数理解析研究所講究録
第 845 巻 1993 年 8-12

8

A conjugacy class ofregular operators
Emmanuel Andronikof (*)(univ. Paris XIII)

1- Let $X=C_{(t,x)}^{1+n}$ with coordinates $t\in C,$ $x=(x_{1}, \ldots x_{n})\in C^{n},$ $(t, x;\tau, \xi)$the sheaf of microdifferential operators
the symplectic coordinate of $T^{*}X$ andon $T^{*}X$ . In Kashiwara and Oshima’s study of regular systems (cf. [K-O]), thefollowing definition occurs (with a slightly different vocabulary): a matrix of microdifferential Qperators $A(x, D_{x})$ is essentially of nonpositive order if there exists$\iota/>0$ such that the coefficients of any power ofare microdifferential operators of$KO|$system of microdifFerential
[
any
shown
in
that
at
is
regularorder
most . It
equationswith regular singularities along $V=\{t=\xi_{1}=\cdots=\xi_{r}=0\}$ , is aquotient of a system of the form $(tD_{t}-A(x, D_{x}))u=D_{x_{1}}=\cdots=D_{x_{r}}=0$ , with
A essentially of nonpositive order, which enables one to define the monodromy of
may be chosen such that it has the following. Actually, it is shown thatslightly more precise form:$\mathcal{E}x$

$A$

$\nu$

$\mathcal{M}$$A$$\mathcal{M}$(1)$\{\begin{array}{l}[A,D_{t}]=[A,t]=0,andAconsistsofblocksA_{ij}ofN_{i}\cross N_{j}matricesofdiff erentialoperatorssuchthatA_{ij}=0fori>jandA_{ii}=A_{ii}(x)isamatrixofholomorphicfunctions\end{array}$

When investigating e.g. distribution solutions of regular systems from a microloof tempered
cal point of view, that is, solutions with values in the sheaf
to
for
the
,
simplest
one
is
led
look
normalform
over somemicrofunctions on(see remark 2) below).
extension of
to a ring of operators acting on, the ring of tempered microlocal
In particular one has onan action ofoperators (cf. [An]) and the purpose of this note is to prove the$C_{IR^{n}}^{f}$$\mathbb{R}^{n}$$C_{IR^{n}}^{f}$$\mathcal{E}_{X}$$C_{IR^{n}}^{f}$$\mathcal{E}_{X}^{IR,f}$

Theorem 1. Let $A=A(x, D_{x})$ be a matrix of differential operators such that (1)holds. Then
, with(i) $(D_{t}I_{N})^{A(x,D_{x})}$ is a well defined invertible matrix operator over
inverse $(D_{t}I_{N})^{-A(x,D_{x})}$ , and(ii) one has $(D_{1}I_{N})^{A(xD_{x})})(tD_{t}I_{N}-A(x, D_{x}))(D_{t}I_{N})^{-A(x,D_{x})}=tD_{t}I_{N}$ .
$\mathcal{E}_{X,(0;dt)}^{IR,f}$$(^{*})$

Universit\’e Paris-Nord, D\’epartement de Math\’ematiques, Laboratoire C.N.R.S. “Analyse, G\’eom\’etrie et Applications”, URA 742, F93430 Villetaneuse.
Typeset by$\mathcal{A}_{\mathcal{M}^{S-}}$TEX

92- Before going into the proof let us recall the following facts about. Let$X$ be a n-dimensional complex manifold. The sheaf of rings
is the tempered$T^{*}X$
version defined in [An] of the sheaf of ringsonof holomorphic microlocal
operators of [S-K-K], of which it is a subsheaf.
is faithfully flat over
. AlsoOne has.
We are going to make use ofa few topics of the theory of symbols of holomorphic microlocal operators as developped by Kataoka and Aoki (see [Ao] and theliterature quoted there), which we adapt to the framework of.
$\mathcal{E}_{X}^{\mathbb{R},f}$$\mathcal{E}_{X}^{\mathbb{R},f}$$\mathcal{E}_{X}^{IR}$$\gamma^{-1}\gamma_{*}\mathcal{E}_{X}^{\mathbb{R},f}=\mathcal{E}_{X}$$\mathcal{E}_{X}^{\mathbb{R},f}$$\mathcal{E}_{X}$$\mathcal{E}_{X}^{\mathbb{R},f}$Let
Denote by$x_{0}^{*}=(x_{0}; \xi_{0})\in\mathring{T}^{*}X$$S(U)$

and

$U$

a

$\mathbb{R}>0$-conical open neighborhood of$x_{0}^{*}$.

