Bounds for spectral projectors on tori

Abstract

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of $L^{p}$ norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it.

L2 to Lp bounds for spectral projectors on the Euclidean two-dimensional torus

“…• If p = 4, the conjecture was proved for the full range λ −1 < δ < 1 by Bourgain-Burq-Zworski [6]. • If p ≤ 6, the conjecture with loss is a consequence of the 2 decoupling of Bourgain-Demeter [9] as was observed in [15]. • If p = ∞, the conjecture for the full range λ −1 < δ < 1 with loss follows immediately from the bound λ for the L ∞ norm of eigenfunctions.…”

“…If , the conjecture with ϵ loss is a consequence of the decoupling of Bourgain–Demeter [9] as was observed in [15].…”

“…Remark 1.1. The justification for this conjecture can be found in [15], where two basic examples are considered: the Knapp example, and the spherical example. They lead to the two terms on the right-hand side of (1.2).…”

“…Conjecture A is known to hold in a number of cases: If δ = 1, the conjecture corresponds to the fundamental result of Sogge [23], which holds on any Riemannian manifold (see also [1] for a recent extension to logarithmically small spectral windows for general non-positively curved manifolds, including in particular the torus). If p = 4, the conjecture was proved for the full range by Bourgain–Burq–Zworski [6]. If , the conjecture with ϵ loss is a consequence of the decoupling of Bourgain–Demeter [9] as was observed in [15]. If , the conjecture for the full range with ϵ loss follows immediately from the bound λ ϵ for the norm of eigenfunctions. If , the conjecture without ϵ loss would be a consequence of the estimate for the number N ( r ) of integer points in the disc with radius r .…”

Bounds for spectral projectors on the Euclidean cylinder

“…The best result in this direction is due to Bourgain and Demeter [3]. More recently, the authors of the present paper [4] considered the problem for general values of λ and δ, conjectured the bound for general tori…”