1 Introduction
The extension problem for differentiable maps naturally arises in the context of manifolds with boundary or corners. In the finite-dimensional context, Whitney’s extension theorem [Reference Whitney20] guarantees more generally the extendability of Whitney jets (families of continuous functions that define formal Taylor expansions) on closed subsets of euclidean spaces. A characterization of closed subsets that admit continuous linear extension operators on $C^\infty$ - Whitney jets was given by Tidten in [Reference Tidten18] (see [Reference Frerick2] for further investigations). Recent research into Whitney-type extension operators [Reference Michor, Segev and Epstein11, Reference Roberts and Schmeding16] is concerned with generalizations to maps on closed subsets of finite-dimensional manifolds (Whitney germs in [Reference Michor, Segev and Epstein11], and in [Reference Roberts and Schmeding16], subsets that satisfy the so-called cusp condition) with values in vector bundles or (infinite-dimensional) manifolds. In [Reference Michor, Segev and Epstein11], the smooth category in the context of the convenient calculus [Reference Kriegl and Michor7] is considered, and in [Reference Roberts and Schmeding16], the smooth category within Bastiani’s differential calculus [Reference Glöckner3]. Throughout this paper, we work in Bastiani’s setting that is recalled in Section 2.1. We refer to [Reference Hamilton4, Reference Neeb15] for self-contained introductions into Bastiani’s calculus.
Besides Whitney’s approach, there is an alternative (significantly simpler) extension construction available that works for maps defined on half spaces. This approach is due to Seeley [Reference Seeley17]. He constructs a continuous linear map that extends such smooth maps $(-\infty ,0)\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}$ ( $n\in {\mathbb {N}}$ ) to ${\mathbb {R}}\times {\mathbb {R}}^n$ , whose partial derivatives extend continuously to $(-\infty ,0]\times {\mathbb {R}}^n$ . In this paper, we generalize Seeley’s result into several directions:
Let $E,F$ be Hausdorff locally convex vector spaces, and denote the system of continuous seminorms on F by ${\textsf {Sem}(F)}$ . For
and $U\subseteq E$ nonempty open, let $C^k(U,F)$ denote the set of all k-times continuously differentiable maps $U\rightarrow F$ . Let $\Omega (E)$ denote the set of all pairs $(V,{\mathfrak {V}})$ such that $V\subseteq E$ is nonempty open, and ${\mathfrak {V}}\subseteq E$ is contained in the closure of V in E with $V\subseteq {\mathfrak {V}}$ . For $-\infty \leq a<b\leq \infty $ , let $\mathcal {C}^k_{\mathfrak {V}}((a,b)\times V,F)$ denote the set of all $f\in C^k((a,b)\times V,F)$ , such that for each $\ell \in {\mathbb {N}}$ with $\ell \leq k$ , the $\ell $ th differential of f extends to a continuous map
(we set $(a,b]:= (a,\infty )$ if $b=\infty $ holds). Our main result Theorem 3.1 (stated to the full extent in Section 3.1) inter alia implies that, for $-\infty \leq a<\tau <0$ fixed, there exists a linear (extension) map
such that for $f\in \mathcal {C}_{{\mathfrak {V}}}^k((a,0)\times V,F)$ and $0\leq \ell \leq k$ , we have
For $E={\mathbb {R}}^n$ , $F={\mathbb {R}}$ , and $a=-\infty $ , this implies Seeley’s original theorem from [Reference Seeley17]. We mention, but do not present the details at this point, that Theorem 3.1 is formulated more generally in terms of families of extension operators indexed by triples $(E,V,{\mathfrak {V}})$ , where E runs over the class of Hausdorff locally convex vector spaces and $(V,{\mathfrak {V}})\in \Omega (E)$ holds (a and $\tau $ are thus fixed parameters). Theorem 3.1 additionally contains continuity estimates, as well as compatibility conditions that can be used, e.g., to construct extensions of maps by gluing together local extensions. This is demonstrated in Example 3.10 for the unit ball in a real pre-Hilbert space. In Application 3.6 in Section 3.2, we carry over the extension result (in the form stated above) to quadrants, which is of relevance in the context of (infinite-dimensional) manifolds with corners [Reference Margalef-Roig and Outerelo Dominguez8]. Specifically, given $k\in {\mathbb {N}}$ and $(V,{\mathfrak {V}})\in \Omega (E)$ , we construct an extension operator for $C^k$ -maps
(with $-\infty \leq a_1,\ldots ,a_n<0$ ) whose $\ell $ th differential, for $0\leq \ell \leq k$ , extends continuously to
. We remark that in the convenient setting (for $k=\infty $ and $V={\mathfrak {V}}=E=\{0\}$ ) the existence of a continuous extension operator was already shown in Proposition 24.10 in [Reference Kriegl and Michor7]. The proof given there also works in Bastiani’s setting, but still only for $k =\infty $ as the exponential law for smooth mappings is explicitly applied.Footnote 1
We finally want to emphasize that our extension result can also be used to extend $C^k$ -maps on subsets in infinite dimensions that admit a certain kind of geometry. Indeed, we have already mentioned that Example 3.10 covers the (real) pre-Hilbert unit ball. In Application 3.8 in Section 3.3, we consider subsets of Hausdorff locally convex vector spaces that are defined by a particular kind of distance function (e.g., nonzero $C^k$ -seminorms). The (real) pre-Hilbert unit ball is an example for this, but the construction in Example 3.10 differs from the construction in Application 3.8 that gets along without explicit use of the compatibility property admitted by the extension operators.
