Bessel potential action on a product of function

Question

Let's consider the Bessel potential $$J^s:=(I-\Delta)^{\frac{s}{2}}$$ Does it exist some kind of Leibniz rule for $$J^s(fg)$$?

Answer 1

Yes there is an estimate on the product, which says for $1<p<\infty$, $s,\alpha,\beta \ge 0$ and $\frac1p = \frac1p_i+\frac1q_i$ with $i=1,2$, $1<q_1\le \infty$ and $1<p_2\le \infty$ one has $$\|J^s(fg)\|_p \le c\left( \|J^{s+\alpha}(f)\|_{p_1}\| J^{-\alpha}(g)\|_{q_1}+\|J^{-\beta}(f)\|_{p_2}\|J^{s+\beta}(g)\|_{q_2}\right).$$

It has been proved in this paper https://www.jstor.org/stable/25098514?seq=1#metadata_info_tab_contents