Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909445134483456 |
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| author | Toft, Joachim Pfeuffer, Christine Teofanov, Nenad |
| author_facet | Toft, Joachim Pfeuffer, Christine Teofanov, Nenad |
| contents | Let $\mathscr B$ be a normal quasi-Banach function space with respect to $r_0 \in (0,1]$ and $v_0$, $ω$ be $v$-moderate, and let $r\in [r_0,\infty ]$. Then we prove that $f$ belongs to the modulation space $M(ω,\mathscr B )$, iff $V_ϕf$ belongs to the Wiener amalgam space $W ^r(ω,\mathscr B )$, and $$ \| f \| _{M(ω, \mathscr B)} \asymp \| V _ϕf \, ω\| _{\mathscr B} \asymp \| V _ϕf\| _{W ^r(ω, \mathscr B)}. $$
We also use the results to deduce continuity for pseudo-differential operators with symbols in weighted $M^{\infty,r_0}$-spaces, with $r_0\le 1$, when acting on $M(ω,\mathscr B )$-spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10503 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators Toft, Joachim Pfeuffer, Christine Teofanov, Nenad Functional Analysis Let $\mathscr B$ be a normal quasi-Banach function space with respect to $r_0 \in (0,1]$ and $v_0$, $ω$ be $v$-moderate, and let $r\in [r_0,\infty ]$. Then we prove that $f$ belongs to the modulation space $M(ω,\mathscr B )$, iff $V_ϕf$ belongs to the Wiener amalgam space $W ^r(ω,\mathscr B )$, and $$ \| f \| _{M(ω, \mathscr B)} \asymp \| V _ϕf \, ω\| _{\mathscr B} \asymp \| V _ϕf\| _{W ^r(ω, \mathscr B)}. $$ We also use the results to deduce continuity for pseudo-differential operators with symbols in weighted $M^{\infty,r_0}$-spaces, with $r_0\le 1$, when acting on $M(ω,\mathscr B )$-spaces. |
| title | Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2407.10503 |