A brief outline of the paper is as follows. In Section 2, we fix the notations, recall Bastiani’s differential calculus, and provide some elementary facts and definitions concerning locally convex vector spaces (and maps) that we shall need in the main text. In Section 3, we state our main result, Theorem 3.1, and discuss various applications to it. Section 4 is dedicated to the proof of Theorem 3.1.
2 Preliminaries
Let ${\textsf {hlcVect}}$ denote the class of Hausdorff locally convex vector spaces, and let $E\in {\textsf {hlcVect}}$ be given. We denote the completion of E by ${\textsf {comp}({E})}\in {\textsf {hlcVect}}$ . The system of continuous seminorms on E is denoted by ${\textsf {Sem}(E)}$ . For ${\mathfrak {p}}\in {\textsf {Sem}(E)}$ , we let $\hat {{\mathfrak {p}}}$ denote the continuous extension of ${\mathfrak {p}}$ to ${\textsf {comp}({E})}$ . For a subset $V\subseteq E$ , we let ${\textsf {clos}(V)}\subseteq E$ denote the closure of V in E. A subset ${\mathcal {B}}\subseteq E$ is said to be bounded if $\sup \{{\mathfrak {p}}(X)\:|\: X\in {\mathcal {B}}\}<\infty $ holds for each ${\mathfrak {p}}\in {\textsf {Sem}(E)}$ . Let $-\infty \leq a < b\leq \infty $ be given:
-
• For $a=-\infty $ , we set $[a,b]:=(-\infty ,b]$ and $[a,b):=(-\infty ,b)$ .
-
• For $b=\phantom {-}\infty $ , we set $[a,b]:=[a,\infty )$ and $(a,b]:=(a,\infty )$ .
-
• For $a=-\infty $ , $b=\infty $ , we set $[a,b]:=(-\infty ,\infty )$ .
Let be given. We write $0\leq \ell \preceq k$ ,
-
• for $k\in {\mathbb {N}}$ if ${\mathbb {N}}\ni \ell \leq k$ holds,
-
• for $k=\infty $ if $\ell \in {\mathbb {N}}$ holds.
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
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
-
• 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
In particular, we have $C^\infty (U,F)=\mathcal {F}C^\infty (U,F)$ .
We have the following differentiation rules [Reference Glöckner3].
Proposition 2.2
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(a) A map is of class $C^k$ for $k\geq 1$ if and only if ${\mathrm{d}} f$ is of class $C^{k-1}$ when considered as a map $E \times E \supseteq U\times E \rightarrow F$ .
-
(b) Let be linear and continuous. Then, f is smooth, with ${\mathrm{d}}^1_xf=f$ for each $x\in E$ , as well as ${\mathrm{d}}^pf=0$ for each $p\geq 2$ .
