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Nonlinear Analysis

Volume 200, November 2020, 112063
Nonlinear Analysis

A version of Calderón–Mityagin theorem for the class of rearrangement invariant groups☆

https://doi.org/10.1016/j.na.2020.112063Get rights and content

Abstract

Let l0 be the group (with respect to the coordinate-wise addition) of all sequences of real numbers x=(xk)k=1∞ that are eventually zero, equipped with the quasi-norm ‖x‖0=card{suppx}. A description of orbits of elements in the pair (l0,l1) is given, which complements (in the sequence space setting) the classical Calderón–Mityagin theorem on a description of orbits of elements in the pair (l1,l∞). As a consequence, we obtain that the pair (l0,l1) is K-monotone.

Section snippets

Introduction, preliminaries and main results

According to the classical Calderón–Mityagin theorem (see [5], [8]), if b=(bk)k=1∞∈l∞, then a sequence a=(ak)k=1∞ is representable in the form a=Tb for some linear operator T bounded both in l∞ and l1 if and only if ∑i=1kai∗≤C∑i=1kbi∗,k=1,2,…for some C>0, where (ui∗)i=1∞ is the nonincreasing rearrangement of the sequence (|ui|)i=1∞. Here, we give a constructive proof of a counterpart of this result for the class of rearrangement invariant groups, intermediate between l1 and the group l0 of

Proofs

We begin with proving some auxiliary results.

Proposition 1

Let b=(bi)i=1∞∈l1. If a=(ai)i=1∞∈Orb(b;l0,l1), then ∑i=k∞ai∗≤‖a‖Orb⋅∑i=[k∕‖a‖Orb]∞bi∗,k=1,2,…

Proof

First, let (X0,X1) be a pair of quasi-normed groups, and let T be a bounded homomorphism in X0 and X1. Then, E(t,Tx;X0,X1)=inf{‖Tx−y0‖X1:‖y0‖X0≤t}≤inf{‖Tx−Tx0‖X1:‖x0‖X0≤t∕‖T‖}≤‖T‖E(t∕‖T‖,x;X0,X1).

Since a=(ai)i=1∞∈Orb(b;l0,l1), for every ε>0 there exists a homomorphism T such that Tb=a and ‖T‖≔‖T‖(l0,l1)≤‖a‖Orb+ε. Then, applying the preceding estimate to the

Groups of Marcinkiewicz type that are interpolation with respect to the pair (l0,l1)

Let (αk)k=1∞ be an increasing sequence of positive numbers such that for some constants R1 and R2 we have α2k≤R1αk,k=1,2,…,and ∑i=k∞αi−1≤R2kαk−1,k=1,2,…Denote by Mα the set of all sequences x=(xk)k=1∞ such that ‖x‖α≔supk=1,2,…αkxk∗<∞.To prove that this functional defines a r.i. quasi-norm on Mα, it suffices to check that for every x,y∈Mα we have ‖x+y‖α≤R12(‖x‖α+‖y‖α).Indeed, by [2, Proposition 2.1.7], (x+y)i∗≤x[i∕2]∗+y[i∕2]∗,i=2,3,…Hence, for all i=2,3,… from (22) it follows that αi(x+y)i∗≤αix[i

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The work was completed as a part of the implementation of the development program of the Scientific and Educational Mathematical Center Volga Federal District, agreement no. 075-02-2020-1488/1.

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