$L^p$ boundedness of wave operators for higher order schr\"odinger operators with threshold eigenvalues

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when $n>4m$ to lower dimensions $2m<n\leq 4m$ to show that when $H$ has a threshold eigenvalue and no resonances, the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{2n}{n-1}$ when $n$ is odd and $1\leq p<\frac{2n}{n-2}$ when $n$ is even. We further show that if the zero energy eigenfunctions are orthogonal to $x^\alpha V(x)$ for all $|\alpha|<k_0$, then the wave operators are bounded on $1\leq p<\frac{n}{2m-k_0}$ when $k_0<2m$ in all dimensions $n>2m$. The range is $p\in [1,\infty)$ and $p\in[1,\infty]$ when $k_0=2m$ and $k_0>2m$ respectively. The proofs apply in the classical $m=1$ case as well and streamlines existing arguments in the eigenvalue only case, in particular the $L^\infty(\mathbb R^n)$ boundedness is new when $n>3$.