I will discuss recent joint work with Chris Sogge, obtaining best possible bounds on the ratio of the L^p to L^2 norm of spectral clusters in 2 dimensional manifolds with boundary, with Dirichlet boundary conditions. (A spectral cluster is a function with frequency spread of size one.) This ratio can grow as a power of the frequency, with exponent depending on p. For boundaryless manifolds, the sharp exponents were otained by Sogge in 1988. For manifolds with boundary, such as the unit disk, it is known that for some p the exponents can be strictly larger. Using successive multiscale decompositions of the function, we are able to establish the sharp exponenents in the case of 2 dimensions.
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