The advanced scalar response

For $0\le t\le T$,

The propagator estimates imply

The equation and its derivatives give

For ${{\partial }_t}^a{{\partial }_x}^{\beta }{u}_f$ with $a+|\beta |\le 4$, apply (eq:Sobolev) to its spatial $H^2$ norm. If $a=0$ or $a=1$, (eq:u-basic) suffices with spatial orders at most six. If $a=2,3,4$, use (eq:u-time); each additional ${D_m}^2$ costs two spatial derivatives, but the allowed spatial order decreases with $a$. Every resulting derivative of $f$ has total order at most six. For example, when $a=4,\beta =0$, the largest spatial term is ${D_m}^4{u}_f$ in $H^2$, controlled by $f$ in $H^6$. The term ${{\partial }_t}^{2}f$ in $H^2$ uses total order four. Consequently