  We consider the strictly hyperbolic Cauchy problem
  \begin{align*}
  &D_t^m u - \sum\limits_{j = 0}^{m-1} \sum\limits_{|\gamma|+j = m}
a_{m-j,\,\gamma}(t,\,x) D_x^\gamma D_t^j u = 0, \newline
  &D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots,\,m,
  \end{align*}
  for $(t,\,x) \in [0,\,T]\times \mathbb{R}^n$ with coefficients belonging to
the Zygmund class $C^s_\ast$ in $x$ and having a modulus of continuity below
Lipschitz in $t$. Imposing additional conditions to control oscillations, we
obtain a global (on $[0,\,T]$) $L^2$ energy estimate without loss of
derivatives for $s \geq \{1+\varepsilon,\,\frac{2m_0}{2-m_0}\}$, where $m_0$ is
linked to the modulus of continuity of the coefficients in time.
