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 Poincaré-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 $\gamma \in (1\mathrm{/}2,1)$, 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.