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

Let $(Ω,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$ and $u_λ:= ϕ_λ |_{\partial Ω}$ the associated Dirichlet data (ie. boundary restriction of $ϕ_λ$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary corner points) and any $δ\in [0,1),$ $\| ϕ_h \|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$ 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 $Γ$ (possibly including corner points), the boundary restrictions $u_h:= ϕ_h |_{\partial Ω}$ satisfy the bounds $\|u_λ \|_{L^2(Γ)} = O_ε(λ^{1/4 + ε}),$ for any $ε>0.$ The exponent $1/4$ is sharp and the result improves on the $O(λ^{1/3})$ universal $L^2$-restriction bound for Neumann eigenfunctions due to Tataru \cite{Ta}. The $O(λ^{1/4})$ -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.

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