Lemma 4.2: smoothing of the regularized adjoint

For $r\in [0,T]$ and $h\in H$, define the time row $Q_{r}h(t)={\mathbf{1}}_{\{r<t<T\}}\eta W(t-r)h$. Its $L^1(0,T;H)$ norm is at most $T\mathrm{\parallel }\eta {\mathrm{\parallel }}_{H\to H}\mathrm{\parallel }h{\mathrm{\parallel }}_H$, by unitarity and the multiplier bound. Its $L^2(0,T;Y)$ norm is at most $C_s\mathrm{\parallel }h{\mathrm{\parallel }}_H$: after changing variables it is the homogeneous norm on $(0,T-r)$. Both values represent the same $H$-valued time class under $Y\hookrightarrow H$.

For fixed $h$, the row is continuous in $r$ in $L^1(0,T;H)$ by strong continuity of $W$, the bounded multiplier, and the vanishing measure of the moving interval. It is continuous in $L^2(0,T;Y)$ because it is the restriction of a translate of the zero extension of $\eta W(\cdot )h$ on $(0,T)$; use the finite-exponent translation theorem on a larger finite interval. The two time spaces embed continuously into $L^1(0,T;H)$, so their compatible-pair space is Banach. Consequently $r\mapsto Q_{r}f(r)$ is strongly measurable in that pair: approximate the continuous $f$ uniformly by finite-valued functions and use row continuity. Its pair norm is bounded by a constant times $\mathrm{\parallel }f(r){\mathrm{\parallel }}_H$, hence it is Bochner integrable.

Integrating the rows in the compatible-pair space and using the Bochner norm inequality in its $L^2(0,T;Y)$ coordinate gives a class $q$ with $\mathrm{\parallel }q{\mathrm{\parallel }}_{L^{2}Y}\le C_s{\int }_0^T\mathrm{\parallel }f(r){\mathrm{\parallel }}_{H}dr$. To identify it, first take a continuous piecewise-linear approximation to $f$ and Riemann sums in the parameter: boundedness of $\eta$, the group law and the $H$-valued Bochner integral identify the $L^1(0,T;H)$ coordinate with $t\mapsto \eta {\int }_0^{t}W(t-r)f(r)dr$. The identification extends to the given continuous $f$ by uniform approximation, the row bound and the Bochner norm inequality. Compatibility therefore identifies $q$ with this localized Duhamel term as an $H$-valued time class.

Apply homogeneous smoothing to $y_0$ and combine it with step 3.1 and the Duhamel identity. The triangle inequality gives $\mathrm{\parallel }\eta y{\mathrm{\parallel }}_{L^{2}Y}\le C_s\mathrm{\parallel }{y}_0{\mathrm{\parallel }}_H+C_s{\int }_0^T\mathrm{\parallel }f(r){\mathrm{\parallel }}_{H}dr\le C_s(1+T{C}_f)\mathrm{\parallel }{y}_0{\mathrm{\parallel }}_H$. Squaring proves the displayed estimate. The compatible embedding also establishes the claimed almost-everywhere higher-regularity representative.