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