Log-Lipschitz spatial energy estimate with logarithmically varying Sobolev loss

Establish whether, for strictly hyperbolic operators whose coefficients satisfy the spatial log-Lipschitz assumptions (H1′)–(H2′), there exist constants \(\beta^\ast,K,C>0\) and \(T_0\in(0,T]\) with \(\theta+\beta^\ast T_0<1\) such that every smooth solution satisfies the stated energy estimate with Sobolev orders \(1-\theta-\beta^\ast t\) and \(-\theta-\beta^\ast t\), measured in the weighted spaces \(H^s_{\alpha,K}\).

Background

The main theorem assumes spatial Lipschitz continuity of the coefficients and their first time derivatives. The second concluding problem asks whether these spatial assumptions can be weakened to log-Lipschitz continuity while retaining the time-oscillation bounds governed by $h(t;\alpha)$, symmetry, and strict hyperbolicity.

The proposed estimate allows a loss of derivatives that increases linearly with time, with rate $\beta^\ast$, and is required only on a sufficiently short interval $[0,T_0]$. The paper notes that the corresponding estimate is already known when $\alpha=0$, but leaves the question unresolved for the full range $\alpha\in[0,1)$ under (H1′)–(H2′).