mu \in (0,\delta )$ . Hence, we have .▪

4. (iv) Let $H={\mathbb {R}}\times E$ for $E\in {\mathsf {hlcVect}}$ , $p\in {\mathbb {R}}$ , as well as $U={\mathbb {R}}\times V$ with $\emptyset \neq V\subseteq E$ open. Then, $\{p\}\times V\subseteq U$ is harmonic.

   Proof $A:=\{p\}\subseteq {\mathbb {R}}$ is harmonic by (ii). The claim thus follows from (iii) (with $H,U\equiv\mathbb{R},\ F\equiv E\ \mathrm{and}\ W\equiv V $ ).▪

5. (v) If $p\in (0,\infty )$ and $0\neq {\mathfrak {h}}\in {\mathsf {Sem}(H)}$ , then is harmonic.

   Proof $B:={\mathfrak {h}}^{-1}(p)$ is closed in H as ${\mathfrak {h}}$ is continuous, as well as nonempty as ${\mathfrak {h}}\neq 0$ . Hence, is closed in U. For $z\in B$ and $w\in H$ , the reverse triangle inequality yields

   (2.5)

   
   for all $\lambda \in (0, 1)$ . Let now $x\in A$ and $0\neq v\in H$ be given:

   • • Let ${\mathfrak {h}}(v)=0$ . Then, (2.5) applied to $z=x$ and for $\mu \in (0,\infty )$ yields

     
     hence . Since U is open with $x\in U$ , there exists $\varepsilon>0$ with , hence . The condition (2.4) thus holds for $\delta :=\varepsilon $ and .

   • • Let ${\mathfrak {h}}(v)>0$ . Since ${\mathfrak {h}}(x)=p>0$ holds, there exists (by continuity) $0<\sigma <\min \big (1,\frac {p}{{\mathfrak {h}}(v)}\big )$ with

     
     Then, given $\lambda ,\mu \in (0,\sigma )$ , (2.5) applied to $z=x$ and yields
     
     We obtain . Since U is open with $x\in U$ , there exists $0<\varepsilon <\sigma ^2$ with , hence . The condition (2.4) thus holds for $\delta :=\varepsilon $ and .▪

6. (vi) If $0\neq {\mathfrak {h}}\in {\mathsf {Sem}(H)}$ , then is harmonic.

   Proof $B:={\mathfrak {h}}^{-1}(0)$ is closed in H as ${\mathfrak {h}}$ is continuous. Hence, is closed in U. The reverse triangle inequality yields (observe $|{\mathfrak {h}}(z+w)- {\mathfrak {h}}(w)|\leq {\mathfrak {h}}(z)$ for all $z,w\in H$ )

   (2.6) $$ \begin{align} {\mathfrak{h}}(z + w)={\mathfrak{h}}(w)\qquad\quad \forall\: z\in B,\: w\in H. \end{align} $$
   Since ${\mathfrak {h}}\neq 0$ holds, there exists some $u\in H\setminus B$ . Let now $x\in A$ and $0\neq v\in H$ be given:

   • • Let ${\mathfrak {h}}(v)>0$ . Then, holds, by (2.6) applied to $z=x$ and for $\mu \in (0,\infty )$ . Since U is open with $x\in U$ , there exists $\varepsilon>0$ with , hence . Condition (2.4) thus holds for $\delta :=\varepsilon $ and .

   • • Let ${\mathfrak {h}}(v)=0$ . We obtain for $t,\mu \in (0,\infty )$ that

     
     holds, hence . Since U is open with $x\in U$ , there exists $\varepsilon>0$ with , so that (2.4) holds for $\delta :=\varepsilon $ and . ▪

Notably, the statement in (iv) also follows from (i), (v), and (vi):

Proof Let for $H={\mathbb {R}}\times E$ . Then, $0\neq {\mathfrak {h}}\in {\mathsf {Sem}(H)}$ holds, with ${\mathfrak {h}}^{-1}(p)=\{-p,p\}\times E$ . Hence, we have for $U={\mathbb {R}}\times V$ , so that (v) ( $p\neq 0$ ) and (vi) ( $p=0$ ) show that $A\subseteq U$ is harmonic. By (i), then also $B:= \{p\}\times V\subseteq A$ is harmonic, as nonempty and closed in U.▪

We have the following statement.

Lemma 2.7 Let $H,F\in {\mathsf {hlcVect}}$ , $U\subseteq H$ nonempty open, $A\subseteq U$ harmonic, and $S\subseteq H$ a subset with $U\subseteq S$ . Let $f\in C^k(U\setminus A,F)$ for be given. For each $0\leq \ell \preceq k$ , let be continuous with

$$ \begin{align*}\Phi^{\ell}|_{(U\setminus A)\times H^{\ell}}={\mathrm{d}}^{\ell} f. \end{align*} $$

Then, we have $\tilde {f}:=\Phi ^0|_{U}\in C^k(U,F)$ , with ${\mathrm {d}}^{\ell } \tilde {f}=\Phi ^{\ell }|_{U\times H^{\ell }}$ for all $0\leq \ell \preceq k$ .

Proof By definition, we have $\tilde {f}\in C^0(U,F)$ with ${\mathrm {d}}^0 \tilde {f}=\tilde {f}=\Phi ^0|_{U}$ . We thus can assume that there exists $0\leq q<k$ , such that $\tilde {f}$ is of class $C^q$ with ${\mathrm {d}}^{\ell } \tilde {f}=\Phi ^{\ell }|_{U\times H^{\ell }}$ for all $0\leq \ell \leq q$ . The claim then follows by induction once we have shown thatFootnote ⁷

(2.7)

holds for all $x\in A$ and $v_1,\ldots ,v_{q},v\in H$ . To show (2.7), we choose $\delta>0$ and

as in (2.4), and consider the maps

• • By assumption, we have

  

• • By compactness and continuity, we have for each ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ .

Since the Riemann integral is $C^0$ -continuous (used in the second step), and since $\Phi ^q$ is continuous (used in the last step), we obtain for $0<h< \delta $ that (in the fourth step, we apply (2.2) as well as Proposition 2.2.(d))

holds. Together with (2.3) this implies (2.7).▪

2.3 Particular mapping spaces

Let $H,F\in {\mathsf {hlcVect}}$ and be given. Let ${\Omega }(H)$ denote the set of all pairs $(U,{\mathfrak {U}})$ that consist of a nonempty open subset $U\subseteq H$ , and a subset ${\mathfrak {U}}\subseteq {\mathsf {clos}(U)}$ with $U\subseteq {\mathfrak {U}}$ . Let $\mathcal {C}_{{\mathfrak {U}}}^k(U,F)$ denote the set of all $f\in C^k(U,F)$ , such that ${\mathrm {d}}^{\ell } f$ extends for $0\leq \ell \preceq k$ to a continuous map .

Remark 2.8 Let $1\leq \ell \preceq k$ , $(U,{\mathfrak {U}})\in \Omega (H)$ , and $f\in \mathcal {C}_{{\mathfrak {U}}}^k(U,F)$ be given. By continuity, the map

is necessarily $\ell $ -multilinear and symmetric for each fixed $z\in {\mathfrak {U}}$ . Thus, given ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ and ${\mathrm{K}}\subseteq {\mathfrak {U}}$ compact, Lemma 2.5 provides ${\mathfrak {h}}\in {\mathsf {Sem}(H)}$ as well as $O\subseteq H$ open with ${\mathrm{K}}\subseteq O$ , such that

holds for all

and ${\underline {w}}=(w_1,\ldots ,w_{\ell })\in H^{\ell }$ .▪

We have the following corollary to Lemma 2.7.

