Bounds for spectral projectors on the Euclidean cylinder

We prove essentially optimal bounds for norms of spectral projectors on thin spherical shells for the Laplacian on the cylinder (R/Z)*R. In contrast to previous investigations into spectral projectors on tori, having one unbounded dimension available permits a compact self-contained proof.

1. Introduction 1.1.Spectral projectors on general manifolds and tori.Given a Riemannian manifold with Laplace-Beltrami operator ∆, consider the spectral projector P λ,δ on (perhaps generalized) eigenfunctions with eigenvalues within O(δ) of λ.It is defined through functional calculus by the formula where χ is a cutoff function, which is irrelevant for our purposes.An interesting question is to determine the operator norm from L 2 to L p , with p > 2, of this operator.A theorem of Sogge [5] gives an optimal answer for any Riemannian manifold if δ = 1 While this completely answers the question if δ > 1, the case δ < 1 is still widely open.Understanding the case δ < 1 requires a global analysis on the Riemannian manifold, which makes it very delicate.
In the case of the rational torus R d /Z d , L p bounds on eigenfunctions attracted a lot of attention; this corresponds to the choice δ = 1/λ.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 and were able to establish this bound for a range of the parameters δ, λ, p.
A full proof of this conjecture seems very challenging in every dimension d.Restricting to the case d = 2, consider the case (R/Z) × R = T × R instead of T 2 .The conjecture remains identical, but a short proof, relying on ℓ 2 decoupling, can be provided; this is the main observation of the present paper.Generalizations to higher dimensions are certainly possible.
1.2.The Euclidean cylinder.On T×R = (R/Z)×R, we choose coordinates (x, y), with x ∈ [0, 1] and y ∈ R. The Laplacian operator is given by ∆ = ∂ 2 x + ∂ 2 y .A function f on T × R can be expanded through Fourier series in x and Fourier transform in y: The spectral projector can then be expressed as Furthermore, this estimate is optimal, up to the subpolynomial factor λ ǫ δ −ǫ and the multiplicative constant.

Strichartz estimates.
It is interesting to draw a parallel with Strichartz estimates in dimension 2 for the Schrödinger equation, in which case the critical exponent equals 4. It was proved in the foundational paper of Bourgain [2] that for s > 0.
Takaoka and Tzvetkov [6] proved that the above inequality fails for s = 0, but that, on T × R, Finally, Barron, Christ and Pausader [1] determined the correct global (in time) estimate, for which a further summation index is needed.These examples suggest that optimal estimates might differ by subpolynomial factors between T 2 and T × R.
Acknowledgements.PG was supported by the Simons collaborative grant on weak turbulence.SLRM was supported by a Leverhulme Early Career Fellowship.

Proof of the main theorem
Proof.By Plancherel's theorem, it suffices to prove (2.1) for f a function whose Fourier transform is localized in the corona C λ,δ of radius λ and with thickness δ/λ: By symmetry, one can furthermore assume that f (k, η) is localized in the first quadrant k, η ≥ 0. The function f can be split into two pieces, which will correspond to the two terms on the right-hand side of (2.1).
δ The Fourier support of f 1 is made up of a collection of segments.We will see in Lemma 3.1 below that the added length of these segments can be bounded by Therefore, by the Cauchy-Schwartz inequality, Interpolating with L 2 , this gives The case |k − λ| > 1 δ We start by choosing a function φ ∈ S which is > 1/2 on [−1, 1], and has Fourier support in [−1, 1].We use periodicity in the x variable to expand the range of x from x ∈ [0, 1] to x < δ −1 , so that .
We now change variables as follows: The effect of this change of variables is that the function of (X, Y ) whose L p norm we want to evaluate has Fourier transform supported in the corona C 1,3δ/λ of radius 1 and width δ/λ, and also in the first quadrant X, Y ≥ 0. This enables us to apply the ℓ 2 decoupling theorem of Bourgain and Demeter [3]: for a smooth partition of unity (χ θ ) corresponding to a suitable almost disjoint , where χ θ (D) is the Fourier multiplier with symbol χ θ .We now apply the inequality , which follows by applying in turn the Hausdorff-Young and Hölder inequalities, and then the Plancherel theorem.Since by Lemma 3.1 below .
By almost orthogonality, this becomes Finally, undoing the change of variables and using once again periodicity in the x variable gives Optimality.The optimality of the statement of the theorem is proved through two examples.The first one is an analog of the Knapp example: assume λ ∈ N, and consider the function g given by its Fourier transform Here, 1 λ is the indicator function of {λ} and χ is a cutoff function with a sufficiently small support, so that Supp g ⊂ C λ,δ .In physical space, .
We now consider the function h given by its Fourier transform here, 1 C λ,δ is the indicator function of the annulus, and 1 [0,λ/2] the indicator function of the interval.
It is easy to check that | Supp h| ∼ λδ, so that h L ∞ ∼ λδ and h L 2 ∼ √ λδ, and finally By the Bernstein inequality, The examples g and h show that the statement of the theorem is optimal, up to subpolynomial losses.Proof.(i) Consider f as in the proof of Theorem 1.1, namely with Fourier support in C λ,δ .Since (k, η) range in Z × R with k, η ≥ 0, the Fourier support of f is contained in ∪ k∈Z {k} × E λ k , where

Bounds for the Fourier support
Recalling that f1 is just f restricted to |k − λ| ≤ 1 δ , one can then add up these pieces to get the bound and as y → 1/ √ y is decreasing this is (ii) Turning to F , it has Fourier support in (3.1) Consider χ θ (D)F , for a cap θ with dimensions ∼ δ λ × δ λ adapted to the corona C 1,3δ/λ .Given such a cap, there is j ∈ N with 2 j > 1 δ such that every point in the intersection of θ with the set (3.1) satisfies |λ − k| ∼ 2 j .

Lemma 3 . 1 (
Bound on the size of Fourier support).With the notations of the proof of Theorem 1.1, (i) The function f 1 is a function on T × R. As such, its Fourier transform is supported on a union of lines, and has one-dimensional measure | Supp f 1 | √ λδ .(ii) The function χ θ (D)F is a function on R 2 .Its Fourier transform is defined on R 2 , and has two-dimensional measure | Supp χ θ F | δ 5/2 λ −3/2 .