Local $l^\infty$ bounds for eigenfunctions of complex elliptic operators via diophantine problems

TL;DR

This paper establishes local bounds on eigenfunctions of complex elliptic operators using Diophantine inequalities, improving classical results for operators with complex coefficients and extending to higher derivatives.

Contribution

It introduces new bounds for eigenfunctions of complex elliptic operators based on Diophantine problems, surpassing classical estimates in certain cases.

Findings

Bounds depend on solutions to Diophantine inequalities.

Improved exponents for complex coefficient operators.

Extension to higher-order derivatives of eigenfunctions.

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 \"ormander'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.