Corollary 2.9 Let $H,F\in {\mathsf {hlcVect}}$ , $(U,{\mathfrak {U}})\in \Omega (H)$ , $A\subseteq U$ harmonic, and $f\in C^k(U\setminus A,F)$ for be given. For each $0\leq \ell \preceq k$ , let be continuous with

$$ \begin{align*} \Phi^{\ell}|_{(U\setminus A)\times H^{\ell}}={\mathrm{d}}^{\ell} f. \end{align*} $$

Then, we have $\tilde {f}:=\Phi ^0|_{U}\in \mathcal {C}^k_{\mathfrak {U}}(U,F)$ , with ${\mathrm{Ext}({\tilde {f}},{\ell })}=\Phi ^{\ell }$ for all $0\leq \ell \preceq k$ .

Corollary 2.3 provides the following statement.

For ${\mathrm{K}}\subseteq {\mathfrak {U}} $ compact, ${\mathscr {B}}\subseteq H$ bounded, ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ , $f\in \mathcal {C}_{\mathfrak {U}}^k(U,F)$ , we define

(2.8) $$ \begin{align} {\mathfrak{p}}^0_{{\mathrm{K}}}\equiv {\mathfrak{p}}[0]_{{\mathrm{K}}\times {\mathscr{B}}}(f)&:= \sup\{{\mathfrak{p}}({\mathrm{Ext}({f},{0})}(z))\:|\: z \in {\mathrm{K}}\}, \notag \\ {\mathfrak{p}}[\ell]_{{\mathrm{K}}\times {\mathscr{B}}}(f)&:= \sup\{{\mathfrak{p}}({\mathrm{Ext}({f},{\ell})}(z,{\underline{w}}))\:|\: z \in {\mathrm{K}},\: {\underline{w}}\in {\mathscr{B}}^{\ell}\}\qquad\quad\forall\: 1\leq \ell\preceq k, \notag \\ {\mathfrak{p}}^{\mathrm{s}}_{{\mathrm{K}}\times {\mathscr{B}}}(f)&:= \max\big(0\leq \ell\leq {\mathrm{s}}\:\big|\: {\mathfrak{p}}[\ell]_{{\mathrm{K}}\times {\mathscr{B}}}(f)\big)\hspace{84.4pt}\forall\: 0\leq {\mathrm{s}}\preceq k. \end{align} $$

Finally assume $H=P\times E$ with $P,E\in {\mathsf {hlcVect}}$ . Then,

$$ \begin{align*} &(U,{\mathfrak{U}}):=(W\times V, {\mathfrak{W}}\times {\mathfrak{V}}) \in {\Omega}(H)\quad\text{ holds for all }\quad \\ &\quad (W,{\mathfrak{W}})\in {\Omega}(P)\quad\text{ and }\quad (V,{\mathfrak{V}})\in {\Omega}(E).\\[-15pt] \end{align*} $$

In the following, we will rather denote

$$ \begin{align*} \mathcal{C}_{{\mathfrak{V}}}^k(W\times V,F):=\mathcal{C}_{{\mathfrak{U}}}^k(W\times V,F),\\[-15pt] \end{align*} $$

as it will be clear from the context, which ${\mathfrak {W}}\subseteq {\mathsf {clos}(W)}$ has to be assigned to some given $W\subseteq P$ .

3.1 Statement of the main result

Let $F\in {\mathsf {hlcVect}}$ and be fixed. For each $E\in {\mathsf {hlcVect}}$ , we set $H[E]:={\mathbb {R}}\times E$ , and defineFootnote ⁸

(3.1) $$ \begin{align} \mathcal{C}_{{\mathfrak{V}}}^k((a,b)\times V,F)&:=\mathcal{C}_{(a,b]\times {\mathfrak{V}}}^k((a,b)\times V,F)\qquad\quad\forall\: -\infty\leq a < b\leq \infty, \\[-15pt]\nonumber\end{align} $$

for each $(V,{\mathfrak {V}})\in {\Omega (E)}$ . For a bounded subset ${\mathcal {B}}\subseteq E$ , we set

(3.2)

Let $R\subseteq {\mathbb {R}}$ be a subset, and $\mathcal {W}\subseteq E$ a linear subset.

• • For each $x\in E$ and $\ell \in {\mathbb {N}}$ , we define

  
  hence .

• • Given $\bar {E}\in {\mathsf {hlcVect}}$ , $\bar {x}\in \bar {E}$ , $\ell \in {\mathbb {N}}$ , and a linear map , we define

  
  hence .

Our main result states the following.

Theorem 3.1 Let $-\infty \leq a<\tau <b< \infty $ be fixed. There exist linear (extension) maps

for $E\in {\mathsf {hlcVect}}$ and $(V,{\mathfrak {V}})\in {\Omega (E)}$ , such that the following conditions are fulfilled:

1. (1) For $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in \Omega (E)$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((a,b)\times V,F)$ , $0\leq \ell \preceq k$ , we have

   $$ \begin{align*} {\mathrm{Ext}({{\mathcal{E}}_{a,\tau,b}(E,V,{\mathfrak{V}})(f)},{\ell})}|_{(a,b]\times {\mathfrak{V}}\times{H}[E]^{\ell}}\hspace{14pt}&={\mathrm{Ext}({f},{\ell})},\\ {\mathrm{Ext}({{\mathcal{E}}_{a,\tau,b}(E,V,{\mathfrak{V}})(f)},{\ell})}|_{[2b-\tau,\infty)\times {\mathfrak{V}}\times {H}[E]^{\ell}}&=0. \end{align*} $$

2. (2) There exist constants $\{C_{\mathrm {s}}\}_{0\leq {\mathrm {s}}\preceq k}\subseteq [1,\infty )$ , such that the following assertions hold for each $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $t\in (b,\infty )$ , $x\in {\mathfrak {V}}$ , ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ , and $f\in \mathcal {C}_{{\mathfrak {V}}}^k((a,b)\times V,F)$ :

   • • We have .

   • • For $1\leq \ell \leq {\mathrm {s}}\preceq k$ , ${\mathcal {B}}\subseteq E$ bounded, and ${\underline {w}}=((\lambda _1, X_1),\ldots ,(\lambda _{\ell }, X_{\ell }))\in ({\mathbb {R}}\times {\mathcal {B}})^{\ell }$ we have

     

3. (3) Let $E,\bar {E}\in {\mathsf {hlcVect}}$ , $\mathcal {W}\subseteq E$ a linear subspace, a linear map, $(V,{\mathfrak {V}})\in {\Omega }(E)$ , $(\bar {V},\bar {{\mathfrak {V}}})\in {\Omega }(\bar {E})$ , as well as

   $$ \begin{gather*} f\in \mathcal{C}^k_{{\mathfrak{V}}}((a,b)\times V,F),\quad\:\: \bar{f}\in \mathcal{C}_{\bar{{\mathfrak{V}}}}^k((a,b)\times \bar{V},F),\\ \quad\:\: x\in {\mathfrak{V}},\quad\:\: \bar{x}\in \bar{{\mathfrak{V}}},\quad\:\: 0\leq {\mathrm{s}}\preceq k. \end{gather*} $$
   Then, the first line implies the second line:
   

Remark 3.2 The extension operator in Theorem 3.1 and the constants $\{C_{\mathrm {s}}\}_{0\leq {\mathrm {s}}\preceq k}$ in Part (2), only depend on the choice of some fixed $\varrho \in C^\infty ({\mathbb {R}},{\mathbb {R}})$ with

$$ \begin{align*} |\varrho|\leq 1,\qquad\varrho|_{(-\infty,\tau]}=0,\qquad\varrho|_{[\upsilon,b]}=1, \end{align*} $$

for some $\tau <\upsilon < b$ . Specifically, see (4.1) for the case $a=-\infty $ and $b=0$ as well as (4.3) for an ad hoc definition of the extension $\tilde {f}\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,\infty )\times V,F)$ of some given $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ . See also (4.27) and (4.29) for the definition of the constants $\{C_{\mathrm {s}}\}_{0\leq {\mathrm {s}}\preceq k}$ via the constants (4.2), i.e.,

$$ \begin{align*} M_p= \sup \big\{ {\hspace{1pt}} \big| {\hspace{1pt}}\varrho^{(j)}(t){\hspace{1pt}}\big| \:\: \Big| \: \:t\in [\tau,0],\: 0\leq j\leq p\:\big\}\qquad\quad\forall\: p\in {\mathbb N}. \end{align*} $$

