The Gradient Estimate of a Neumann Eigenfunction on a Compact Manifold with Boundary

Let e(lambda)(x) be a Neumann eigenfunction with respect to the positive Laplacian Delta on a compact Riemannian manifold M with boundary such that Delta e(lambda) = lambda(2)e(lambda) in the interior of M and the normal derivative of e(lambda) vanishes on the boundary of M. Let chi(lambda) be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M. The authors show the following gradient estimate for chi(lambda) f as lambda >= 1: parallel to del chi(lambda) f parallel to(infinity) <= C(lambda parallel to chi(lambda) f parallel to(infinity) + lambda(-1)parallel to Delta chi(lambda) f parallel to(infinity)), where C is a positive constant depending only on M. As a corollary, the authors obtain the gradient estimate of e(lambda): For every lambda >= 1, it holds that parallel to del e(lambda)parallel to(infinity) <= C lambda parallel to e(lambda)parallel to(infinity).