论文标题
决定性的大会以及郭键和Abja-olive的工作
Determinant majorization and the work of Guo-Phong-Tong and Abja-Olive
论文作者
论文摘要
本说明的目的是为所有操作员$ n $ n $ n $ n $ N $ N $ on Symmetric $ n \ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ N $确定的所有操作员确定的所有操作员$ f $的决定符量化公式$ f（a）^{1 \ f（a）^{1 \ forn} \ geq \ det（a）^{1 \ over n} $。当矩阵分别为对称，对称对称或quaternionic对称时，这里的“不变”是指o $ $（n）$，u $（n）$或sp $（n）$的组下。这大大扩展了Gu-Phong-Tong和Gu-Phong最近在复杂歧管上的微分方程的适用性。它还与Abja-Olive对内部规律性的工作有关。 根据有序的特征值，对对角线运营商和操作员的进一步申请。显示了显示结果准确性的例子。为了申请Abja-Olive的作品以及本文中的其他评论，我们为附录中的Garding-Dirichlet操作员建立了一些结果。一个是花园锥体的疲惫引理。另一个给出了高阶衍生品的界限，这是由于其优雅的表达方式作为花园特征值的功能而产生的。还讨论了中央射线假设的关键假设。
The objective of this note is to establish the Determinant Majorization Formula $F(A)^{1\over N} \geq \det(A)^{1\over n}$ for all operators $F$ determined by an invariant Garding-Dirichlet polynomial of degree $N$ on symmetric $n \times n$ matrices. Here "invariant" means under the group O$(n)$, U$(n)$ or Sp$(n)$ when the matrices are real symmetric, Hermitian symmetric, or quaternionic symmetric respectively. This greatly expands the applicability of the recent work of Guo-Phong-Tong and Guo-Phong for differential equations on complex manifolds. It also relates to the work of Abja-Olive on interior regularity. Further applications to diagonal operators and to operators depending on the ordered eigenvalues are given. Examples showing the preciseness of the results are presented. For the application to Abja-Olive's work, and other comments in the paper, we establish some results for Garding-Dirichlet operators in appendices. One is an exhaustion lemma for the Garding cone. Another gives bounds for higher order derivatives, which result from their elegant expressions as functions of the Garding eigenvalues. There is also a discussion of the crucial assumption of the Central Ray Hypothesis.
