Boundary Operators Associated With the Sixth-Order GJMS Operator

• Published in: International mathematics research notices Vol. 2021; no. 14; pp. 10600 - 10653
• Main Authors: Case, Jeffrey S, Luo, Weiyu
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01-07-2021
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rnz121

Abstract
We describe a set of conformally covariant boundary operators associated with the 6th-order Graham--Jenne--Mason--Sparling (GJMS) operator on a conformally invariant class of manifolds that includes compactifications of Poincaré–Einstein manifolds. This yields a conformally covariant energy functional for the 6th-order GJMS operator on such manifolds. Our boundary operators also provide a new realization of the fractional GJMS operators of order one, three, and five as generalized Dirichlet-to-Neumann operators. This allows us to prove some sharp Sobolev trace inequalities involving the interior $W^{3,2}$-seminorm, including an analogue of the Lebedev–Milin inequality on six-dimensional manifolds.