$\mathbb{R}^d$中凸域的Gagliardo-Nirenberg-Sobolev不等式
Gagliardo-Nirenberg-Sobolev inequalities for convex domains in $\mathbb{R}^d$
摘要：
$\mathbb{R}^d$中一类特殊的Gagliardo-Nirenberg-Sobolev（GNS）不等式在Lieb-Thirring不等式的几个证明中起了关键作用.最近，需要GNS不等式的凸域的$\mathbb{R}^d$，特别是立方体，已经出现。这份手稿的目的是双重的。首先，我们证明了一个GNS不等式的凸域，显式常数依赖于域的几何形状。然后，利用Rumin方法的离散化形式，证明了具有改进常数的立方体上的GNS不等式。
英文摘要：
A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in $\mathbb{R}^d$ has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of $\mathbb{R}^d$, in particular for cubes, has arised. The purpose of this manuscript is two-fold. First we prove a GNS inequality for convex domains, with explicit constants which depend on the geometry of the domain. Later, using the discrete version of Rumin's method, we prove GNS inequalities on cubes with improved constants.
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