On Kato–Ponce and fractional Leibniz
By Dong Li. Revista matemática iberoamericana, 2016
212 citations recorded by the source corpus.
We show that in the Kato-Ponce inequality $\|J^s(fg)-fJ^s g\|_p \lesssim \| \partial f \|_{\infty} \| J^{s-1} g \|_p + \| J^s f \|_p \|g\|_{\infty}$, the $J^s f$ term on the RHS can be replaced by $J^{s-1} \partial f$. This solves a question raised in Kato-Ponce \cite{KP88}. We propose and prove a new fractional Leibniz rule for $D^s=(-\Delta)^{s/2}$ and similar operators, generalizing the Kenig-Ponce-Vega estimate \cite{KPV93} 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 $L^{\infty}$-norm on the RHS cannot be replaced by the weaker BMO norm. Some divergence-free counterexamples are also included.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/d1884a7182aa633b91d9739010af7339ea7ecece
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In Applied Mathematics
Citations in the corpus
- Sharp commutator estimates via harmonic extensions (2016, background): “In particular it would be interesting to see if one obtains sharp limit space estimates as in [3, 28].”
- Paralinearization of the Muskat Equation and Application to the Cauchy Problem (2019, unknown): “Λs1uk L p1 kΛ s2vk L 2 (3.8) where s = s1+s2 and 1/r = 1/p1+1/p2. Kenig, Ponce and Vega considered the case s < 1. Since, for our purpose we need s > 1, we will use the recent improvement by Li [36] (see also D’Ancona [25]) showing that (3.8) holds under the assumptions s = s1+s2 ∈ (0,2), sj ∈”