数理解析研究所講究録
第 1540 巻 2007 年 35-4235

D-closedness of paratangent space
近畿大学大学院総合理工学研究科 伊藤洋忠 (Hirotada Ito)

Interdisciplinary Graduate School
for Comprehensive Science and Engineering, Kinki University
We consider a kind of higher order tangent space of a closed subset.
There is an easy algebraic definition of the higher order tangent bundle 7“ of following the construction of the Zariski tangent space in algebraic geometry, whichis called $the$ &riski paratangent bundle by Bierstone, Milmannn and Pawbucki[BMPI]. In 1934, Whitney posed the extension problem of finding the condition
for a function on of to be extendable to afunction on . As an approachto this problem, Bierstone, Milmann and Pawtucki [BMPI] have introduced thehigher order paratangent bundle of a subset of a Euclidean space, generalising Glaeser’s idea. (We often omit the term “higer order“.) The paratangentsof order of form a fibred space over . They [BMPI] and Izumi [I] used
this notion to solve Whitney’s problem in some restricted cases [BMPI]. We areinterested in characteristing the paratangent bundle as a subset of the higher ordertangent bundle of the ambient Euclidean space.
A general higer order tangent of at can be expressed as a linear combination of higher order derivatives of the Dirac delta function . Hence we express
it by the total symbol of the linear combination, a polynomial in the dual coordinates , The purpose of this note is to show that the paratangent space of a set $X$
is D- closed which implies that the space is invariant with respect to partial differentiations by the components of . We concider that this is an important property
which characterises the paratangent bundle.
Recently Fefferman has given answer to the Whitney problem above and given
its refinement. According to Bierstone, Milman, Pawluki [BMPI], Fefferman’s
result is a solution to a variant of their conjecture [BMP2]. In this variant they use
the third paratangent bundle . It is known that. Their results implies$\tau=T=\mathcal{T}$([BMP2]). It is easy to see that Zariski paratangent apace isthatD-closed. Hence the result $\tau=T=\mathcal{T}$ implies that they are D-closed. Since thisargument requires the difficult theory of Fefferman [F], we show a direct simple
proof.
$X\subset R^{\prime 1}$

$X$

$X$

$C^{c\prime}$$X$$\tau$$d$$R^{n}$$X$$X$$R^{n}$$a$$\delta_{a}$$\xi_{a}$$\xi_{\sigma}$$T$$\tau\subset T\subset \mathcal{T}$36

Zariski paratangent bundle

1

We use the following notations$\alpha=(\alpha_{1},\alpha_{2}, \cdots,\alpha_{n})$$e_{t}$,$(\alpha_{j}\in NU\{0\})$,$:=(0,$ $\ldots,0,\underline{1},0,\ldots,0\rangle$ .
$j$$|\alpha|=\alpha_{1}+\alpha_{2}+\cdots$

十$x=(x_{1},x_{2}, \ldots x_{n})$$\alpha_{n}$,,!

$\alpha!=\alpha_{1}!\alpha_{2}$$\cdots\alpha_{n}!$,$x^{a}=x_{1}^{\alpha_{I}}x_{2}^{\alpha_{2}}\cdots x_{l}^{a_{n}}$for multi exponent and multi indeces. Let be a subset of a Euclidean spaceof degree equal to or. Let
be the vector space of polynomials on
be its dual vector space. We define functionalbyless than and let
$X$

$d$

$\xi_{ai}f$$:= \frac{\partial f}{\partial x_{i}}(a)$$R^{n}$$\mathbb{R}$$P_{d}$$\mathbb{R}^{n}$$r_{d}$$\xi_{ai}\in \mathcal{P}_{d}^{*}$. We put$\xi_{a}$Let$f\in \mathcal{P}_{d}$and$\xi_{a}^{(p)}$:$P_{d}arrow \mathbb{R}$$:=(\xi_{a1},$ $\ldots,\xi_{an}\rangle$,

denote the assignment.

$f \}arrow\frac{\partial^{|\alpha|}f}{\partial x^{\Phi}}(a)=\frac{\partial^{|\alpha|}.f}{\partial x_{1}^{\alpha_{1}}\cdot\cdot X_{n}^{\alpha,}’}(a)$This is an element of$\mathcal{P}_{l}^{*}$. For example,$1_{a}(f)=\xi_{a}^{(0)}(f):=f(a)$ ,$\xi_{a}^{\{\epsilon_{1})}(f)=\xi_{a1}$$:= \frac{\partial f}{\partial x_{1}}(a)$$\xi_{a}^{(e\downarrow+e_{2})}(f);=\xi_{a1}\xi_{a2}\zeta f):=\frac{\partial^{2}f}{\partial x_{1}\partial x_{2}}(a)$,.

