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Article: On Kato–Ponce and fractional Leibniz
Title | On Kato–Ponce and fractional Leibniz |
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Authors | |
Keywords | Fractional Leibniz Kato–Ponce |
Issue Date | 2019 |
Citation | Revista Matematica Iberoamericana, 2019, v. 35, n. 1, p. 23-100 How to Cite? |
Abstract | We show that in the Kato–Ponce inequality J s (fg)−fJ s gp ∂f∞ J s − 1 gp + J s fp g∞, the J s f term on the right-hand side can be replaced by J s − 1 ∂f. This solves a question raised in Kato–Ponce [14]. We propose a new fractional Leibniz rule for D s = (−Δ) s/2 and similar operators, generalizing the Kenig–Ponce–Vega estimate [15] to all s > 0. We also prove a family of generalized and refined Kato–Ponce type inequalities which include many commutator estimates as special cases. To showcase the sharpness of the estimates at various endpoint cases, we construct several counterexamples. In particular, we show that in the original Kato–Ponce inequality, the L ∞ -norm on the right-hand side cannot be replaced by the weaker BMO norm. Some divergence-free counterexamples are also included. |
Persistent Identifier | http://hdl.handle.net/10722/327225 |
ISSN | 2023 Impact Factor: 1.3 2023 SCImago Journal Rankings: 1.458 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Li, Dong | - |
dc.date.accessioned | 2023-03-31T05:29:50Z | - |
dc.date.available | 2023-03-31T05:29:50Z | - |
dc.date.issued | 2019 | - |
dc.identifier.citation | Revista Matematica Iberoamericana, 2019, v. 35, n. 1, p. 23-100 | - |
dc.identifier.issn | 0213-2230 | - |
dc.identifier.uri | http://hdl.handle.net/10722/327225 | - |
dc.description.abstract | We show that in the Kato–Ponce inequality J s (fg)−fJ s gp ∂f∞ J s − 1 gp + J s fp g∞, the J s f term on the right-hand side can be replaced by J s − 1 ∂f. This solves a question raised in Kato–Ponce [14]. We propose a new fractional Leibniz rule for D s = (−Δ) s/2 and similar operators, generalizing the Kenig–Ponce–Vega estimate [15] to all s > 0. We also prove a family of generalized and refined Kato–Ponce type inequalities which include many commutator estimates as special cases. To showcase the sharpness of the estimates at various endpoint cases, we construct several counterexamples. In particular, we show that in the original Kato–Ponce inequality, the L ∞ -norm on the right-hand side cannot be replaced by the weaker BMO norm. Some divergence-free counterexamples are also included. | - |
dc.language | eng | - |
dc.relation.ispartof | Revista Matematica Iberoamericana | - |
dc.subject | Fractional Leibniz | - |
dc.subject | Kato–Ponce | - |
dc.title | On Kato–Ponce and fractional Leibniz | - |
dc.type | Article | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.4171/rmi/1049 | - |
dc.identifier.scopus | eid_2-s2.0-85062045887 | - |
dc.identifier.volume | 35 | - |
dc.identifier.issue | 1 | - |
dc.identifier.spage | 23 | - |
dc.identifier.epage | 100 | - |
dc.identifier.isi | WOS:000465204500002 | - |