$L^\infty$ bounds for eigenfunctions on rough domains via diophantine problems

Abstract: We introduce a new method to estimate the $L^\infty$ norm for eigenfunctions of partial differential operators on an arbitrary open set $\Omega$ in $\mathbb{R}^d$. We establish a general inequality which estimates the local $L^\infty$ norm by the number of integer solutions to a certain diophantine problem arising from the principal symbol of the operator. The novelty of our method lies in its complete independence from the regularity of the boundary, the boundary condition or the validity of a local Weyl law. Our result applies to domains with fractal boundary, unbounded domains, as well as to non-self-adjoint operators.