Abstract
We use microlocal and paradifferential techniques to obtain L 8 norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal L q bounds, in the range 2⩽q⩽∞, 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 q̅n⩽q⩽∞.
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The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.
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Smith, H.F., Sogge, C.D. On the L p norm of spectral clusters for compact manifolds with boundary. Acta Math 198, 107–153 (2007). https://doi.org/10.1007/s11511-007-0014-z
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DOI: https://doi.org/10.1007/s11511-007-0014-z