arXiv subjects
Publications and source records attributed to Jan Kisyński.
The original M.Valdivia proof of his theorem on representation of the space $O_M$ uses results of S.Rolewicz concerning metric linear spaces and results of A.Grothendieck from his theory of topological tensor product. We present a more direct proof.
Let $\Mmm$ denote the set of $\mm$ matrices with complex entries, and let $\calG(\partial_1,...,\partial_n)$ be an $\mm$ matrix whose entries are partial differential operators on $\Rn$ with constant complex coefficients. It is proved that $\calG(\partial_1,...,\partial_n)\otimes δ$ is the generating distribution of a smooth one-parameter convolution semigroup of $\Mmm$-valued rapidly decreasing distributions on $\Rn$ if and only if $$\sup_{(ξ_1,...,ξ_n)\in\Rn}\hReσ(\calG(iξ_1,...,iξ_n))<\infty.$$ Applications to systems of partial differential operators with constant coefficients are considered.
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.
Let $P(\partial/\partial x)$ be an $m\times n$ matrix whose entries are PDO on $\bbR^n$ with constant coefficients, and let $\calS(\bbR^n)$ be the space of infinitely differentiable rapidly decreasing functions on $\bbR^n$. It is proved that $P(\partial/\partial x)|_{(\calS(\bbR^n))^m}$ is the infinitesimal generator of a $(C_0)$-semigroup $(S_t)_{t\ge0}\subset L((\calS(\bbR^n))^m)$ if and only if $P(\partial/\partial x)$ satisfies the Petrovski\uıcorrectness condition. Moreover, if it is the case, then $(S_t)_{t\ge0}$ is an exponential semigroup whose characteristic exponent is equal to the stability index of $P(\partial/\partial x)$. Similar statements are also proved for some other function spaces on $\bbR^n$, and for the space of tempered distributions.