Average L^q growth and nodal sets of eigenfunctions of the Laplacian on surfaces

2015/10/08 by Guillaume Roy-Fortin, Roy-Fortin, Guillaume
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Spectral Theory (math.SP) #math.AP #math.SP

paper · pdf · doi:10.48550/arxiv.1510.02376

11 pages

arxiv created 2015/10/08 · openalex publication_date 2015/10/08 · arxiv updated 2015/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a recent paper, we exhibit a link between the average local growth of Laplace eigenfunctions on surfaces and the size of their nodal set. In that paper, the average local growth is computed using the uniform - or L^∞ - growth exponents on disks of wavelength radius. The purpose of this note is to prove similar results for a broader class of L^q growth exponents with q ∈ (1, ∞). More precisely, we show that the size of the nodal set is bounded above and below by the product of the average local L^q growth with the frequency. We briefly discuss the relation between this new result and Yau's conjecture on the size of nodal sets.

Citations

• L^p norms, nodal sets, and quantum ergodicity

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