具有全支撑的分布零因子
Distributional zero divisors with full support
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中文总结 AI 辅助
该研究证明正维光滑流形上存在满足特定波前集条件的全支撑分布零因子对，给出正性准则并说明其在特定拓扑中的累积性，一维情形采用Kozma-Olevskiǐ的傅里叶零级数。
中文摘要 AI 辅助
我们证明，每一个正维光滑流形$M$都存在全支撑分布$u,v\in \mathcal D'(M)$，满足$\mathrm{WF}(u)\cap(-\mathrm{WF}(v))=\emptyset$且$uv=0$。两个因子都不能在任何非空开集上连续，但$u$可被选为最优Sobolev阶$\dim M/2$，其波前集包含在任意指定的闭锥集$\Gamma\subset T^*M\setminus 0$中，该锥集具有非空且非极大的纤维。同时，$v$的波前集可被限制在任意相容的开锥集，或在几何情形下限制在闭锥集，如叶状结构的法丛或由局部共形闭1-形式生成的射线丛。我们给出正性准则，并证明全支撑零因子对在Sobolev-微局部拓扑中累积在$(1,0)$处。一维情形使用Kozma-Olevskiǐ的傅里叶零级数。
英文摘要
We show that every smooth manifold $M$ of positive dimension admits fully supported distributions $u,v\in \mathcal D'(M)$ with $\mathrm{WF}(u)\cap({-}\mathrm{WF}(v))=\emptyset$ and $uv=0$. Neither factor can be continuous on any non-empty open set, but $u$ may be chosen of optimal Sobolev order $\dim M/2$, with wavefront set contained in any prescribed closed conical set $Γ\subset T^\ast M\setminus 0$ with non-empty and non-maximal fibers. Simultaneously, the wavefront set of $v$ may be confined to any compatible open conical set or, in geometric situations, to closed conical sets such as conormal bundles of foliations or ray bundles generated by locally conformally closed one-forms. We give positivity criteria and show that full-support zero divisor pairs accumulate at $(1,0)$ in Sobolev-microlocal topologies. The one-dimensional case uses a Fourier null series of Kozma-Olevski\uı.
