L1 in time estimate for Lp Duhamel forcing

Let $1\le p<\mathrm{\infty }$. The map $\mathrm{\Phi }(s,g):=H_{t-s}g$ is jointly norm continuous on $[0,t]\times L^p({\mathbb{R}}^n)$: $\mathrm{\parallel }{H}_{t-s}g-H_{t-s^{\mathrm{\prime }}}{g}^{\mathrm{\prime }}{\mathrm{\parallel }}_p\le \mathrm{\parallel }g-g^{\mathrm{\prime }}{\mathrm{\parallel }}_p+\mathrm{\parallel }(H_{t-s}-H_{t-s^{\mathrm{\prime }}})g^{\mathrm{\prime }}{\mathrm{\parallel }}_p$ by [F2], and the second term tends to $0$ as $s\to s^{\mathrm{\prime }}$ by strong continuity at zero applied to the semigroup difference. If $f_k$ are the simple approximations of [F1], the functions $s\mapsto \mathrm{\Phi }(s,f_k(s))$ are strongly measurable: for each of the finitely many values $g$ of $f_k$, the continuous curve $s\mapsto H_{t-s}g$ on $[0,t]$ is uniformly approximated by finite-valued mesh functions; multiply these approximations by the measurable level-set indicators of $f_k$ and add them, Choosing mesh error at most $1\mathrm{/}k$ for the finitely many curves associated with $f_k$ yields a single sequence of finite-valued measurable approximations; contractions and $f_k\to f$ show that this sequence converges pointwise off the original null set to $H_{t-s}f(s)$. This proves strong measurability directly from [F1].

Let $p=\mathrm{\infty }$ and let $0<\delta <t$. For fixed $g\in L^{\mathrm{\infty }}({\mathbb{R}}^n)$ and $\delta \le \tau ,{\tau }^{\mathrm{\prime }}\le t$, [F3], [F4] and ${\mathrm{\partial }}_{\tau }\mathrm{\Gamma }=\mathrm{\Delta }\mathrm{\Gamma }$ give $\mathrm{\parallel }{\mathrm{\Gamma }}_{\tau }-{\mathrm{\Gamma }}_{{\tau }^{\mathrm{\prime }}}{\mathrm{\parallel }}_1\le {\int }_0^1\mathrm{\parallel }{\mathrm{\partial }}_{\tau }{\mathrm{\Gamma }}_{{\tau }^{\mathrm{\prime }}+u(\tau -{\tau }^{\mathrm{\prime }})}{\mathrm{\parallel }}_{1}du\mathrm{\mid }\tau -{\tau }^{\mathrm{\prime }}\mathrm{\mid }\le C_{\delta }\mathrm{\mid }\tau -{\tau }^{\mathrm{\prime }}\mathrm{\mid }$ with $C_{\delta }<\mathrm{\infty }$, so $\tau \mapsto H_{\tau }g$ is norm continuous on $[\delta ,t]$ by [F4] (Young's inequality with $\mathrm{\parallel }K{\mathrm{\parallel }}_1$); consequently $\mathrm{\Phi }$ is jointly norm continuous on $[0,t-\delta ]\times L^{\mathrm{\infty }}$ and the argument of step 1.1 makes $s\mapsto H_{t-s}f(s)$ strongly measurable on $[0,t-\delta ]$. Take the countable exhaustion $[0,t-t\mathrm{/}(m+1)]$, $m\ge 1$. For each simple approximation $f_k$, approximate its finitely many continuous flow curves uniformly on these intervals by finite mesh functions. On the $k$-th interval choose error at most $1\mathrm{/}k$ and set the approximation to zero on the omitted tail. At every $s<t$ off the original null set, these finite-valued measurable functions tend to $H_{t-s}f(s)$, because contractions also give $\mathrm{\parallel }{H}_{t-s}(f_k(s)-f(s)){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{f}_k(s)-f(s){\mathrm{\parallel }}_{\mathrm{\infty }}$. The single endpoint has measure zero. This proves strong measurability on $[0,t]$ directly from [F1].

In both cases the contraction bound [F2] gives $\mathrm{\parallel }{H}_{t-s}f(s){\mathrm{\parallel }}_p\le \mathrm{\parallel }f(s){\mathrm{\parallel }}_p$, so ${\int }_0^t\mathrm{\parallel }{H}_{t-s}f(s){\mathrm{\parallel }}_{p}ds\le {\int }_0^t\mathrm{\parallel }f(s){\mathrm{\parallel }}_{p}ds\le {\int }_0^T\mathrm{\parallel }f(s){\mathrm{\parallel }}_{p}ds<\mathrm{\infty }$; the Bochner integrability criterion [F5] therefore makes $Df(t)={\int }_0^t{H}_{t-s}f(s)ds$ a well-defined element of $L^p({\mathbb{R}}^n)$, and the norm inequality for Bochner integrals Bochner integral norm inequality gives $\mathrm{\parallel }Df(t){\mathrm{\parallel }}_p\le {\int }_0^t\mathrm{\parallel }f(s){\mathrm{\parallel }}_{p}ds$.

The construction claims no continuity of the integrand at $s=t$ nor of $H$ at zero when $p=\mathrm{\infty }$: strong measurability of the integrand, which is all that integrability needs, was proved only up to null sets; and for forcing that is merely weakly measurable, no strong measurability of $s\mapsto H_{t-s}f(s)$ is available, so no existence of the Bochner integral is asserted in that case. This proves the theorem with the stated caveats.