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Gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

Let $e_ł(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $Δ$ on a compact Riemannian manifold $M$ with boundary such that $Δ\, e_ł=ł^2 e_ł$ in the interior of $M$ and the normal derivative of $e_ł$ vanishes on the boundary of $M$. Let $χ_λ$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $χ_λ\,f$ as $λ\geq 1$: $\|\nabla\ χ_ł f\|_\infty\leq Cł\|χ_ł\f\|_\infty+ł^{-1}\|Δ χ_ł f\|_\infty$, where $C$ is a positive constant depending only on $M$. As a corollary, we obtain the gradient estimate of $e_ł$: for every $ł\geq 1$, there holds $\|\nabla e_ł\|_\infty\leq C\,ł\, \|e_ł\|_\infty$.