A fractional Hardy-Sobolev inequality of Michael-Simon type on convex hypersurfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910053100945408 |
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| author | Csató, Gyula Roy, Prosenjit |
| author_facet | Csató, Gyula Roy, Prosenjit |
| contents | In this paper we prove a fractional version of a Caffarelli-Kohn-Nirenberg type interpolation inequality on hypersurfaces $M\subset\R^{n+1}$ which are boundaries of convex sets. The inequality carries a universal constant independent of $M$ and involves the fractional mean curvature of $M.$ In particular, it interpolates between the fractional Micheal-Simon Sobolev inequality recently obtained by Cabré, Cozzi, and the first author, and a new fractional Hardy inequality on $M$. Our method, when restricted to the plane case $M=\R^n$, gives a new simple proof of the fractional Hardy inequality. To obtain the fractional Hardy inequality on a hypersurface, we establish an inequality which bounds a weighted perimeter of $M$ by the standard perimeter of $M$ (modulo a universal constant), and which is valid for all convex hypersurfaces $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A fractional Hardy-Sobolev inequality of Michael-Simon type on convex hypersurfaces Csató, Gyula Roy, Prosenjit Analysis of PDEs In this paper we prove a fractional version of a Caffarelli-Kohn-Nirenberg type interpolation inequality on hypersurfaces $M\subset\R^{n+1}$ which are boundaries of convex sets. The inequality carries a universal constant independent of $M$ and involves the fractional mean curvature of $M.$ In particular, it interpolates between the fractional Micheal-Simon Sobolev inequality recently obtained by Cabré, Cozzi, and the first author, and a new fractional Hardy inequality on $M$. Our method, when restricted to the plane case $M=\R^n$, gives a new simple proof of the fractional Hardy inequality. To obtain the fractional Hardy inequality on a hypersurface, we establish an inequality which bounds a weighted perimeter of $M$ by the standard perimeter of $M$ (modulo a universal constant), and which is valid for all convex hypersurfaces $M$. |
| title | A fractional Hardy-Sobolev inequality of Michael-Simon type on convex hypersurfaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.12439 |