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(c) Let $F_1,\ldots ,F_m$ be Hausdorff locally convex vector spaces, and let be of class $C^k$ for $k\geq 1$ and $q=1,\ldots ,m$ . Then,
-
(d) 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, is of class $C^k$ , with
-
(e) Let $F_1,\ldots ,F_m,E\in {\textsf {hlcVect}}$ , and be of class $C^0$ . Then, f is of class $C^1$ if and only if for each $p=1,\ldots ,m$ , the partial derivative
$$ \begin{align*} \textstyle{\mathrm{d}} f((x_1,\ldots,x_m),v_1,\ldots,v_m)&\textstyle=\sum_{p=1}^m\partial_p f((x_1,\ldots,x_m),v_p),\\ \big(\!&\textstyle= \sum_{p=1}^m \hspace{4.5pt}{\mathrm{d}} f((x_1,\ldots,x_m),(0,\ldots,0, v_p,0,\ldots,0)){\hspace{1pt}}\big), \end{align*} $$for each $(x_1,\ldots ,x_m)\in U$ , and $v_p\in F_p$ for $p=1,\ldots ,m$ .
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
such that the following conditions are fulfilled:
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• 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
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
The Riemann integral is linear, with
It follows that the Riemann integral is $C^0$ -continuous, i.e., continuous w.r.t. the seminorms
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
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
2.2.3 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
Example 2.6 (Harmonic Subsets)
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(i) If $A\subseteq U$ is harmonic and $\emptyset \neq B\subseteq A$ closed in U, then $B\subseteq U$ is harmonic.
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(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
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(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
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• 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 $\lambda \in (0,1)$ )
-
-
(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 $ ).▪
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(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
-
• 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
-
-
(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
-
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
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
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))
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
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 $0\leq \ell \preceq k$ .
Proof Set $S\equiv {\mathfrak {U}}$ in Lemma 2.7.▪
Corollary 2.3 provides the following statement.
Lemma 2.10 Let $H,\bar {H}, F\in {\mathsf {hlcVect}}$ , $O\subseteq H$ , $\bar {O}\subseteq \bar {H}$ both nonempty open, and $\psi \in C^k(O,\bar {O})$ be fixed. Let $(U,{\mathfrak {U}})\in \Omega (H)$ with ${\mathfrak {U}}\subseteq O$ be given, as well as $(\bar {U},\bar {{\mathfrak {U}}})\in \Omega (\bar {H})$ with $\psi (U)\subseteq \bar {U}$ and $\psi ({\mathfrak {U}})\subseteq \bar {{\mathfrak {U}}}$ . Then, for $f\in \mathcal {C}_{\bar {{\mathfrak {U}}}}^k(\bar {U},F)$ we have . Specifically, the following assertions hold:
-
(i) We have .
-
(ii) For $1\leq \ell \preceq k$ , we have
$$ \begin{align*} &(x,v_1,\ldots,v_{\ell}) \\ &\quad \mapsto {\mathrm{Ext}({f},{q})}(\psi(x), {\mathrm{d}}^{p_1} \psi(x,v_1,\ldots,v_{p_1}),\ldots, {\mathrm{d}}^{p_q} \psi(x,v_{\ell-p_q+1},\ldots,v_{\ell})), \end{align*} $$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 Part (i) is clear from the continuity properties of the involved maps. Now, we have , as $\psi $ is of class $C^k$ . Moreover, $\psi $ is defined on ${\mathfrak {U}}\subseteq O$ with $\psi ({\mathfrak {U}})\subseteq \bar {{\mathfrak {U}}}$ . Part (ii) is thus clear from Corollary 2.3, as well as from continuity of the occurring differentials and their extensions.▪
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
Finally assume $H=P\times E$ with $P,E\in {\mathsf {hlcVect}}$ . Then,
In the following, we will rather denote
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 Statement of the results
In this section, we state our main result Theorem 3.1, and discuss several applications. Theorem 3.1 is proven in Section 4.
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
for each $(V,{\mathfrak {V}})\in {\Omega (E)}$ . For a bounded subset ${\mathcal {B}}\subseteq E$ , we set
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
-
• Given $\bar {E}\in {\mathsf {hlcVect}}$ , $\bar {x}\in \bar {E}$ , $\ell \in {\mathbb {N}}$ , and a linear map , we define
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) 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) 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) 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
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.,
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 $a<\tau <b$ .▪
Remark 3.4 The second point in Theorem 3.1 shows that the extension operators constructed admit considerable continuity properties. Seeley already mentioned in [Reference Seeley17] that his extension operator is continuous in many functional topologies. Expectably, the same holds true for their infinite-dimensional counterparts. However, it would go far beyond the scope of this article to investigate all possible continuity properties of the extension operators provided here—they have to be extracted on demand from the explicit construction performed in Section 4. At this point, we only want to emphasize the following:
-
• The second estimate in Theorem 3.1.(2) can be sharpened if $\lambda _j=0$ holds for $j=1,\ldots ,\ell $ . Specifically, on the right side of this estimate, the set ${\mathscr {B}}({\mathcal {B}})$ then can just be replaced by $\{0\}\times {\mathcal {B}}$ .
