On the $L^p$ norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain $L^8$ norm bounds for spectral clusters associated to elliptic second order operators on two-dimensional manifolds with boundary. The result leads to optimal $L^q$ bounds, in the range $2\le q\le \mathrm{\infty }$, for $L^2$-normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp $L^q$ estimates in higher dimensions for a range of exponents ${\bar{q}}_n\le q\le \mathrm{\infty }$.