Rearrangement-invariant norms commuting with dilations

TL;DR

This paper characterizes rearrangement-invariant spaces over [0,∞) that have norms commuting with dilations, showing such spaces are p-homogeneous and exploring their embedding properties.

Contribution

It identifies the specific form of norms that commute with dilations in rearrangement-invariant spaces and analyzes their structural properties.

Findings

Only p-homogeneous norms satisfy the dilation commutation property.

p-homogeneous spaces exhibit particular embedding characteristics.

The paper classifies which r.i. spaces meet the dilation invariance condition.

Abstract

We study rearrangement-invariant spaces $X$ over $[0,\infty)$ for which there exists a function $h:(0,\infty)\to (0,\infty)$ such that \[ \|D_rf\|_X = h(r)\|f\|_X \] for all $f\in X$ and all $r>0$, where $D_r$ is the dilation operator. It is shown that this may hold only if $h(r)=r^{-\frac1p}$ for all $r>0$, in which case the norm $\|\cdot\|_X$ is called $p$-homogeneous. We investigate which types of r.i. spaces satisfy this condition and show some important embedding properties.