Spatial smoothing of forcing separated from the observation time

Let $a=\epsilon \mathrm{/}2$ and put $g:={\int }_0^{t-\epsilon }{H}_{t-s-a}f(s)ds\in L^p$. This integral exists by the forcing estimate [F1], applied at observation time $t-a$ to the forcing cut off after $t-\epsilon$. Since $f=0$ a.e. on the omitted interval and $H_a{H}_{t-s-a}=H_{t-s}$, [F2] gives $Df(t)=H_{a}g$. By [F3], it therefore has a spatial $C^{\mathrm{\infty }}$ representative, without choosing joint scalar representatives of the original forcing.

The bounded map $D^{\alpha }{H}_a:L^p\to L^q$ commutes with the integral defining $g$. Differentiating $H_a{H}_{b}h=H_{a+b}h$ in space (both sides have the smooth representatives of [F3]) gives $D^{\alpha }{H}_a{H}_{b}h=D^{\alpha }{H}_{a+b}h$. Consequently $D^{\alpha }Df(t)=D^{\alpha }{H}_{a}g={\int }_0^{t-\epsilon }{D}^{\alpha }{H}_{t-s}f(s)ds$ in $L^q$. Strong measurability of this integrand follows by applying the bounded map to the strongly measurable integrand defining $g$, and its norm is integrable by the next estimate.

For $s\le t-\epsilon$, [F3] gives $\mathrm{\parallel }{D}^{\alpha }{H}_{t-s}f(s){\mathrm{\parallel }}_q\le C_{n,p,q,\alpha }{\epsilon }^{-\mathrm{\mid }\alpha \mathrm{\mid }\mathrm{/}2-n(1\mathrm{/}p-1\mathrm{/}q)\mathrm{/}2}\mathrm{\parallel }f(s){\mathrm{\parallel }}_p$. Integrating and applying [F2] proves the stated bound. This includes $p=\mathrm{\infty }$ and $q=\mathrm{\infty }$, since every map used is bounded between the indicated Banach spaces; if $t=\epsilon$ the interval is empty and $Df(t)=0$. The argument proves spatial regularity at the chosen time and makes no assertion across an active forcing diagonal.