A linear operator bounded in all Besov but not in Triebel-Lizorkin spaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917647635972096 |
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| author | Yao, Liding |
| author_facet | Yao, Liding |
| contents | We construct a linear operator $T:\mathscr S'(\mathbb R^n)\to \mathscr S'(\mathbb R^n)$ such that $T:\mathscr B_{pq}^s(\mathbb R^n)\to\mathscr B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but $T(\mathscr F_{pq}^s(\mathbb R^n))\not\subset \mathscr F_{pq}^s(\mathbb R^n)$ unless $p=q$. As a result Triebel-Lizorkin spaces cannot be interpolated from Besov spaces unless $p=q$. In the appendix we purpose a question for the interpolation framework via structured Banach spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_05813 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A linear operator bounded in all Besov but not in Triebel-Lizorkin spaces Yao, Liding Functional Analysis Classical Analysis and ODEs 46E35 (primary) 46B70, 42B35 and 42B25 (secondary) We construct a linear operator $T:\mathscr S'(\mathbb R^n)\to \mathscr S'(\mathbb R^n)$ such that $T:\mathscr B_{pq}^s(\mathbb R^n)\to\mathscr B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but $T(\mathscr F_{pq}^s(\mathbb R^n))\not\subset \mathscr F_{pq}^s(\mathbb R^n)$ unless $p=q$. As a result Triebel-Lizorkin spaces cannot be interpolated from Besov spaces unless $p=q$. In the appendix we purpose a question for the interpolation framework via structured Banach spaces. |
| title | A linear operator bounded in all Besov but not in Triebel-Lizorkin spaces |
| topic | Functional Analysis Classical Analysis and ODEs 46E35 (primary) 46B70, 42B35 and 42B25 (secondary) |
| url | https://arxiv.org/abs/2404.05813 |