In the book Non-Homogeneous Boundary Value Problems and Applications I by Lions and Magenes, the Intermediate Derivative Theorem is stated as follows:
Intermediate Derivative Theorem: Let $X\subset Y$ be a dense continuous injection of separable (complex) Hilbert spaces. Define $$W^m(a,b,X,Y)=\{u\in L^2(a,b;X)\ |\ \partial_t^m u\in L^2(a,b;Y)\}$$ where we define $\mathcal{D}'((a,b);X)=\mathscr{L}(C_c^\infty((a,b)),X)$, with the norm on $W^m$ given by $$\lVert u\rVert_{W^m(a,b,X,Y)}^2=\lVert u\rVert_{L^2(a,b;X)}^2+\lVert u\rVert_{L^2(a,b;Y)}^2 .$$ Let $u\in L^2(a,b;X)$, and $\partial_t^m u\in L^2(a,b;Y)$ distributionally. Then $$\partial_t^j u\in L^2(a,b;[X,Y]_{j/m})$$ for all $0\le j\le m$, and $$\partial_t^j:W^m(a,b,X,Y)\to L^2(a,b;[X,Y]_{j/m})$$ is continuous for all $0\le j\le m$.
In the book, $[X,Y]_{j/m}$ is defined using domains of unbounded operators, but I believe in this case that complex interpolation is equivalent.
However, the proof as given in the book is incredibly long and unwieldily, using many techniques such as measurable Hilbertian sums, and interpolation via domains of operators, and I suspect that a better proof should exist using complex interpolation. However, I have been unable to find a reference for any "Intermediate Derivative Theorem" in the literature, which suggests that this result may go by another name?
Any modern reference for this theorem would be greatly appreciated.