Remark 3.3 Let $E,F$ be normed spaces, and recall the definitions made in Remark 2.1. Given $(V,{\mathfrak {V}})\in \Omega (E)$ , $a<b$ and $k\in {\mathbb {N}}$ , let $\mathcal {FC}^k_{\mathfrak {V}}((a,b)\times V,F)$ denote the set of all $f\in \mathcal {F}C^k((a,b)\times V, F)$ , such that $D^{(\ell )}f$ extends for $0\leq \ell \preceq k$ to a continuous map

. Seeley already mentioned in [Reference Seeley17] that his construction also works for smooth ${\mathbb {R}}$ -valued functions defined on half Banach spaces. Expectably, the same holds true for the construction made in Section 4, then leading to extension operators

for $a<\tau <b$ , $(V,{\mathfrak {V}})\in \Omega (E)$ , $k\in {\mathbb {N}}$ that admit properties analogous to that in Theorem 3.1. We will not provide the details in this paper, but mention that Theorem 3.1 together with Remark 2.1 already provides the extension operatorsFootnote ⁹

for $V\subseteq E$ non-empty open and $a<\tau <b$ .▪

We close this section with the following summarizing corollary to Lemma 2.4 and Remark 2.8 that we shall need for our estimates in Section 4.3.

Corollary 3.5 Let $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in \Omega (E)$ , $-\infty \leq a<c<d \leq b<\infty $ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((a,b)\times V,F)$ , ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ , $1\leq \ell \preceq k$ , and ${\mathrm{K}}\subseteq {\mathfrak {V}}$ be compact. There exist ${\widetilde {C}}_{\ell }\geq 1$ , ${\mathfrak {q}}\in {\mathsf {Sem}(E)}$ , and $U\subseteq E$ open with ${\mathrm{K}}\subseteq U$ , such thatFootnote ¹⁰

holds for all

and ${\underline {w}}=(w_1,\ldots ,w_{\ell })\in H[E]^{\ell }$ .

Proof According toFootnote ¹¹ Lemma 2.4 and Remark 2.8, there exist ${\widetilde {C}}_{\ell }\geq 1$ , ${\mathfrak {q}}\in {\mathsf {Sem}(E)}$ , as well as $O\subseteq H[E]$ open with $[c,d]\times {\mathrm {K}}\subseteq O$ , such that

(3.3)

holds for all and ${\underline {w}}=(w_1,\ldots ,w_{\ell })\in H[E]^{\ell }$ . By compactness, there exists $U\subseteq E$ open with ${\mathrm {K}}\subseteq U$ , such that $[c,d]\times U\subseteq O$ holds. Then, (3.3) holds for each and ${\underline {w}}=(w_1,\ldots ,w_{\ell })\in H[E]^{\ell }$ , which proves the claim.▪

3.2 Multiple variables

Let $F\in {\mathsf {hlcVect}}$ and

be fixed. For $n\geq 1$ and $E\in {\mathsf {hlcVect}}$ , we define ${H}[E,n]:={\mathbb {R}}^n\times E$ . Given ${\underline {a}}=(a_1,\ldots ,a_n)$ , ${\underline {{\tau }}}=(\tau _1,\ldots ,\tau _n)$ , ${\underline {b}}=(b_1,\ldots ,b_n)$ with $-\infty \leq a_i<\tau _i<b_i\leq \infty $ for $i=1,\ldots ,n$ , we set

If $b_1,\ldots ,b_n=\infty $ holds, we also denote ${\underline {b}}={\underline {{\infty }}}$ , and observe that then ${\mathcal {Q}}({\underline {a}},{\underline {b}})={\breve {\mathcal {Q}}}({\underline {a}},{\underline {b}})$ holds according to our conventions concerning intervals. For each $(V,{\mathfrak {V}})\in {\Omega (E)}$ , we set

$$ \begin{align*} \mathcal{C}_{{\mathfrak{V}}}^k({\mathcal{Q}}({\underline{a}},{\underline{b}})\times V,F):=\mathcal{C}_{{\breve{\mathcal{Q}}}({\underline{a}},{\underline{b}})\times {\mathfrak{V}}}^k({\mathcal{Q}}({\underline{a}},{\underline{b}})\times V,F). \end{align*} $$

Theorem 3.1 provides the following statement.

Application 3.6 Let $n\geq 1$ , $E\in {\mathsf {hlcVect}}$ , and $(V,{\mathfrak {V}})\in {\Omega (E)}$ . Let ${\underline {a}}=(a_1,\ldots ,a_n)$ , ${\underline {{\tau }}}=(\tau _1,\ldots ,\tau _n)$ , ${\underline {b}}=(b_1,\ldots ,b_n)$ be given with $-\infty \leq a_i<\tau _i<b_i< \infty $ for $i=1,\ldots ,n$ . There exists a linear (extension) map

that admits the following two properties:

1. (a) For $f\in \mathcal {C}_{{\mathfrak {V}}}^k({\mathcal {Q}}({\underline {a}},{\underline {b}})\times V,F)$ and $0\leq \ell \preceq k$ , we have

   $$ \begin{align*} {\mathrm{Ext}({{\mathcal{E}}_{{\underline{a}},{\underline{{\tau}}},{\underline{b}}}(E,V,{\mathfrak{V}})(f)},{\ell})}|_{{\breve{\mathcal{Q}}}({\underline{a}},{\underline{b}})\times {\mathfrak{V}}\times{H}[E,n]^{\ell}}={\mathrm{Ext}({f},{\ell})}. \end{align*} $$

2. (b) Let ${\underline {y}}=(y_1,\ldots ,y_n)\in {\mathcal {Q}}({\underline {a}},{\underline {{\infty }}})$ be given, with $y_i\geq 2b_i-\tau _i$ for some $1\leq i\leq n$ . Then,

   $$ \begin{align*} {\mathrm{Ext}({{\mathcal{E}}_{{\underline{a}},{\underline{{\tau}}},{\underline{b}}}(E,V,{\mathfrak{V}})(f)},{\ell})}|_{\{{\underline{y}}\}\times {\mathfrak{V}}\times H[E,n]^{\ell}}=0 \end{align*} $$
   holds for each $f\in \mathcal {C}_{{\mathfrak {V}}}^k({\mathcal {Q}}({\underline {a}},{\underline {b}})\times V,F)$ and $0\leq \ell \preceq k$ .

