On Strichartz and eigenfunction estimates for low regularity metrics

Abstract

A b s t r a c t .We produce, for dimensions n ≥ 3, examples of wave operators for which the Strichartz estimates fail.The examples include both Lipschitz and C 1,α metrics, for each 0 < α < 1, where by the latter we mean that the gradient satisfies a Hölder condition of order α.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 ≥ 2, second order elliptic operators with Lipschitz or C 1,α coefficients for which certain eigenfunction estimates, established by the second author for operators with C ∞ coefficients, fail to hold.

Locations

• Mathematical Research Letters - View - PDF