$$C^\infty $$ C ∞ Scaling Asymptotics for the Spectral Projector of the Laplacian

Abstract

This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant size is a normalized Bessel function depending only on n.

Topologies of Nodal Sets of Random Band‐Limited Functions

“…and this holds uniformly together with any fixed number of derivatives with respect to u and v. They establish this for M real analytic in [9] and for the general C 1 case in [10]. Our discussion above does not apply directly to the interesting special case of M being the standard round sphere S n and E M;1 .T / being the space of spherical harmonics of a given degree, since for these Á is bounded.…”

“…Specifically, as pointed out in equation (5) of [10], the remainder R.x; y; T / (our T is their ) in their equation (4) is O.T n 1 /, and this is proved without any assumptions on the geodesic flow (see [17, theorem 4.4]). In their analysis of the main term in (4) leading to their theorem 1, there is a parameter , which they allow to go to 0 and which makes use of the non-self-focal condition.…”

“…For the spectral window that we treat, a parametrix for a small fixed time interval suffices. The recent papers [9,10], which we will use as a reference, go beyond what we need in that they allow Á.T / to be bounded in the case˛D 1. Their goal is a remainder term of o.T n 1 /, and for that they assume some properties of the geodesic flow.…”

“…Since we assume that Á.T / ! 1 (in fact, that Á.T / D Tˇfor some 0 <ˇ< 1 2 ) and we are content with a bound of O.T n 1 / for the remainder, the analysis from [9,10] is simpler, and we don't need to impose any conditions on M D .S n ; g/.…”

Convex Hull Property and Exclosure Theorems for H-Minimal Hypersurfaces in Carnot Groups

“…The proof of this fact, which is elementary, can be given as in the Riemannian case; see [47] Proof. The proof is an adaptation of the classical one valid for the Laplace-Beltrami operator; see [5] or [11].…”