Proof According to Theorem 3.1, we can assume that the claim holds for some $n\geq 1$ . Let thus $-\infty \leq a_0,\ldots ,a_n,\tau _0,\ldots ,\tau _n,b_0,\ldots ,b_n<\infty $ be given with $a_i<\tau _i<b_i$ for $i=0,\ldots ,n$ , and define

$$ \begin{align*} {\underline{a}}&:=(a_1,\ldots,a_{n}),\qquad{\underline{a}}_0:=(a_0,\ldots,a_n),\\ {\underline{{\tau}}}&:=(\tau_1,\ldots,\tau_{n}),\qquad\hspace{1.8pt}{\underline{{\tau}}}_0:=(\tau_0,\ldots,\tau_n),\\ {\underline{b}}&:=(b_1,\ldots,b_{n}),\qquad\hspace{1.25pt}{\underline{b}}_0:=(b_0,\ldots,b_n). \end{align*} $$

Let $E\in {\mathsf {hlcVect}}$ and $(V,{\mathfrak {V}})\in {\Omega (E)}$ be given. We set ${\widetilde {E}}:={\mathbb {R}}\times E$ , ${\widehat {E}}:={\mathbb {R}}^{n}\times E$ , as well as

$$ \begin{align*} ({\widetilde{V}},{\widetilde{\mathfrak{V}}})&:=\hspace{4pt}((a_{0},b_{0})\times V, (a_{0},b_{0}]\hspace{0.9pt}\times {\mathfrak{V}})\in {\Omega}({\widetilde{{E}}}),\\ (\widehat{V}, \widehat{{\mathfrak{V}}} ) &:=({\mathcal{Q}}({\underline{a}},{\underline{{\infty}}})\times V,{\mathcal{Q}}({\underline{a}},{\underline{{\infty}}})\times {\mathfrak{V}})\in {\Omega} (\widehat{E}). \end{align*} $$

The induction hypotheses provides the extension operator

(3.4)

Theorem 3.1 provides the extension operator

(3.5)

We consider the linear map

so that under the identifications made we have

Let now $f\in \mathcal {C}^k_{{\mathfrak {V}}}({\mathcal {Q}}({\underline {a}}_0,{\underline {b}}_0)\times V,F)$ and ${\underline {y}}=(y_0,\ldots ,y_n)\in {\mathcal {Q}}({\underline {a}}_0,{\underline {b}}_0)$ be given. The induction hypotheses provides the following statements:

• • Up to the identifications in (3.4), we have

  $$ \begin{align*} {\mathrm{Ext}({{\mathcal{E}}_{{\underline{a}},{\underline{{\tau}}},{\underline{b}}}({\widetilde{E}},{\widetilde{{V}}},{\widetilde{{{\mathfrak{V}}}}})(f)},{\ell})}|_{(a_0,b_0]\times ({\breve{\mathcal{Q}}}({\underline{a}},{\underline{b}})\times {\mathfrak{V}})\times{H}[{\widehat{{E}}},1]^{\ell}}&={\mathrm{Ext}({f},{\ell})}\!\!\!\!\qquad\quad\forall\: 0\leq\ell\preceq k. \end{align*} $$

• • Let $y_i\geq 2b_i-\tau _i$ for some $1\leq i\leq n$ , as well as $y_0\in (a_0,b_0)$ . Then, up to the identifications in (3.4), we have

  (3.6)

  

Theorem 3.1.(1) (for ${\mathcal {E}}_{a_0,\tau _0,b_0}({\widehat {E}},{\widehat {V}},{\widehat {{{\mathfrak {V}}}}})$ ), provides the following statements:

• • For $0\leq \ell \preceq k$ , we have (under the identification in (3.4))

  $$ \begin{align*} &{\mathrm{Ext}({{\mathcal{E}}_{a_0,\tau_0,b_0}({\widehat{E}},{\widehat{V}},{\widehat{\mathfrak{V}}})({\mathcal{E}}_{{\underline{a}}, {\underline{{\tau}}},{\underline{b}}}({\widetilde{E}},{\widetilde{{V}}},{\widetilde{{{\mathfrak{V}}}}})(f))},{\ell})}|_{(a_0,b_0] \times {\widehat{{{\mathfrak{V}}}}}\times{H}[{\widehat{{E}}},1]^{\ell}}\\&\quad ={\mathrm{Ext}({{\mathcal{E}}_{{\underline{a}}, {\underline{{\tau}}},{\underline{b}}}({\widetilde{E}},{\widetilde{{V}}},{\widetilde{{{\mathfrak{V}}}}})(f)},{\ell})}. \end{align*} $$

• • If $y_0\geq 2b_0-\tau _0$ holds, then for $0\leq \ell \preceq k$ , we have

  (3.7)

  

We obtain for $0\leq \ell \preceq k$ (under the identification in (3.5) in the first step) that

$$ \begin{align*} \mathrm{Ext} &({\mathcal E}_{\underline{a}_{0}, \underline{\tau}_{0}, \underline{b}_{0}} (E, V,{\mathfrak{V}})(f), {\ell})|_{{\breve{\mathcal{Q}}}({\underline{a}}_0,{\underline{b}}_0)\times {\mathfrak{V}}\times {H}[E,n+1]^{\ell}}\\ &\qquad ={\mathrm{Ext}({{\mathcal{E}}_{a_0,\tau_0,b_0}({\widehat{E}},{\widehat{V}},{\widehat{\mathfrak{V}}})({\mathcal{E}}_{{\underline{a}}, {\underline{{\tau}}},{\underline{b}}}({\widetilde{E}},{\widetilde{{V}}},{\widetilde{{{\mathfrak{V}}}}})(f))},{\ell})}|_{(a_0,b_0] \times ({\breve{\mathcal{Q}}}({\underline{a}},{\underline{b}})\times {\mathfrak{V}})\times{H}[{\widehat{E}},1]^{\ell}} \\ &\qquad ={\mathrm{Ext}({{\mathcal{E}}_{{\underline{a}},{\underline{{\tau}}},{\underline{b}}}({\widetilde{E}},{\widetilde{{V}}}, {\widetilde{{{\mathfrak{V}}}}})(f)},{\ell})}|_{(a_0,b_0] \times ({\breve{\mathcal{Q}}}({\underline{a}},{\underline{b}})\times {\mathfrak{V}})\times{H}[{\widehat{E}},1]^{\ell}} \\ &\qquad = {\mathrm{Ext}({f},{\ell})} \end{align*} $$

holds, which proves Part (a). Finally, assume that $y_i\geq 2b_i-\tau _i$ holds for some $1\leq i\leq n$ . Then, Theorem 3.1.(2) together with (3.6) shows

for each ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ and $0\leq \ell \preceq k$ . Together with (3.7), this proves Part (b).▪

Remark 3.7 Domains as considered in Application 3.6 occur, e.g., in the context of manifolds with corners as (open subsets of) quadrants in Hausdorff locally convex vector spaces. Specifically, let $H\in {\mathsf {hlcVect}}$ , and

with $n\geq 1$ be linearly independent continuous linear maps. Consider the closed subspace

, and let ${\underline {a}}:=(-\infty ,\dots ,-\infty )$ , ${\underline {b}}:=(0,\dots ,0)$ (both n-times). Then, we have

, and the corresponding open and closed quadrants $\mathrm {Q}\subseteq H$ and $\breve {\mathrm {Q}}\subseteq H$ , respectively, are given by

Proof Let $e_1,\dots ,e_n\in H$ be linearly independent with ${\mathcal {L}}_i(e_j)=\delta _{ij}$ for $1\leq i,j\leq n$ , and set

. Then,

is a continuous projection operator, with

,

,

as topological spaces. Moreover, the following maps are continuous, linear, and inverse to each other:

Since

,

, and

are homeomorphic, the claim follows.▪

3.3 Particular subsets in infinite dimensions

Let $P,E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in \Omega (E)$ , $0< \tau <1$ , be fixed, and set $H:=P\times E$ . Let be a continuous map that admits the following properties:

1. (1) $\xi $ is of class $C^k$ on ${\mathcal {W}}:= \xi ^{-1}((0,\infty ))$ ,

2. (2) $\xi (Z)\neq 0$ for some $Z\in P$ ,

3. (3) for each $\lambda \in [0,\infty )$ and $Z\in P$ , and

4. (4) $\mathcal {A}:=\xi ^{-1}(1)\subseteq {\mathcal {W}}$ is harmonic.

We consider the sets:

• • ${\mathcal {O}}:=\xi ^{-1}((0,1))$ ,

• • ${\breve {\mathcal {O}}}:=\xi ^{-1}((0,1])$ ,

• • $\mathcal {U}:=\xi ^{-1}((1,\infty ))$ ,

• • $\mathcal {T}:= \xi ^{-1}([2-\tau ,\infty ))$ ,

• • $\mathcal {V}:=\xi ^{-1}((2-\tau ,\infty ))$ ,

• • ${\mathcal {J}}:= {\mathcal {W}}\setminus \mathcal {A}$ , and

• • $A:= \mathcal {A}\times V$ .

