关于Kato-Ponce与分数阶莱布尼茨
On Kato-Ponce and fractional Leibniz
摘要：
证明了在Kato-Ponce不等式$\|J^s(fg)-fJ^s g\|_p \lesssim \| \partial f \|_{\infty} \| J^{s-1} g \|_p + \| J^s f \|_p \|g\|_{\infty}$中，RHS上的$J^s f$项可以用$J^{s-1} \partial f$代替.这解决了Kato-Ponce \cite{KP88}中提出的一个问题。我们提出并证明了一个新的分数Leibniz规则的$D^s=(-Δ)^{s/2}$和类似的运营商，推广的Kenig-Ponce-Vega估计\cite{KPV93}的所有$s>0$。我们还证明了一族广义和精化Kato-Ponce型不等式，其中包括许多交换子估计作为特殊情况。为了展示在各种端点情况下估计的尖锐性，我们构造了几个反例。特别地，我们证明了在原来的Kato-Ponce不等式中，RHS上的$L^{\infty}$ -范数不能被较弱的BMO范数所取代。还包括一些无发散的反例。
英文摘要：
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 $D^s=(-Δ)^{s/2}$ 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.
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