Commuted wave energy ($A_m(t)={\sum }_{|I|\le m}\Vert \partial Z^{I}u(t){\Vert }_2$)

A commuted wave energy controls derivatives of a solution after applying the commutation vector fields for the wave equation. A convenient equivalent norm is $A_m(t)={\sum }_{|I|\le m}\Vert \partial Z^{I}u(t){\Vert }_2$. The wave energy estimate controls its growth through commuted sources, while the Klainerman-Sobolev inequality gives pointwise decay for low-order derivatives.