Plancherel-type estimates and sharp spectral multipliers |
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Authors: | Xuan Thinh Duong |
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Institution: | a School of Mathematics, Physics, Computing and Electronics, Macquarie University, NSW 2109, Australia b Institut de Mathematiques, Universite de Bordeaux 1 351, Cours de la Liberation, 33405 Talence cedex, France c Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003-8001, USA |
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Abstract: | We study general spectral multiplier theorems for self-adjoint positive definite operators on L2(X,μ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the L2 norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R3 and new spectral multiplier theorems for the Laguerre and Hermite expansions. |
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Keywords: | primary 42B15 secondary 35P99 |
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