Discrete logarithmic Sobolev inequalities in Banach spaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910337210515456 |
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| author | Cordero-Erausquin, Dario Eskenazis, Alexandros |
| author_facet | Cordero-Erausquin, Dario Eskenazis, Alexandros |
| contents | Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $σ_n$. We prove that if $(E,\|\cdot\|_E)$ is a Banach space of finite cotype and $p\in[1,\infty)$, then every function $f:\mathscr{C}_n\to E$ satisfies the dimension-free vector-valued $L_p$ logarithmic Sobolev inequality $$\|f-\mathbb{E} f\|_{L_p(\log L)^{p/2}(E)} \leq \mathsf{K}_p(E) \left( \int_{\mathscr{C}_n} \Big\| \sum_{i=1}^n δ_i \partial_i f\Big\|_{L_p(E)}^p \, dσ_n(δ)\right)^{1/p}.$$ The finite cotype assumption is necessary for the conclusion to hold. This estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss space and the optimal vector-valued version of a deep inequality of Talagrand (1994). As an application, we use such vector-valued $L_p$ logarithmic Sobolev inequalities to derive new lower bounds for the bi-Lipschitz distortion of nonlinear quotients of the Hamming cube into Banach spaces with prescribed Rademacher type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_03878 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Discrete logarithmic Sobolev inequalities in Banach spaces Cordero-Erausquin, Dario Eskenazis, Alexandros Functional Analysis Metric Geometry Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $σ_n$. We prove that if $(E,\|\cdot\|_E)$ is a Banach space of finite cotype and $p\in[1,\infty)$, then every function $f:\mathscr{C}_n\to E$ satisfies the dimension-free vector-valued $L_p$ logarithmic Sobolev inequality $$\|f-\mathbb{E} f\|_{L_p(\log L)^{p/2}(E)} \leq \mathsf{K}_p(E) \left( \int_{\mathscr{C}_n} \Big\| \sum_{i=1}^n δ_i \partial_i f\Big\|_{L_p(E)}^p \, dσ_n(δ)\right)^{1/p}.$$ The finite cotype assumption is necessary for the conclusion to hold. This estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss space and the optimal vector-valued version of a deep inequality of Talagrand (1994). As an application, we use such vector-valued $L_p$ logarithmic Sobolev inequalities to derive new lower bounds for the bi-Lipschitz distortion of nonlinear quotients of the Hamming cube into Banach spaces with prescribed Rademacher type. |
| title | Discrete logarithmic Sobolev inequalities in Banach spaces |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2304.03878 |