Let be a Poincaé-Einstein manifold and let be any boundary defining function such that is a metric of the compact manifold . For and , there are different ways to define the fractional derivative of . Denote .

In Jeffrey and Alice’s paper, they use the notion of smooth metric measure space ,

here is the weighted Schouten scalar defined by

Suppose is the solution of the extension problem

then

here and is the **geodesic** defining function.

In Maria and Alice’s paper, one can define as the solution of the extension problem

here the derivative are taken with respect to the metric ,

or equivalently

One may wonder what the difference of the two ways of extension is? The leading operator are both second order. It attempts to compare the error term and . Since has constant scalar curvature , one has the following identity

If is the geodesic defining function, that is , then

Therefore, in this case the two error term are the same. Consequently the equations of extension are the same.

It seems for general defining function, the two ways of extension are different.