Characterization of the restricted type spaces R(X)

TL;DR

This paper investigates the properties of the restricted type spaces R(X), focusing on their relationship with Lorentz spaces and the Hardy operator, providing conditions for their characterization within rearrangement invariant spaces.

Contribution

It establishes conditions under which R(X) equals a given Lorentz space and shows how to identify the minimal r.i. Banach space for the Hardy operator in this context.

Findings

Conditions for R(X) to equal a Lorentz space are provided.

The minimal r.i. Banach space for the Hardy operator can be explicitly characterized.

The functorial properties of R(X) are analyzed within the framework of rearrangement invariant spaces.

Abstract

We study functorial properties of the spaces $R(X)$, introduced in [Studia Math. 197 (2010), 69-79] as a central tool in the analysis of the Hardy operator minus the identity on decreasing functions. In particular, we provide conditions on a minimal Lorentz space $\Lambda_{\varphi}$ so that the equation $R(X)=\Lambda_{\varphi}$ has a solution within the category of rearrangement invariant (r.i.) spaces. Moreover, we show that if $R(X)=\Lambda_{\varphi}$, then we can always take $X$ to be the minimal r.i. Banach range space for the Hardy operator defined in $\Lambda_{\varphi}$.