Let $H$, $A$ be self-adjoint operators on a Hilbert space. Moreover, let $I$ be a bounded open interval contained in the spectrum of $H$. Assume that $H$,$A$ satisfy the following positive commutator estimate: $$E(I) [H, \mathrm{i}A]E(I) \geq \alpha E(I),$$ where $E(\cdot)$ are the spectral projection operators of $H$ and $\alpha>0$ is a positive constant. Leaving a few technical assumptions aside, this positive commutator estimate implies the limiting absorption principle: For every closed interval $J\subset I$ and $s>1/2$, $$\sup_{z\in J^\pm} \| \langle A\rangle^{-s} (H-z)^{-1} \langle A\rangle^{-s} \| < \infty,$$ where $J^\pm = \{z\in\mathbb{C} \mid \Re z \in J, \pm \Im z > 0 \}$.
Question: Does the limiting absorption principle also hold for higher powers of the resolvent? That is, is it true that, for $n\in\mathbb{N}$, the estimate $$\sup_{z\in J^\pm} \| \langle A\rangle^{-s} (H-z)^{-n} \langle A\rangle^{-s} \| < \infty$$ holds for suitable $s$?
I expect that if I choose $s$ large enough, then the limiting absorption principle should still hold. However, I was not able to find anything like this in the literature.
