Upper bounds on homogeneous fractional Gagliardo-Nirenberg-Sobolev constants via parabolic estimates
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914898126045184 |
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| author | Hott, Michael |
| author_facet | Hott, Michael |
| contents | Common proofs of the Gagliardo-Nirenberg-Sobolev (GNS) do not provide explicit bounds on the involved constants, unless a sharp constant is being determined. GNS inequalities naturally occur in error estimates for numerical approximations. In particular, bounds on GNS constants allow us to provide explicit a priori estimates. We provide an algorithm that determines upper bounds on the non-endpoint homogeneous GNS inequalities in terms of explicit upper bounds for Young's convolution inequality and parabolic estimates. Our method is based on the heat-kernel representation of the inverse Laplacian, from which we deduce interpolation estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_07515 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Upper bounds on homogeneous fractional Gagliardo-Nirenberg-Sobolev constants via parabolic estimates Hott, Michael Functional Analysis Analysis of PDEs 46E35, 46B70 Common proofs of the Gagliardo-Nirenberg-Sobolev (GNS) do not provide explicit bounds on the involved constants, unless a sharp constant is being determined. GNS inequalities naturally occur in error estimates for numerical approximations. In particular, bounds on GNS constants allow us to provide explicit a priori estimates. We provide an algorithm that determines upper bounds on the non-endpoint homogeneous GNS inequalities in terms of explicit upper bounds for Young's convolution inequality and parabolic estimates. Our method is based on the heat-kernel representation of the inverse Laplacian, from which we deduce interpolation estimates. |
| title | Upper bounds on homogeneous fractional Gagliardo-Nirenberg-Sobolev constants via parabolic estimates |
| topic | Functional Analysis Analysis of PDEs 46E35, 46B70 |
| url | https://arxiv.org/abs/2303.07515 |