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arXiv · 2501.15565

Rearrangement-invariant norms commuting with dilations

Santiago Boza·Martin Křepela·Javier Soria
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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.

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Santiago Boza, Martin Křepela, Javier Soria. 2025-01-26. Rearrangement-invariant norms commuting with dilations. https://arxiv.org/abs/2501.15565

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