Sharp $L^q$ bounds on spectral clusters for Holder metrics

Abstract

Abstract. We establish L q bounds on eigenfunctions, and more generally on spectrally localized functions (spectral clusters), associated to a self-adjoint elliptic operator on a compact manifold, under the assumption that the coefficients of the operator are of regularity C s , where 0 ≤ s ≤ 1. We also produce examples which show that these bounds are best possible for the case q = ∞, and for 2 ≤ q ≤ qn.

Spectral cluster estimates for metrics of Sobolev regularity

“…We remark that this theorem yields an improvement over what would be obtained by embedding the metric into a space C s and applying the current results in [9], [5]. This is because it allows us to control the operator norm of Π λ by a smaller power of λ than would otherwise be needed.…”

“…We also remark that a recent work of Koch-Smith-Tataru [5] has developed a version of these estimates for Hölder C s metrics below Lipschitz regularity which are sharp for 2 ≤ q ≤ q n and q = ∞.…”

“…In light of recent examples of Koch-Smith-Tataru [5], the assumption that g is a Lipschitz metric is crucial for large values of q in Theorem 1.3. In their work, for each 0 < s < 1, they produce functions ρ λ , ψ λ over R n such that…”

$L^p$-bounds on spectral clusters associated to polygonal domains

“…where a ± are smooth and bounded. When |z| ∈ (δ/2, 4δ), (14) follows by using that the fact that χ is Schwartz allows one to essentially replace r by λ. Seeing the rapid decay in λ when |z| / ∈ (δ/2, 4δ) takes some extra work.…”

Sharp L p Bounds on Spectral Clusters for Lipschitz Metrics

“…The same idea occurs in this paper in the λ − 1 3 time scale expansion of u in terms of simple tube solutions. For metrics of Hölder regularity C s with s < 1, the optimal bounds, and corresponding examples, were obtained in [5] for 2 ≤ p ≤ p d , as well as p = ∞. For s < 1 there can occur exponentially localized eigenfunctions, and as a result the p = ∞ bounds are strictly worse than in the case of Lipschitz coefficients.…”