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Weighted GJMS operators on smooth metric measure spaces

Ayush Khaitan

Abstract

We construct weighted GJMS operators on smooth metric measure spaces, and prove that they are formally self-adjoint. We also provide factorization formulas for them in the case of quasi-Einstein spaces and under Gover--Leitner conditions.

Weighted GJMS operators on smooth metric measure spaces

Abstract

We construct weighted GJMS operators on smooth metric measure spaces, and prove that they are formally self-adjoint. We also provide factorization formulas for them in the case of quasi-Einstein spaces and under Gover--Leitner conditions.
Paper Structure (9 sections, 10 theorems, 51 equations)

This paper contains 9 sections, 10 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. The weighted ambient space
  3. Commutation relations
  4. Weighted conformally invariant operators
  5. The leading order term
  6. Factorization formulas
  7. Quasi-Einstein conditions
  8. Gover--Leitner conditions
  9. Formal self-adjointness

Key Result

Theorem 1.1

If $d+m\notin 2\mathbb{N}$, then for each positive integer $k$ there is a conformally invariant operator with leading term $(\Delta_\phi)^k$. If $d+m\in 2\mathbb{N}$, the same result holds with the restriction $1\leq k\leq \frac{1}{2}(d+m)$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 9 more