Frequency-localized dispersive estimates for the wave equation on flat cones

Let $C(\mathbb{S}^1_\rho)$ be the flat cone with metric $\mathrm{g}$, let $\Delta_{\mathrm{g}}$ be its Laplace operator, and let $\nabla_{\mathrm{g}}$ denote the associated gradient. For a smooth cutoff $\beta$ supported in $\left(\frac{1}{4},4\right)$ and a frequency parameter $\mu>0$, define

$${\mathcal{U}}(t)=\frac{\sin\left(t\sqrt{\Delta_{\mathrm{g}}}\right)}{\sqrt{\Delta_{\mathrm{g}}}},\qquad {\dot{\mathcal{U}}}(t)=\cos\left(t\sqrt{\Delta_{\mathrm{g}}}\right).$$

Frequency-localized dispersive estimate. If

$$\beta\!\left(\mu^{-1}\sqrt{\Delta_{\mathrm{g}}}\right)f=f,\qquad \beta\!\left(\mu^{-1}\sqrt{\Delta_{\mathrm{g}}}\right)g=g,$$

then

$$\left\|{\mathcal{U}}(t)g\right\|_{L^\infty(C(\mathbb{S}^1_\rho))}\lesssim \mu(1+\mu|t|)^{-1/2}\left\|g\right\|_{L^1(C(\mathbb{S}^1_\rho))},$$

and

$$\left\|{\dot{\mathcal{U}}}(t)f\right\|_{L^\infty(C(\mathbb{S}^1_\rho))}\lesssim \mu(1+\mu|t|)^{-1/2}\left(\mu\left\|f\right\|_{L^1(C(\mathbb{S}^1_\rho))}+\left\|\nabla_{\mathrm{g}}f\right\|_{L^1(C(\mathbb{S}^1_\rho))}\right).$$

These frequency-localized $L^1\to L^\infty$ bounds are intended to provide the dispersive input for Strichartz estimates for the wave equation on a flat cone. The supplied text presents them as the estimates one needs to prove, but gives no resolution status, so the claim is recorded as open.