On weak associated reflexivity of weighted Sobolev spaces of the first order on real line

TL;DR

This paper investigates the structure of associate and double associate spaces of weighted Sobolev spaces on the real line, revealing a nuanced division into strong and weak cases and establishing weak reflexivity properties.

Contribution

It introduces a novel classification of associativity in weighted Sobolev spaces, distinguishing between strong and weak cases, and proves weak reflexivity of certain Sobolev spaces.

Findings

Weak associated reflexivity of Sobolev spaces of compactly supported functions.

Double weak-strong associate space is vacuous.

Characterization of power weights via Cesàro and Copson spaces.

Abstract

We study associate and double associate spaces of two-weighted Sobolev spaces of the first order on real half-line and we show that unlike the notion of duality the associativity is divided into two cases which we call "strong" and "weak" ones with the division of the second associativity into four cases. On the way we prove that the Sobolev space of compactly supported functions possess weak associated reflexivity and the double weak-strong associate space is vacuous. The case of power weights was recently characterized by reduction to Ces\`{a}ro or Copson type spaces [18].