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Transactions of the American Mathematical Society

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Global versions of the Gagliardo-Nirenberg-Sobolev inequality and applications to wave and Klein-Gordon equations


Authors: Leonardo Enrique Abbrescia and Willie Wai Yeung Wong
Journal: Trans. Amer. Math. Soc. 374 (2021), 773-802
MSC (2010): Primary 35A23, 35L71, 35B45
DOI: https://doi.org/10.1090/tran/8277
Published electronically: November 25, 2020
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove global, or space-time weighted, versions of the Gagliardo-Nirenberg interpolation inequality, with $ L^p$ ( $ p < \infty $) endpoint, adapted to a hyperboloidal foliation. The corresponding versions with $ L^\infty $ endpoint was first introduced by Klainerman and is the basis of the classical vector field method, which is now one of the standard techniques for studying long-time behavior of nonlinear evolution equations. We were motivated in our pursuit by settings where the vector field method is applied to an energy hierarchy with growing higher order energies. In these settings the use of the $ L^p$ endpoint versions of Sobolev inequalities can allow one to gain essentially one derivative in the estimates, which would then give a corresponding gain of decay rate. The paper closes with the analysis of one such model problem, where our new estimates provide an improvement.


References [Enhancements On Off] (What's this?)

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Additional Information

Leonardo Enrique Abbrescia
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: leonardo.abbrescia@vanderbilt.edu

Willie Wai Yeung Wong
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: wongwwy@math.msu.edu

DOI: https://doi.org/10.1090/tran/8277
Keywords: Sobolev inequality, space-time weighted, hyperboloidal foliation, vector field method, energy hierarchy, wave and Klein-Gordon equations
Received by editor(s): September 26, 2019
Received by editor(s) in revised form: August 11, 2020
Published electronically: November 25, 2020
Additional Notes: The first author was supported by an NSF Graduate Research Fellowship (DGE-1424871).
The second author was supported by a Collaboration Grant from the Simons Foundation, #585199.
Article copyright: © Copyright 2020 American Mathematical Society