Rearrangement-invariant norms commuting with dilations
Santiago Boza, Martin Křepela, Javier Soria
TL;DR
The paper identifies when rearrangement-invariant spaces on $([0,\infty),\lambda)$ admit a dilation-homogeneous norm, showing the dilation scaling must be of the form $h(r)=r^{-1/p}$ for some $p\in[1,\infty]$, making the norm $p$-homogeneous. It then maps out the landscape of such spaces, proving Lorentz spaces $L^{p,q}$ are the canonical $p$-homogeneous examples and that duality and interpolation preserve homogeneity; it also provides constructions of non-Lorentz $p$-homogeneous spaces via an extrapolation-style framework and broadens the family with Orlicz–Lorentz spaces $L^{p,\Phi}$, for which the fundamental function scales as $\varphi_{L^{p,\Phi}}(t)=C_0 t^{1/p}$ with $C_0$ determined by a simple integral condition. The results give precise conditions under which these spaces exhibit $p$-homogeneity, clarify the embeddings within the Lorentz scale, and demonstrate the existence of rich families of homogeneous r.i. spaces beyond Lorentz, with explicit constructions. Overall, the work advances understanding of how dilation invariance interacts with the structure of rearrangement-invariant spaces and their interpolation and duality properties.
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.