(1) $$ex sen{D^k f}_{dot B^s_{p,q}}sim sen{f}_{dot B^{s+k}_{p,q}}. eex$$
(2) $$eex ea &quad s>0, qin [1,infty],quad p_1,r_1in [1,infty], cfrac{1}{p}=cfrac{1}{p_1}+cfrac{1}{p_2}=cfrac{1}{r_1}+cfrac{1}{r_2}\ & a sen{fg}_{dot B^s_{p,q}}leq Csex{ sen{f}_{L^{p_1}}sen{g}_{dot B^s_{p_2,q}} +sen{g}_{L^{r_1}}sen{f}_{dot B^s_{r_2,q}} }. eea eeex$$
(3) $$eex ea &quad s_1,s_2leq cfrac{n}{p},quad s_1+s_2>0\ & a sen{fg}_{dot B^{s_1+s_2-frac{n}{p}}_{p,1}} leq Csen{f}_{dot B^{s_1}_{p,1}}sen{g}_{dot B^{s_2}_{p,1}}. eea eeex$$
(4) $$eex ea &quad -cfrac{n}{p}-1<sleq cfrac{n}{p}\ & a sen{[u,lap_q]w}_{L^p} leq c_q 2^{-q(s+1)}sen{u}_{dot B^{-frac{n}{p}+1}_{p,1}}sen{w}_{dot B^s_{p,1}}quadsex{sum_{qin{mathbb{Z}}} c_qleq 1}. eea eeex$$
(5) $$eex ea &quad s,s_1>0, s= t s_1, 0< t<1\ & a sen{f}_{dot B^s_{2,1}}leq Csen{f}_{dot B^{s_1}_{2,1}}^ t sen{f}_{L^2}^{1- t}. eea eeex$$
(6) [to be determined...the definition of Triebel-Lizorkin space $dot F^s_{infty,q}$ for $1leq q<infty$...] $$ex sen{f}_{BMO}leq Csex{sen{ f}_{BMO}+sen{f}_{L^2}}. eex$$
(7) $$ex sen{f}_{L^infty}leq Csen{f}_{L^2}^frac{1}{4} sen{lap f}_{L^2}^frac{3}{4}. eex$$ see [D. Chae, J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations, 256 (2014), 3835--3858].