In this thesis, a pseudodifferential calculus for a degenerate hyperbolic Cauchy
problem is developed. The model for this problem originates from a certain observation
in fluid mechanics, and is then extended to a more general class of hyperbolic
Cauchy problems where the coefficients degenerate like a power of $t + |x|^2$ as
$(t,x) \to (0,0)$.
Symbol classes and pseudodifferential operators are introduced. In this process,
it becomes apparent that exactly in the origin, these operators are of type (1,1).
Although these operators are not $L^2$-continuous in general, a proof of continuity in
$\mathscr C([0,T],L^2(\mathbb R^d))$ is given for a suitable subclass.
An adapted scale of function spaces is defined, where at $t = 0$ these spaces coincide
with 2-microlocal Sobolev spaces with respect to the Lagrangian $\dot T^*_0\mathbb R^d$. In these
spaces, energy estimates are derived, so that a symbolic approach can be applied to
prove wellposedness of the Cauchy problem.