-
• Let $0\leq {\mathrm {s}}\preceq k$ , $x\in {\mathfrak {V}}$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((a,b)\times V,F)$ be given. Then, Theorem 3.1.(2) shows
$$ \begin{align*} {\ \mathrm{Ext}({f},{\ell})}|_{[\tau,b]\times \{x\}\times H[E]^{\ell}}\hspace{3.5pt}&=0\qquad\quad \forall\: 0\leq \ell\leq{\mathrm{s}},\\ \Longrightarrow\qquad{\mathrm{Ext}({{\mathcal{E}}_{a,\tau,b}(E,V,{\mathfrak{V}})(f)},{\ell})}|_{[\tau,\infty)\times \{x\}\times H[E]^{\ell}}&=0\qquad\quad \forall\: 0\leq \ell\leq{\mathrm{s}}.\qquad\quad \end{align*} $$ -
• Let $f,g\in \mathcal {C}_{{\mathfrak {V}}}^k((a,b)\times V,F)$ be given, such that
$$ \begin{align*} {\mathrm{C}}:={\mathsf{clos}({\{z\in (a,b)\times {\mathfrak{V}}\:|\: {\mathrm{Ext}({f},{0})}(z)\neq {\mathrm{Ext}({g},{0})}(z)\}})} \end{align*} $$is compact. Then, ${\mathrm {C}}\subseteq [c,b]\times {\mathrm{K}}$ holds for certain $-\infty <c\leq b$ as well as ${\mathrm{K}}\subseteq {\mathfrak {V}}$ compact. Then, is compact, and the parts (1) and (3) of Theorem 3.1 imply$$ \begin{align*} &\mathsf{clos}(\{z\in (a,b)\times V\:|\: {\mathrm{Ext}({{\mathcal{E}}_{a,\tau,b}(E,V,{\mathfrak{V}})(f)},{0})}(z)\\ &\quad \neq {\mathrm{Ext}({{\mathcal{E}}_{a,\tau,b}(E,V,{\mathfrak{V}})(g)},{0})}(z) \})\subseteq {\widetilde{{{\mathrm{C}}}}}. \end{align*} $$This might be of relevance, e.g., in the context of spaces of smooth mappings between manifolds $M,N$ (N possibly infinite-dimensional), where the $\mathscr {D}$ -topology [Reference Michor10] (called very strong topology in [Reference Hjelle and Schmeding6]) is refined to the $\mathscr {FD}$ -topology [Reference Michor, Segev and Epstein11] (called fine very strong topology in [Reference Hjelle and Schmeding6]) by additionally considering the classes defined by the equivalence relation$$ \begin{align*} f\sim g\qquad\Longleftrightarrow\qquad {\mathsf{clos}({\{x\in M\:|\: f(x)\neq g(x)\}})}\subseteq M\quad\text{is compact} \end{align*} $$with $f,g\in C^\infty (M,N)$ .▪
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
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
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:
-
(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
$$ \begin{align*} {\mathrm{Ext}({{\mathcal{E}}_{{\underline{a}},{\underline{{\tau}}},{\underline{b}}}(E,V,{\mathfrak{V}})(f)},{\ell})}|_{{\breve{\mathcal{Q}}}({\underline{a}},{\underline{b}})\times {\mathfrak{V}}\times{H}[E,n]^{\ell}}={\mathrm{Ext}({f},{\ell})}. \end{align*} $$ -
(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
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
The induction hypotheses provides the extension operator
Theorem 3.1 provides the extension operator
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
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) $\xi $ is of class $C^k$ on ${\mathcal {W}}:= \xi ^{-1}((0,\infty ))$ ,
-
(2) $\xi (Z)\neq 0$ for some $Z\in P$ ,
-
(3) for each $\lambda \in [0,\infty )$ and $Z\in P$ , and
-
(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
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
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 $0\leq \ell \preceq k$ .▪
Example 3.10 Let $0\neq \xi \in {\mathsf {Sem}(P)}$ be of class $C^k$ on ${\mathcal {W}}$ . Then, (1)–(3) are evident, and (4) holds by Example 2.6.(v). For instance.