These sets are nonempty by (2) and (3); and, by continuity of $\xi $ , the sets ${\mathcal {W}},{\mathcal {O}},\mathcal {U},\mathcal {V},{\mathcal {J}}$ are open. Moreover, $A\subseteq {\mathcal {W}}\times V$ is harmonic by Example 2.6.(iii), and the condition (3) implies:

• $-$ ${\breve {\mathcal {O}}}\subseteq {\mathsf {clos}({{\mathcal {O}}})}$ , hence $({\mathcal {O}},{\breve {\mathcal {O}}})\in \Omega (P)$ and $({\mathcal {O}}\times V,{\breve {\mathcal {O}}} \times {\mathfrak {V}})\in {\Omega }(H)$ ,

• $-$ $\mathcal {T}\subseteq {\mathsf {clos}({\mathcal {V}})}$ , and

• $-$ ${\mathcal {J}}\subseteq {\mathsf {clos}({{\mathcal {W}}})}$ , hence $({\mathcal {J}},{\mathcal {W}})\in \Omega (P)$ and $({\mathcal {J}}\times V,{\mathcal {W}} \times {\mathfrak {V}})\in {\Omega }(H)$ .

We define

$$ \begin{align*} \mathcal{C}_{{\mathfrak{V}}}^k({\mathcal{O}}\times V,F)\hspace{3pt}:=&\:{\hspace{1pt}}\mathcal{C}_{{\breve{\mathcal{O}}}\times {\mathfrak{V}}}^k({\mathcal{O}}\times V,F),\\ \mathcal{C}_{{\mathfrak{V}}}^k({\mathcal{W}}\times V,F):= &\:{\hspace{1pt}}\mathcal{C}_{{\mathcal{W}}\times {\mathfrak{V}}}^k({\mathcal{W}}\times V,F). \end{align*} $$

In this section, we prove the following statement.

Application 3.8 There exists a linear (extension) map

such that for all $f\in \mathcal {C}_{{\mathfrak {V}}}^k({\mathcal {O}}\times V,F)$ and $0\leq \ell \preceq k$ , we have

(3.8) $$ \begin{align} {\mathrm{Ext}({{\mathcal{E}}(f)},{\ell})}|_{{\breve{\mathcal{O}}}\times {\mathfrak{V}}\times H^{\ell}}={\mathrm{Ext}({f},{\ell})}\qquad\quad\text{and}\qquad\quad {\mathrm{Ext}({{\mathcal{E}}(f)},{\ell})}|_{\mathcal{T}\times {\mathfrak{V}}\times H^{\ell}}=0. \end{align} $$

Remark 3.9 Application 3.8 holds in the same form if ${\mathcal {O}}$ is replaced by ${\mathcal {S}}:=\xi ^{-1}([0,1))$ , ${\breve {\mathcal {O}}}$ is replaced by ${\breve {\mathcal {S}}}:=\xi ^{-1}([0,1])$ , and ${\mathcal {W}}$ is replaced by P.

Proof We have $({\mathcal {S}},{\breve {\mathcal {S}}})\in \Omega (P)$ by continuity of $\xi $ as well as by (2) and (3). Let now $f\in \mathcal {C}_{{\mathfrak {V}}}^k({\mathcal {S}}\times V,F)$ be given. Then, $f|_{{\mathcal {O}}\times V}\in \mathcal {C}_{{\mathfrak {V}}}^k({\mathcal {O}}\times V,F)$ holds by Lemma 2.10 (with $O \equiv P$ and $\psi \equiv {\mathrm {id}}_P$ ). Hence, we have ${\mathcal {E}}(f|_{{\mathcal {O}}\times V})\in \mathcal {C}_{{\mathfrak {V}}}^k({\mathcal {W}}\times V,F)$ , with ${\mathcal {E}}$ as in Application 3.8. We define

• • By construction, we have $\tilde {{\mathcal {E}}}(f)|_{\mathcal {W}\times V}={\mathcal {E}}(f|_{{\mathcal {O}}\times V})|_{\mathcal {W}\times V}$ , as well as $\tilde {{\mathcal {E}}}(f)|_{{\mathcal {S}}\times V}=f$

  $$ \begin{align*} \text{by}\qquad \tilde{{\mathcal{E}}}(f)|_{{\mathcal{O}}\times V}={\mathcal{E}}(f|_{{\mathcal{O}}\times V})|_{{\mathcal{O}}\times V}\stackrel{(3.8)}{=}f|_{{\mathcal{O}}\times V}\qquad\text{and}\qquad \tilde{{\mathcal{E}}}(f)|_{\{0\}\times V}=f|_{\{0\}\times V}. \end{align*} $$
  Since $\mathcal {W}\times V$ , ${\mathcal {S}}\times V$ are open with , the map $\tilde {{\mathcal {E}}}(f)$ is of class $C^k$ with
  $$ \begin{align*} ({\mathrm{d}}^{\ell} \tilde{{\mathcal{E}}}(f))|_{{\mathcal{W}}\times V\times H^{\ell}} &= {\mathrm{d}}^{\ell} {\mathcal{E}}(f|_{{\mathcal{O}}\times V})= {\mathrm{Ext}({{\mathcal{E}}(f|_{{\mathcal{O}}\times V})},{\ell})}|_{{\mathcal{W}}\times V\times H^{\ell}} \qquad\quad\hspace{-2pt} \forall\: 0\leq \ell\preceq k,\\ ({\mathrm{d}}^{\ell} \tilde{{\mathcal{E}}}(f))|_{{\mathcal{S}}\times V\times H^{\ell}}\hspace{3.6pt} &=\hspace{19pt} {\mathrm{d}}^{\ell} f\hspace{15pt}= {\mathrm{Ext}({f},{\ell})}|_{{\mathcal{S}}\times V\times H^{\ell}} \qquad\quad\hspace{36pt}\forall\: 0\leq \ell\preceq k. \end{align*} $$
  Then, continuity implies $\tilde {{\mathcal {E}}}(f)\in \mathcal {C}_{{\mathfrak {V}}}^k(P\times V,F)$ , with (observe $\xi ^{-1}(0)\subseteq {\breve {\mathcal {S}}}$ )
  $$ \begin{align*} {\mathrm{Ext}({\tilde{{\mathcal{E}}}(f)},{\ell})}|_{{\mathcal{W}}\times {\mathfrak{V}}\times H^{\ell}}&={\mathrm{Ext}({{\mathcal{E}}(f|_{{\mathcal{O}}\times V})},{\ell})} \qquad\:\:\text{and}\qquad\:\: \\ {\mathrm{Ext}({\tilde{{\mathcal{E}}}(f)},{\ell})}|_{{\breve{\mathcal{S}}}\times {\mathfrak{V}}\times H^{\ell}}&={\mathrm{Ext}({f},{\ell})}, \end{align*} $$
  for $0\leq \ell \preceq k$ . We thus have the linear (extension) map
  

• • By construction, we have $\tilde {{\mathcal {E}}}(f)|_{\mathcal {T}\times V}= {\mathcal {E}}(f|_{{\mathcal {O}}\times V})|_{\mathcal {T}\times V}\stackrel {(3.8)}{=}0$ . Since $\mathcal {T}\times V$ is open, we obtain

  $$ \begin{align*} ({\mathrm{d}}^{\ell} \tilde{{\mathcal{E}}}(f))|_{\mathcal{T}\times V\times H^{\ell}} = 0\qquad\stackrel{\text{continuity}}{\Longrightarrow}\qquad {\mathrm{Ext}({\tilde{{\mathcal{E}}}(f)},{\ell})}|_{\mathcal{T}\times {\mathfrak{V}}\times H^{\ell}}=0 \end{align*} $$
  for $0\leq \ell \preceq k$ .▪

