On the stability of the Lions-Peetre method of real interpolation with functional parameter. (English) Zbl 1486.46025
Summary: Let \(\vec{X}=(X_0,X_1)\) be a compatible couple of Banach spaces, \(1\le p\le\infty\) and let \(\varphi\) be positive quasi-concave function. Denote by \(\overline{X}_{\varphi,p}=(X_0,X_1)_{\varphi,p}\) the real interpolation spaces defined by S. Janson [J. Funct. Anal. 44, 50–73 (1981; Zbl 0492.46059)]. We give necessary and sufficient conditions on \(\varphi_0\), \(\varphi_1\) and \(\varphi\) for the validity of
\[
(\overline{X}_{\varphi_0,1},\overline{X}_{\varphi_1,1})_{\varphi,p}=(\overline{X}_{\varphi_0,\infty},\overline{X}_{\varphi_1,\infty})_{\varphi,p}
\]
for all \(1\le p\le\infty\), and all Banach couples \(\overline{X}\).
MSC:
| 46B70 | Interpolation between normed linear spaces |
| 46M35 | Abstract interpolation of topological vector spaces |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
