Lemma: Uniform high-frequency asymptotics

Let $Y\subset M$ be compact. Uniformly for $y\in Y$, $\mathbf{n}\in S^2$, $r\ge 1$, and $0<\epsilon \le 1$,

where

For $|q|\le 1$, $F_{f,\epsilon }(y,q)$ is uniformly bounded on $Y$, independently of $0<\epsilon \le 1$.

Proof. Separate (eq:K) into its direct and reflected terms.

term. After integration against $f$, the direct term contains the Fourier transform of $f$, or of $tf$, at temporal frequency $|p|+{\omega }_m(r)$ and spatial frequency $p+r\mathbf{n}$. Since $f$ is smooth and compact,

and the same holds with $tf$. Integrating against $|p{|}^{-1}+|p{|}^{-2}$ proves arbitrary rapid decrease in $r$, uniformly in $\epsilon$ and in $y\in Y$.

term. Write

On $|a|\le r/2$,

The constants in the last estimate may depend on the fixed mass. The phase is

uniformly for $t$ in the support of $f$ and $y\in Y$.

The spatial Fourier transform of $f(t,\cdot )$ decreases rapidly in $a$, uniformly in $t$, with all required derivatives. The errors above therefore contribute $O(r^{-1}+\epsilon )$ after extracting $r^{-2}$.

The leading $a$-integral is Fourier inversion:

Its $t$-integral is $A_f$.

On $|a|>r/2$, arbitrary rapid decrease in $a$ dominates the locally integrable factor $|p{|}^{-2}$. After extracting $r^{-2}$, that tail is $O(r^{-N})$ for any prescribed $N$, by taking sufficiently many Fourier-decay powers. The direct term can be absorbed into the same error. This proves (eq:Fasymptotic)–(eq:Ebound).

For bounded $q$, integrate the spatial Fourier-decay bounds against $|p{|}^{-1}+|p{|}^{-2}$. Both are locally integrable, and the large-$|p|$ tails are rapidly decreasing. This proves the last assertion.