Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

TL;DR

This paper derives precise asymptotic expansions for the Poisson kernel and Green's functions of the fractional conformal Laplacian on asymptotically hyperbolic manifolds, advancing understanding of the fractional Yamabe problem.

Contribution

It provides sharp asymptotic expansions that resolve part of a conjecture and facilitate solving the fractional Yamabe problem in specific geometric settings.

Findings

Sharp expansions of Green's functions near singularities.

Resolution of a conjecture for locally flat conformal infinities.

Application of asymptotics to solve the fractional Yamabe problem.

Abstract

We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei[22] in the case of locally flat conformal infinities of Poincar\'e-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [30] to show solvability of the fractional Yamabe problem for conformal infinities of dimension $3$ and fractional parameter in $(\frac{1}{2} , 1)$ to a global case left by previous works.