(resp. $S^{f}(U)$ , resp. $R(U)$ ),the space of holomorphic functions $p(x, \xi)$ on
erated cone $U’\subset U$ one has :$U$such that for any compactly gen-$\{\begin{array}{l}forany\epsilon>0p(x,\xi)=O(e^{e|\xi|})onU’(resp.thereexistsm>0suchthatp(x,\xi)=O(|\xi|^{m})onU’)(resp.thereexists\delta>0suchthatp(x,\xi)=O(e^{-\delta|\xi|})onU’)\end{array}$

The notations $S(U),$ $R(U)$ are borrowed from [Ao].

Proposition 2.
(i) (cf. [Ao]) There is an isomorphism of vector spaces$U^{arrow_{*}}h_{\ni}m_{x_{0}}S(U)/R(U)arrow^{\sim}\mathcal{E}_{X,x_{0}^{*}}^{\mathbb{R}}$.

(ii) The isomorphism of (i) induces an isomorphism$U \ni x_{0}\lim_{arrow_{*}}S^{f}(U)/R(U)arrow^{\sim}\mathcal{E}_{X,x_{0}^{f_{*}}}^{1R}$.

In a local coordinate system $(x_{1}, \cdots x_{n})$ , the above morphisms taketo ,and. In fact (ii) is easily deduced from the calculation in
to
to
[Ao].
A representative $p(x, \xi)\in S(U)$ of an operator
for a suitable neigh$P$
is called a symbol of , and the lower bound of theborhood ofsuchthat $p(x, \xi)=O(|\xi|")$ in a conical neighborhood of
is called the order of $P$ at, e.g. $m<\infty$ iff. As noted by Aoki, (i) of Proposition 2 entails that
if $p(x, \xi)\in S(U)$ satisfies $p(x, \xi)=o(|\xi|)$ in a conical neighborhood ofthen$\exp(p(x, \xi))$ is a symbol of an operator of. By (ii) we get also:$x_{i}$$\xi_{i}$$x_{i}\xi_{i}$$D_{\xi:}$$x_{i}D_{\xi_{1}}.$$P\in \mathcal{E}_{X,x_{0}^{*}}^{\mathbb{R}}$$U$$m\in \mathbb{R}$$x_{0}^{*}$$x_{0}^{*}$$x_{0}^{*}$$P\in \mathcal{E}_{X,x^{*}}^{IR,f_{0}}$$x_{0}^{*}$$\mathcal{E}_{X,x_{0}^{*}}^{IR}$(2)$\{\begin{array}{l}ifp(x,\xi)\in S(U)satisfi esp(x,\xi)=O(log|\xi|)inaconicalneighborhoodofx_{0}^{*}thenexp(p(x,\xi))isasymbolofanoperatorof\mathcal{E}_{X^{R},x_{0}^{f_{*}}}^{]}\end{array}$$x_{i}$

10
For example, if $X=C_{x}^{n}$ with coordinates $x=(x_{1}, \ldots x.)$ , one has
whereas.

$\exp(x_{1}\log D_{x_{1}})\in \mathcal{E}_{X,d^{f}x_{1}}^{IR}$$\exp(x_{1}\sqrt{D_{x_{1}}})\in \mathcal{E}_{X,dx_{1}}^{IR}\backslash \mathcal{E}_{X,d^{f}x_{1}}^{\mathbb{R}}$3- Proof of theorem 1. We have to estimate the growth order in of the symbolof $\exp(A(x;\xi)\log\tau)$ . This is done by a straightforward use of norms of matrices of
microdifferential operators. (Of course, if $N=1$ , the theorem is already impliedby (2).)
we denote as usually by
For a differential operator $P(x, D_{x})$ of order$N_{1}(P, s)=N_{l}(P(x;\xi), s)$ the Boutet de Monvel and Kr\’ee formal norm of $P$ definedas the series$\tau$$\leq l$,$N_{l}(P, s)= \sum_{\alpha,\beta,k}\frac{2}{(2n)^{k}}\frac{k!}{(|\alpha|+k)!(|\beta|+k)!}|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}P_{l-k}(x;\xi)|s^{2k+|\alpha+\beta|}$where $P_{l-k}(x;\xi)$ denotes the symbol of the homogenous part of order $l-k$ of $P$ ,and is an independant variable.
If $P=(P_{ij})$ is $N\cross N$ matrix whose entries are (micro-) differential operators oforder at most , we denote, as usually by $N_{1}(P, s)$ the matrix whose entries are the$N_{l}(P_{ij}, s)$ . Recall then, that if
is another $N\cross N$ matrix of (micro-) differential‘
operators of order at most one has
$s$

