Homogeneous bounds transfer to L1 retarded forcing

Assume Countable Choice (The Axiom of Countable Choice (AC_ω)). Let $H$ be a Hilbert space and $W$ a strongly continuous unitary group (Strongly continuous one-parameter unitary group). Fix $0<T<\mathrm{\infty }$ and $1\le p<\mathrm{\infty }$. Suppose the homogeneous bound $\mathrm{\parallel }W(\cdot )h{\mathrm{\parallel }}_{L^p(0,T;Y)}\le C_T\mathrm{\parallel }h{\mathrm{\parallel }}_H$ is given, with one of the following precise norm and compatibility conventions.

Abstract Bochner convention: $Y$ is Banach, the orbit is a Bochner $L^p(Y)$ class, and the spaces $L_t^{1}H$ and $L_t^{p}Y$ are continuously and injectively mapped into a common Hausdorff space of time classes or distributions, agreeing on the truncated homogeneous rows. These identifications must commute with time restriction and time translation when passing to a shorter or translated interval. This is the closed-pair compatibility of Compatible Banach pairs and their integrals–Abstract retarded integration in compatible time spaces. It applies, for example, to $H,Y$ continuously embedded in a common Banach value space; it does not assume arbitrary Hilbert vectors are scalar functions.

Scalar Banach-function convention: $H$ has continuous injective realization by scalar locally integrable functions on a sigma-finite spatial space, bounded into $L^1$ on a fixed finite-measure exhaustion. The scalar Banach function space $Y$ satisfies the local-integrability, lattice, sequential Fatou, measurable-section-norm and explicit spatial integral-triangle hypotheses in Scalar Banach-function retarded estimates by Fatou approximation, and the orbit bound uses its scalar mixed norm. These hypotheses include every $Y=L^q(X)$, $1\le q\le \mathrm{\infty }$, by Spatial Lq classes satisfy the scalar mixed-norm hypotheses. In particular $H\hookrightarrow L^2(X)$ supplies the required local $L^1$ compatibility by Holder on finite-measure pieces.

Under either convention, for every strongly measurable $f\in L^1(0,T;H)$ the $H$-valued Bochner integral $Df(t)={\int }_0^{t}W(t-r)f(r)dr$ belongs to $C([0,T];H)$ and has the specified compatible $L_t^{p}Y$ value, with $\mathrm{\parallel }Df{\mathrm{\parallel }}_{C_{t}H}+\mathrm{\parallel }Df{\mathrm{\parallel }}_{L_t^{p}Y}\le (1+C_T)\mathrm{\parallel }f{\mathrm{\parallel }}_{L_t^{1}H}$. The $C_{t}H$ bound follows from Bochner integral norm inequality, while the finite-time mixed bound uses Minkowski's integral inequality in the scalar convention and Bochner norm integration in the abstract convention. The shifted estimate uses ${\mathbf{1}}_{t>r}W(t-r)$ and restriction to $(0,T-r)$. The same constant works on every translated interval of length at most $T$, with its correspondingly translated norm. At spatial $L^{\mathrm{\infty }}$ the conclusion is a scalar mixed-norm assertion and makes no claim of strong measurability into that nonseparable Banach space.