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Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means

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Abstract

We consider abstract non-negative self-adjoint operators on L 2(X) which satisfy the finite-speed propagation property for the corresponding wave equation. For such operators, we introduce a restriction type condition, which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next, we show that in the considered abstract setting, our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on Rn, the harmonic oscillator, elliptic operators on compact manifolds, and Schr¨odinger operators on asymptotically conic manifolds.

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Authors and Affiliations

  1. Department of Mathematics, Sun Yat-Sen (Zhongshan)University, Guangzhou, 510275, P.R. China

    Peng Chen & Lixin Yan

  2. Institut de Mathématiques de Bordeaux Université Bordeaux, Umr Cnrs 5251, 351, Cours de la Libération, 33405, Talence, France

    El Maati Ouhabaz

  3. Department of Mathematics, Macquarie University, Sydney, Nsw, 2109, Australia

    Adam Sikora

Authors
  1. Peng Chen
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  2. El Maati Ouhabaz
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  3. Adam Sikora
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  4. Lixin Yan
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Correspondence to Peng Chen.

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P. Chen is supported by NNSF of China 11501583, Guangdong Natural Science Foundation 2016A030313351 and the Fundamental Research Funds for the Central Universities 161gpy45.

A. Sikora was partly supported by Australian Research Council Discovery Grant DP 110102488.

L. Yan was supported by NNSF of China (Grant Nos. 11371378 and 11521101), Guangdong Province Key Laboratory of Computational Science.

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Chen, P., Ouhabaz, E.M., Sikora, A. et al. Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means. JAMA 129, 219–283 (2016). https://doi.org/10.1007/s11854-016-0021-0

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  • DOI: https://doi.org/10.1007/s11854-016-0021-0

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