$l$

$Q$

$l$

$N_{l+l’}(PQ, s)<<N_{l}(P, s)N_{l’}(Q, s)$ ,where the symbol $<<means$ that each entry of the matrix on the right side is amajorant series of the corresponding entry of the matrix on the left side.
We will need the following two estimates. Let $P(x, D_{x})$ be a matrix of differential
operators of order at most .
Fix. By using Cauchy inequalities, it is easy to see that one mayfind:, a constant $M>0$ and a matrix function $c(s)=$a conic neighborhood ofa constant matrix with nonnegative
holomorphic near $s=0$ , where
entries,such that$l$$(x_{0} ; \xi_{0})\in T^{*}C^{n}$$V$$\sum a_{j}s^{j}$(3)$(x_{0} ; \xi_{0})$$a_{j}$$N_{l}(P, s)<<M(1+|\xi|^{l})c(s)$ ,uniformly in $(x;\xi)\in V$.

If $a=(a_{ij})$ is a $N\cross N$ matrix of complex numbers we use the notation$\Vert a\Vert=Sup_{i,j}|a_{ij}|$.

Then if $P$ is a matrix of differential operators of order at most as before, and if$s=r$ with $0<r<1$ , we get from the definition the obvious estimate
$l$

(4)

$\Vert P(x;\xi)\Vert\leq\sum_{0\leq k\leq l}\Vert P_{l-k}(x;\xi)\Vert\leq\frac{l!}{2}\frac{(2n)^{l}}{r^{2l}}N_{l}(P, r)$,11where $P_{l-k}(x;\xi)$ is the matrix of symbols of order $l-k$ of $P$ and $P(x;\xi)=$$\sum_{k}P_{l-k}(x;\xi)$ is the total symbol matrix.

Proof of (i). The matrix $A=A(x, D_{x})$ being as in (1), we may write $A=A_{0}+B$

where $A_{0}=A_{0}(x)$ is the matrix of holomorphic functions consisting of the diagonal
blocksof , and $:=A-A_{0}$ consists of the blocks
with $B_{ij}=A_{ij}$ if $i<j$and zero otherwise. Let be the least integer $1\leq v\leq N-1$ such that $B_{ij}=0$
for all $i,j$ such that $i+v\leq j$ .
For multi-indices, we set$A_{ii}$$A$$B$$B_{ij}$$\nu$$\alpha=(\alpha_{1}, \alpha_{2}, \ldots),$$\beta=(\beta_{1}, \beta_{2}, \ldots)\in \mathbb{N}^{(N)}$$\varphi^{\alpha,\beta}(A_{0}, B)=A_{0}^{\alpha_{1}}B^{\beta_{1}}A_{0}^{\alpha_{2}}B^{\beta_{2}}\cdots\in M_{N}(\mathcal{E}_{X})$Now because of the particuliar forms ofand $B$ we get that$|\beta|>l/$ . Hence for any integer $m\geq 0$ , we have$A_{0}$.

$\varphi^{\alpha,\beta}(A_{0}, B)=0$$A^{m}=(A_{0}+B)^{m}= \sum_{|\alpha+\beta|=m,|\beta|\leq\nu}\varphi^{\alpha,\beta}(A_{0}, B)$for.

Let be the maximum order of the entries of $B$ , then the above formula implies
that for any integer $m\geq 0$ , the matrix
has entries of order, and we have$l$$A^{m}$$\leq\iota/l$$N_{\nu l}(A^{m}, s)<< \sum_{0\leq k\leq\nu}(km)N_{0}(A_{0}, s)^{m-k}N_{l}(B, s)^{k}$.

Fixing $(0;\xi_{0})\in T^{*}C^{n}$ , and making use of (3) and of its notations, one may find
$M>0$ and a matrix function $c(s)$ holomorphic near $s=0$ such that
$N_{l}(B, s)^{k}<<M(1+|\xi|^{\iota})^{\nu}c(s)$holds for $0\leq k\leq v$ and $(x;\xi)$ in a conic neighborhood
Thus
(5)

$V$

of

$(0;\xi_{0})$$N_{\nu l}(A^{m}, s)<<M(1+|\xi|^{l})^{\nu}c(s)(1+N_{0}(A_{0}, s))^{m}$.

