Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations

In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{\Delta u}\sts{x} -{\bm\alpha}\sts{N,\lambda} \int_{\R^N} { \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}{\lambda}} }\diff{y}\, u^{p-1}\sts{x} =0,\quad x\in \R^N \end{equation*} where $N\geq 3$, $0<\lambda<N$, $p=\frac{2N-\lambda}{N-2}$ and ${\bm\alpha}\sts{N,\lambda}$ is a normalized constant such that $ u(x)=\left(1+|x|^2\right)^{-\frac{N-2}{2} }$ is a bubble solution of the equation \eqref{NLH}. It solves an open nondegeneracy problem in \cite{MWX:Hartree, GMYZ2022cvpde} and generalizes the partial nondegeneracy results in \cite{DY2019dcds, GWY2020na, LTX2021} to the full range $0<\lambda<N$. The key observation is that by use of the stereographic projection $\mathcal{S}$, the weighted pushforward map $\mathcal{S}_*$ is one-to-one map between the null space of the linearized operator and the spherical harmonic function subspace $\mathcal{H}_1^{N+1}$ of degree one.