Abstract
Let P be a non-negative, self-adjoint differential operator of degree d on ℝn. Assume that the associated Bochner-Riesz kernel s δR satisfies the estimate, |s δR (x, y)| ≤ C Rn/d(1+R1/d|x - y|-αδ+β)for some fixed constants a>0 and β. We study Lp boundedness of operators of the form m(P), m coming from the symbol class S −αp . We prove that m(P) is bounded on LP if\(\alpha > \frac{{n(1 - \rho )}}{\alpha }|\frac{1}{p} - \frac{1}{2}|\). We also study multipliers associated to the Hermite operator H on ℝn and the special Hermite operator L on ℂn given by the symbols\(m_\alpha (\lambda ) = \lambda ^{ - \alpha /2} J_\alpha (t\sqrt \lambda )\). As a special case we obtain Lp boundedness of solutions to the Wave equation associated to H and L.
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Communicated by Hans G. Feichtinger
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Narayanan, E.K., Thangavelu, S. Oscillating multipliers for some eigenfunction expansions. The Journal of Fourier Analysis and Applications 7, 373–394 (2001). https://doi.org/10.1007/BF02514503
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DOI: https://doi.org/10.1007/BF02514503