The $functiona11_{a}$ is nothing but the Dirac delta function supported at .
In the dual space , any higher partial derivatives of can be expressed as alinear combination of those of . This is an immediate consequence of Taylor’s$a$$r_{l}$$\xi_{b}$$\xi_{a}$formula.

Lemma 1.1.

$\xi_{b}^{\alpha}=\sum_{|\alpha|+\beta|\leq d}\frac{1}{\beta!}(b-a\rangle^{\beta}\xi_{a}^{\alpha+\beta}$

37

Definition 1.2 Let be a real vector space of dimension $d(<\infty)$ . A bundle (oflinear subspaces of ) over is a subset of $X\cross V$ such that, for all $a\in X$ , the$:=\{u\in V : (a, u)\in E\}$ is a linear subspace of. We don’t require local
fibletriviality.
$V$

$V$

$E$

$X$

$V$

$E_{a}$

Dednition 1.3 If$V\subset P,$, let$V^{\perp}:=\{\eta\in P_{d}$: $\eta(f)=0(f\in V)$

denote the subspace of annihilators of in the dual space . Let
denote the set of thedenote the ideal of functions that vanish on $X$ and. Then the Zariski paratangent
atTaylor series of degrees of,with fibre, is the subbundle ofbundle of orderfor each $a\in X$ .
$V$$\mathcal{P}_{d}$$T_{a}^{d}I^{d}(X)$$C^{d}$$d$$d,$$I^{d}(X)\subset C^{d}(R^{n})$$a\in X\subset \mathbb{R}^{n}$$I^{tl}(X)$$\mathcal{T}’(X)$$\mathcal{T}_{a}^{d}=(T_{a}^{l}I^{d}(X))^{\perp}$$X\cross r_{\iota l}$

2 D-closedness of Zariski paratangent bundleof a fibre of a bundle $E\subset X\cross V$ at a DDefinition 2.1 We call a subspaceclosed if it is invariant with respect to partial differentiations with respect to theof .
components$E_{\sigma}$$\xi_{a}$$\xi_{ai}$In view of 1. I, the property D-closedness is independent of the choice of thepoint .
$a$Theorem 2.2

We put$\eta$$\mathcal{T}^{p\prime}(X)$is D-closed.

$:= \sum_{a}c_{\alpha}\xi_{a}^{\alpha}\in \mathcal{T}^{d}(X)(c_{\alpha}\in \mathbb{R})$series of degree
We have$d$of$C^{d}$

function $f$ at

and$a\in X\subset \mathbb{R}^{n}$$g$$:=T_{a}^{d}f$. Here is the Taylor$g$.

$(c_{\beta}^{f}= \frac{1}{\beta!}f^{(\beta)}(a))$$g:= \sum_{\beta|\leq d}\frac{1}{\beta!}(x-a)^{\beta}f^{(\beta\rangle}(a)=\sum_{\beta|\leq d}c_{\beta}’(x-a\rangle^{\beta}$,,$\frac{\partial\eta}{\partial\xi_{a1}}(g)=(\frac{\partial}{\partial\xi_{a1}}\sum_{\alpha}c_{\sigma}\xi_{a}^{\alpha})\sum_{\beta 1\leq\prime}c_{\beta}’(x-a)^{\beta}=\sum_{\alpha}c_{\alpha}\sum_{\beta|\leq d}c_{\beta}’\frac{\partial\xi_{a}^{\alpha}}{\partial\xi_{a1}}(x-a)^{\beta}$

38$\frac{\partial\xi_{a}^{\alpha}}{\partial\xi_{a1}}(x-a)^{\beta}=\alpha_{1}\xi_{a1}^{\alpha_{I}- 1}\xi_{a2}^{\alpha_{2}}\cdots\xi_{an}^{\alpha,\prime}(x_{1}-a_{1}\rangle^{\beta_{1}}(x_{2}’-a_{2})^{\beta_{\underline{7}}}\cdots(x_{n}-a_{n})^{\beta_{n}}$$=\alpha_{1}(\alpha_{1}-1)!\alpha_{2}$!

$\cdots\alpha_{\iota}!=\{\begin{array}{ll}\alpha!

