2017-2018-2偏微分方程复习题解析2

Problem: Show the Bony decomposition $$\bex uv=\dot T_uv+\dot T_vu+\dot R(u,v), \eex$$ where $$\bex \dot T_uv=\sum_j \dot S_{j-1} u\dot \lap_jv,\quad \dot R(u,v)=\sum_{|k-j|\leq 1} \dot \lap_k u\dot \lap_j v. \eex$$  

Proof: $$\beex \bea uv&=\sum_{j',j}\dot \lap_{j'}u\dot\lap_j v\\ &=\sum_{j'\leq j-2} \dot\lap_{j'}u\dot \lap_jv +\sum_{j'\geq j+2} \dot \lap_{j'}u\dot \lap_jv +\sum_{|j'-j|\leq 1} \dot \lap_{j'}u \dot \lap_jv\\ &=\sum_j \dot S_{j-1} u\dot \lap_jv +\sum_{j'} \dot S_{j'-1} v\dot \lap_{j'}u +\sum_{|j'-j|\leq 1} \dot \lap_{j'}u\dot \lap_jv\\ &=\dot T_uv+\dot T_vu+\dot R(u,v). \eea \eeex$$