常秩算子同调性质的初等探讨
An elementary approach to the homological properties of constant-rank operators
摘要：
我们给出了Raien-Rang关于每个常秩算子都有一个正合势和一个正合零化子的结果的一个简单的构造性推广。我们的建设是完全自包含的，并提供了一个改进的顺序的运营商所构建的Raietzeren，以及顺序的显式零化子椭圆算子由于Van Schaftingen。我们还给出了常秩算子的最优零化子的一个抽象构造，推广了Van Schaftingen关于椭圆算子的最优构造。最后，我们建立了$\mathbb{R}^d$上常秩算子和齐性空间的一个推广的Poincaré引理，并证明了周期映射空间上位势的存在性所需的条件严格弱于常秩性质.
英文摘要：
We give a simple and constructive extension of Raiţă's result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Raiţă, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincaré lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.
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