Rearrangement-invariant norms commuting with dilations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912847135506432 |
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| author | Boza, Santiago Křepela, Martin Soria, Javier |
| author_facet | Boza, Santiago Křepela, Martin Soria, Javier |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rearrangement-invariant norms commuting with dilations Boza, Santiago Křepela, Martin Soria, Javier Functional Analysis 46E30, 46B42 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. |
| title | Rearrangement-invariant norms commuting with dilations |
| topic | Functional Analysis 46E30, 46B42 |
| url | https://arxiv.org/abs/2501.15565 |