The action of pseudo-differential operators on functions harmonic outside a smooth hyper-surface

TL;DR

This paper investigates how pseudo-differential operators act on harmonic functions outside a smooth hyper-surface in a Riemannian manifold, deriving the principal symbol of the resulting operator.

Contribution

It demonstrates that the conjugation of a pseudo-differential operator by the Poisson operator yields a pseudo-differential operator on the hyper-surface and computes its principal symbol.

Findings

B = P^* A P is a pseudo-differential operator on Z

Explicit formula for the principal symbol of B

Extension of pseudo-differential calculus to harmonic extensions

Abstract

We consider a smooth hyper-surface Z of a closed Riemannian manifold X. Let P be the Poisson operator associating to a smooth function on Z its harmonic extension on X\Z. If A is a pseudo-differential operator on X of degree <3, we prove that B=P^* A P is a pseudo-differential operator on Z and calculate the principal symbol of B.