(\alpha_{1}=\beta_{1}+1, \alpha_{2}=\beta_{2}, \ldots, \alpha_{n}=\beta_{n})0 (otherwise).\end{array}$Since$P_{a}(x_{1}-a_{1})(x-a)^{\beta}=\xi_{a}^{\alpha}(x_{1}-a_{1})^{\beta_{1}+1}(x_{2}-a_{2}\rangle^{\beta_{2}}\cdots(x_{n}-a_{n}\rangle^{\beta_{n}}$$=\{\begin{array}{ll}\alpha!

(\alpha_{1}=\beta_{1}+1, \alpha_{2}=\beta_{2}, \ldots, \alpha_{n}=\beta_{l})0 (otherwise).\end{array}$

We have.

$\frac{\partial\xi_{a}^{\alpha}}{\partial\xi_{a1}}(x-a\rangle^{\beta}=\xi_{a}^{\alpha}(x_{1}-a_{1})(x-a\rangle^{\beta}$

Therefore we have$\frac{\partial\eta}{\partial\xi_{a1}}(g)=\sum_{\alpha}c_{\alpha}\sum_{\beta|\leq d}c_{\beta}’\frac{\partial\xi_{a}^{\alpha}}{\partial\xi_{a1}}(x-a)^{\beta}$$= \sum_{\alpha}c^{\alpha}\xi_{a}^{\alpha}f\sum_{fl\leq 1}c_{\beta}’(x_{1}-a_{1})(x-a)^{\beta}$$=\eta((x_{1}-a_{1})g)$.

If $g\in I^{d}(X)$ , we have $(x_{1}-a_{1})g\in I^{d}(X)$ .
This implies$\frac{\partial\eta}{\partial\xi_{a1}}(g)=0$

Thus$\mathcal{T}^{d}(X)$is D-closed with respect to.

$\frac{\partial}{\partial\xi_{a1}}$

$0$

39

3 Paratangent space [BMPI]Let us put$E_{0}$$:=\{(a, \lambda\xi_{a}) ; a\in X, \lambda\in \mathbb{R}, \xi_{a}\in \mathcal{P}_{d}^{*}\}$Let us fix a natural number $N\in N\cup\{0\}$ . We define
is defined, we put,

inductiveIy as follows. If$E_{k}$$E_{k}$$\Delta E_{k}$$:=\{(a_{0}^{k}, a_{1}^{k},;.

$|at-a_{0}^{k}|^{d-|\alpha|}|\eta t(x-a_{v}^{k})^{\alpha}|\leq 1,$

and$:=\pi(\overline{\Delta E_{k}}\cap\{(a, \ldots,a, \eta)$

where
the first factor times
We put$\eta_{v}^{k}\in E_{k,a_{\nu}}$,$0\leq v\leq N\rangle$ $\}$.
: $a\in X,$$\eta\in \mathcal{P}_{d}I)$,

denotes the canonical projection onto$\pi:X\cross X\cross\cdots\cross X\cross \mathcal{P}_{\iota l}^{*}arrow X\cross P_{d}^{*}$$\mathcal{P}^{*},$$a_{v}^{k}\in X,$$(|\alpha|\leq d,$$\subset X^{N+1}\cross \mathcal{P}_{d}^{*}$$E_{k}’$:., a_{N}^{k}, \eta_{0}^{k}+\cdots+\eta_{N}^{k})$.

$E_{A+1}$$;= \bigcup_{a\epsilon X}(\{a\}\cross SpanE_{k.a}’)\subset XxP_{d}$where Span denotes the liner span ofstabilizes and we have$E_{k}’$$E_{k}’$

in the fibre. The sequence$E_{0}\subset E_{1}\subset$$E_{2}\subset\cdots$$E_{k}=E_{2di_{lt1’}P_{d}^{*}}$( $k\geq 2$ dim$r_{d}$)as a general property of Glaeser operation [G], [BMPI]. The saturation. is called the paratangent bundle of order of$d$$E_{2\dim \mathcal{P}_{d}}$called the paratangent space of order

$d$of

$X$

at .
$a$

Remark 3.1 We can replace the control condition$|a_{v}^{k}-a_{0}^{k}|^{l-|\alpha|}|\eta_{v}^{k}(x-a_{l}^{k})^{\alpha}|\leq 1$$by$$|al-a_{0}^{A}|^{d-|\alpha|}\}\eta_{\nu}^{k}(x-a_{\nu}^{k})^{\alpha}|\leq c$with any $c>0$ independent of .
$k$

$X$

and the fibre$\tau_{N}^{d}(X)$$:=$$\tau_{N.a}^{d}(X)$

is

40

4 D-closedness of$\tau_{N}^{d}$

We have defined that a liner subspace$A\subset \mathcal{P}_{d}^{*}=\mathbb{R}[\xi_{a}]$closed with respect to all the derivations

asseItion that$\gamma_{N}^{d}$is called D-closed if it is$\frac{\partial}{\partial\xi_{ai}}(1\leq i\leq n)$. Our main result is theis D-closed.

