Comparison Theorems for Fractional GJMS Operators

In this paper, we mainly focus on the fractional GJMS operators $P_{2\gamma}$ which are defined on the conformal infinity of a Poincar\'{e}-Einstein manifold. We derive two comparison inequalities of the fractional Yamabe constants associated to the fractional GJMS operators. One is between $P_{1}$ and $P_{2\gamma}$ for $\gamma\in (1/2,1)$, and the other is between $P_{2}$ and $P_{2\gamma}$ for $\gamma\in (1,2)$. They both imply the rigidity theorems by characterizing the equalities. Together with the result in \cite{WZ1}, we partially provide some evidence for the monotonicity of the fractional Yamabe constants.