$\mathbb{C}$上的自然零化子和常秩算子
Natural annihilators and operators of constant rank over $\mathbb{C}$
摘要：
即使两个常秩微分算子的傅立叶符号对于每个非平凡相空间变量具有相同的零空间，这些微分算子的零空间也可能相差无限维空间。在$\mathbb{C}$上常秩的自然条件下，我们建立了零空间在傅立叶符号水平上的相等性已经蕴涵了$\mathscr{D}'$模多项式中的微分算子零空间的相等性.特别是，这个条件允许在微分算子的复体的框架内谈论自然零化子。作为应用，我们建立了二维常复秩微分算子的Poincaré型引理.
英文摘要：
Even if the Fourier symbols of two constant rank differential operators have the same nullspace for each non-trivial phase space variable, the nullspaces of those differential operators might differ by an infinite dimensional space. Under the natural condition of constant rank over $\mathbb{C}$, we establish that the equality of nullspaces on the Fourier symbol level already implies the equality of the nullspaces of the differential operators in $\mathscr{D}'$ modulo polynomials of a fixed degree. In particular, this condition allows to speak of natural annihilators within the framework of complexes of differential operators. As an application, we establish a Poincaré-type lemma for differential operators of constant complex rank in two dimensions.
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