Theorem 4.1 The fibre$(’\mu_{N}f)_{a}$over$a$is D-closed.

This theorem follows from the following lemmas.

Lemma 4.2 Since an element of
we have$\Delta E_{k}$can be expressed as a linear combination of$\xi_{a_{\nu^{f}}}^{\alpha}$$(A_{\alpha}^{k}\in \mathbb{R})$$\eta_{\nu}^{k}$$:= \sum_{\alpha}A_{\alpha}^{k}\xi_{u_{\mathfrak{i}}^{k}}^{\alpha}\in\tau_{N}^{c\prime}(X)$.

If the control condition$|a_{\nu}^{k}-a_{0}^{k}|^{d-|\alpha|}|\eta_{t^{r}}^{k}(x-a_{\nu}^{k})^{\alpha}|\leq 1$

is satisfied, we have$|a_{\nu}^{k}-a_{0}^{k}|^{d-|a|}| \frac{\partial\eta_{\nu}^{k}}{\partial\xi_{a_{0}^{k}i}}(x-a_{\nu}^{k})^{\alpha}|\leq c$

in a sufficiently small neighborhood of
only on and .

for$a_{v}^{k}$$n$$a_{0}^{k}$and for constant $c>0$ dependent$d$

We have only to prove that in the case $i=1$ without losing generality. Wehave$\frac{\partial\eta_{t’}^{k}}{\partial\xi_{a_{0}^{k}1}}=\frac{\partial}{\partial\xi_{a_{0}^{k}1}}\sum_{\alpha}A_{\alpha}^{k}\xi_{a_{\gamma}^{k}}^{\alpha}=\sum_{\alpha}A_{\sigma}^{k}\alpha+\beta-e_{I}\underline{\succ}0\sum_{|\alpha|+\beta|\leq d}\frac{]}{\beta!}(a_{\nu}^{k}-a_{0}^{k})^{\beta}(\alpha_{1}+\beta_{1}\rangle\xi_{a_{0}^{k}}^{\alpha+\beta-e_{1}}$meansby Lemma 1.1. Here). Accordingly we have$\alpha\succeq\beta$$\beta=\phi_{1},$$\alpha_{j}\geq\beta_{i}(1\leq\forall i\leq n\rangle$

for$\alpha=(\alpha_{1}, \ldots , \alpha_{n})$and$\ldots,\beta_{n}$$| \frac{\partial\eta_{\nu}^{k}}{\partial\xi_{a_{(\}}^{\lambda}1}}(x-al)^{\alpha}|=|\sum_{a}A_{a}^{k}\alpha+\beta-e_{1}\succeq 0\sum_{|\alpha|+\beta|\leq d}\frac{1}{\beta!}(ai-a_{0}^{k})^{\beta}(\alpha_{1}+\beta_{1})\xi_{a_{t\}}^{k}}^{\alpha+\beta-e_{1}}(x-a_{v}^{k})^{\alpha 1}$

41.
$\leq\sum_{\alpha}|A_{\alpha}^{k}|\alpha+\beta-e_{1}\succeq 0\sum_{|\alpha|+\beta|\underline{<}cl}\frac{\alpha_{1}+\beta_{1}}{\beta!}|a_{v}^{k}-a_{0}^{k}|^{\beta}\alpha!(1+|a_{01}^{k}-a_{\backslash \prime 1}^{k}|)$Since$|a_{v}^{k}-a_{0}^{k}|^{l-|\alpha|}|\eta_{v}^{k}(x-a_{v})^{\alpha}|=|a_{v}^{k}-a_{0}^{k}|^{d-|\alpha|}|A_{\alpha}^{k}|\alpha!\leq]$,

we have$|a_{v}^{k}-a_{0}^{k}|^{d-|\alpha|}| \frac{\partial\eta_{\nu}^{k}}{\partial\xi_{a_{()}^{k}1}}(x-a_{\gamma}^{k})^{\alpha}|\leq\sum_{\alpha}\sum_{\beta}\frac{\alpha_{1}+\beta_{1}}{\beta i}|(at-a_{0}^{k}\rangle^{\beta}|\{\alpha!(1+|a_{01}^{k}-a_{v1}^{k}|)\}$

When we consider thatis sufficiently close to$a_{v}^{k}$$a_{0}^{k}$.

