Higher-order affine Sobolev inequalities
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911312002416640 |
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| author | Bullion-Gauthier, Tristan |
| author_facet | Bullion-Gauthier, Tristan |
| contents | Zhang refined the classical Sobolev inequality $\|f\|_{L^{Np/(N-p)}} \lesssim \| \nabla f \|_{L^p}$, where $1\leq p \lt N$, by replacing $\|\nabla f\|_{L^p}$ with a smaller quantity invariant by unimodular affine transformations. The analogue result in homogeneous fractional Sobolev spaces $\mathring{W}^{s,p}$, with $0 \lt s \lt 1$ and $sp \lt N$, was obtained by Haddad and Ludwig. We generalize their results to the case where $s \gt 1$. Our approach, based on the existence of optimal unimodular transformations, allows us to obtain various affine inequalities, such as affine Sobolev inequalities, reverse affine inequalities, and affine Gagliardo-Nirenberg type inequalities. In a different but related direction, we also answer a question concerning reverse affine inequalities, raised by Haddad, Jiménez, and Montenegro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10473 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher-order affine Sobolev inequalities Bullion-Gauthier, Tristan Functional Analysis Zhang refined the classical Sobolev inequality $\|f\|_{L^{Np/(N-p)}} \lesssim \| \nabla f \|_{L^p}$, where $1\leq p \lt N$, by replacing $\|\nabla f\|_{L^p}$ with a smaller quantity invariant by unimodular affine transformations. The analogue result in homogeneous fractional Sobolev spaces $\mathring{W}^{s,p}$, with $0 \lt s \lt 1$ and $sp \lt N$, was obtained by Haddad and Ludwig. We generalize their results to the case where $s \gt 1$. Our approach, based on the existence of optimal unimodular transformations, allows us to obtain various affine inequalities, such as affine Sobolev inequalities, reverse affine inequalities, and affine Gagliardo-Nirenberg type inequalities. In a different but related direction, we also answer a question concerning reverse affine inequalities, raised by Haddad, Jiménez, and Montenegro. |
| title | Higher-order affine Sobolev inequalities |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2506.10473 |