Optimal interpolation theorem for quasi-Banach spaces \(\ell_ p\) with weights and for operators from von Neumann-Schatten classes in Hilbert pairs
In [the author, Math. Sb., Nov. Ser. 136(178), No. 2(6), 227-240 (1988; Zbl 0657.46056)] an optimal theorem was proved for weighted \(L_ p\) spaces which improves theorems of Marcinkiewicz and Riesz-Torin. It was assumed here that the operators act on Banach pairs of \(L_ p\) or Lorentz spaces. The proof of this optimal theorem has two steps. Firstly, reduction to discrete pairs of spaces which is only possible in the Banach case, and secondly, proof of the so-called model optimal interpolation theorem for discrete spaces \(\ell_ p\). This proof which is based on properties of the imbedding operator \(\ell_ p\subset \ell_ \infty\) also does not extend to the case of quasi-Banach spaces \(\ell_ p\), i.e., \(p<1\). In the present note we give a proof of an optimal interpolation theorem for discrete spaces \(\ell_ p\) for arbitrary positive \(p\). It turns out that this interpolation theorem is closely connected with a comparatively old interpolation theorem for operators from the von Neumann-Schatten classes \({\mathfrak S}_ p\) in Hilbert pairs [the author, Mat. Zametki 27, 273-282 (1980; Zbl 0429.46047)]. In particular, one is able to prove an analog of the model interpolation theorem in the quasi-Banach case precisely with its help. In addition, thanks to the optimality of the model theorem one is able to establish optimality in the corresponding theorem for operators in Hilbert pairs which remained open in the general case in the second cited paper.
theorems of Marcinkiewicz and Riesz-Torin
scientific article; zbMATH DE number 567877