论文标题
卷积和差分运算符内核的线性拓扑不变
Linear topological invariants for kernels of convolution and differential operators
论文作者
论文摘要
我们为各种类型的卷积和差分运算符的光滑内核建立了条件$（ω）$。由$（dn）$ - $（ω）$分配的沃格特和瓦格纳理论，这意味着这些运算符在矢量价值的平稳功能的相应空间上过滤，并在Montel $（df）$的产物中具有值的价值 - $（dn）$（DN）$（DN）$（DN）$（例如Dn）$（例如，dn）$（例如）设置$ y \ subseteq \ mathbb {r}^n $或space $ \ mathscr {s}'（\ mathbb {r}^n）$ prepend Distribution的$。最值得注意的是，我们表明： $） \，p（d）f = 0 \} $满足任何差分运算符$ p（d）$的$（ω）$和任何开放式凸套装$ x \ subseteq \ subseteq \ mathbb {r}^d $。 $（ii）$让$ p \ in \ mathbb {c} [ξ_1，ξ_2] $和$ x \ subseteq \ subseteq \ mathbb {r}^2 $打开，以至于$ p（d）：\ mathscr {e}（e}（x）（x）\ rightArrow \ rightArrow \ rightArrow \ m m mathscr {e}（x）$ surective。然后，$ \ mathscr {e} _p（x）$满足$（ω）$。 $（iii）$ let $μ\ in \ mathscr {e}'（\ mathbb {r}^d）$，以至于$ \ mathscr {e}（\ mathbb {r}^d）\ rightArrow \ rightArrow \ rightArrow \ mathscr \ mathscr \ mathscr {e}（e}（e}（e}）冲销。然后，$ \ {f \ in \ mathscr {e}（\ mathbb {r}^d）\，| \，μ\ ast f = 0 \} $满足$（ω）$。 本文的中心结果指出，当一般卷积方程的平滑零解决方案的空间时，当时方程式$（ω）$时，方程式零解决方案的空间才能满足条件$（pΩ）$。然后，上述结果遵循有关卷积和差分运算符的分布内核的已知结果（PΩ）$。
We establish the condition $(Ω)$ for smooth kernels of various types of convolution and differential operators. By the $(DN)$-$(Ω)$ splitting theorem of Vogt and Wagner, this implies that these operators are surjective on the corresponding spaces of vector-valued smooth functions with values in a product of Montel $(DF)$-spaces whose strong duals satisfy the condition $(DN)$, e.g., the space $\mathscr{D}'(Y)$ of distributions over an open set $Y \subseteq \mathbb{R}^n$ or the space $\mathscr{S}'(\mathbb{R}^n)$ of tempered distributions. Most notably, we show that: $(i)$ $\mathscr{E}_P(X) = \{ f \in \mathscr{E}(X) \, | \, P(D)f = 0 \}$ satisfies $(Ω)$ for any differential operator $P(D)$ and any open convex set $X \subseteq \mathbb{R}^d$. $(ii)$ Let $P\in\mathbb{C}[ξ_1,ξ_2]$ and $X \subseteq \mathbb{R}^2$ open be such that $P(D):\mathscr{E}(X)\rightarrow\mathscr{E}(X)$ is surjective. Then, $\mathscr{E}_P(X)$ satisfies $(Ω)$. $(iii)$ Let $μ\in \mathscr{E}'(\mathbb{R}^d)$ be such that $ \mathscr{E}(\mathbb{R}^d) \rightarrow \mathscr{E}(\mathbb{R}^d), \, f \mapsto μ\ast f$ is surjective. Then, $ \{ f \in \mathscr{E}(\mathbb{R}^d) \, | \, μ\ast f = 0 \}$ satisfies $(Ω)$. The central result in this paper states that the space of smooth zero solutions of a general convolution equation satisfies the condition $(Ω)$ if and only if the space of distributional zero solutions of the equation satisfies the condition $(PΩ)$. The above and related results then follow from known results concerning $(PΩ)$ for distributional kernels of convolution and differential operators.
