Spectral cluster estimates for metrics of Sobolev regularity

Abstract

Abstract. 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 L p estimates for the wave equation which are shown by employing wave packet techniques.

From spectral cluster to uniform resolvent estimates on compact manifolds

“…Here we take the branch of the square root with Im √ • > 0. After possibly multiplying the metric by an innocuous factor, we may always assume that δ = 1, in accordance with (2). Because σ(q) is decreasing in 1/q, the exponent of |z| is always nonpositive.…”

“…γ(q) = σ(q), as "perfect" estimates. 2 For the resolvent, we distinguish two types of perfect estimates. The first we call estimates with "perfect exponents".…”

“…Most modern proofs of spectral cluster and resolvent estimates rely on wave equation techniques. One popular approach is via square function estimates for the wave equation [34,39,2,9,40], another one is via Strichartz estimates for a half wave equation [28,7,31,27]. The latter is purely stationary, i.e.…”

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

“…Note, we place no restrictions here on the polygon in terms of convexity or rationality. Suppose {φ j }, φ j : Ω −→ C is an orthonormal L 2 (Ω) eigenbasis for the (positive) Laplacian operator on Ω with either Dirichlet or Neumann boundary conditions on ∂Ω, (1) ∆φ j = λ 2 j φ j 0 ≤ λ 0 < λ 1 ≤ · · · ≤ λ j ≤ λ j+1 ≤ · · · , φ j L 2 (Ω) = 1.…”

“…Indeed, L p bounds on the eigenfunctions can be viewed via the Stein-Tomas restriction theorem as a version of the adjoint restriction estimate on the sphere. The authors have previously treated the analogs of adjoint restriction estimates for polygonal domains in cases of the parabola in [3,10] and the cone in [4] by proving Strichartz estimates for the Schrödinger equation and wave equation respectively 1 in the setting polygonal domains. Arguably, the sphere presents unique challenges since Strichartz bounds for the Schrödinger and wave equations rely only on fixed time bounds for the corresponding kernel, whereas the spectral cluster bounds typically require integrating/averaging the wave kernel and estimating the contributions of the jumps in the transition from geometric to diffracted wave fronts.…”