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Duality for Garding operators and a new inequality of GM/AM type
Abstract
This article concerns Garding Dirichlet operators, and also the theory of Garding polynomials. Garding Dirichlet operators are those which arise from homogeneous polynomials on the space S(n) of symmetric n × n polynomials, which are hyperbolic with respect to any positive definite matrix. (In particular, any polynomial on ℝn, which is hyperbolic with respect to elements in (ℝ+)n, gives rise to a Garding operator on S(n), in many ways.) Here a new duality theorem is developed with many special cases and examples. The dual g* of a Garding operator g is used to prove a new basic inequality for g. An important Lemma of Guo-Phong-Tong is improved upon as a final application.
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BibTeX
@misc{harvey2026duality,
title = {{Duality for Garding operators and a new inequality of GM/AM type}},
author = {F. Reese Harvey and H. Blaine Lawson Jr},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28773},
eprint = {2609.28773},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28773},
}
RIS
TY - UNPB
TI - Duality for Garding operators and a new inequality of GM/AM type
AU - F. Reese Harvey
AU - H. Blaine Lawson Jr
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28773
UR - https://arxiv.org/abs/2609.28773
AB - This article concerns Garding Dirichlet operators, and also the theory of Garding polynomials. Garding Dirichlet operators are those which arise from homogeneous polynomials on the space S(n) of symmetric n × n polynomials, which are hyperbolic with respect to any positive definite matrix. (In particular, any polynomial on ℝ^n, which is hyperbolic with respect to elements in (ℝ_+)^n, gives rise to a Garding operator on S(n), in many ways.) Here a new duality theorem is developed with many special cases and examples. The dual g^* of a Garding operator g is used to prove a new basic inequality for g. An important Lemma of Guo-Phong-Tong is improved upon as a final application.
ER -