Interpolation of Lorentz spaces, the diagonal case (Q2707665)

The authors consider function spaces \(E^p(w)\) defined as collections of all functions \(f\) with \(\int _{0}^{\infty }y^{p-1}w^p (|\{|f(x)|>y\}|) dy<\infty \), where \(1\leq p<\infty \) and \(w\) is a~positive non-decreasing continuous function on \([0,\infty)\) satisfying \(\Delta _2\) condition and such that \(w(0)=0\). Such spaces present a~natural generalization of the well-known classical Lorentz spaces. The main result states that the interpolation space \((E^p(w_0),E^p(w_1))_{\theta ,p}\), where \(\theta \in (0,1)\), can be identified as \(E^p(w_0^{1-\theta }w_1^{\theta })\). The result extends the earlier work of \textit{J. Cerdà} and \textit{J. Martín} [Proc. Edinb. Math. Soc., II. Ser. 42, No. 2, 243-256 (1999; Zbl 0932.46017)] and \textit{C. Merucci} [C. R. Acad. Sci., Paris, Sér. I 294, 131-133 (1982; Zbl 0486.46053)].NEWLINENEWLINEFor the entire collection see [Zbl 0933.00035].