-
(a) Let $P={\mathbb {R}}^2$ and . Then, $\xi $ is smooth on ${\mathcal {W}}=\{(x,y)\in {\mathbb {R}}^2\:|\: x\neq 0\}$ .
-
(b) Let be a real or complexFootnote 12 pre-Hilbert space, and set $P:={\mathscr {H}}$ as well as . Then, $\xi $ is smooth on $\mathcal {W}=P\setminus \{0\}$ by Proposition 2.2. We mention that in the real case, an extension operator can also be obtained by explicit application of Theorem 3.1.(3). More details are provided in Appendix A.▪
Let $\rho \in C^\infty ({\mathbb {R}},{\mathbb {R}})$ be given withFootnote 13
-
• 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))
-
– $\mu ({\mathcal {W}}\times Y)\subseteq (0,\infty )\times {\breve {\mathcal {O}}} \times Y$ ,
-
– $\mu ( {\mathcal {O}}\times Y)\subseteq \hspace {5.5pt}(0,1)\times {\mathcal {O}}\times Y$ , 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)
(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:
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
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:
-
(a) $\psi |_{\{t\}\times S_\alpha }$ is constant for each $t\in (a,\infty )$ and $\alpha \in I$ .
-
(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 $\alpha (z)\in I$ .
-
(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
We obtain the linear extension map
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 ${\mathbb {R}}^2$ )
Then, (3.15) provides the linear extension map
. Let
We obtain a linear extension map
if we set
for each $f\in \mathcal {C}^k_{{\breve {\mathscr {D}}}}({\mathscr {D}},F)$ .▪
4 The proof of Theorem 3.1
In this section, we prove Theorem 3.1. For this, we let $F\in {\mathsf {hlcVect}}$ and be fixed, and recall the definitions made in the beginning of Section 3.1. We make the following simplifications to our argumentation:
-
• It suffices to prove Theorem 3.1 for the case $a=-\infty $ , as the general case then follows by cutoff arguments. Specifically, let $-\infty <a<\tau <b<\infty $ be given, and fix $a<\kappa <\kappa '< \tau $ as well as $\rho \in C^\infty ({\mathbb {R}},{\mathbb {R}})$ with
$$ \begin{align*} \rho|_{(-\infty,\kappa]}=0\qquad\quad \text{and}\qquad\quad \rho|_{[\kappa',\infty)}=1. \end{align*} $$For each $E\in {\mathsf {hlcVect}}$ and $(V,{\mathfrak {V}})\in \Omega (E)$ , we define the linear map byfor $f\in \mathcal {C}_{{\mathfrak {V}}}^k((a,b)\times V,F)$ . We obtain extension operators as in Theorem 3.1 if for $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in \Omega (E)$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((a,b)\times V,F)$ , we set
$$ \begin{align*}&{\mathcal{E}}_{a,\tau,b}(E,V,{\mathfrak{V}})(f)(t,x)\\&\quad := \begin{cases} f(t,x)&\:\:\text{for}\quad\:\: (t,x)\in\hspace{3.7pt} (a,b)\times V,\\ {\mathcal{E}}_{-\infty,\tau,b}(E,V,{\mathfrak{V}})(\xi(E,V,{\mathfrak{V}})(f))(t,x) &\:\:\text{for}\quad\:\: (t,x)\in [b,\infty)\times V. \end{cases} \end{align*} $$ -
• To simplify the notations, in the following we restrict to the case $b=0$ . The case $b\neq 0$ follows in the same way, and can alternatively be obtained from the statement for $b=0$ via application of translations.
For the rest of this section, let thus $\tau \in (-\infty ,0)$ be fixed (i.e., we have $a=-\infty $ and $b=0$ ). We choose $\tau <\upsilon < 0$ and $\varrho \in C^\infty ({\mathbb {R}},{\mathbb {R}})$ , such that
and define the constants
According to [Reference Seeley17], there exists a sequence $\{c_n\}_{n\in {\mathbb {N}}}\subseteq {\mathbb {R}}$ with
-
(i) for each $q\in {\mathbb {N}}$ and
-
(ii) for each $q\in {\mathbb {N}}$ .
