Gradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander Multiplier Theorem

Abstract

On compact Riemannian manifolds (M,g) of dimension n ≥ 2 with smooth boundary, the gradient estimates for the eigenfunctions of the Dirichlet Laplacian are proved by the maximum principle. Using the L∞ estimates and gradient estimates, the Hörmander multiplier theorem is shown for the eigenfunction expansion of Dirichlet Laplacian on compact manifolds with boundary.