Let $\rho \in C^\infty ({\mathbb {R}},{\mathbb {R}})$ be given withFootnote ¹³

(3.9) $$ \begin{align} {\mathrm{id}}_{(0,1)}<\rho|_{(0,1)}<1,\qquad \rho(1)=1,\qquad {\mathrm{id}}_{(1,\infty)}<\rho|_{(1,\infty)}. \\[-28pt]\nonumber\end{align} $$

• • We consider the smooth map . Lemma 2.10 and (3) imply

  

• • Theorem 3.1 provides the extension operator $\hat {{\mathcal {E}}}\equiv {\mathcal {E}}_{0,\tau ,1}(H,{\mathcal {O}}\times V,{\breve {\mathcal {O}}}\times {\mathfrak {V}})$ , hence

  

• • We consider the $C^k$ -map (recall (1), (3) and (3.9))

  
  By construction, we have , and furthermore for $Y\subseteq E$ :

  • – $\mu ({\mathcal {W}}\times Y)\subseteq (0,\infty )\times {\breve {\mathcal {O}}} \times Y$ ,

  • – $\mu ( {\mathcal {O}}\times Y)\subseteq \hspace {5.5pt}(0,1)\times {\mathcal {O}}\times Y$ , and

  • – $\mu (\mathcal {U}\times Y)\subseteq (1,\infty )\times {\mathcal {O}}\times Y$ .

Let now $f\in \mathcal {C}_{{\breve {\mathcal {O}}}\times {\mathfrak {V}}}^k({\mathcal {O}}\times V,F)$ be given. We set

• • We have $\alpha \in \mathcal {C}^k_{{\mathcal {W}}\times {\mathfrak {V}}}( {\mathcal {J}} \times V,F)$ by Lemma 2.10, because

  $$ \begin{align*} \mu({\mathcal{J}}\times V)&\subseteq (0,\infty)\times {\mathcal{O}}\times V\qquad\text{and}\qquad \mu({\mathcal{W}}\times {\mathfrak{V}})\subseteq (0,\infty)\times {\breve{\mathcal{O}}}\times {\mathfrak{V}}\qquad\text{holds}. \end{align*} $$

• • Since $\mu ( {\mathcal {O}}\times V)\subseteq (0,1)\times {\mathcal {O}}\times V$ holds, we have by Theorem 3.1.(1)

  (3.10)

  

We consider the continuous maps

and proceed as follows:

• • By construction, we have

  (3.11) $$ \begin{align} \Phi^{\ell}|_{{\mathcal{J}}\times V\times H^{\ell}}={\mathrm{Ext}({\alpha},{\ell})}|_{{\mathcal{J}}\times V\times H^{\ell}}={\mathrm{d}}^{\ell}\alpha \qquad\quad\forall\: 0\leq \ell\preceq k. \end{align} $$
  Corollary 2.9 (with $f\equiv \alpha $ , $A\equiv \mathcal {A}\times V$ , $U\equiv {\mathcal {W}}\times V$ , ${\mathfrak {U}}\equiv {\mathcal {W}}\times {\mathfrak {V}}$ , i.e., $U\setminus A={\mathcal {J}}\times V$ ) thus shows
  $$ \begin{align*} &\tilde{f}:=\Phi^0|_{{\mathcal{W}}\times V}\in \mathcal{C}^k_{{\mathcal{W}}\times {\mathfrak{V}}}({{\mathcal{W}}\times V},F)\\ &\qquad\text{with} \qquad {\mathrm{Ext}({\tilde{f}},{\ell})}=\Phi^{\ell}\qquad\text{for all}\qquad 0\leq \ell\preceq k. \end{align*} $$

• • We obtain from (3.10) and (3.11) that

  $$ \begin{align*} \tilde{f}|_{{\mathcal{O}}\times V}=(\Phi^0|_{{\mathcal{W}}\times V})|_{{\mathcal{O}}\times V}\stackrel{(3.11)}{=}{\mathrm{Ext}({\alpha},{0})}|_{{\mathcal{O}}\times V}=\alpha|_{{\mathcal{O}}\times V}\stackrel{(3.10)}{=}f \end{align*} $$
  holds. Since ${\mathcal {O}}\times V$ is open, we obtain
  $$ \begin{align*} {\mathrm{d}}^{\ell}\tilde{f}|_{{\mathcal{O}}\times V\times H^{\ell}} ={\mathrm{d}}^{\ell} (\tilde{f}|_{{\mathcal{O}}\times V})={\mathrm{d}}^{\ell} f={\mathrm{Ext}({f},{\ell})}|_{{\mathcal{O}}\times V\times H^{\ell}} \qquad\quad\forall\: 0\leq \ell\preceq k, \end{align*} $$
  so that continuity yields
  (3.12) $$ \begin{align} {\mathrm{Ext}({\tilde{f}},{\ell})}|_{{\breve{\mathcal{O}}}\times {\mathfrak{V}}\times H^{\ell}}={\mathrm{Ext}({f},{\ell})}\qquad\quad\forall\: 0\leq \ell\preceq k. \end{align} $$

• • We obtain from (3.11)

  
  Since $\rho |_{(1,\infty )}>{\mathrm {id}}_{(1,\infty )}$ holds, Theorem 3.1.(1) yields
  
  Since $\mathcal {V}\times V$ is open, this implies ${\mathrm {d}}^{\ell }\tilde {f}|_{\mathcal {V}\times V\times H^{\ell }}=0$ for all $0\leq \ell \preceq k$ , so that continuity implies
  (3.13) $$ \begin{align} {\mathrm{Ext}({\tilde{f}},{\ell})}|_{\mathcal{T}\times {\mathfrak{V}}\times H^{\ell}}=0\qquad\quad\forall\: 0\leq \ell\preceq k. \end{align} $$

We are ready for the proof of Application 3.8:

Proof of Application 3.8 Obviously, the assignment

is linear; and the rest is clear from (3.12) and (3.13).▪

3.4 Partially constant maps and parametrizations

Let $E,F\in {\mathsf {hlcVect}}$ ,

, and $S\equiv \{S_\alpha \}_{\alpha \in I}$ be a family of disjoint subsets of E with $E=\bigcup _{\alpha \in I} S_\alpha $ . For $-\infty \leq a<b\leq \infty $ , we define

Let now $-\infty \leq a<\tau <b<\infty $ be fixed. Theorem 3.1 provides the extension operator

Theorem 3.1.(3) (for ${\mathrm {s}}\equiv 0$ ) implies

(3.14)

We can apply this in the following way. Let $H\in {\mathsf {hlcVect}}$ , and $\psi \in C^k((a,\infty )\times E,H)$ an open map, such that the following conditions are fulfilled:

1. (a) $\psi |_{\{t\}\times S_\alpha }$ is constant for each $t\in (a,\infty )$ and $\alpha \in I$ .

2. (b) For each $z\in {\mathrm {im}}[\psi ]$ , we have $\psi ^{-1}(z)=\{t(z)\}\times S_{\alpha (z)}$ , for certain $t(z)\in (a,\infty )$ and $\alpha (z)\in I$ .

3. (c) For each $z\in {\mathrm {im}}[\psi ]$ , there exist $U_z\subseteq (a,\infty )\times E$ and $W_z\subseteq {\mathrm {im}}[\psi ]$ open with $z\in W_z$ , such that is a $C^k$ -diffeomorphism, i.e., we have $(\psi |_{U_z})^{-1}\in C^k(W_z,U_z)$ .

Let $U:=\psi ((a,b)\times E)$ and ${\mathfrak {U}}:=\psi ((a,b]\times E)$ .

• • Since $\psi $ is continuous and open, we have $(U,{\mathfrak {U}})\in \Omega (H)$ .

