Determinant majorization and the work of Guo-Phong-Tong and Abja-Olive

Abstract

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.