Year: 2026
Type: preprint
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 $L^{2}$ 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.
Source: arXiv (Cornell University)
Author: Takeuchi, Yuya
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214112398