2016/09/30 by Enno Lenzmann, Armin Schikorra · 1 citation
Mathematics · #Acoustics #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Algebra over a field #Commutator #Harmonic #Mathematical Analysis and Transform Methods #Mathematics #Physics #Pure mathematics #math.AP #math.CA
paper · pdf · doi:10.1016/j.na.2018.10.017
published as Nonlinear Analysis 2019 · typos fixed, references added. Statement (6.2) of the C^σ-estimate in Theorem 6.1 weakened
arxiv created 2016/12/07 · openalex publication_date 2018/11/27 · arxiv updated 2018/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We give an alternative proof of several sharp commutator estimates involving Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only involve harmonic extensions to the upper half-space, integration by parts, and trace space characterizations. The commutators we investigate are Jacobians, more generally Coifman-Rochberg-Weiss commutators, Chanillo's commutator with the Riesz potential, Coifman-Meyer or Kato-Ponce-Vega type commutators, and the Da Lio-Rivière three-term commutators. We also give a new limiting L1-estimate for a double commutator of Coifman-Rochberg-Weiss-type, and several intermediate estimates. The beauty of our method is that all those commutator estimates, which are originally proven by various specific methods or by general paraproduct arguments, can be obtained purely from integration by parts and trace theorems. Another interesting feature is that in all these cases the cancellation effect responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way.