Let $T>0$, $X,Y$ be Banach spaces, $f:[0,T]\times X\to Y$ be Fréchet differentiable in the second argument and such that ${\rm D}_2f$ is (jointly) continuous, $H$ be a subspace of all Borel measurable functions $[0,T]\to X$ and $$g(x):=\int_0^Tf(t,x(t))\:{\rm d}t\;\;\;\text{for }x\in H.$$
I would like to know for which choice of $H$ the function $g$ is well-defined and Fréchet differentiable.
I'm mostly interested in the choice $H=C([0,T],X)$, $H=C^1([0,T],X)$ and $H=\mathcal L^p([0,T],X)$, $p\ge1$.
But I really to struggle to check the desired claim. If it exists, the Fréchet derivative of $g$ at $x\in H$ should be given by $$Ah:=\int_0^T{\rm D}_2f(t,x(t))h(t)\;{\rm d}t\;\;\;\text{for }h\in H.$$
(Maybe we need to impose a linear growth assumption on $f$ to ensure that the integrals are well-defined.)