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

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.