Problem: Show the Bony decomposition $$ex uv=dot T_uv+dot T_vu+dot R(u,v), eex$$ where $$ex dot T_uv=sum_j dot S_{j-1} udot lap_jv,quad dot R(u,v)=sum_{|k-j|leq 1} dot lap_k udot lap_j v. eex$$
Proof: $$eex ea uv&=sum_{j',j}dot lap_{j'}udotlap_j v\ &=sum_{j'leq j-2} dotlap_{j'}udot lap_jv +sum_{j'geq j+2} dot lap_{j'}udot lap_jv +sum_{|j'-j|leq 1} dot lap_{j'}u dot lap_jv\ &=sum_j dot S_{j-1} udot lap_jv +sum_{j'} dot S_{j'-1} vdot lap_{j'}u +sum_{|j'-j|leq 1} dot lap_{j'}udot lap_jv\ &=dot T_uv+dot T_vu+dot R(u,v). eea eeex$$