Sharp Counterexamples for Strichartz Estimates for Low Regularity Metrics

On the other hand, in [3] there were constructed for each α < 1 examples of A with coefficients of regularity C for which the same estimates fail to hold. The first author then showed in [1] that, in space dimensions 2 and 3, the estimates do hold if the coefficients of A are C. The second author subsequently showed in [4] that the estimates hold for C metrics in all space dimensions, and that for operators with C coefficients, such estimates hold provided that γ is replaced by γ + σ/p, where σ = 1−α 3+α . Indeed, [5] showed that such estimates hold under the condition that 1 + α derivatives of the coefficients belong to LtL ∞ x , which is important for applications to quasilinear wave equations. The counterexamples of [3] do not coincide with the estimates established by [4], however. In this paper we remedy this gap, by producing examples of time independent C metrics which show that the results established in [4] are indeed best possible. (Strictly speaking, we produce a family of examples, one at each frequency, which show that if the Strichartz estimates hold with a constant C depending only on the C norm of the coefficients, and the assumption that the coefficients are pointwise close to the euclidean metric, then the loss of σ/p derivatives is sharp.) We remark that this construction also produces examples of C metrics which show that the closely related spectral projection estimates for C metrics established by the first author [2] are best possible. For the spectral projection estimates, however, the counterexamples of [3] were already sharp.