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We prove sharp uniform $L^p$-bounds for low-lying eigenfunctions ofnon-self-adjoint semiclassical pseudodifferential operators $P$ on$\mathbb{R}^{n}$ whose principal symbols are doubly-characteristic at theorigin 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 tothe principal symbol of $P$ at the origin is elliptic along its singular space.Most notably, our assumptions on the quadratic approximation are lessrestrictive than those made in prior works, and our main theorem improves thealready known results in the case when the symbol of $P$ is analytic.

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