Logarithmic singularities of Schwartz kernels and local invariants of conformal and CR structures

TL;DR

This paper establishes that the logarithmic singularities of Schwartz and Green kernels of conformal and CR invariant pseudodifferential operators are expressed as local invariants derived from ambient metrics, with special cases in even dimensions and above critical weights.

Contribution

It provides explicit invariant formulas for the logarithmic singularities of Green kernels of GJMS operators in both conformal and CR geometries, extending previous understanding of local invariants.

Findings

Logarithmic singularities are linear combinations of Weyl conformal invariants in odd and certain even dimensions.

Invariant expressions for Green kernels of GJMS operators are derived.

CR analogues of these invariants are established, including for CR GJMS operators.

Abstract

This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl conformal invariants, i.e., of local conformal invariants arising from complete tensorial contractions of the covariant derivatives of the Lorentz ambient metric of Fefferman-Graham. In even dimension and above the critical weight exceptional local conformal invariants may further come into play. As a consequence, this allows us to get invariant expressions for the logarithmic singularities of the Green kernels of the GJMS operators (including the Yamabe and Paneitz operators). In the second part, we prove analogues of these results in CR geometry. Namely, we prove that the logarithmic singularities of Schwartz kernels and Green kernels of CR invariant Heisenberg pseudodifferential operators give rise to local CR invariants, and below the critical weight are linear combinations of complete tensorial contractions of the covariant derivatives of Fefferman's K\"alher-Lorentz ambient metric. As a consequence, we can obtain invariant descriptions of the logarithmic singularities of the Green kernels of the CR GJMS operators of Gover-Graham (including the CR Yamabe operator of Jerison-Lee).