,for any integer $m\geq 0$ and any $(0;\xi_{0})\in V$ .
Reducing the conic neighborhood if necessary, we may choose $0<r<1$ anda constant $M_{1}>0$ such thatuniformly in $(0;\xi_{0})\in V$ . Then,
since
$V$

$\Vert N_{0}(A_{0}, r)\Vert\leq M_{1}$$\Vert A^{m}(x;\xi)\Vert\leq\frac{(\nu l)!}{2}\frac{(2n)^{\nu l}}{r^{2\nu l}}N_{\nu l}(A^{m}, r)$by (4), we find by (5) that there is a constant $M_{2}>0$ such that
(6)$\Vert A^{m}(x;\xi)\Vert\leq M_{2}(1+|\xi|^{l})^{\nu}(1+M_{1})^{m}$,for any $m\geq 0$ and $(0;\xi_{0})\in V$ .
Fix the determination ofinsuch that $\log 1=0$ . We have to estimate the growth of the the matrix-valued holomorphic function $\exp(A(x, \xi)\log\tau)$$\log\tau$${\rm Re}\tau>{\rm Im}\tau$12in a conic neighborhood of
as $|\tau|arrow\infty,$. Using this and (6) we get$\tau$$dt\infty$. Let $|\tau|>e^{\pi}$ , thus $|\log\tau|\leq$$\sqrt{2}\log|\tau$$\Vert\tau^{A(x;\xi)}\Vert=\Vert\sum_{m\geq 0}(A(x;\xi)\log\tau)^{m}/m!\Vert\leq\sum_{m\geq 0}\Vert A^{m}(x;\xi)\Vert|\log\tau|^{m}/m!$$\leq M_{2}(1+|\xi|^{l})^{\nu}\sum_{m\geq 0}(1+M_{1})^{m}2^{m/2}(\log|\tau|)^{m}/m!$$\leq M_{2}(1+|\xi|^{l})^{\nu}|\tau|^{\sqrt{2}(1+M_{1})}$,and this holds for $|\tau|>e^{\pi}$ , uniformly for $(x;\xi)$ in a small enough conic neighhaving been made arbitrarily, this proves
borhood of. The choice ofofthat
is a well defined symbol an operator of $X,(0;dt)$ that we denote by$(0;\xi_{0})$$\xi_{0}$$\mathcal{E}^{1R,f}$$\tau^{A(x;\xi)}$$D_{t}^{A(x,D_{x})}$(we omit the notation$I_{N}$).

Since $[A, D_{t}]=0$ one has $[(D_{t})^{A}, t]=A(D_{t})^{A-I_{N}}$ .
$(D_{t})^{A}(tD_{t}-A)=t(D_{t})^{A+I_{N}}+A(D_{t})^{A}-(D_{t})^{A}A=tD_{t}(D_{t})^{A}$ .

Proof of (ii).

Hence$\square$Remarks.
1- The theorem should be more generally true when the matrix $A(x, D_{x})$ isessentially of nonpositive order, but the proof given above breaks down for thismore general case.
2- Distribution solutions of regular operators are investigated in [A-M. ]; theabove theorem provides an alternate proof of proposition 3.1 of that paper.
$F$

References[An] Andronikof, E. : Microlocalisation temp\’er\’ee des distributions et des fonctions holomorphes I, II. C. R. Acad. Sci. 303, 347-350 (1986) and 304, 511-514 (1987).
[Ao] Aoki, T. : Calcul exponentiel des op\’erateurs microdiff\’erentiels d’ordre infini. Ann.
Inst. Fourier, Grenoble, 33, 4, 227-250 (1983).

[A-M.F.] Andronikof, E., Monteiro Fernandes T.: On the tempered solutions of regular systems.
Proc. Intern. Conf. “D-modules and microlocal geometry” Lisbon (1990). de Gruyter,
Berlin, New-York (to appear).

[K-O] Kashiwara, M., Oshima, T. : Systems of differential equation with regular singularities
and their boundary value problems. Ann. of Math., 106, 145-200 (1977).
[S-K-K] Sato, M., Kawai, T., Kashiwara, M., : Microfunctions and pseudo-differential equations. Proc. Conf. at Katata, 1971, Lect. Notes in Math., 287, Springer 265-529(1973).