演化偏微分方程的基本解和快速下降分布
Fundamental solutions of evolutionary PDOs and rapidly decreasing distributions
摘要：
令$P(\partial_0,\partial_1,...,\partial_n)$是$\symR^{1+n}$上的具有常系数的PDO。本文证明了：（i）当$(ξ_1,...,ξ_n)$在$\symR^n$上时，多项式$P(λ,iξ_1,...,iξ_n)$的$λ$ -根的实部是上有界的当且仅当（ii）$P$有一个支撑在$H_+=\{(x_0,x_1,\allowbreak..., x_n)\in \symR^{1+n}:x_0\ge0\}$中的基本解，该基本解具有用L. Schwartz空间$\calO^{\prime}_C$的快速减少的分布。证明了具有这些特殊性质的$H_+$中具有支撑的基本解是唯一的.
英文摘要：
Let $P(\partial_0,\partial_1,...,\partial_n)$ be a PDO on $\symR^{1+n}$ with constant coefficients. It is proved that (i) the real parts of the $λ$-roots of the polynomial $P(λ,iξ_1,...,iξ_n)$ are bounded from above when $(ξ_1,...,ξ_n)$ ranges over $\symR^n$ if and only if (ii) $P$ has a fundamental solution with support in $H_+=\{(x_0,x_1,\allowbreak..., x_n)\in \symR^{1+n}:x_0\ge0\}$ having some special properties expressed in terms of the L. Schwartz space $\calO^{\prime}_C$ of rapidly decreasing distributions. Moreover, it is proved that the fundamental solution with support in $H_+$ having these special properties is unique.
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