Linear topological invariants for kernels of convolution and differential operators
Linear topological invariants for kernels of convolution and differential operators
复制标题
卷积核和微分算子的线性拓扑不变量
DOI：
10.1016/j.jfa.2023.109886
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发表时间：
2022
影响因子：
1.7
通讯作者：
T. Kalmes
中科院分区：
文献类型：
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作者：
A. Debrouwere;T. Kalmes
我们建立了各种类型的卷积和微分算子的光滑核的条件（Ω）。根据沃格特和瓦格纳的（DN）-（Ω）分裂定理，这意味着这些算子在相应的向量值光滑函数空间上是满射的，这些函数的值在Montel（DF）-空间的乘积中，而Montel（DF）-空间的强模满足条件（DN），例如开集Y <$Rn上的分布空间D′（Y）或调和分布空间S′（Rn）.最值得注意的是，我们证明了：（i）E P（X）={f∈ E（X）|对任意微分算子P（D）和任意开凸集X <$Rd，P（D）f= 0}满足（Ω）.（i i）设P∈ C [<$1，<$2]，X <$R2开，使得P（D）：E（X）→ E（X）是满射.则E P（X）满足（Ω）。（i i i）设μ∈ E′（Rd）使得E（Rd）→ E（Rd），f <$μ <$f是满射.则{f∈ E（Rd）|μ ∈ f= 0}满足（Ω）。本文的主要结果是：一般卷积方程的光滑零解空间满足条件（Ω）当且仅当该方程的分布零解空间满足条件（PΩ）.上述和相关的陈述然后从已知的关于卷积和微分算子的分布核的（PΩ）的结果[3]，[15]，[16]得出。
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), eg, the space D′(Y) of distributions over an open set Y⊆ R n or the space S′(R n) of tempered distributions. Most notably, we show that:(i) E P (X)={f∈ E (X)| P (D) f= 0} satisfies (Ω) for any differential operator P (D) and any open convex set X⊆ R d.(i i) Let P∈ C [ξ 1, ξ 2] and X⊆ R 2 open be such that P (D): E (X)→ E (X) is surjective. Then, E P (X) satisfies (Ω).(i i i) Let μ∈ E′(R d) be such that E (R d)→ E (R d), f↦ μ⁎ f is surjective. Then,{f∈ E (R d)| μ⁎ f= 0} satisfies (Ω). The central result in this paper is 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 statements then follow from known results concerning (PΩ) for distributional kernels of convolution and differential operators [3],[15],[16].