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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 tothe Zygmund class $C^s_\ast$ in $x$ and having a modulus of continuity belowLipschitz in $t$. Imposing additional conditions to control oscillations, weobtain a global (on $[0,\,T]$) $L^2$ energy estimate without loss ofderivatives for $s \geq \{1+\varepsilon,\,\frac{2m_0}{2-m_0}\}$, where $m_0$ islinked to the modulus of continuity of the coefficients in time.

Strictly hyperbolic Cauchy problems with coefficients low-regular in time and space