Scalar Banach-function retarded estimates by Fatou approximation

By the forcing convention and [F5], choose a strongly measurable representative of $f$, with integrable norm and integrable simple approximants. All integrals below are independent of this choice by [F5]. The function $W(-r)f(r)$ is strongly measurable by pointwise simple approximation of $f$ and continuous orbit approximation for each of its finitely many values. It has integrable norm $\mathrm{\parallel }f(r){\mathrm{\parallel }}_H$. Thus $G(t)={\int }_0^{t}W(-r)f(r)dr$ is continuous, since increments have norm bounded by the integral of $\mathrm{\parallel }f{\mathrm{\parallel }}_H$ over the increment interval. The function $Df(t)=W(t)G(t)$ is continuous and has $C_{t}H$ norm at most $\mathrm{\parallel }f{\mathrm{\parallel }}_1$. A continuous $H$ path has a jointly measurable scalar representative: approximate it uniformly in $H$ by finite interval-step paths, restrict using each bounded $J_n$, and pass to a common almost-everywhere limit on the countable product exhaustion. To justify that last passage choose errors summable in the uniform $H$ norm; their product $L^1$ errors on each $A_n$ are summable, so Tonelli gives almost-everywhere convergence there. The limits agree on overlaps and represent the path in local $L^1$ for almost every time. This construction also supplies measurable orbit representatives with Borel time parameter, allowing the shifts $t-r$.

For a simple forcing $f(r)={\sum }_{j=1}^N{\mathbf{1}}_{E_j}(r)h_j$, use those representatives to form the measurable kernel $K(r,t,x)={\mathbf{1}}_{0<r<t<T}W(t-r)f(r)(x)$. On each finite spatial piece, its absolute triple integral is finite: $\mathrm{\parallel }{J}_{n}W(t-r)h_j{\mathrm{\parallel }}_{L^1(A_n)}\le \mathrm{\parallel }{J}_n\mathrm{\parallel }\mathrm{\parallel }{h}_j{\mathrm{\parallel }}_H$ by unitarity. Scalar Fubini and bounded-map interchange show that ${\int }_0^{t}K(r,t,x)dr$ is the local scalar representative of the $H$ integral for almost every $(t,x)$. The $Y$ integral-triangle property and then time Minkowski give $\mathrm{\parallel }Df{\mathrm{\parallel }}_{L_t^{p}Y}\le {\int }_0^T\mathrm{\parallel }{\mathbf{1}}_{t>r}W(t-r)f(r){\mathrm{\parallel }}_{L_t^{p}Y}dr\le C_T\mathrm{\parallel }f{\mathrm{\parallel }}_{L_t^{1}H}$. The last inequality is the original homogeneous bound restricted after the translation $\tau =t-r$ to $(0,T-r)$; the section norms and all scalar integrals here are measurable by the stated axioms and [F2].

Using [F5] and Countable Choice, choose integrable simple $f_j\to f$ with $\mathrm{\parallel }{f}_j-f{\mathrm{\parallel }}_{L^1(H)}\le 2^{-j}$. Step 1.1 gives uniform $H$ convergence $D{f}_j\to Df$, hence product $L^1((0,T)\times A_n)$ convergence with summable errors for every $n$. Use the measurable representative of $Df$ constructed there; Tonelli applied to the error series shows $D{f}_j(t,x)\to Df(t,x)$ outside a single product-null set in the countable exhaustion. Fubini then gives spatial almost-everywhere convergence for almost every $t$. The sequential Fatou hypothesis yields $\mathrm{\parallel }Df(t){\mathrm{\parallel }}_Y\le {\mathrm{lim\hspace{0.17em}inf}}_j\mathrm{\parallel }D{f}_j(t){\mathrm{\parallel }}_Y$, and scalar Fatou applied to the $p$th powers proves $\mathrm{\parallel }Df{\mathrm{\parallel }}_{L_t^{p}Y}\le {\mathrm{lim\hspace{0.17em}inf}}_j\mathrm{\parallel }D{f}_j{\mathrm{\parallel }}_{L_t^{p}Y}\le C_T\mathrm{\parallel }f{\mathrm{\parallel }}_1$.

For completeness, the assumed spatial integral property has a well-defined left side: local-integrability bounds give $\int {\int }_{A_n}\mathrm{\mid }K(r,x)\mathrm{\mid }d\mu dr\le b_n\int \mathrm{\parallel }K(r,\cdot ){\mathrm{\parallel }}_{Y}dr<\mathrm{\infty }$, so Tonelli implies absolute scalar integrability for almost every $x$ on every $A_n$. The mixed-norm limit has already been identified with the actual $H$ solution in step 2.1, rather than postulated. Combining that estimate with step 1.1 proves the asserted sum estimate.