  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 H\"ormander, 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.
