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Optimal Forward and Reverse Estimates of Morawetz and Kato–Yajima Type with Angular Smoothing Index

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Abstract

For the solution of the free Schrödinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the space variable, and incorporate a certain angular regularity. This will follow from a more general result which permits analogous sharp forward and reverse smoothing estimates and a characterisation of extremisers for the solution of the free Klein–Gordon and wave equations. The nature of extremisers is shown to be sensitive to both the dimension and the size of the smoothing index relative to the dimension. Furthermore, in four spatial dimensions and certain special values of the smoothing index, we obtain an exact identity for each of these evolution equations.

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Acknowledgments

The first author was partially supported by JSPS Kakenhi grant number 26887008.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama, 338-8570, Japan

    Neal Bez

  2. Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-8602, Japan

    Mitsuru Sugimoto

Authors
  1. Neal Bez
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  2. Mitsuru Sugimoto
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Correspondence to Neal Bez.

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Communicated by Luis Vega.

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Bez, N., Sugimoto, M. Optimal Forward and Reverse Estimates of Morawetz and Kato–Yajima Type with Angular Smoothing Index. J Fourier Anal Appl 21, 318–341 (2015). https://doi.org/10.1007/s00041-014-9371-0

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  • DOI: https://doi.org/10.1007/s00041-014-9371-0

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