# Endpoint in commutator estimate

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), $$$$\left\|J^{s}(f g)-f J^{s} g\right\|_{p} \leq c\left\{\|\nabla f\|_{p_{1}}\|g\|_{s-1, p_{2}}+\|f\|_{s, p_{3}}\|g\|_{p_{4}}\right\}\ \ \ \ (*)$$$$ where $$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 $$$$\sum_{|\alpha| \leq m}\left\|\partial^{\alpha}(fg)-f\partial^{\alpha}g\right\|_{L^{p}},$$$$ we can directly use the Leibniz rule and the following $$$$\|uv\|_{L^{p}} \leqslant C\left(\|u\|_{L^{q_1}}\|v\|_{L^{q_1}}+\|v\|_{L^{q_3}}\|u\|_{L^{q_4}}\right),$$$$ where $$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!

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