Schauder estimates for germs of distributions on smooth manifolds
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918350331838464 |
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| author | Costeri, Beatrice Dappiaggi, Claudio Rinaldi, Paolo Savasta, Matteo |
| author_facet | Costeri, Beatrice Dappiaggi, Claudio Rinaldi, Paolo Savasta, Matteo |
| contents | We discuss germs of distributions on $d-$dimensional smooth Riemannian manifolds and, in particular, we derive \emph{multi-level Schauder estimates} without making any further assumptions on the underlying geometry. As a preliminary step, we define the notions of coherence and homogeneity for germs of distributions on open subsets of $\mathbb{R}^d$, $d \ge 1$. Subsequently, we formulate both the reconstruction theorem, cf., [CZ20], and the Schauder estimates, cf., [BCZ24], in this setting. Leveraging the properties of the exponential map, we extend these results to Riemannian manifolds. Specifically, we devise a counterpart of the reconstruction theorem previously established in the literature [RS21], while additionally proving the regularity of the reconstructed distribution in suitable Hölder-Zygmund spaces. Finally, by introducing a novel concept of $β$-regularizing kernels on Riemannian manifolds, we establish Schauder estimates for coherent and homogeneous germs in this context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19593 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Schauder estimates for germs of distributions on smooth manifolds Costeri, Beatrice Dappiaggi, Claudio Rinaldi, Paolo Savasta, Matteo Mathematical Physics Analysis of PDEs Probability We discuss germs of distributions on $d-$dimensional smooth Riemannian manifolds and, in particular, we derive \emph{multi-level Schauder estimates} without making any further assumptions on the underlying geometry. As a preliminary step, we define the notions of coherence and homogeneity for germs of distributions on open subsets of $\mathbb{R}^d$, $d \ge 1$. Subsequently, we formulate both the reconstruction theorem, cf., [CZ20], and the Schauder estimates, cf., [BCZ24], in this setting. Leveraging the properties of the exponential map, we extend these results to Riemannian manifolds. Specifically, we devise a counterpart of the reconstruction theorem previously established in the literature [RS21], while additionally proving the regularity of the reconstructed distribution in suitable Hölder-Zygmund spaces. Finally, by introducing a novel concept of $β$-regularizing kernels on Riemannian manifolds, we establish Schauder estimates for coherent and homogeneous germs in this context. |
| title | Schauder estimates for germs of distributions on smooth manifolds |
| topic | Mathematical Physics Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2602.19593 |