* $$\bex \p_i=\f{\p}{\p x_i},\quad \lap=\p_1\p_1+\cdots+\p_n\p_n. \eex$$
* Fourier multiplier (Fourier 乘子) $$\bex m(D)f(x)=\calF^{-1}(m(\cdot)\hat f(\cdot))(x). \eex$$
* 分数阶 Laplacian $$\bex \vLm=(-\lap)^\f{1}{2}:\ \vLm f(x)=\calF^{-1}(|\cdot|\calF f(\cdot))(x); \eex$$ $$\bex \vLm^s =(-\lap)^{\f{s}{2}}:\ \vLm^s f(x)=\calF^{-1}(|\cdot|^s\calF f(\cdot))(x). \eex$$
* Commutator estimate (交换子估计) $$\bex \sen{\vLm^s(fg)-f\vLm^s g}_{L^p} \leq C\sez{ \sen{\n f}_{L^{p_1}} \sen{\vLm^{s-1}g}_{L^{p_2}} +\sen{\vLm^s f}_{L^{p_3}} \sen{g}_{L^{p_4}} }, \eex$$ if $$\bex s>0,\quad 1<p,p_2,p_3<\infty,\quad 1\leq p_1,p_4\leq\infty,\quad \f{1}{p}=\f{1}{p_1}+\f{1}{p_2}=\f{1}{p_3}+\f{1}{p_4}. \eex$$ See [Kato, Tosio; Ponce, Gustavo. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), no. 7, 891--907] Lemma X1.
* $$\bex \sen{\vLm^s(fg)}_{L^p} \leq C\sex{ \sen{f}_{L^{p_1}}\sen{\vLm^s g}_{L^{p_2}} +\sen{\vLm^s f}_{L^{p_3}} \sen{g}_{L^{p_4}} }, \eex$$ if $$\bex s>0,\quad 1<p,p_2,p_3<\infty,\quad 1\leq p_1,p_4\leq\infty,\quad \f{1}{p}=\f{1}{p_1}+\f{1}{p_2}=\f{1}{p_3}+\f{1}{p_4}. \eex$$ See [Kato, Tosio; Ponce, Gustavo. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), no. 7, 891--907] Lemma X4.