Some energy inequalities involving fractional gjms operators

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Under a spectral assumption on the Laplacian of a Poincaré-Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order 2γ ∈ 2 (0,2) or 2γ ∈ 2 (2,4) and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption is necessary and sufficient for such an inequality to hold. We prove the energy inequalities by introducing conformally covariant boundary operators associated to the weighted conformal Laplacian and weighted Paneitz operator which generalize the Robin operator. As an application, we establish a new sharp weighted Sobolev trace inequality on the upper hemisphere.

Original languageEnglish (US)
Pages (from-to)253-280
Number of pages28
JournalAnalysis and PDE
Volume10
Issue number2
DOIs
StatePublished - Jan 1 2017

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Energy Inequality
Fractional
Operator
Trace Inequality
Einstein Manifold
Sobolev Inequality
Hemisphere
Energy
Sufficient
Generalise
Necessary

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

Cite this

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Some energy inequalities involving fractional gjms operators. / Case, Jeffrey Steven.

In: Analysis and PDE, Vol. 10, No. 2, 01.01.2017, p. 253-280.

Research output: Contribution to journalArticle

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AB - Under a spectral assumption on the Laplacian of a Poincaré-Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order 2γ ∈ 2 (0,2) or 2γ ∈ 2 (2,4) and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption is necessary and sufficient for such an inequality to hold. We prove the energy inequalities by introducing conformally covariant boundary operators associated to the weighted conformal Laplacian and weighted Paneitz operator which generalize the Robin operator. As an application, we establish a new sharp weighted Sobolev trace inequality on the upper hemisphere.

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