On Strichartz and eigenfunction estimates for low regularity metrics

We produce, for dimensions $n\ge 3$, examples of wave operators for which the Strichartz estimates fail. %The metrics constructed are either Lipschitz or of the class %$C^{1,\alpha}$ for $0<\alpha<1$, The examples include both Lipschitz and $C^{1,\alpha}$ metrics, for each $0<\alpha<1$, where by the latter we mean that the gradient satisfies a Hölder condition of order $\alpha$. %The metric %is of the class $C^{1,\alpha}$ (that is, the gradient %satisfies a Hölder condition of order $\alpha\,$), where $\alpha$ %can be any number strictly less than 1. We thus conclude that, on the scale of Hölder regularity, an assumption of at least 2 bounded derivatives for the metric (i.e., $C^{1,1}$) is necessary in order to assure that the Strichartz estimates hold. The same construction also yields, for dimensions $n\ge 2$, second order elliptic operators with Lipschitz or $C^{1,\alpha}$ coefficients for which certain eigenfunction estimates, established by the second author for operators with $C^\infty$ coefficients, fail to hold.

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Published 1 January 1994