关于凸超曲面的Michael-Simon型分数Hardy-Sobolev不等式
A fractional Hardy-Sobolev inequality of Michael-Simon type on convex hypersurfaces
AI总结：
本文基于凸超曲面，推导出一种分数Michael-Simon型Hardy-Sobolev不等式，该不等式包含通用常数和分数平均曲率，连接了新获得的分数Hardy不等式与已有不等式，并在平面情况下提供新证明。
摘要：
在本文中，我们证明了在超曲面$M\subset\R^{n+1}$上的一种分数版本的Caffarelli-Kohn-Nirenberg型插值不等式，其中$M$是凸集的边界。该不等式包含一个与$M$无关的通用常数，并涉及$M$的分数平均曲率。特别是，它连接了最近由Cabré、Cozzi和第一作者获得的分数Michael-Simon Sobolev不等式和在$M$上新获得的分数Hardy不等式。我们的方法在平面情况$M=\R^n$下给出分数Hardy不等式的新简单证明。为了在超曲面上获得分数Hardy不等式，我们建立了一个不等式，该不等式将$M$的加权周长与$M$的标准周长（模一个通用常数）联系起来，并且对所有凸超曲面$M$都成立。
英文摘要：
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 $M$.
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