, we can assume$|a^{k})’-a_{0}^{k}| \leq\frac{1}{d}\frac{(n!d!)^{2}}{\{(n+d)!\}^{2}}$.

Hence we have$|a_{v}^{k}-a_{0}^{k}|^{d-|\alpha|}| \frac{M_{\nu}}{\partial\xi_{a_{0}^{k}1}}(x-a_{v}^{k})^{\alpha}|\leq\frac{((n+d)!)^{2}}{(n!d!)^{2}}d\{\frac{1}{d}\frac{(n!d!)^{2}}{((n+d)^{f}\rangle^{\angle}}\}^{d}\{]+\frac{1}{d}\frac{(n!d!)^{2}}{((n+d)!)^{2}}\}$

Let

$c$

denotes the right constant. Then we obtain the desired estimate.

Lemma 4.3 If $barrow a$ and$\etaarrow\eta^{r}$, we have$\frac{\partial\eta}{\partial\xi_{bi}}arrow\frac{\partial\eta’}{\partial\xi_{ai}}$.
$0$.

Proof. We have only to prove that in the case $i=1$ without losing generality. We
know that$\frac{\partial\eta}{\partial\xi_{b1}}=\sum_{j}\frac{\partial\xi_{aj}}{\partial\xi_{b1}}\frac{\partial\eta}{\partial\xi_{aj}}=\sum_{j}\beta+e_{j}-e_{I}\succeq 0\beta+|\leq cl\sum_{j}\frac{1}{\beta!}(a-b)^{\beta}(\beta_{1}+5_{j1})P_{b}^{+\ell_{j}-e_{I}}\frac{\partial\eta}{\partial\xi_{a\dot{\text{ノ}}}}$

because of$\xi_{aj}=\sum_{\beta+e_{j}|\leq c1}\frac{]}{\beta i}(a-b\rangle^{\beta}?_{b^{+e_{i}}}$We have$\sum_{j}\delta_{j1}\xi_{a}^{e_{j}-e_{1}}=1$$0(i\neq j))$. Here. Therefore we have$\delta_{tj}$.

denotes the Kronecker delta$(\delta_{ji}=1,$$\delta_{i\dot{\text{ノ}}}=$. Since the linear map$\frac{\partial\eta}{\partial\xi_{b1}}arrow\frac{\partial\eta}{\partial\xi_{a1}}(barrow a)$42$\frac{\partial}{\partial\xi_{a1}}$:$\mathcal{P}_{d}^{*}arrow \mathcal{P}_{d}^{*}$

is continuous as a linear map for the vector space of the finite

dimension, we have$\etaarrow\eta’$$\frac{\partial\eta}{\partial\xi_{a1}}arrow\frac{\partial\eta’}{\partial\xi_{a1}}(\etaarrow\eta’)$are satisfied, we have$\frac{\partial\eta}{\partial\xi_{b1}}arrow\frac{\partial\eta’}{\partial\xi_{a1}}$

is satisfied. Thus, if $barrow a$ and.

$0$

References[BMPI] E. Bierstone, P. Milman, W. Pawlucki, Differentiable functions definedin closed sets, A problem of Whitney, Inventiones Math. 151, 329-352(2003).
[BMP2] E. Bierstone, P Milman, W. Pawlucki, Higher-order tangents and Fef-fermans paper on Whtney’s extension problem. Annals of Mathematics.
164, 361-370 (2006).

. Annals of Mathemat-[F]C. Fefferman, Whitney’s extension problem forics. 164, 313-359 (2006).

[G]

G. Glaeser, \’Etudes de quelques alg\‘ebles tayloriennes, J. Analyse Math. 6.
1-124 ($C^{m}$$1958\rangle$[I]$[W_{1}]$$[W_{2}]$

S. Izumi, Restrictions of smooth functions to a closed subset, Annales del’Institute Fourier, Grenoble. 541811-1826 (2005).
H. Whitney, Analytic extensions of differetiable functions defined inclosed sets, Trans. Amer. Math. Soc. 36, 63-89 (1934).

H. Whitney, Differentiable functions defined in closed sets. I, Trans.
Amer. Math. Soc. 36-2, 369-387 (1934).