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

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 $L^{\infty }$ - 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 $q\in (1,\infty )$. 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.

11 pages

References in corpus (3)

• Small scale quantum ergodicity in negatively curved manifolds
• $L^p$ norms, nodal sets, and quantum ergodicity
• Nodal sets and growth exponents of Laplace eigenfunctions on surfaces