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On compact Riemannian manifolds with boundary, the L∞ estimates and gradient estimates for the eigenfunctions of the Neumann Laplacian are proved. Applying the Lp estimates and gradient estimates to multiplier problems on eigenfunction expansions for the Neumann Laplacian, some new estimates for Bochner Riesz means and the sharp Hormander Multiplier Theorem are obtained.

Eigenfunction estimates for Neumann Laplacian and applications to multiplier problems