Norm estimates for the fractional derivative of multiple factors
We extend the Kato-Ponce inequality to a product of m functions, proving an estimate currently missing from the literature. This study is motivated by the fact that the 3-factor Kato-Ponce does not follow directly from the 2-factor version in the full range of permissible indices. Our methodology is...
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| Published in | Journal of mathematical analysis and applications Vol. 537; no. 2; p. 128409 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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| Abstract | We extend the Kato-Ponce inequality to a product of m functions, proving an estimate currently missing from the literature. This study is motivated by the fact that the 3-factor Kato-Ponce does not follow directly from the 2-factor version in the full range of permissible indices. Our methodology is based upon that in [12] but our extension entails a novel decomposition that elegantly and effectively handles the technical difficulties that arise from the combinatorial complexity of the possibly large number of factors. |
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| AbstractList | We extend the Kato-Ponce inequality to a product of m functions, proving an estimate currently missing from the literature. This study is motivated by the fact that the 3-factor Kato-Ponce does not follow directly from the 2-factor version in the full range of permissible indices. Our methodology is based upon that in [12] but our extension entails a novel decomposition that elegantly and effectively handles the technical difficulties that arise from the combinatorial complexity of the possibly large number of factors. |
| ArticleNumber | 128409 |
| Author | Douglas, Sean Grafakos, Loukas |
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| Cites_doi | 10.5802/aif.708 10.4310/MRL.1999.v6.n1.a1 10.1006/aima.2001.2028 10.1002/cpa.3160410704 10.1007/BF02392566 10.4171/rmi/1049 10.1007/s00028-011-0113-3 10.1090/tran/7866 10.1080/03605302.2013.822885 10.1007/s00041-017-9541-y 10.4310/MRL.2020.v27.n4.a8 10.1007/s12220-012-9367-4 10.1016/0022-1236(91)90103-C 10.4310/MAA.2015.v22.n4.a3 10.1353/ajm.1996.0050 |
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| Keywords | Littlewood-Paley theory Paraproducts Fractional derivatives Kato-Ponce Normed fractional Leibniz rules Multi-factored |
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| References | Muscalu, Pipher, Tao, Thiele (br0190) 2004; 193 Naibo, Thomson (br0210) 2019; 372 Christ, Weinstein (br0040) 1991; 100 Bae, Biswas (br0010) 2015; 22 Grafakos, Naibo, Maldonado (br0110) 2014; 27 Li (br0180) 2019; 35 Grafakos (br0100) 2014; vol. 250 Bernicot, Maldonado, Moen, Naibo (br0020) 2014; 24 Grafakos (br0090) 2014; vol. 249 Grafakos, Oh (br0120) 2014; 39 Fujiwara, Georgiev, Ozawa (br0080) 2018; 24 Oh, Wu (br0220) 2020; 27 Kenig, Stein (br0170) 1999; 6 Coifman, Meyer (br0050) 1978; 28 Kato, Ponce (br0160) 1988; 41 Grafakos, Torres (br0130) 2002; 165 Sean Douglas, Loukas Grafakos, Weighted Kato-Ponce inequalities for multiple factors, submitted for publication. Hale, Naibo (br0150) 2023; 29 Muscalu, Schlag (br0200) 2013; vol. 138 Bourgain, Li (br0030) 2014; 27 Cordero, Zucco (br0060) 2011; 11 Gulisashvili, Kon (br0140) 1996; 118 Bae (10.1016/j.jmaa.2024.128409_br0010) 2015; 22 Grafakos (10.1016/j.jmaa.2024.128409_br0090) 2014; vol. 249 Kenig (10.1016/j.jmaa.2024.128409_br0170) 1999; 6 Fujiwara (10.1016/j.jmaa.2024.128409_br0080) 2018; 24 Grafakos (10.1016/j.jmaa.2024.128409_br0110) 2014; 27 10.1016/j.jmaa.2024.128409_br0070 Muscalu (10.1016/j.jmaa.2024.128409_br0190) 2004; 193 Oh (10.1016/j.jmaa.2024.128409_br0220) 2020; 27 Christ (10.1016/j.jmaa.2024.128409_br0040) 1991; 100 Gulisashvili (10.1016/j.jmaa.2024.128409_br0140) 1996; 118 Cordero (10.1016/j.jmaa.2024.128409_br0060) 2011; 11 Bourgain (10.1016/j.jmaa.2024.128409_br0030) 2014; 27 Coifman (10.1016/j.jmaa.2024.128409_br0050) 1978; 28 Kato (10.1016/j.jmaa.2024.128409_br0160) 1988; 41 Grafakos (10.1016/j.jmaa.2024.128409_br0120) 2014; 39 Muscalu (10.1016/j.jmaa.2024.128409_br0200) 2013; vol. 138 Grafakos (10.1016/j.jmaa.2024.128409_br0100) 2014; vol. 250 Grafakos (10.1016/j.jmaa.2024.128409_br0130) 2002; 165 Li (10.1016/j.jmaa.2024.128409_br0180) 2019; 35 Bernicot (10.1016/j.jmaa.2024.128409_br0020) 2014; 24 Naibo (10.1016/j.jmaa.2024.128409_br0210) 2019; 372 Hale (10.1016/j.jmaa.2024.128409_br0150) 2023; 29 |
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Anal. doi: 10.1007/s12220-012-9367-4 contributor: fullname: Bernicot – volume: 100 start-page: 87 issue: 1 year: 1991 ident: 10.1016/j.jmaa.2024.128409_br0040 article-title: Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(91)90103-C contributor: fullname: Christ – volume: 22 start-page: 377 issue: 4 year: 2015 ident: 10.1016/j.jmaa.2024.128409_br0010 article-title: Regularity for a class of dissipative equations with analytic nonlinearity publication-title: Methods Appl. Anal. doi: 10.4310/MAA.2015.v22.n4.a3 contributor: fullname: Bae – volume: 118 start-page: 1215 issue: 6 year: 1996 ident: 10.1016/j.jmaa.2024.128409_br0140 article-title: Exact smoothing properties of Schrödinger semigroups publication-title: Am. J. Math. doi: 10.1353/ajm.1996.0050 contributor: fullname: Gulisashvili |
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| Title | Norm estimates for the fractional derivative of multiple factors |
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