Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 29 min ago
PreprintarXiv licencearXiv

Lower bounds for the critical CR GJMS operator

  • Yuya Takeuchi
Open at sourcePDFDOICite

Abstract

We establish lower bounds for the critical CR GJMS operator on closed pseudo-Hermitian manifolds. In dimension 2n+1, assuming embeddability, we obtain a lower bound involving the Heisenberg Sobolev norm of order n+1, modulo an L2 term, on the orthogonal complement of the kernel. In dimension three, we prove that the CR Paneitz operator is bounded below without assuming embeddability. The proofs combine the Heisenberg calculus with a square-root construction based on the joint functional calculus of the sub-Laplacian and the Reeb vector field on the Heisenberg group. As an application, we show that the infinitely many negative eigenvalues of the CR Paneitz operator on the Rossi sphere have zero as their only accumulation point. Consequently, this operator does not have closed range.

Related

  1. Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions
    Inbo Gottlieb Fenves, Jiahao Tan
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  2. Total scalar curvature under a curvature operator lower bound
    Jian Ge, Chuanhuan Li, Ronggang Li
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  3. High-Probability Minimax Lower Bounds
    Tianyi Ma, Kabir Aladin Verchand, Richard J. Samworth
    Statistical SciencePeer-reviewedLicence not statedFull text24 citedStatisticsOpenAlex
  4. Lower curvature bounds on metric surfaces via the Bishop--Gromov inequality
    Alexander Lytchak, Stephan Stadler
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  5. Lower Bounds for Moments of L-functions
    Sanoli Gun, Gaurav Kumar, Deep Thakur
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{takeuchi2026lower,
  title = {{Lower bounds for the critical CR GJMS operator}},
  author = {Yuya Takeuchi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26371},
  eprint = {2609.26371},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26371},
}
RIS
TY  - UNPB
TI  - Lower bounds for the critical CR GJMS operator
AU  - Yuya Takeuchi
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26371
UR  - https://arxiv.org/abs/2609.26371
AB  - We establish lower bounds for the critical CR GJMS operator on closed pseudo-Hermitian manifolds. In dimension 2n+1, assuming embeddability, we obtain a lower bound involving the Heisenberg Sobolev norm of order n+1, modulo an L^2 term, on the orthogonal complement of the kernel. In dimension three, we prove that the CR Paneitz operator is bounded below without assuming embeddability. The proofs combine the Heisenberg calculus with a square-root construction based on the joint functional calculus of the sub-Laplacian and the Reeb vector field on the Heisenberg group. As an application, we show that the infinitely many negative eigenvalues of the CR Paneitz operator on the Rossi sphere have zero as their only accumulation point. Consequently, this operator does not have closed range.
ER  -