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Abstract The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$where$$f: X \to {\Bbb R}$$,Xa set, finite or infinite, andKand$$\mu $$denote a suitable kernel and a measure, respectively. Given a connected ordered graphGonnvertices, consider the multi-linear form$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$where$${\mathcal E}(G)$$is the edge set ofG. Define$$\Lambda _G(p_1, \ldots , p_n)$$as the smallest constant$$C>0$$such that the inequality(0.1)$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$holds for all nonnegative real-valued functions$$f_i$$,$$1\le i\le n$$, onX. The basic question is, how does the structure ofGand the mapping properties of the operator$$T_K$$influence the sharp exponents in (0.1). In this paper, this question is investigated mainly in the case$$X={\Bbb F}_q^d$$, thed-dimensional vector space over the field withqelements,$$K(x^i,x^j)$$is the indicator function of the sphere evaluated at$$x^i-x^j$$, and connected graphsGwith at most four vertices.
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This content will become publicly available on June 6, 2026
Asymptotics for the spectral function on Zoll manifolds
On a smooth, compact, Riemannian manifold without boundary(M,g), let\Delta_{g}be the Laplace–Beltrami operator. We define the orthogonal projection operator \Pi_{I_\lambda}\colon L^{2}(M)\to \bigoplus_{\mathclap{\lambda_j\in I_\lambda}}\ker(\Delta_{g}+\lambda_{j}^{2}) for an intervalI_{\lambda}centered around\lambda\in\Rof a small, fixed length. The Schwartz kernel,\Pi_{I_\lambda}(x,y), of this operator plays a key role in the analysis of monochromatic random waves, a model for high energy eigenfunctions. It is expected that\Pi_{I_\lambda}(x,y)has universal asymptotics as\lambda \to \inftyin a shrinking neighborhood of the diagonal inM\times M(providedI_{\lambda}is chosen appropriately) and hence that certain statistics for monochromatic random waves have universal behavior. These asymptotics are well known for the torus and the round sphere, and were recently proved to hold near points inMwith few geodesic loops by Canzani–Hanin. In this article, we prove that the same universal asymptotics hold in the opposite case of Zoll manifolds (manifolds all of whose geodesics are closed with a common period) under an assumption on the volume of loops with length incommensurable with the minimal common period.
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- Award ID(s):
- 2045494
- PAR ID:
- 10620728
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Journal of Spectral Theory
- ISSN:
- 1664-039X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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