A factorization of the GJMS operators of special Einstein products and applications

We show that the GJMS operators of a special Einstein product factor as a composition of second- and fourth-order differential operators. In particular, our formula applies to the Riemannian product $H^{\mathrm{\ell }}\times S^{d-\mathrm{\ell }}$. We also show that there is an integer $D=D(k,\mathrm{\ell })$ such that if $d\ge D$, then for any special Einstein product $N^{\mathrm{\ell }}\times M^{d-\mathrm{\ell }}$, the Green’s function for the GJMS operator of order $2k$ is positive. As a result, these products give new examples of closed Riemannian manifolds for which the $Q_{2k}$-Yamabe problem is solvable.