On the stability of the Lions-Peetre method of real interpolation with functional parameter

Let $\vec{X}=(X_0, X_1)$ be a compatible couple of Banach spaces, $1\le p \le \infty$ and let $φ$ be positive quasi-concave function. Denote by $\overline{X}_{φ,p}=(X_0,X_1)_{φ,p}$ the real interpolation spaces defined by S. Janson (1981). We give necessary and sufficient conditions on $φ_{0}$, $φ_{1}$ and $φ$ for the validity of \begin{equation*} \left(\overline{X}_{φ_{0},1},\overline{X}_{φ_{1},1} \right) _{φ,p}= \left(\overline{X}_{φ_{0},\infty},\overline{X}_{φ_{1},\infty}\right)_{φ,p} \end{equation*} for all $1\le p\le \infty$, and all Banach couples $\overline{X}.$

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