数理解析研究所講究録 1261 巻 2002 年 150-152

150

LACUNAS AND SINGULARITIES OF
FUNDAMENTAL SOLUTIONS OF
HYPERBOLIC DIFFERENTIAL OPERATORS
MOTOO UCHIDA (内田素夫)
Osaka University, Graduate School of Science (大阪大学理学研究科)
RIMS, Kyoto, 22 October 2001

Introduction
In this talk, we dust off and treat again an out-0f-date problem ofpartial differential equations; in fact, we will consider the supports
and the singularities of the fundamental solutions of hyperbolic operators with constant coefficients. The former is known as the problemof lacunas of the fundamental solutions. The latter is known also as
an interesting (and difficult to answer in generall) problem (see [H]for afully detailed study in the case of at most double characteristics).
We have two famous classical papers on this subject (in particular,for the first problem). It was Petrowsky [P] who first systematicallystudied the fundamental solutions of hyperbolic differential operators with constant coefficients. In [ABG], Atiyah, Bott and Girding
clarified and even generalized the Petrowsky theory of lacunas for hyperbolic differential operators. Their theory allows us to draw someconclusions on (non)-existence of lacunas of fundamental solutions.
For example, it is proved in [ABG] (Theorem 7.7 of Part ) that thefundamental solution of ahyperbolic operator in variables has nostrong lacunas if $n\leq 3$ . After their works, the local lacunas of hyperbolic operators has been studied in several papers, in which the$\mathrm{I}\mathrm{I}$$n$except for the strictly hyperbolic case151
sharpness of the fundamental solution is related in detail to the singularity of the wave front surface. For this, see [V] and the references
cited there. It seems however that any explicit theorem on (non)existence of strong lacunas for hyperbolic operators (in particular, inthe case of more than 3variables) is not known.

The purpose of these notes is to give acriterion of non-existence oflacunas (and also for the equality in the general inclusionon the singularities of the fundamental solutions) for hyperbolic
differential operators in variables, $n\geq 4$ , in asimple explicit form
(even in weakly hyperbolic cases).
$\mathrm{W}\mathrm{F}(E)\subset$$\mathrm{W}$$n$Main Result

with. Let $P(D)$ be a differential operator on
constant coefficients (i.e., apolynomial in ), and assume $P(D)$ to$K$ be
be hyperbolic in the direction 0in the sense of Garding. Let
the propagation cone of with respect to 0, and let
Let$\mathrm{R}^{n}$$\theta\in \mathrm{R}^{n}\backslash \{0\}$

$D$

$P$

$\mathrm{W}$$=\{(x, \xi)\in T^{*}\mathrm{R}^{n}|x\in K_{\xi}, \xi\neq 0\}$,denotes the local propagation cone of $P$ at (i.e., the propwhereof $P$ ) with respect to . Letting
agation cone of the localization$E$ be the forward fundamental solution of $P(D)$ , we know in general
that.
andSupp(E)denote the wave front set and the analytic wave
andfront set of $E$ respectively.
defined by $P(z)=$Let $V(P)$ denote the closed algebraic set in. Remark that $P(z)$ denotes the total symbol of $P(D)$ .
Theorem 1.1. Assume that $V(P)$ is irreducible and everywhere nonsingular. Then we have$\xi$

$K_{\xi}$$\theta$$P_{\xi}$$\subset K$$\mathrm{W}\mathrm{F}(E)$$\mathrm{W}\mathrm{F}(E)\subset \mathrm{W}\mathrm{F}_{A}(E)\subset \mathrm{W}$$\mathrm{W}\mathrm{F}_{A}(E)$$\mathrm{C}^{n}$

$0$

(1.1)

Supp(E) $=K$

and
(1.2)

.

$\mathrm{W}\mathrm{F}(E)=\mathrm{W}\mathrm{F}_{A}(E)=\mathrm{W}$This theorem is acorollary of amore general result. The proof istheory
based on the$\mathrm{A}\mathrm{t}\mathrm{i}\mathrm{y}\mathrm{a}\mathrm{h}- \mathrm{B}\mathrm{o}\mathrm{t}\mathrm{t}- \mathrm{G}[mathring]_{\mathrm{a}}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$152
REFERENCES, M. F., Bott, R., and Girding, L., Lacunas for hyperbolic differential operators with constant coefficients , Acta
Math. 124 (1970), 109-189; $II$, Acta Math. 131 (1973), 145-$[\mathrm{A}\mathrm{B}\mathrm{G}]\mathrm{A}\mathrm{t}\mathrm{i}\mathrm{y}\mathrm{a}\mathrm{h}$

$I$

206.
[H]
[P]
[V]
[U]

Hormander, L., The wave front set of the fundamental solutionof a hyperbolic operator with double characteristics, J. Anal.
Math. 59 (1992), 1-36.

Petrowsky, I. G., On the diffusion of waves and the lacunas for
hyperbolic equations, Mat. Sb. 17 (59) (1945), 289-370.
Vassiliev, V. A., Ramified Integrals, Singularities and Lacunas,
Kluwer Acad. Publ., Dordrecht, 1995.
Uchida, M., A non-eistence theorem of lacunas for hyperbolic
differential operators with constant coefficients, RRM 00-14,
Osaka Univ., 2000 (to appear in Ark. Mat.)