### 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 language | English (US) |
---|---|

Pages (from-to) | 253-280 |

Number of pages | 28 |

Journal | Analysis and PDE |

Volume | 10 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Applied Mathematics

### Cite this

}

*Analysis and PDE*, vol. 10, no. 2, pp. 253-280. https://doi.org/10.2140/apde.2017.10.253

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Some energy inequalities involving fractional gjms operators

AU - Case, Jeffrey Steven

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85014628859&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014628859&partnerID=8YFLogxK

U2 - 10.2140/apde.2017.10.253

DO - 10.2140/apde.2017.10.253

M3 - Article

AN - SCOPUS:85014628859

VL - 10

SP - 253

EP - 280

JO - Analysis and PDE

JF - Analysis and PDE

SN - 2157-5045

IS - 2

ER -