Non-concentration and restriction bounds for Neumann eigenfunctions of piecewise $C^{\mathrm{\infty }}$ bounded planar domains

Let $(\mathrm{\Omega },g)$ be a piecewise-smooth, bounded convex domain in ${\mathbb{R}}^2$ and consider $L^2$-normalized Neumann eigenfunctions ${\varphi }_{\lambda }$ with eigenvalue ${\lambda }^2$ and $u_{\lambda }:={\varphi }_{\lambda }{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ the associated Dirichlet data (ie. boundary restriction of ${\varphi }_{\lambda }$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overline{\Omega},$(includingboundarycornerpoints)andany$\delta \in [0,1),$ $$\mathrm{\parallel }{\varphi }_h{\mathrm{\parallel }}_{B(x_0,{\lambda }^{-\delta })\cap \mathrm{\Omega }}=O({\lambda }^{-\delta \mathrm{/}2})\mathrm{.}$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for $L^2$ restrictions of boundary eigenfunctions that are valid up to boundary corners. In particular, in Theorem \ref{dirichlet} we prove that for any {\em flat} boundary edge $\mathrm{\Gamma }$ (possibly including corner points), the boundary restrictions $u_h:= \phi_h |{\partial \Omega}$satisfythebounds$$ |u{\lambda} |{L^2(\Gamma)} = O{\epsilon}(\lambda^{1/4 + \epsilon}),$$forany$\epsilon >0.$Theexponent$1/4$issharpandtheresultimprovesonthe$O(\lambda^{1/3})$universal$L^2$-restrictionboundforNeumanneigenfunctionsduetoTataru\text{cite}Ta\mathrm{.}The$O(\lambda^{1/4})$ -bound is also an extension to the boundary (including corner points) of well-known interior $L^2$ restriction bounds of Burq-Gerard-Tzvetkov \cite{BGT} along totally-geodesic hypersurfaces.