Given some $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ , its extension will be defined (see (4.12) in Section 4.2) in analogy to [Reference Seeley17] by
The sum in the second line is locally finite as $\varrho $ is zero on $(-\infty ,\tau ]$ , hence $\tilde {f}$ is defined and of class $C^k$ on $({\mathbb {R}}\setminus \{0\})\times V$ . We basically will have to show that $\tilde {f}$ is of class $C^k$ on whole ${\mathbb {R}}\times V$ , and that its $\ell $ th differential extends continuously to ${\mathbb {R}}\times {\mathfrak {V}}\times H[E]^{\ell }$ for each $0\leq \ell \preceq k$ . For this, we need to construct these extensions explicitly, which will be done in analogy to the definition of $\tilde {f}$ . For our argumentation, we shall need the following corollary to Lemma 2.7 (Corollary 2.9) and Example 2.6.(iv).
Corollary 4.1 Let $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , as well as $f_{-}\in C^k((-\infty ,0)\times V,F)$ and $f_{+}\in C^k((0,\infty )\times V,F)$ be given. Assume furthermore that for each $0\leq \ell \preceq k$ , there exists a continuous map that restricts to ${\mathrm {d}}^{\ell } f_{\pm }$ . Then, we have
(the third line implies the second line).
Proof Let $U:= {\mathbb {R}}\times V\subseteq H:=H[E]$ , $A:=\{0\}\times V\subseteq U$ , ${\mathfrak {U}}:= {\mathbb {R}}\times {\mathfrak {V}}\subseteq H$ ,
observe that A is harmonic by Example 2.6.(iv) and apply Corollary 2.9.▪
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
-
• 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
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)
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
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
In Section 4.3, we prove the following statement.
Lemma 4.3 Let $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ be given. Then, $\Phi [f]^{\ell }$ is continuous for each $0\leq \ell \preceq k$ .
Together with Lemma 4.2 and Corollary 4.1, Lemma 4.3 impliesFootnote 15
For $E\in {\mathsf {hlcVect}}$ and $(V,{\mathfrak {V}})\in \Omega (E)$ , we define the map
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
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 ${\boldsymbol {\varsigma }}\leq -1$ :
-
(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) $$ \begin{align} {\mathfrak{p}}(\Psi^0[f,{\boldsymbol{\varsigma}}](t,x'))\leq C_0\qquad\quad \forall\: t\in (0,\infty),\: x'\in U_x. \hspace{4pt} \end{align} $$ -
(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
(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
(4.20)
-
We are ready for the proof of Lemma 4.3.
Proof of Lemma 4.3 Let $E\in {\mathsf {hlcVect}}$ , $(V,{\mathfrak {V}})\in {\Omega (E)}$ , $f\in \mathcal {C}_{{\mathfrak {V}}}^k((-\infty ,0)\times V,F)$ , $x\in {\mathfrak {V}}$ , ${\mathfrak {p}}\in {\mathsf {Sem}(F)}$ , and $\varepsilon>0$ be given. We discuss the cases $\ell =0$ and $1\leq \ell \preceq k$ separately:
-
• Let $\ell =0$ . We choose $C_0\geq 1$ and $U_x\subseteq {\mathfrak {V}}$ as in (a). By Property (ii), there exists $N\in {\mathbb {N}}$ with $\sum _{j=N+1}^\infty |c_j|<\frac {\varepsilon }{4C_0}$ . We obtain from (4.14), (4.15) and the triangle inequality that
(4.21)for $t\in (0,\infty )$ and $x'\in U_x$ . We observe the following:-
∘ By (4.1) and (4.7), we have for $0\leq j\leq N$ :
-
∘ Since ${\mathrm {Ext}({f},{0})}$ is continuous, we can shrink $U_x\subseteq {\mathfrak {V}}$ around x and fix $0<\delta <|\upsilon |/2^N$ , such that
Combining both points with (4.21), we obtain
-
-
• Let $1\leq \ell \preceq k$ and ${\underline {w}}\in {H}[E]^{\ell }$ be fixed. We choose $C_{\ell }\geq 1$ , $U_x\subseteq {\mathfrak {V}}$ , and $O_{{\underline {w}}}\subseteq H[E]^{\ell }$ as in (b), and define
-
∘ Given $\Delta>0$ , Property (i) provides some $N_\Delta \in {\mathbb {N}}$ with
(4.22) -
∘ By Property (ii), there exists some $N\geq \tilde {N}$ with
(4.23)holds for all $t\in (0,\infty )$ , $x'\in U_x$ , ${\underline {w}}'\in O_{{\underline {w}}}$ .
