Does anyone know how to prove or has an explicit reference for interpolation inequalities of the form
$$ \|f\|_{H^{l}(0,T;H^{(1-l)}(\Omega))} \leq C \|f\|_{H^{1}(0,T;L^2(\Omega))}^l\|f\|_{L^2(0,T;H^1(\Omega))}^{1-l}$$
and
$$ \|f\|_{H^{l}(0,T;H^{(2(1-l))}(\Omega))} \leq C \|f\|_{H^{1}(0,T;L^2(\Omega))}^l\|f\|_{L^2(0,T;H^2(\Omega))}^{1-l}$$ ?
I know how to do interpolation for Sobolev spaces $H^s(\Omega)$ and there is lots of literature. I am also aware of many posts here regarding interpolation, none of which I found answered my question. I often see citations to books like those of Lions & Magenes, but I never found such interpolation estimates explicitly inside. Maybe it is hidden in there behind several abstract constructions, or I overlooked something. I have the impression that often such books get quoted without any pointer to the page of the several hundred ones that is relevant for constructing a special case out of the abstract arguments given.
The above estimates I naively created (they are guessed) by applying the interpolation results for $H^s(\Omega)$ Sobolev spaces separately in the time and space variables. I want to validate my guess, but I do not see how to extend the proofs for interpolation of the spaces $H^s(\Omega)$ (which I know) to the case of Bochner-Sobolev spaces $H^r(0,T;H^s(\Omega))$. If the time spaces would be $L^p$, then I would use Holder's inequality combined with the interpolation for $H^s(\Omega)$ spaces. But I am interested in the examples above, particular in something like the second one.