A local Bernstein inequality for Laplace eigenfunctions

Abstract

A powerful heuristics in the study of Laplace eigenfunctions is that they behave like polynomials of degree proportional to the square root of the eigenvalue. We present an instance in which this polynomial behaviour can be made precise, namely we prove a version of the classical (for polynomials) Bernstein inequality. Along the way we show that analogous inequalities hold for solutions to nice elliptic PDEs, which also resemble polynomials once an appropriate notion of degree is introduced. Based on joint work with Eugenia Malinnikova.