• • Let $f\in \mathcal {C}_{{\mathfrak {U}}}^k(U,F)$ be fixed. By Lemma 2.10 and (a), we have , hence $\tilde {g}:=\mathcal {E}_S(g)\in C^k(a,\infty ,S)$ by (3.14).

• • We fix with $\iota (z)\in \psi ^{-1}(z)$ for each $z\in {\mathrm {im}}[\psi ]$ , and set

  
  This is defined by (b) and $\tilde {g}\in C^k(a,\infty ,S)$ . In particular, for each $z\in {\mathrm {im}}[\psi ]$ , we have , which shows that $\tilde {f}$ is of class $C^k$ .

We obtain the linear extension map

(3.15)

We consider the following example.

Example 3.11 Let $E={\mathbb {R}}$ , $H:={\mathbb {R}}^2$ , $a=0$ , $b=1$ , $I:=[0,2\pi )$ ,

for $\alpha \in I$ , and

According to the above definitions, we have (

denotes the euclidean norm on ${\mathbb {R}}^2$ )

$$ \begin{align*} U=\{x\in {\mathbb{R}}^2\:|\: 0<\|x\|<1\},\qquad {\mathfrak{U}}=\{x\in {\mathbb{R}}^2\:|\: 0<\|x\|\leq 1\},\qquad {\mathrm{im}}[\psi]={\mathbb{R}}^2\setminus \{0\}. \end{align*} $$

Then, (3.15) provides the linear extension map

. Let

$$ \begin{align*} {\mathscr{D}}:=\{x\in {\mathbb{R}}^2\:|\: \|x\|<1\}\qquad\quad \text{and}\qquad\quad{\breve{\mathscr{D}}}:=\{x\in {\mathbb{R}}^2\:|\: \|x\|\leq 1\}. \end{align*} $$

We obtain a linear extension map

if we set

for each $f\in \mathcal {C}^k_{{\breve {\mathscr {D}}}}({\mathscr {D}},F)$ .▪

4.1 Elementary facts and definitions

For $E\in {\mathsf {hlcVect}}$ , we define ${\mathbf {1}}:=(1,0)\in {H}[E]$ as well as ${\mathbf {1}}_p:=({\mathbf {1}},\dots ,{\mathbf {1}})\in H[E]^p$ for $p\geq 1$ , and consider the maps

We furthermore define the following:

• • For $1\leq \ell \preceq k$ and $1\leq j\leq \ell $ , we set

  

• • For $1\leq \ell \preceq k$ , $p\geq 1$ , and $1\leq z_1,\dots ,z_p\leq \ell $ , we set

  
  It helps to simplify the notations, in the following just to denote ${\boldsymbol {\Lambda }}_{\ell ,z_1,\dots ,z_p}({\underline {w}}):=1$ for the case that $p=0$ holds.

• • For $1\leq \ell \preceq k$ and $0\leq p\leq \ell $ , we let $\mathrm {I}_{\ell ,p}$ denote the set of all

  $$ \begin{align*} {\underline{{\sigma}}}=(z_1,\dots,z_p,o_1,\dots,o_{\ell-p}) \in \{1,\dots,\ell\}^{\ell},\\[-16pt] \end{align*} $$
  such that the following conditions are fulfilled:Footnote ¹⁴

  • – $z_i<z_{i+1}$ for $1\leq i \leq p-1$ ,

  • – $o_j<o_{j+1}$ for $1\leq j\leq \ell -p-1$ , and

  • – $z_i\neq o_j$ for $1\leq i \leq p$ and $1\leq j\leq \ell -p$ .

Let $V\subseteq E$ be nonempty open, $\Gamma \in C^k((0,\infty )\times V,F)$ , and $1\leq \ell \preceq k$ . By symmetry (and multilinearity) of the $\ell $ th differential, we have

(4.4)

for each $t\in (0,\infty )$ , $x\in V$ , ${\underline {w}}\in {H}[E]^{\ell }$ .

Similarly, if $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ holds, we obtain for $q=0,\dots ,\ell $ (recall Remark 2.8)

(4.5)

for each $t\in (-\infty ,0)$ , $x\in {\mathfrak {V}}$ , ${\underline {w}}\in H[E]^{\ell }$ .

4.2 Construction of the extension operators

Let $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ , and ${\boldsymbol {\varsigma }}\leq -1$ be given.

• • We define the $C^k$ -map and its $C^0$ -extension:

  (4.6)

  
  (4.7)

  

• • For $1\leq \ell \preceq k$ , we obtain from (4.4) that

  (4.8)

  
  holds for all $t\in (0,\infty )$ , $x\in V$ , ${\underline {w}}\in {H}[E]^{\ell }$ .

• • For $1\leq \ell \preceq k$ and $q=0,\dots ,\ell $ , we define the continuous map

  (4.9)

  
  for all $t\in (0,\infty )$ , $x\in {\mathfrak {V}}$ , and ${\underline {w}}\in {H}[E]^{\ell }$ . Then by (4.8), the map
  (4.10) $$ \begin{align} \Psi^{\ell}[f,{\boldsymbol{\varsigma}}]&\textstyle:= \sum_{q=0}^{\ell} {\boldsymbol{\Theta}}_{\ell,q}[f,{\boldsymbol{\varsigma}}] \in C^0((0,\infty)\times {\mathfrak{V}}\times {H}[E]^{\ell},F) \end{align} $$
  continuously extends ${\mathrm {d}}^{\ell } \Gamma [f,{\boldsymbol {\varsigma }}]$ for $1\leq \ell \preceq k$ , i.e.,
  (4.11) $$ \begin{align} \Psi^{\ell}[f,{\boldsymbol{\varsigma}}]|_{(0,\infty)\times V\times H[E]^{\ell}}={\mathrm{d}}^{\ell} \Gamma[f,{\boldsymbol{\varsigma}}]\qquad\quad\forall\: 1\leq \ell\preceq k. \end{align} $$

For $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ , we define the maps

We have the following statement.

Lemma 4.2 Let $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ be given. Then, $f_+\in C^k((0,\infty )\times V,F)$ holds, as well as

$$ \begin{align*} \Phi[f]_+^{\ell} \in C^0((0,\infty)\times {\mathfrak{V}}\times {H}[E]^{\ell},F)\qquad\quad\forall\:0\leq \ell\preceq k. \end{align*} $$

Moreover, $\Phi [f]_+^{\ell }$ restricts to ${\mathrm{d}}^{\ell } f_+$ for $0\leq \ell \preceq k$ , with $\Phi [f]_+^{\ell }|_{[-\tau ,\infty )\times {\mathfrak {V}}\times H[E]^{\ell }}=0$ .

Proof Let $s\in (0,\infty )$ as well as $0<\varepsilon < s$ be given, and set $I:=(s-\varepsilon ,s+\varepsilon )$ . There exists $N\in {\mathbb {N}}$ , such that

holds for each $j\geq N$ . Since $\varrho |_{(-\infty ,\tau ]}=0$ , we have (the first line implies the second line)

for $0\leq \ell \preceq k$ . Thus, $\Phi [f]_+^{\ell }$ is defined and continuous for $0\leq \ell \preceq k$ , $f_+$ is defined and of class $C^k$ , and $\Phi [f]_+^{\ell }$ restricts to ${\mathrm {d}}^{\ell } f_+$ for $0\leq \ell \preceq k$ by (4.11). Since

holds for each $j\in {\mathbb {N}}$ (with $\varrho |_{(-\infty ,\tau ]}=0$ ), we have $\Phi [f]_+^{\ell }|_{[\tau ,\infty )\times {\mathfrak {V}}\times H[E]^{\ell }}=0$ for $0\leq \ell \preceq k$ .▪

For $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ , and $0\leq \ell \preceq k$ , we define the map by

$$ \begin{align*} \Phi[f]^{\ell}|_{(-\infty,0]\times {\mathfrak{V}}\times H[E]^{\ell}}&:={\mathrm{Ext}({f},{\ell})}\qquad\quad\text{as well as}\\\Phi[f]^{\ell}|_{(0,\infty)\times {\mathfrak{V}}\times H[E]^{\ell}}&:=\Phi[f]^{\ell}_+. \end{align*} $$

In Section 4.3, we prove the following statement.

