Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that \begin{equation} V \subset H \subset V' \end{equation} where $V'$ is the dual of $V$ and the inclusions are continuous with dense ranges.
Also, let $1<q \leq p<\infty$ be such that $\frac{1}{p}+\frac{1}{q}=1$ and consider an arbitrary element $\phi \in L^{q}\bigl(0,1 ; V'\bigr)$.
My question: Can we always find some $\Phi \in L^{p}\bigl(0,1 ;V\bigr)$ such that $\partial_t \Phi = \phi$?
Here, $\partial_t\Phi$ is defined as a continuous linear operator from $\mathcal{D}$ into $V'$ with the formula \begin{equation} \partial_t\Phi(f):=-\int_0^1 \Phi(t) f'(t)dt \end{equation} where $\mathcal{D}$ is the Fréchet space of real-valued, compactly supported smooth functions on $[0,1]$ and $V'$ is equipped with the weak$^*$ topology.
I am pretty sure that we can always find such $\Phi$ but cannot explicitly construct one myself. Could anyone please help me?
