Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914692399628288 |
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| author | Bellido, José Carlos Mora-Corral, Carlos Schönberger, Hidde |
| author_facet | Bellido, José Carlos Mora-Corral, Carlos Schönberger, Hidde |
| contents | We address the study of nonlocal gradients defined through general radial kernels $ρ$. Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function. Specifically, even with minimal assumptions on $ρ$, we establish Poincaré inequalities and compact embeddings into Lebesgue spaces. Additionally, we present a fundamental theorem of calculus that enables us to recover a function from its nonlocal gradient through a convolution. This is used to demonstrate embeddings into Orlicz spaces and spaces of continuous functions that mirror the well-known Sobolev and Morrey inequalities for classical gradients. Finally, we establish conditions for inclusions and equality of spaces associated to different kernels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16487 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings Bellido, José Carlos Mora-Corral, Carlos Schönberger, Hidde Analysis of PDEs 26A33, 46E35, 47G20 (Primary) 42B35, 74A70 (Secondary) We address the study of nonlocal gradients defined through general radial kernels $ρ$. Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function. Specifically, even with minimal assumptions on $ρ$, we establish Poincaré inequalities and compact embeddings into Lebesgue spaces. Additionally, we present a fundamental theorem of calculus that enables us to recover a function from its nonlocal gradient through a convolution. This is used to demonstrate embeddings into Orlicz spaces and spaces of continuous functions that mirror the well-known Sobolev and Morrey inequalities for classical gradients. Finally, we establish conditions for inclusions and equality of spaces associated to different kernels. |
| title | Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings |
| topic | Analysis of PDEs 26A33, 46E35, 47G20 (Primary) 42B35, 74A70 (Secondary) |
| url | https://arxiv.org/abs/2402.16487 |