-
∘ By (4.1), (4.9), (4.10), for $0\leq j\leq N$ , $t\in (0,|\upsilon |/2^N)$ , $x'\in {\mathfrak {V}}$ , and ${\underline {w}}'\in H[E]^{\ell }$ we have
(4.24)for all $t\in (0,\delta )$ , $x'\in U_x$ , ${\underline {w}}'\in O_{{\underline {w}}}$ .
Since $\sum _{j=0}^\infty c_j=1$ holds by Property (i), the triangle inequality together with (4.22)–(4.24) yields
-
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 ${\boldsymbol {\varsigma }}\leq -1$ :
-
(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).
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 $f\in \mathcal {C}_{{\mathfrak {V}}}^k((0,\infty )\times V,F)$ .
-
• Let ${\mathrm {s}}=0$ . By Property (ii) we have $C_0:=\sum _{j=0}^\infty |c_j|<\infty $ . We obtain from (4.25) and the triangle inequality that
-
• Let $1\leq {\mathrm {s}}\preceq k$ . We choose $Q_{\mathrm {s}}\geq 1$ as in (4.27), and define
(4.29)Then $C_{\mathrm {s}}\geq 1$ holds, as we have $Q_{\mathrm {s}}\geq 1$ as well as for $1\leq \ell \leq {\mathrm {s}}$ by Property (i). We obtain from (4.28) that
Appendix A.1. Some details to Example 3.10.(b)
Let
be a real pre-Hilbert space, $E\in {\mathsf {hlcVect}}$ and $H={\mathscr {H}} \times E$ . We set
, fix $0<\tau <1$ , and define ${\mathcal {S}}$ and ${\breve {\mathcal {S}}}$ as in Remark 3.9. Given $Z\in \mathcal {A}=\xi ^{-1}(\{1\})$ , we set
The following maps are smooth and inverse to each other:
Now, given $g\in \mathcal {C}_{{\breve {\mathcal {S}}}\times {\mathfrak {V}}}^k({\mathcal {S}}\times V,F)$ , the same arguments as in Remark 3.9 show that it suffices to construct
in order to obtain an extension $\tilde {g}\in \mathcal {C}_{ {\mathfrak {V}}}^k({\mathscr {H}}\times V,F)$ of g. For this, we proceed as follows:
-
• We have by Lemma 2.10
-
• First applying the extension operator ${\mathcal {E}}_{0,\tau ,1}(Z_\bot \times E,{\mathscr {D}}(Z)\times V,{\mathscr {D}}(Z)\times {\mathfrak {V}})$ from Theorem 3.1, and then composing with $\psi _Z\times {\mathrm {id}}_V$ , we obtain (from Lemma 2.10) an extension
-
• Given $Z'\in \mathcal {A}$ and , then the definitions (and continuity) ensure that for and , we have
$$ \begin{align*} {\mathrm{Ext}({f_Z},{0})}(t,Y,x)&= {\mathrm{Ext}({f},{0})}(t/\xi(X),X,x)\\&= {\mathrm{Ext}({f_{Z'}},{0})}(t,Y',x)\qquad\quad\forall\: t\in [\tau,1],\: x\in {\mathfrak{V}}. \end{align*} $$Theorem 3.1.(3) implies , and we conclude$$ \begin{align*} {\mathrm{d}}^{\ell}{\tilde{f}}_Z|_{U\times V\times H^{\ell}}&={\mathrm{d}}^{\ell}{\tilde{f}}_{Z'}|_{U\times V\times H^{\ell}}\\\stackrel{\text{continuity}}{\Longrightarrow}\quad\:\: {\mathrm{Ext}({\tilde{f}}_Z,{\ell})}|_{U\times {\mathfrak{V}}\times H^{\ell}} & = {\mathrm{Ext}({{\tilde{f}}_{Z'}},{\ell})}|_{U\times {\mathfrak{V}}\times H^{\ell}}. \end{align*} $$
It follows that the maps $\{{\tilde {f}}_Z\}_{Z\in \mathcal {A}}$ glue together to an extension $\tilde {f}\in \mathcal {C}^k_{{\mathfrak {V}}}({\mathcal {W}}\times V,F)$ of (4.30).
Acknowledgment
The author thanks Helge Glöckner for general remarks and discussions. This research was supported by the Deutsche Forschungsgemeinschaft, DFG, project number HA 8616/1-1.