Real interpolation of vector-valued spaces in non-diagonal case

Abstract:

It is shown that the formula \begin{equation*} (l_{p_{0}}^{s_{0}}(A_{0}),...,l_{p_{n}}^{s_{n}}(A_{n})) _{\vec {\theta },q} =l_{q}^{s}((A_{0},...,A_{n})_{\vec {\theta },q}), \end{equation*} where $\vec {\theta }=(\theta _{0},...,\theta _{n})$ and $s=\theta _{0}s_{0}+...+\theta _{n}s_{n}$ is correct under the restrictions $A_{n-1}=A_{n}$ and $s_{n-1}\neq s_{n}.$ It is also true if we suppose that \begin{equation*} A_{n}=(A_{0},A_{1},...,A_{n-1})_{\vec {\lambda },p},s_{n}\neq \lambda _{0}s_{0}+\lambda _{1}s_{1}+...+\lambda _{n-1}s_{n-1}, \end{equation*} and the spaces $A_{0},A_{1},...,A_{n-1}$ are functional Banach or quasi-Banach lattices on the same measure space $(\Omega ,\mu ).$