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The purpose of this paper is to give a simple proof of sharp $L^\infty$ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ with boundary $\partial M$ and then to use these estimates to prove new estimates for Bochner Riesz means in this setting. Our eigenfunction estimates involve estimating the $L^2\to L^\infty$ mapping properties of the operators $\chi_\lambda$ which project onto unit bands of the spectrum of the square root of the Laplacian. These generalize the recent estimates of D. Grieser for individual eigenfunctions. We use an idea of Grieser of estimating away from the boundary using earlier estimates of Seeley, Pham The Lai, and Hormander, and then proving estimates near the boundary using a maximum principle. Our proof of the Bochner Riesz estimates is related to an argument for the Euclidean case of C. Fefferman, and one for the boundaryless Riemannian case that is due to the author. Here, unlike in the boundaryless Riemannian case we cannot use parametrices for the wave equation, and instead lie on a more economical argument that only uses the eigenfunction estimates and the finite propagation speed of the wave equation.

Eigenfunction and Bochner Riesz estimates on manifolds with boundary