Lemma: Finite-slab Cauchy estimate

For $b\in L^1(I;H^s({\mathbb{R}}^3))$, $s\ge 0$, and a compact time interval $I$ containing $t_0$, the zero-data solution of $({{\partial }_t}^2-\Delta +m^2)u=b$ is the sine integral in (eq:S) with $b$ in place of $L_h\varphi$, and

It is unique in this class. Smooth data and source give a smooth solution. Propagation has speed at most one.

Proof. The Fourier multipliers are $\sin ((t-s){\omega }_p)/{\omega }_p$ and $\cos ((t-s){\omega }_p)$, with ${\omega }_p=\sqrt{|p{|}^2+m^2}$. Their weighted bounds give the estimate and differentiation proves the equation. Uniqueness follows from the homogeneous energy identity. For finite propagation integrate

over a truncated causal cone. In a source-free cone the outward characteristic flux is nonnegative; zero initial energy implies zero solution there. Smoothness follows by applying the estimates at all orders.