Consider a Cauchy problem
$$
Lu(t,x)=0, \quad u(0,x)=g(x), \quad(t,x)\in[0,T]\times \mathbb{R}^n
$$
for a symmetric linear hyperbolic operator $L$. Suppose that we can prove an energy estimate of the form
$$
\Vert u(t,\cdot) \Vert_{H^s} \leq C \Vert g \Vert_{H^s}, \quad t \in [0,T].
$$
How does one infer the following
(i). Existence of solution
(ii). Finite propagation speed of the solution
from the energy estimate. Once the existence is established the uniqueness follows immediately.