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

2022/03/20 by Doktorski, Leo R. Ya., Fernández-Martínez, Pedro, Signes, Teresa
#26A12 #46B70 #46E30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2203.10500

Abstract

We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces, and the so-called L, R, LL, RL, RL, and 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.

Related

• Absolute continuity of the (quasi)norm in rearrangement-invariant spaces
• Isometric classification of norms in rearrangement-invariant function spaces
• Lorentz--Karamata spaces
• Every rearrangement-invariant quasi-Banach function space is an interpolation space between two Lorentz spaces
• Homogeneity of rearrangement-invariant norms