Together with Lemma 4.2 and Corollary 4.1, Lemma 4.3 impliesFootnote ¹⁵

(4.12) $$ \begin{align} \mathcal{C}_{\mathfrak{V}}^k({\mathbb{R}}\times V,F)\ni \tilde{f}:=&\:\Phi[f]^{0}|_{{\mathbb{R}}\times V}, \notag \\ {\mathrm{d}}^{\ell} \tilde{f}=&\:\Phi[f]^{\ell}\hspace{0.8pt}|_{{\mathbb{R}}\times V\times {H}[E]^{\ell}}\qquad\quad 0\leq \ell\preceq k,\\ {\mathrm{Ext}({\tilde{f}},{\ell})}=&\:\Phi[f]^{\ell}\qquad\qquad\qquad\quad\hspace{9pt} 0\leq \ell\preceq k. \notag \end{align} $$

For $E\in {\mathsf {hlcVect}}$ and $(V,{\mathfrak {V}})\in \Omega (E)$ , we define the map

(4.13)

We observe the following:

• • It is clear from the construction that (4.13) is a linear map, with

  $$ \begin{align*} {\mathrm{Ext}({{\mathcal{E}}_{-\infty,\tau,0}(E,V,{\mathfrak{V}})(f)},{\ell})}|_{(-\infty,0]\times {\mathfrak{V}}\times{H}[E]^{\ell}}&={\mathrm{Ext}({f},{\ell})},\\ {\mathrm{Ext}({{\mathcal{E}}_{-\infty,\tau,0}(E,V,{\mathfrak{V}})(f)},{\ell})}|_{[-\tau,\infty)\times {\mathfrak{V}}\times {H}[E]^{\ell}}&=0, \end{align*} $$
  for each $f\in \mathcal {C}_{\mathfrak {V}}^k((-\infty ,0)\times V,F)$ and $0\leq \ell \preceq k$ (for the second line use the last statement in Lemma 4.2).

• • Let $E,\bar {E}\in {\mathsf {hlcVect}}$ , $\mathcal {W}\subseteq E$ a linear subspace and a linear map. Let $(V,{\mathfrak {V}})\in {\Omega }(E)$ , $(\bar {V},\bar {{\mathfrak {V}}})\in {\Omega }(\bar {E})$ , $x\in {\mathfrak {V}}$ , $\bar {x}\in \bar {{\mathfrak {V}}}$ , $f\in \mathcal {C}^k_{{\mathfrak {V}}}({\mathbb {R}}\times V,F)$ , $\bar {f}\in \mathcal {C}_{\bar {{\mathfrak {V}}}}^k({\mathbb {R}}\times \bar {V},F)$ , and $0\leq {\mathrm {s}}\preceq k$ be given with

  
  Then, it is clear from the construction that
  
  holds for each $j\in {\mathbb {N}}$ , hence
  

To establish Theorem 3.1, it thus remains to prove Lemma 4.3 (see Section 4.3), as well as the continuity estimates in Part (2) of Theorem 3.1 (see Section 4.4).

4.3 The proof of Lemma 4.3

Let $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ , $x\in {\mathfrak {V}}$ , ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ be given. The following estimates hold for each ${\boldsymbol {\varsigma }}\leq -1$ :

1. (a) Since ${\mathrm {Ext}({f},{0})}$ is continuous, and since $[\tau ,0]$ is compact, there exists $C_0\geq 1$ and a neighbourhood $U_x\subseteq {\mathfrak {V}}$ of x, with

   (4.14) $$ \begin{align} {\mathfrak{p}}({\mathrm{Ext}({f},{0})}(t,x'))\hspace{1.1pt}\leq\hspace{1.5pt} C_0\qquad\quad\forall\: t\in [\tau,0],\: x'\in U_x.\hspace{22.5pt} \end{align} $$
   We obtain from (4.1), (4.7), and (4.14) that
   (4.15) $$ \begin{align} {\mathfrak{p}}(\Psi^0[f,{\boldsymbol{\varsigma}}](t,x'))\leq C_0\qquad\quad \forall\: t\in (0,\infty),\: x'\in U_x. \hspace{4pt} \end{align} $$

2. (b) Let $1\leq \ell \preceq k$ and ${\underline {w}}=(w_1,\dots ,w_{\ell })\in {H}[E]^{\ell }$ be given.

   • ∘ According to Point (a) and Corollary 3.5, there exists a neighborhood $U_x\subseteq {\mathfrak {V}}$ of x, ${\widetilde {{C}}}_{\ell }\geq 1$ , and ${\mathfrak {q}}\in {\mathsf {Sem}(E)}$ , such that we have

     (4.16) $$ \begin{align} {\mathfrak{p}}({\mathrm{Ext}({f},{0})}(t,x'))\leq {\widetilde{{C}}}_{\ell}\qquad\quad\forall\: t\in [\tau,0],\: x'\in U_x, \end{align} $$
     as well as
     (4.17)

     
     for each $t\in [\tau ,0]$ , $x'\in U_x$ , $1\leq q\leq \ell $ , and ${\underline {w}}'=(w^{\prime }_1,\dots ,w^{\prime }_q)\in {H}[E]^q$ .

   • ∘ We obtain for $0\leq q\leq \ell $ from (4.1), (4.2), (4.9), (4.16), (4.17) that

     
     holds for each $t\in (0,\infty )$ , $x'\in U_x$ , and ${\underline {w}}'=(w^{\prime }_1,\dots ,w^{\prime }_{\ell })\in {H}[E]^{\ell }$ . We define
     (4.18)

     
     and obtain for $t\in (0,\infty )$ , $x'\in U_x$ , ${\underline {w}}'=(w^{\prime }_1,\dots ,w^{\prime }_{\ell })\in {H}[E]^{\ell }$ from (4.10) that
     (4.19)

     

   • ∘ We define , as well as

     
     We have by (4.17), (4.18) (used for the first line), and (4.19) (used for the second line) that
     (4.20)

     

We are ready for the proof of Lemma 4.3.

4.4 The proof of Theorem 3.1.(2)

Let $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $t\in (-\infty ,0)$ , $x\in {\mathfrak {V}}$ , ${\mathcal {B}}\subseteq E$ bounded, ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ , and $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ . We recall (3.2) as well as the seminorms in (2.8). The following estimates hold for each ${\boldsymbol {\varsigma }}\leq -1$ :

• • By (4.1) and (4.7), we have

  (4.25) $$ \begin{align} {\mathfrak{p}}(\Psi^0[f,{\boldsymbol{\varsigma}}](t,x))\leq{\mathfrak{p}}^0_{[\tau,0]\times \{x\}}(f). \end{align} $$

• • Let $1\leq {\mathrm {s}}\preceq k$ . Then, for $1\leq \ell \preceq {\mathrm {s}}$ and $0\leq q\leq \ell $ , we have (recall (4.1), (4.2), (4.9), (4.25))

  (4.26)

  
  for each ${\underline {w}}\in ({\mathbb {R}}\times {\mathcal {B}})^{\ell }$ . We define
  (4.27)

  
  and obtain for $1\leq \ell \leq {\mathrm {s}}$ from (4.10) and (4.26) that
  (4.28)

  
  holds for each ${\underline {w}}\in ({\mathbb {R}}\times {\mathcal {B}})^{\ell }$ .

We are ready for the proof of Theorem 3.1.(2).