Optimal interpolation theorem for quasi-Banach spaces \(\ell_ p\) with weights and for operators from von Neumann-Schatten classes in Hilbert pairs
type: Item
instance of: scholarly article
title: Optimal interpolation theorem for quasi-Banach spaces \(\ell_ p\) with weights and for operators from von Neumann-Schatten classes in Hilbert pairs
zbMATH Open document ID: 0807.46084
DOI: 10.1007/BF01208323
published in: Mathematical Notes
publication date: 1994-07-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46M35
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A16
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 567877
zbMATH Keywords: weighted \(L_ p\) spaces
zbMATH Keywords: theorems of Marcinkiewicz and Riesz-Torin
zbMATH Keywords: Lorentz spaces
zbMATH Keywords: model optimal interpolation theorem
zbMATH Keywords: quasi-Banach spaces
zbMATH Keywords: von Neumann-Schatten classes
author: V. I. Ovchinnikov
MaRDI profile type: Publication
cites work: Q3805305
cites work: Q3866674
cites work: Interpolation Theorems for the Pairs of Spaces (L p , L ∞ ) and (L 1 L q )
description: scientific article; zbMATH DE number 567877