On an endpoint Kato-Ponce inequality. (English) Zbl 1340.42021

Let \(D^s=(-\Delta)^{s/2}\), \(J^s=(1-\Delta)^{s/2}\) be the fractional Laplacian operators for \(s\in \mathbb R\). The Kato-Ponce commutator estimate \(\|J^s(fg)-fJ^sg\|_p \leq C(\|\nabla f\|_\infty\|J^{s-1}g\|_p+\|J^sf\|_p\|g\|_\infty)\) \((s>0, 1<p<\infty)\) plays an important role in the wellposedness of Navier-Stokes and Euler equations in Sobolev spaces. Relating to this, Kenig, Ponce and Vega showed (1) \(\|D^s(fg)-fD^sg-gD^s f\|_r\leq C(\|D^{s_1} f\|_p\|D^{s_2}g\|_q\) \((0<s,s_1,s_2<1, s=s_1+s_2, 1<p,q,r<\infty\) with \(1/r=1/p+1/q)\). One variant of the Kato-Ponce inequality is the following fractional Leibniz rule: (2) \(\|D^s(fg)\|_r\leq C (\|D^s f\|_{p_1}\|g\|_{q_1}+\|D^s g\|_{p_2}\|f\|_{q_2})\), where \(s>0\), \(1/r=1/p_j+1/q_j)\), \(1/2<r<\infty\), \(1<p_j,q_j\leq\infty)\) \((j=1,2)\).
The authors’ first result is the endpoint case \(r=\infty\) in (2): \(\|D^s(fg)\|_\infty\leq C (\|D^s f\|_{\infty}\|g\|_{\infty}+\|D^s g\|_{\infty}\|f\|_{\infty})\). About (1), they give: \(\|D^s(fg)-fD^sg-gD^s f\|_\infty\leq C (\|f\|_\infty\|g\|_{\dot B_{\infty,\infty}^s}+\|f\|_{\dot B_{\infty,\infty}^s}\|g\|_\infty)\), \(s>0\).
Similarly, for \(1\leq p<\infty\) and any \(s>n/p\), \(\|D^s(fg)-fD^sg-gD^s f\|_p\leq C (\|f\|_p^{1-n/sp}\|f\|_{\dot B_{p,\infty}^s}^{n/sp}\|g\|_{\dot B_{p,\infty}^s}+\|f\|_{\dot B_{p,\infty}^s}\|g\|_p^{1-n/sp}\|g\|_{\dot B_{p,\infty}^s}^{n/sp})\). They get similar estimate in the endpoint case \(s=n/p\).
They further treat refinements of the above and relating inequalities to these. Similar estimates hold also for \(J^s\).


42B15 Multipliers for harmonic analysis in several variables
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems