If $S(t)$ is a $C_0$-semigroup, is $S(t-s)f(s)$ Bochner integrable?

Question

Let $X$ be a Banach space and let $S(t)$, $t \geq 0$, be a $C_0$-semigroup on $X$.

Assume that $f : [0,+\infty) \rightarrow X$ is Bochner integrable.

Is $S(t-s)f(s)$ Bochner integrable on $[0,t]$ and does $t \mapsto \int_0^t S(t-s)f(s)ds \in C^0([0,+\infty),X)$ ?

The function $t \mapsto \int_0^t S(t-s)f(s)ds$ arises when we define the notion of weak solution to an inhomogeneous evolution PDE $$\partial_t u(t) = Au(t) + f(t), \quad u(0) = u_0$$

where $A$ is the infinitesimal generator of $S(t)$.

If $f$ is continuous, I know that the result is true, but I'm interested in the non-continuous case. I would expect this to be true as well.

If needed, one can assume that the semigroup is uniformly bounded.

The tricky part is, I think, to prove that $S(t-s)f(s)$ is Bochner measurable.

Any proof or reference is welcomed.

Answer 1

Following a comment, I think I've come up with a sketch of proof.

First, here are some properties about Bochner integration. Let $\Omega \subset \mathbb R^n$ be open.

1. Dominated Convergence and Fubini-Tonelli hold for Bochner integration.
2. The change of variables formula holds for transformations $t = as + b$ with $a, b \in \mathbb R$. Prove it for simple functions first and then approximate.
3. $C_c(\Omega,X)$ is dense in $L^1(\Omega,X)$. Approximate $f \in L^1(\Omega,X)$ by a simple function $g = \sum_{k=0}^n x_i \chi_{A_i}$ and approximate each $\chi_{A_i}$, where $A_i$ is a Lebesgue measurable set of finite measure, with a $C_c(\Omega,\mathbb R)$ function. For e.g., see https://planetmath.org/compactlysupportedcontinuousfunctionsaredenseinlp
4. The translation operator is strongly continuous on $L^1(\Omega,X)$. The proof is the same as in the Lebesgue case. For e.g., see https://math.stackexchange.com/a/458234/272494
5. $C^{\infty}_c(\Omega,X)$ is dense in $L^1(\Omega,X)$. This is done by convolution with a real-valued mollifier. The smoothness follows from the Dominated Convergence theorem. The rest of the proof is similar to the Lebesgue case. For e.g., see https://people.math.gatech.edu/~heil/7338/fall09/approxid.pdf p.41 and https://math.stackexchange.com/a/597465/272494

Now, approximate $f \in L^1([0,+\infty),X)$ by a sequence $(f_n) \subset C^{\infty}_c((0,+\infty),X)$ of smooth, compactly supported functions. By passing to a subsequence, we can assume that $||f_n(s) - f(s)||_X$, which converges to zero in $L^1([0,+\infty),\mathbb R)$, also converges to zero pointwisely for almost every $s \geq 0$.

Fix $t \geq 0$ and let $s \in [0,t]$. Since $S(t-s)$ is a continuous operator, $S(t-s)f(s)$ is the pointwise limit almost everywhere of $S(t-s)f_n(s)$, a sequence of continuous functions. Therefore, $S(t-s)f(s)$ is Bochner measurable.

The growth estimate $||S(t-s)|| \leq M e^{b(t-s)}$ allows us to conclude that $S(t-s)f(s)$ is Bochner integrable on $[0,t]$.

The proof of continuity is also standard.

Note :

I haven't found any good book where all those properties of the Bochner integral are covered. However, I've found on the web two master theses ("The Bochner integral and an application to singular integrals" by Harry Thompson Potter and "Sobolev Spaces of Vector-Valued Functions" by Marcel Kreuter) where such properties are proved.