Local $l^\infty$ bounds for eigenfunctions of complex elliptic operators via diophantine problems
Omer Friedland, Henrik Ueberschaer
TL;DR
This work derives local L∞ bounds and derivative estimates for eigenfunctions of complex constant-coefficient elliptic operators with smooth potentials by reducing the problem to counting integer lattice points in Diophantine shells around the symbol level set {P(ξ)=λ}. The core method embeds a localized eigenfunction into a fixed torus and analyzes its Fourier coefficients, yielding bounds controlled by a lattice-counting function 𝒩λ that measures shell thickness. For real coefficients, the bounds recover Hörmander’s exponent up to ε, while for complex symbols the shell count can be significantly smaller, enabling improved exponents in many regimes and even explicit improvements in 2D examples. The approach extends to higher derivatives, supports shrinking interior balls, and promises a bridge between spectral geometry and Diophantine geometry, with potential applications to nodal sets and rough domains.
Abstract
We prove local bounds on the amplitude of eigen- functions of complex constant-coefficient elliptic operators with a smooth potential on an arbitrary open subset of \R^d by estimating it in terms of the number of solutions of a diophantine inequality arising from the symbol of the operator. In the special case of positive elliptic operators, we recover H örmander's classical exponent up to an arbitrarily small loss. We show that a much better exponent may be obtained when the principal symbol of the oper- ator has complex coefficients. We generalize our estimate to any higher-order derivatives of eigenfunctions.