Local $L_1$ wave propagator conjecture for uniformly elliptic approximations

For $\epsilon\leq 1/4$, let

$$G_\epsilon=-\partial_x^2-(1+2\epsilon x+\epsilon^2x^2)\partial_u^2.$$

Fix $c>0$, and let $f$ be supported in

$$\{(x,u):\|(x,u)\|\leq c\}.$$

Local $L_1$ wave propagator conjecture. If $\alpha>1/2$, then there exists a constant $C_{\alpha,c}$ such that, for all $\epsilon\leq1/4$ and sufficiently small $t$,

$$\left\|\frac{\exp(it\sqrt{G_\epsilon})}{(1+G_\epsilon)^{\alpha/2}}f\right\|_{L_1}\leq C_{\alpha,c}\|f\|_{L_1}.$$

This is proposed as an $L_1$ version of the standard Fourier-integral-operator estimate, uniformly over the elliptic operators $G_\epsilon$. The source describes the corresponding estimates as expected from known elliptic wave-equation methods.