Abstract retarded integration in compatible time spaces

Let $Q_{r}h$ denote the pair of row values in $V$. Its norm is at most $(T+C_T)\mathrm{\parallel }h{\mathrm{\parallel }}_H$ and $r\mapsto Q_{r}h$ is continuous, by [F1] and the sum norm. Approximate the strongly measurable $f$ pointwise by finite-valued simple functions. For each of their finitely many values, the corresponding continuous pair-valued orbit is uniformly approximable by finite-valued functions on the compact parameter interval. This proves strong measurability of $r\mapsto Q_{r}f(r)$. Its integrable norm and [F2] give $U={\int }_0^T{Q}_{r}f(r)dr\in V$, whose second coordinate has norm at most $C_T\mathrm{\parallel }f{\mathrm{\parallel }}_1$.

Independently, $r\mapsto W(-r)f(r)$ is strongly measurable by the same simple-approximation argument and has norm $\mathrm{\parallel }f(r){\mathrm{\parallel }}_H$. Its primitive $G(t)={\int }_0^{t}W(-r)f(r)dr$ satisfies $\mathrm{\parallel }G(t)-G(s)\mathrm{\parallel }\le {\int }_{\min (s,t)}^{\max (s,t)}\mathrm{\parallel }f(r){\mathrm{\parallel }}_{H}dr$ by the Bochner norm inequality in [F2]. The scalar function $g(r)=\mathrm{\parallel }f(r){\mathrm{\parallel }}_H$ has a measurable representative in $L^1(0,T)$. Given $\epsilon >0$, apply [F3] to $g$ and let $\delta >0$ be its small-measure threshold. The interval $E=(\min (s,t),\max (s,t))\subset (0,T)$ has Lebesgue measure $\mathrm{\mid }t-s\mathrm{\mid }$, so $\mathrm{\mid }t-s\mathrm{\mid }<\delta$ gives $\mathrm{\parallel }G(t)-G(s)\mathrm{\parallel }<\epsilon$. Thus $G$ is continuous on $[0,T]$, including one-sided continuity at the endpoints. Bounded linear maps commute with the Bochner integral by [F2], and the group law gives $Df(t)=W(t)G(t)$. Hence $Df$ is continuous by strong continuity and isometry, and $\mathrm{\parallel }Df{\mathrm{\parallel }}_C\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$.

For an interval-step forcing $f={\mathbf{1}}_{(a,b)}h$, the integral in step 1.1 is the pair-valued integral of the continuous function $Q_{r}h$ on $[a,b]$. Uniform Riemann-sum approximation on this compact interval converges in $V$, since uniform continuity bounds the norm error by the mesh oscillation times its length. The first coordinates of these sums, evaluated at fixed $t$, are Riemann sums for ${\int }_a^b{\mathbf{1}}_{r<t}W(t-r)hdr$. The integrand has at most the single cutoff discontinuity at $r=t$, so these sums converge in $H$ to $Df(t)$. Their norms are uniformly bounded by $(b-a)\mathrm{\parallel }h{\mathrm{\parallel }}_H$; dominated convergence in the finite $t$ interval gives convergence also in $B_1$. Therefore the first coordinate of $U$ is the class of the continuous function $Df$. Finite sums give the same identification for all interval-step forcings.

Approximate arbitrary $f$ by interval-step forcings in $L^1(H)$ using [F1]. The pair-valued integrals converge in $V$ by their norm bound, while the continuous $H$ solutions converge uniformly by step 1.2. The first coordinate therefore remains the $B_1$ class of $Df$, so closed compatibility identifies its second-coordinate value in the common ambient space. The two coordinate estimates yield the asserted sum bound. No Hilbert vector was identified with a scalar function, and no existence of a limit in an unspecified space was assumed.