Fredholm Theory of Non-Elliptic Operators in the Presence of Normally Hyperbolic Trapping

TL;DR

The paper develops a Fredholm theory for non-elliptic operators in the presence of normally hyperbolic trapping by introducing coisotropic Sobolev spaces that incur weakened regularity at the backward trapped set $\Gamma_u$ and by leveraging a refined microlocal analysis in the b-calculus. It proves a key microlocal estimate (Theorem B) and its semiclassical analogue, showing that $P-iQ$ is Fredholm on these spaces under global NHT and indicial-family constraints, with the normal operator $N(P)$ providing the mechanism to improve decay. The framework is then applied to wave operators on trapping spacetimes, including Kerr–de Sitter and a scattering spacetime, establishing Fredholmness and offering a route to quasinormal-mode expansions and linear stability analysis. The results generalize prior trapping analyses by reducing derivative losses and allowing slight decay, thereby yielding robust solvability and decay statements essential for wave propagation problems in gravitational settings.

Abstract

We present an improved Fredholm theory of non-elliptic operators for when the corresponding classical dynamical system exhibits normally hyperbolic trapping with smooth backward and forward trapped sets. It takes place on coisotropic Sobolev spaces with weak regularity at the backward trapped set $Γ_u$, which are roughly speaking made of distributions $v \in H^s$ satisfying $Φ_u v \in H^{s+1}$ for some quantization $Φ_u$ of the defining function $φ_u$ of $Γ_u$. We then apply it to the case of wave operators on spacetimes.

Figures (7)

• Figure 1: An illustration of the trapped set in the characteristic set $p^{-1}(0) \subset T^{\ast}_b M$. It's intersection with fiber-infinitity $\sigma = \infty$ is $\Gamma$ while it's intersection with finite frequencies $\sigma = const.$ gives the semiclassical trapping. It is assumed to not intersect $\sigma = 0$ hence the 'classical' problem is non-trapping.
• Figure 2: An illustration of the photon sphere (in yellow) around the event horizon of a black hole (in black). Photons starting their path tangentially to the photon sphere will orbit on it forever, while the other ones either escape to infinity or spiral towards the black hole. The behavior of photons normal to the photon sphere is normally hyperbolic. The trajectories shown in this figure are not physical but are shown there for illustrative purposes.
• Figure 3: A sketch of the proof, illustrating the propagation along $H_{\phi_u}$ and $H_p$ we are doing. The arrows are not shown to scale; one should think of $\phi_s$ as growing exponentially fast along $H_p$ and linearly along $H_{\phi_u}$.
• Figure 4: An illustration of the cutoff $\psi$ we use for the quantitative propagation estimate along $\sim H_{\phi_u}$. The rapid increase in $(-\delta, \delta)$ provides the large positive commutator we need for the estimate, while the controlled decay in $(\delta, \epsilon)$ corresponds to the assumed control in $\delta \lesssim \phi_s \lesssim \epsilon$.
• Figure 5: An illustration of the cutoff $\psi$ we use for the quantitative propagation estimate along $H_{p}$. The initial increase in $(0,1)$ corresponds to the region of a priori control $\| \phi_s \| \sim \delta$ while the slow decrease in $(1, \log(\epsilon / \delta))$ is how we will obtain our positive commutator estimate in $\delta \lesssim \| \phi_s \| \lesssim \epsilon$.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10