Commuted wave energy
= Commuted wave energy
{title2=$A_m(t)=\sum_{|I|\leq m}\|\partial Z^Iu(t)\|_2$}
A commuted wave energy controls derivatives of a solution after applying the <commutation vector fields for the wave equation>. A convenient equivalent norm is $A_m(t)=\sum_{|I|\leq m}\|\partial Z^Iu(t)\|_2$. The <wave energy estimate> controls its growth through commuted sources, while the <Klainerman-Sobolev inequality> gives pointwise decay for low-order derivatives.