Small-scale mass estimates for Neumann eigenfunctions: piecewise smooth planar domains

Let $\mathrm{\Omega }$ be a piecewise-smooth, bounded convex domain in ${\mathbb{R}}^2$ and consider $L^2$-normalized Neumann eigenfunctions ${\phi }_{\lambda }$ with eigenvalue ${\lambda }^2$. Our main result is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0\in \bar{\mathrm{\Omega }},$ (including boundary and corner points) and any $\delta \in [0,1),$ $$\mathrm{\parallel }{\phi }_{\lambda }{\mathrm{\parallel }}_{B(x_0,{\lambda }^{-\delta })\cap \mathrm{\Omega }}=O({\lambda }^{-\delta \mathrm{/}2})\mathrm{.}$$ The proof is a stationary vector field argument combined with a small scale induction argument.