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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A sharp $k$-plane Strichartz inequality for the Schrödinger equation
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by Jonathan Bennett, Neal Bez, Taryn C. Flock, Susana Gutiérrez and Marina Iliopoulou PDF
Trans. Amer. Math. Soc. 370 (2018), 5617-5633 Request permission

Abstract:

We prove that \begin{equation*} \|X(|u|^2)\|_{L^3_{t,\ell }}\leq C\|f\|_{L^2(\mathbb {R}^2)}^2, \end{equation*} where $u(x,t)$ is the solution to the linear time-dependent Schrödinger equation on $\mathbb {R}^2$ with initial datum $f$ and $X$ is the (spatial) X-ray transform on $\mathbb {R}^2$. In particular, we identify the best constant $C$ and show that a datum $f$ is an extremiser if and only if it is a gaussian. We also establish bounds of this type in higher dimensions $d$, where the X-ray transform is replaced by the $k$-plane transform for any $1\leq k\leq d-1$. In the process we obtain sharp $L^2(\mu )$ bounds on Fourier extension operators associated with certain high-dimensional spheres involving measures $\mu$ supported on natural “co-$k$-planarity” sets.
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Additional Information
  • Jonathan Bennett
  • Affiliation: School of Mathematics, The Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, England
  • MR Author ID: 625531
  • Email: j.bennett@bham.ac.uk
  • Neal Bez
  • Affiliation: Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan
  • MR Author ID: 803270
  • Email: nealbez@mail.saitama-u.ac.jp
  • Taryn C. Flock
  • Affiliation: School of Mathematics, The Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, England
  • Address at time of publication: Department of Mathematics and Statistics, Lederle Graduate Research Tower, University of Massachusetts, Amherst, Massachusetts 01003-9305
  • MR Author ID: 976421
  • Email: flock@math.umass.edu
  • Susana Gutiérrez
  • Affiliation: School of Mathematics, The Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, England
  • Email: s.gutierrez@bham.ac.uk
  • Marina Iliopoulou
  • Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
  • MR Author ID: 1017823
  • Email: m.iliopoulou@berkeley.edu
  • Received by editor(s): December 9, 2016
  • Published electronically: March 20, 2018
  • Additional Notes: The work of the first, third, and fifth authors was supported by the European Research Council (grant number 307617).
    The work of the second author was supported by a JSPS Grant-in-Aid for Young Scientists (A) (grant number 16H05995)
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 5617-5633
  • MSC (2010): Primary 42B37, 35A23
  • DOI: https://doi.org/10.1090/tran/7309
  • MathSciNet review: 3803144