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Transactions of the American Mathematical Society

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Coifman-Meyer multipliers: Leibniz-type rules and applications to scattering of solutions to PDEs


Authors: Virginia Naibo and Alexander Thomson
Journal: Trans. Amer. Math. Soc. 372 (2019), 5453-5481
MSC (2010): Primary 42B25, 42B15; Secondary 42B20, 46E35
DOI: https://doi.org/10.1090/tran/7866
Published electronically: June 19, 2019
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Abstract:

Leibniz-type rules for Coifman-Meyer multiplier operators are studied in the settings of Triebel-Lizorkin and Besov spaces associated with weights in the Muckenhoupt classes. Even in the unweighted case, improvements on the currently known estimates are obtained. The flexibility of the methods of proofs allows one to prove Leibniz-type rules in a variety of function spaces that include Triebel-Lizorkin and Besov spaces based on weighted Lebesgue, Lorentz, and Morrey spaces as well as variable-exponent Lebesgue spaces. Applications to scattering properties of solutions to certain systems of partial differential equations involving fractional powers of the Laplacian are presented.


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Additional Information

Virginia Naibo
Affiliation: Department of Mathematics, Kansas State University, 138 Cardwell Hall, 1228 North 17th Street, Manhattan, Kansas 66506
Email: vnaibo@ksu.edu

Alexander Thomson
Affiliation: Department of Mathematics, Kansas State University, 138 Cardwell Hall, 1228 North 17th Street, Manhattan, Kansas 66506
Email: thomson521@ksu.edu

DOI: https://doi.org/10.1090/tran/7866
Keywords: Fractional Leibniz rules, Kato--Ponce inequalities, Coifman--Meyer multipliers, weighted Triebel--Lizorkin and Besov spaces, scattering of solutions to PDEs
Received by editor(s): April 8, 2018
Received by editor(s) in revised form: September 27, 2018
Published electronically: June 19, 2019
Additional Notes: The authors were partially supported by the NSF under grant DMS 1500381.
Article copyright: © Copyright 2019 American Mathematical Society