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 $$J^s$$.

MSC:

 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