Some examples of equivalent rearrangement-invariant quasi-norms defined via f* or f**

TL;DR

This paper investigates the equivalence of quasi-norms defined via rearrangements and maximal functions in Lorentz-Karamata and related spaces, using Hardy inequalities to derive interpolation formulas.

Contribution

It provides conditions for the equivalence of quasi-norms in various Lorentz-Karamata spaces and applies these results to establish interpolation formulas.

Findings

Quasi-norms via $f^*$ and $f^{**}$ are equivalent under certain conditions.

Hardy-type inequalities are fundamental in proving norm equivalences.

Interpolation formulas for grand and small Lorentz-Karamata spaces are derived.

Abstract

We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces, and the so-called $\mathcal{L}$, $\mathcal{R}$, $\mathcal{LL}$, $\mathcal{RL}$, $\mathcal{RL}$, and $\mathcal{RR}$ spaces. The quasi-norms for a function $f$ in each of these spaces can be defined via the non-increasing rearrangement $f^*$ or via the maximal function $f^{**}$. We investigate when these quasi-norms are equivalent. Most of the proofs are based on Hardy-type inequalities. As application we demonstrate how our general results can be used to establish interpolation formulae for the grand and small Lorentz-Karamata spaces.