On Multilinear Spectral Cluster Estimates for Manifolds with Boundary

Let (M, g) be a smooth, compact n-dimensional Riemannian manifold with boundary and let ∆ be the corresponding Laplace-Beltrami operator acting on functions. If the boundary is non-empty, we assume that either Dirichlet or Neumann conditions are imposed along ∂M. Consider the operators χλ defined as projection onto the subspace spanned by the Dirichlet (or Neumann) eigenfunctions whose corresponding eigenvalues −λj satisfy λj ∈ [λ − 1, λ]. In the case that ∂M is empty, it was established in [10] that the following, best possible L → L estimates hold for χλ: