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L p estimates for bi-parameter and bilinear Fourier integral operators

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Abstract

Fourier integral operators play an important role in Fourier analysis and partial differential equations. In this paper, we deal with the boundedness of the bilinear and bi-parameter Fourier integral operators, which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators. We consider such FIOs when they have compact support in spatial variables. If they contain a real-valued phase ϕ(x, ξ, η) which is jointly homogeneous in the frequency variables ξ, η, and amplitudes of order zero supported away from the axes and the antidiagonal, we can show that the boundedness holds in the local-L 2 case. Some stronger boundedness results are also obtained under more restricted conditions on the phase functions. Thus our results extend the boundedness results for bilinear and one-parameter FIOs and bilinear and bi-parameter pseudo-differential operators to the case of bilinear and bi-parameter FIOs.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China

    Qing Hong

  2. Department of Mathematical Sciences, Binghamton University, Binghamton, New York, 13902–6000, USA

    Lu Zhang

Authors
  1. Qing Hong
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  2. Lu Zhang
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Correspondence to Lu Zhang.

Additional information

The first author is supported by NNSF of China (Grant No. 11371056); the second author is supported by NSF grant

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Hong, Q., Zhang, L. L p estimates for bi-parameter and bilinear Fourier integral operators. Acta. Math. Sin.-English Ser. 33, 165–186 (2017). https://doi.org/10.1007/s10114-016-6269-6

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  • DOI: https://doi.org/10.1007/s10114-016-6269-6

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