A Inequality in the paper by Kenig, Ponce and Vega

Question

I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle", Communications on Pure and Applied Mathematics 46, No. 4, 527-620 (1993), Zbl 0808.35128, doi/10.1002/cpa.3160460405. The theorem A.8, which states that

Let $\alpha\in (0,1)$ and $\alpha_1,\alpha_2\in [0,\alpha]$ with $\alpha = \alpha_1 + \alpha_2$. If $p,p_1,p_2,q,q_1,q_2 \in (1,\infty)$ be such that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}$. Then $$ \|D_x^{\alpha}(fg)-fD_x^{\alpha}g-gD_x^{\alpha}(f)\|_{L_{x}^{p}L_{T}^{q}}\leq c \|D_x^{\alpha_1}f\|_{L_x^{p_1}L_T^{q_1}}\|D_x^{\alpha_2}g\|_{L_x^{p_2}L_T^{q_2}} $$ Now, here, as $\alpha \in (0,1)$, the mentioned inequality involves fractional powers of the operator $D$. My question is if there are any similar inequalities where $\alpha$ can be chosen from $(0,n)$, where $n$ is some natural number greater than $1$. In other words, can one go beyond fractional powers and still have such inequalities? Any insight is highly appreciated

Answer 1

The inequality you cited is sometimes called a "fractional Leibniz rule", and is related to the Coiffman-Meyer theorem. An extension to all $\alpha \geq 0$ is available in

Fujiwara, Kazumasa; Georgiev, Vladimir; Ozawa, Tohru, Higher order fractional Leibniz rule, J. Fourier Anal. Appl. 24, No. 3, 650-665 (2018). ZBL1400.46027.

(click here for the arXiv version). The formulae are somewhat more complicated due to more different ways one can "distribute" the derivatives among the two functions.

Answer 2

Let me also mention the paper by D.Li, On Kato–Ponce and fractional Leibniz rule, which has quite complete results.