Multiversion of the Hausdorff--Young inequality
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912422466420736 |
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| author | Ivanisvili, Paata Kalantzopoulos, Pavlos |
| author_facet | Ivanisvili, Paata Kalantzopoulos, Pavlos |
| contents | We consider a family of jointly Gaussian random vectors $ξ_j \in \mathbb{R}^{k_j}$, each standard normal but possibly correlated, and investigate when\[ \mathbb{E}\, F\!\Bigl(B\bigl(|T_{z_1} f_1(ξ_1)|,\dots,|T_{z_n} f_n(ξ_n)|\bigr)\Bigr) \;\;\le\;\; F\!\Bigl(\,\mathbb{E}\,B\bigl(|f_1(ξ_1)|,\dots,|f_n(ξ_n)|\bigr)\Bigr) \] holds, where $T_{z}$ is either a Mehler transform $(z \in \mathbb{C})$ or a noise operator $(z \in \mathbb{R})$. This framework unifies and extends real and complex hypercontractivity to multi-function settings, yielding multiversions of the sharp Hausdorff--Young inequality, the log-Sobolev inequality, and a noisy Gaussian--Jensen inequality. Applications include a new covariance-based characterization of the Brascamp--Lieb inequality in the presence of noise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_08494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiversion of the Hausdorff--Young inequality Ivanisvili, Paata Kalantzopoulos, Pavlos Functional Analysis Classical Analysis and ODEs Complex Variables Probability 42B20, 42B35, 47A30, 42A38 We consider a family of jointly Gaussian random vectors $ξ_j \in \mathbb{R}^{k_j}$, each standard normal but possibly correlated, and investigate when\[ \mathbb{E}\, F\!\Bigl(B\bigl(|T_{z_1} f_1(ξ_1)|,\dots,|T_{z_n} f_n(ξ_n)|\bigr)\Bigr) \;\;\le\;\; F\!\Bigl(\,\mathbb{E}\,B\bigl(|f_1(ξ_1)|,\dots,|f_n(ξ_n)|\bigr)\Bigr) \] holds, where $T_{z}$ is either a Mehler transform $(z \in \mathbb{C})$ or a noise operator $(z \in \mathbb{R})$. This framework unifies and extends real and complex hypercontractivity to multi-function settings, yielding multiversions of the sharp Hausdorff--Young inequality, the log-Sobolev inequality, and a noisy Gaussian--Jensen inequality. Applications include a new covariance-based characterization of the Brascamp--Lieb inequality in the presence of noise. |
| title | Multiversion of the Hausdorff--Young inequality |
| topic | Functional Analysis Classical Analysis and ODEs Complex Variables Probability 42B20, 42B35, 47A30, 42A38 |
| url | https://arxiv.org/abs/2506.08494 |