Spectral Cluster Estimates for C Metrics
نویسنده
چکیده
In this paper, we establish Lp norm bounds for spectral clusters on compact manifolds, under the assumption that the metric is C1,1. Precisely, we show that the Lp estimates proven by Sogge [12] in the case of smooth metrics hold under this limited regularity assumption. It is known by examples of Smith-Sogge [11] that such estimates fail for C1,α metrics if α < 1.
منابع مشابه
Spectral Cluster Estimates for C
In this paper, we establish Lp norm bounds for spectral clusters on compact manifolds, under the assumption that the metric is C1,1. Precisely, we show that the Lp estimates proven by Sogge in the case of smooth metrics hold under this limited regularity assumption. It is known by examples of Smith-Sogge that such estimates fail for C1,α metrics if α < 1.
متن کاملSpectral Cluster Estimates for Metrics of Sobolev Regularity
We investigate spectral cluster estimates for compact manifolds equipped with a Riemannian metric whose regularity is determined by its inclusion in a Sobolev space of sufficiently high order. The problem is reduced to obtaining Lp estimates for the wave equation which are shown by employing wave packet techniques.
متن کاملEquivalent a posteriori error estimates for spectral element solutions of constrained optimal control problem in one dimension
In this paper, we study spectral element approximation for a constrained optimal control problem in one dimension. The equivalent a posteriori error estimators are derived for the control, the state and the adjoint state approximation. Such estimators can be used to construct adaptive spectral elements for the control problems.
متن کاملSharp 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 ...
متن کاملL Bounds for Spectral Clusters
In these notes, we review recent results concerning the Lp norm bounds for spectral clusters on compact manifolds. The type of estimates we consider were first established by Sogge [15] in the case of smooth metrics. Recent results of ours in [10] establish the same estimates under the assumption that the metric is C1,1. It is known by examples of Smith-Sogge [12] that such estimates fail for C...
متن کامل