2023/01/01 by Pablo Bhowmick, Bhowmick, Pablo, Alex Iosevich +5 · 2 citations
Computer Science · Mathematics · #42B05 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2301.00463
openalex publication_date 2023/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this paper is to introduce and study the following graph theoretic paradigm. Let TKf(x)=∫ K(x,y) f(y) dμ(y), where f: X → \Bbb R, X a set, finite or infinite, and K and μ denote a suitable kernel and a measure, respectively. Given a connected ordered graph G on n vertices, consider the multi-linear form ΛG(f1,f2, …, fn)=∫x1, …, xn ∈ X ∏_(i,j) ∈ \mathcal E(G) K(xi,xj) ∏l=1n fl(xl) dμ(xl), where \mathcal E(G) is the edge set of G. Define ΛG(p1, …, pn) as the smallest constant C>0 such that the inequality ΛG(f1, …, fn) ≤ C ∏i=1n ||fi||Lpi(X, μ) holds for all non-negative real-valued functions fi, 1≤ i≤ n, on X. The basic question is, how does the structure of G and the mapping properties of the operator TK influence the sharp exponents. In this paper, this question is investigated mainly in the case X=\Bbb Fqd, the d-dimensional vector space over the field with q elements, and K(xi,xj) is the indicator function of the sphere evaluated at xi-xj. This provides a connection with the study of Lp-improving measures and distance set problems.