Loss of derivatives for hyperbolic operators with coefficients oscillating in time

Abstract: We consider second-order strictly hyperbolic operators whose coefficients are Lipschitz continuous in space and may oscillate near the initial time. Their first two time derivatives are bounded by the corresponding powers of $\frac{1}{t}(\log\frac{1}{t})<sup>α$ with $0\leqslantα&lt;1$, and the first time derivative satisfies the corresponding spatial Lipschitz bound. By combining Tarama's energy with frequency-dependent weights, we prove an energy estimate without loss for $α=0$ and with a stretched logarithmic upper bound for the loss when $0<α<1$, which implies an arbitrarily small loss in the Sobolev scale. These estimates extend the analysis of Colombini, Del Santo and Fanelli [\emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5)} \textbf{26} (2025)] for spatially Lipschitz coefficients to time-derivative bounds with an additional logarithmic factor.