Endpoint in commutator estimate

Question

Let $p\in(1,\infty)$ and $J^s=(1-\Delta)^{s / 2}$ with $s>0$. Then we have the following commutator estimate by C. E. Kenig, G.Ponce and L. Vega (1991 JAMS), \begin{equation} \left\|J^{s}(f g)-f J^{s} g\right\|_{p} \leq c\left\{\|\nabla f\|_{p_{1}}\|g\|_{s-1, p_{2}}+\|f\|_{s, p_{3}}\|g\|_{p_{4}}\right\}\ \ \ \ (*) \end{equation} where $p_{2}, p_{3} \in(1, \infty)$ and $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}$.

My Question is: is it possible that $p_2=\infty$ or $p_3=\infty$ (in this case, $p_1=p_4=p$)? Because if we consider the integer case \begin{equation} \sum_{|\alpha| \leq m}\left\|\partial^{\alpha}(fg)-f\partial^{\alpha}g\right\|_{L^{p}}, \end{equation} we can directly use the Leibniz rule and the following \begin{equation} \|uv\|_{L^{p}} \leqslant C\left(\|u\|_{L^{q_1}}\|v\|_{L^{q_1}}+\|v\|_{L^{q_3}}\|u\|_{L^{q_4}}\right), \end{equation} where $q_i$ is allowed to be $\infty$. Therefore I want to know if $p_2=\infty$ or $p_3=\infty$ in $(*)$? Is there any reference??

Thanks!

Answer 1

Check the paper "On an endpoint Kato-Ponce inequality", by Bourgain and Li, Differential and Integral Equations 27 (2014). They prove that the product estimate for $D^s(uv)$ is valid at the endpoint $q_1=q_2=q_3=q_4=\infty$. Concerning the commutator estimate you are asking about, the best result they can prove is with a $\dot B^0_{\infty,\infty}$ norm instead of the $L^\infty$ norm on the derivatives at the RHS. More complete results are in the followup paper by Li alone, "On Kato-Ponce and fractional Leibnitz" arXiv:1609.01780