Finite-loss energy estimate at the critical time-oscillation exponent

Determine whether an energy estimate with a finite loss of derivatives can be obtained for strictly hyperbolic operators with spatially Lipschitz coefficients satisfying (H1)–(H2) at the critical exponent \(\alpha=1\), in a suitable range of Sobolev indices.

Background

The paper proves energy estimates under spatial Lipschitz regularity and time-derivative bounds of order $h(t;\alpha)=t^{-1}(\log(1/t))^\alpha$ for $0\leq\alpha<1$. The result gives no derivative loss when $\alpha=0$ and a stretched logarithmic loss when $0<\alpha<1$.

For coefficients depending only on time, finite-loss estimates are known at $\alpha=1$. In the spatially Lipschitz setting, however, such a loss may reduce the regularity of $\nabla u$ beyond the range where multiplication by Lipschitz functions remains bounded. The authors therefore leave unresolved whether a finite-loss estimate is possible at $\alpha=1$.