Complex Interpolation and Intersection

Question

Does it hold $$[X \cap Y, X \cap Z]_\theta = X \cap [Y,Z]_\theta$$ where $X,Y,Z$ are suitable spaces and $[\cdot,\cdot]_\theta $ denotes the complex interpolation functor of order $\theta \in [0,1]$. The inclusion $\subseteq$ follows directly by the definition. However, I do not see wether the reverse inclusion holds.

Answer 1

The result is not true in general. You can take $X=H^1(\Omega), Y=L^2(\Omega), Z=H^2(\Omega)$ and $\theta=1/2$ to get two different spaces.

Maybe the result you are looking for is Theorem 13.1 of section 13. Intersection Interpolation in

J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. 1, Springer-Verlag (1972).

Edit 1: I found (by chance) a sufficient condition which gives the result you asked for in the following paper, see pp. 3, and Lemma 3.4 (with the comments just before):

C L. Fefferman, K W. Hajduk, J C. Robinson, Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces. Preprint (2019).