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A sharp $ k$-plane Strichartz inequality for the Schrödinger equation


Authors: Jonathan Bennett, Neal Bez, Taryn C. Flock, Susana Gutiérrez and Marina Iliopoulou
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 42B37, 35A23
DOI: https://doi.org/10.1090/tran/7309
Published electronically: March 20, 2018
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Abstract: We prove that

$\displaystyle \Vert X(\vert u\vert^2)\Vert _{L^3_{t,\ell }}\leq C\Vert f\Vert _{L^2(\mathbb{R}^2)}^2,$    

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
Email: j.bennett@bham.ac.uk

Neal Bez
Affiliation: Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan
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
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
Email: m.iliopoulou@berkeley.edu

DOI: https://doi.org/10.1090/tran/7309
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)
Article copyright: © Copyright 2018 American Mathematical Society

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