Reshetnyak-class mappings and composition operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918091131191296 |
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| author | V., Pavlov S. K, Vodopyanov S. |
| author_facet | V., Pavlov S. K, Vodopyanov S. |
| contents | For the Reshetnyak-class homeomorphisms $φ:Ω\to Y$, where~$Ω$ is a~domain in some Carnot group and~$Y$ is a~metric space, we obtain an~equivalent description as the mappings which induce the bounded composition operator $$ φ^*:{\rm Lip}(Y)\to L_q^1(Ω), $$ where $1\leq q\leq \infty$, as $φ^*u=u\circφ$ for $u\in{\rm Lip}(Y)$. We demonstrate the utility of our approach by characterizing the homeomorphisms $φ:Ω\toΩ'$ of domains in some Carnot group~$\mathbb G$ which induce the bounded composition operator $$ φ^*: L^1_p(Ω')\cap {\rm Lip}_{\rm loc}(Ω')\to L^1_q (Ω),\quad 1\leq q \leq p\leq \infty, $$ of homogeneous Sobolev spaces. The new proof is much shorter than the one already available, requires a~minimum of tools, and enables us to obtain new properties of the homeomorphisms in question. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reshetnyak-class mappings and composition operators V., Pavlov S. K, Vodopyanov S. Analysis of PDEs 30L10, 30L99 (Primary) 43A80, 46F10 (Secondary) For the Reshetnyak-class homeomorphisms $φ:Ω\to Y$, where~$Ω$ is a~domain in some Carnot group and~$Y$ is a~metric space, we obtain an~equivalent description as the mappings which induce the bounded composition operator $$ φ^*:{\rm Lip}(Y)\to L_q^1(Ω), $$ where $1\leq q\leq \infty$, as $φ^*u=u\circφ$ for $u\in{\rm Lip}(Y)$. We demonstrate the utility of our approach by characterizing the homeomorphisms $φ:Ω\toΩ'$ of domains in some Carnot group~$\mathbb G$ which induce the bounded composition operator $$ φ^*: L^1_p(Ω')\cap {\rm Lip}_{\rm loc}(Ω')\to L^1_q (Ω),\quad 1\leq q \leq p\leq \infty, $$ of homogeneous Sobolev spaces. The new proof is much shorter than the one already available, requires a~minimum of tools, and enables us to obtain new properties of the homeomorphisms in question. |
| title | Reshetnyak-class mappings and composition operators |
| topic | Analysis of PDEs 30L10, 30L99 (Primary) 43A80, 46F10 (Secondary) |
| url | https://arxiv.org/abs/2507.10254 |