Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem

We deal with the Cauchy problem for a strictly hyperbolic second order operator with non-regular coefficients in the time variable. It is well-known that the problem is well-posed in $L^{2}$ in case of Lipschitz continuous coefficients and that the Log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives. Here we prove that any intermediate modulus of continuity between the Lipschitz and the Log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives. We also provide counterexamples to show that the following classification $$text{modulus of continuity $rightarrow$ loss of derivatives}$$ is sharp: [begin{array}{l} text{Lipschitz $rightarrow$ no loss} text{intermediate $rightarrow$ arbitrary small loss} text{Log-Lipschitz $rightarrow$ finite loss} end{array}]