Identification of some real interpolation spaces

Abstract:

We interrelate the real interpolation spaces associated with the couples $(X,Y), (X+Y,Y), (X,X\cap Y)$, and $(X+Y,X\cap Y)$, proving among others the identities \begin{align*} (X+Y,X)_{\theta ,p} \cap (X+Y, Y)_{\theta ,p} & = (X+Y, X \cap Y)_{\theta ,p}, (X+Y,X)_{\theta ,p} \cap (X+Y, Y)_{1-\theta ,p} & = (X,Y)_{\theta ,p}, (X,X\cap Y)_{\theta ,p} + (Y,X\cap Y)_{\theta ,p} & = (X+Y, X\cap Y)_{\theta ,p}, (X,X\cap Y)_{\theta ,p} + (Y,X\cap Y)_{1-\theta ,p} & = (X,Y)_{\theta ,p} \end{align*} for all $p \in [1,\infty ], \theta \in [0,1]$.