For hyperbolic PDE's, we have two aspects that distinguish them from elliptic and parabolic PDE's and thus *forbid* one from being able to obtain estimates in the form you want:
 
 - Usually the loss of derivatives is worse than in the elliptic case — in fact, even worse than in the more general subelliptic case. Typically, you lose a full (weak = Sobolev) derivative in *a priori* regularity estimates for linear hyperbolic PDE's: the so-called *energy estimates*. This loss is essentially sharp and can be tracked back to the fundamental theorem of Calculus. As pointed by Willie Wong in his [answer](https://mathoverflow.net/a/421891), this can be refined if you take into account the characteristic set of the principal symbol of the operator, by means of techniques collectively called *microlocal analysis*. More precisely, worse than elliptic-type regularity must propagate along the bicharacteristic curves of the principal symbol, which are nonexistent in the elliptic case.
 - Even if you account for the above loss of derivatives, the "optimal" relative shape of the regions $B_1$, $B_2$ is different due to a hallmark property of hyperbolic PDE's — namely, the *finite speed of propagation* of (perturbations of) solutions. Typically, $B_2$ will be a "cone-shaped" shadow region cast by $B_1$, whose precise form is once again dictated by the principal symbol of the operator.