*preprint*

**Inserted:** 23 oct 2023

**Year:** 2021

**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-Wei22 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.