Finite-regularity logarithmic Price's law conjecture for higher angular modes

Consider solutions $\phi$ of the wave equation on a fixed Schwarzschild background. Let $\Gamma$ be the timelike initial hypersurface, let $t$ be a time coordinate, let $u$ and $v$ be retarded and advanced null coordinates, let $\tau$ be proper time along a timelike curve of constant area radius, and let $\mathcal H^+$ and $\mathcal I^+$ denote future null infinity and the future event horizon. Write $\phi_\ell$ for the angular $\ell$-mode. Prescribe data on $\Gamma$ of sufficient but fixed finite regularity satisfying $r^\ell\phi_\ell\sim t^{-1}$ as $t\to-\infty$ for all $\ell$, together with the no incoming radiation condition on $\mathcal I^-$. Finite-regularity logarithmic Price's law conjecture. There exists $\ell_0\in\mathbb N$, increasing with the prescribed regularity, such that for all $\ell\leq\ell_0$ the late-time asymptotics near $i^+$ are

$$r\phi_\ell|_{\mathcal I^+}\sim u^{-2-\ell}\log u,$$ $$\phi_\ell|_{r=\mathrm{constant}}\sim\tau^{-2\ell-3}\log\tau,\qquad \phi_\ell|_{\mathcal H^+}\sim v^{-2\ell-3}\log v.$$

For some $\epsilon>0$, the higher-mode projection satisfies

$$r\phi_{\ell>\ell_0}|_{\mathcal I^+}=\mathcal O(u^{-2-\ell_0-\epsilon}),$$ $$\phi_{\ell>\ell_0}|_{r=\mathrm{constant}}=\mathcal O(\tau^{-2\ell_0-3-\epsilon}),\qquad \phi_{\ell>\ell_0}|_{\mathcal H^+}=\mathcal O(v^{-2\ell_0-3-\epsilon}).$$

If the data are smooth, one may take $\ell_0=\infty$. The conjecture extends the logarithmically modified Price's law suggested by the paper to data of fixed finite regularity; the regularity determines how many angular modes obey the asserted asymptotics, while higher modes satisfy weaker uniform bounds.