Nondegenerate integrated local energy estimate near the Schwarzschild photon sphere

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $(M,g,N)$ be the Schwarzschild exterior datum with $s>0$, $f(r)=1-s\mathrm{/}r$, exterior coordinate $x^0$, and stationary field $T={\mathrm{\partial }}_0$. Let $0<\delta <s\mathrm{/}2$ and put $K_{\delta }:=\{(x^0,r,\omega ):\mathrm{\mid }r-3s\mathrm{/}2\mathrm{\mid }\le \delta \}$ for $\omega \in S^2$. Then $K_{\delta }$ is contained in $r>s$. On the static slice ${\mathrm{\Sigma }}_{x^0}:=\{x^0=\text{constant}\}$ the future unit normal is $n=f^{-1\mathrm{/}2}T$. For a smooth real test wave $\psi$ with finite initial stationary energy, define

$E_T[\psi ](x^0):={\int }_{{\mathrm{\Sigma }}_{x^0}}{Q}_{ab}[\psi ]n^a{T}^{b}d{\mu }_{{\mathrm{\Sigma }}_{x^0}},E_{T,\delta }[\psi ](x^0):={\int }_{{\mathrm{\Sigma }}_{x^0}\cap K_{\delta }}{Q}_{ab}[\psi ]n^a{T}^{b}d{\mu }_{{\mathrm{\Sigma }}_{x^0}}\mathrm{.}$

A nondegenerate integrated local energy estimate with no trapping loss on $K_{\delta }$ means that there is a finite constant $C$, independent of $\psi$, such that every smooth solution of ${\mathrm{\square }}_g\psi =0$ with finite $E_T[\psi ](0)$ satisfies

${\int }_0^{\mathrm{\infty }}{E}_{T,\delta }[\psi ](x^0)d{x}^0\le C{E}_T[\psi ](0)\mathrm{.}$

Here “nondegenerate” means that the estimate integrates the full unweighted first-derivative density, with no weight vanishing at $r=3s\mathrm{/}2$; “no trapping loss” means that the right side contains only the initial first-derivative stationary energy, with no additional derivatives or degenerating weights. This definition asserts no existence or nonexistence of such a constant.

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