Rearrangement-invariant norms commuting with dilations

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 $h(r)=r^{-\frac{1}{p}}$ for all $r>0$, in which case the norm $\Vert \cdot {\Vert }_X$ is called $p$-homogeneous. We investigate which types of r.i. spaces satisfy this condition and show some important embedding properties.