[再寄小读者之数学篇](2014-06-26 Besov space estimates)

(1) $$\bex \sen{D^k f}_{\dot B^s_{p,q}}\sim \sen{f}_{\dot B^{s+k}_{p,q}}. \eex$$  

(2) $$\beex \bea &\quad s>0,\ q\in [1,\infty],\quad p_1,r_1\in [1,\infty],\ \cfrac{1}{p}=\cfrac{1}{p_1}+\cfrac{1}{p_2}=\cfrac{1}{r_1}+\cfrac{1}{r_2}\\ &\ra \sen{fg}_{\dot B^s_{p,q}}\leq C\sex{ \sen{f}_{L^{p_1}}\sen{g}_{\dot B^s_{p_2,q}} +\sen{g}_{L^{r_1}}\sen{f}_{\dot B^s_{r_2,q}} }. \eea \eeex$$  

(3) $$\beex \bea &\quad s_1,s_2\leq \cfrac{n}{p},\quad s_1+s_2>0\\ &\ra \sen{fg}_{\dot B^{s_1+s_2-\frac{n}{p}}_{p,1}} \leq C\sen{f}_{\dot B^{s_1}_{p,1}}\sen{g}_{\dot B^{s_2}_{p,1}}. \eea \eeex$$  

(4) $$\beex \bea &\quad -\cfrac{n}{p}-1<s\leq \cfrac{n}{p}\\ &\ra \sen{[u,\lap_q]w}_{L^p} \leq c_q 2^{-q(s+1)}\sen{u}_{\dot B^{-\frac{n}{p}+1}_{p,1}}\sen{w}_{\dot B^s_{p,1}}\quad\sex{\sum_{q\in{\mathbb{Z}}} c_q\leq 1}. \eea \eeex$$  

(5) $$\beex \bea &\quad s,s_1>0, s=\tt s_1, 0<\tt<1\\ &\ra \sen{f}_{\dot B^s_{2,1}}\leq C\sen{f}_{\dot B^{s_1}_{2,1}}^\tt \sen{f}_{L^2}^{1-\tt}. \eea \eeex$$  

(6) [to be determined...the definition of Triebel-Lizorkin space $\dot F^s_{\infty,q}$ for $1\leq q<\infty$...] $$\bex \sen{f}_{BMO}\leq C\sex{\sen{\n f}_{BMO}+\sen{f}_{L^2}}. \eex$$  

(7) $$\bex \sen{f}_{L^\infty}\leq C\sen{f}_{L^2}^\frac{1}{4} \sen{\lap f}_{L^2}^\frac{3}{4}. \eex$$  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].