Problem: (1) Give the definition of the semi-norm $\sen{u}_{\dot H^s}$ and $\sen{u}_{\dot B^s_{p,q}}$, where $s\in\bbR$, $1\leq p,q\leq\infty$. (2) Show that $\sen{u}_{\dot H^s}$ and $\sen{u}_{\dot B^s_{2,2}}$ are equivalent.
Proof: (1) $$\bex \sen{u}_{\dot H^s}=\sez{\int_{\bbR^d} |\xi|^{2s}|\hat u(\xi)|^2\rd \xi}^\f{1}{2},\quad \sen{u}_{\dot B^s_{p,q}}= \sen{2^{js}\sen{\dot\lap_j u}_{L^p}}_{\ell^q}. \eex$$ (2) $$\beex \bea \sen{u}_{\dot H^s}^2 &=\int_{\bbR^d} |\xi|^{2s}|\hat u(\xi)|^2\rd \xi\\ &\approx \int_{\bbR^d} |\xi|^{2s} \cdot \sum_j \phi^2(2^{-j}\xi) |\hat u(\xi)|^2\rd \xi\\ &\approx \sum_j \int_{\bbR^d} |\xi|^{2s} \phi^2(2^{-j}\xi)|\hat u(\xi)|^2\rd \xi\\ &\approx \sum_j 2^{2js} \int_{\bbR^d} |\phi(2^{-j}\xi)\hat u(\xi)|^2\rd \xi\\ &\qx{\f{3}{4}\leq 2^{-j}|\xi|\leq \f{8}{3} \ra |\xi|\approx 2^j}\\ &\approx \sum_j 2^{2js} \int_{\bbR^d} \sev{\calF^{-1}(\phi(2^{-j}\cdot)\hat u)}^2\rd x\\ &=\sum_j 2^{2js} \sen{\dot \lap_ju}_{L^2}^2 =\sen{u}_{\dot B^s_{2,2}}^2. \eea \eeex$$