Riesz transform, function spaces and their applications on infinite dimensional compact groups
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| Format: | Preprint |
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2025
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| _version_ | 1866912340747747328 |
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| author | Bendikov, Alexander Chen, Li Saloff-Coste, Laurent |
| author_facet | Bendikov, Alexander Chen, Li Saloff-Coste, Laurent |
| contents | On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(μ_t)_{t>0}$. We establish several regularity results of the solution to the Poisson equation $LU=F$, both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for $1\le p\le \infty$: $Λ_θ^p$, defined via the associated Markov semigroup, and $\mathrm L_θ^p$, defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of $Λ_θ^p$ space. In the distributional sense, we further show local regularity in the class of $\mathrm L_θ^{\infty}$ space. These results require some strong assumptions on $-L$. Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free $L^p$ ($1<p<\infty$) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15718 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Riesz transform, function spaces and their applications on infinite dimensional compact groups Bendikov, Alexander Chen, Li Saloff-Coste, Laurent Analysis of PDEs Functional Analysis Probability On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(μ_t)_{t>0}$. We establish several regularity results of the solution to the Poisson equation $LU=F$, both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for $1\le p\le \infty$: $Λ_θ^p$, defined via the associated Markov semigroup, and $\mathrm L_θ^p$, defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of $Λ_θ^p$ space. In the distributional sense, we further show local regularity in the class of $\mathrm L_θ^{\infty}$ space. These results require some strong assumptions on $-L$. Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free $L^p$ ($1<p<\infty$) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms. |
| title | Riesz transform, function spaces and their applications on infinite dimensional compact groups |
| topic | Analysis of PDEs Functional Analysis Probability |
| url | https://arxiv.org/abs/2504.15718 |