# partial differential equations – If \$S(t)\$ is a \$C_0\$-semigroup, is \$S(t-s)f(s)\$ Bochner integrable?

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.

Any proof or reference is welcomed.