$L^p$-bounds for eigenfunctions of analytic non-self-adjoint operators with double characteristics

Abstract

We prove sharp uniform $L^p$-bounds for low-lying eigenfunctions of non-self-adjoint semiclassical pseudodifferential operators $P$ on $\mathbb{R}^{n}$ whose principal symbols are doubly-characteristic at the origin of $\mathbb{R}^{2n}$. Our bounds hold under two main assumptions on $P$: (1) the total symbol of $P$ extends holomorphically to a neighborhood of $\mathbb{R}^{2n}$ in $\mathbb{C}^{2n}$, and (2) the quadratic approximation to the principal symbol of $P$ at the origin is elliptic along its singular space. Most notably, our assumptions on the quadratic approximation are less restrictive than those made in prior works, and our main theorem improves the already known results in the case when the symbol of $P$ is analytic.