On a theorem of M. Jodeit Jr. on pushforwards of Fourier multipliers
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909000478490624 |
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| author | Poissel, Patrick |
| author_facet | Poissel, Patrick |
| contents | A classical theorem of M. Jodeit Jr. implies that if a compactly supported distribution on $\mathbf{R}^d$ is the symbol of an $L^p(\mathbf{R}^d)$-$L^q(\mathbf{R}^d)$ Fourier multiplier, then its pushforward by the canonical homomorphism from $\mathbf{R}^d$ to $\mathbf{T}^d$ is the symbol of an $\ell^p(\mathbf{Z}^d)$-$\ell^q(\mathbf{Z}^d)$ Fourier multiplier. In the present work, we generalise this result to the setting of locally compact groups, including those non-abelian, by characterising the continuous homomorphisms of locally compact groups by which, for every $p,q\in[1,\infty]$, the pushforward of a compactly supported distribution symbol of an $L^p$-$L^q$ Fourier multiplier is a symbol of the same type as those which are open. Motivated by a simple proof in the abelian case, we also investigate pushforwards of positive definite distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15877 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a theorem of M. Jodeit Jr. on pushforwards of Fourier multipliers Poissel, Patrick Functional Analysis 43A22, 43A30 (primary), 43L35, 46L51, 46L52 (secondary) A classical theorem of M. Jodeit Jr. implies that if a compactly supported distribution on $\mathbf{R}^d$ is the symbol of an $L^p(\mathbf{R}^d)$-$L^q(\mathbf{R}^d)$ Fourier multiplier, then its pushforward by the canonical homomorphism from $\mathbf{R}^d$ to $\mathbf{T}^d$ is the symbol of an $\ell^p(\mathbf{Z}^d)$-$\ell^q(\mathbf{Z}^d)$ Fourier multiplier. In the present work, we generalise this result to the setting of locally compact groups, including those non-abelian, by characterising the continuous homomorphisms of locally compact groups by which, for every $p,q\in[1,\infty]$, the pushforward of a compactly supported distribution symbol of an $L^p$-$L^q$ Fourier multiplier is a symbol of the same type as those which are open. Motivated by a simple proof in the abelian case, we also investigate pushforwards of positive definite distributions. |
| title | On a theorem of M. Jodeit Jr. on pushforwards of Fourier multipliers |
| topic | Functional Analysis 43A22, 43A30 (primary), 43L35, 46L51, 46L52 (secondary) |
| url | https://arxiv.org/abs/2603.15877 |