论文标题
加权离散不平等的应用
An application of the weighted discrete Hardy inequality
论文作者
论文摘要
G. H. Hardy在1925年发布的注释中指出了不等式\ begin {equation*} \ sum_ {n = 1}^\ infty \ left（\ frac {1} {n} {n} \ sum_ {k = 1} \ sum_ {n = 1}^\ infty a_n^p，\ end end {qore*}对于任何非负序列$ \ {a_n \} _ {n \ geq 1} $和$ p> 1 $。这种不平等在文献中被称为古典离散不平等。它已经进行了广泛的研究，并显示了一些应用程序和新版本。 在这项工作中，我们使用这种不平等的加权版本的表征，表现出足够的条件，可以在某个不规则平面域中在加权Sobolev空间中存在微分方程$ {\ rm div} \，{\ bf u} = f $。该方程式的可溶性对于分析Stokes方程的基础。 证明是根据局部到全球参数的基于某种功能分解的，这对其他不平等现象或相关结果（例如Korn不平等）的应用也引起了人们的关注。
In a note published in 1925, G. H. Hardy stated the inequality \begin{equation*} \sum_{n=1}^\infty \left(\frac{1}{n}\sum_{k=1}^n a_k \right)^p \leq \left(\frac{p}{p-1}\right)^p \sum_{n=1}^\infty a_n^p, \end{equation*} for any non-negative sequence $\{a_n\}_{n \geq 1}$, and $p>1$. This inequality is known in the literature as the classical discrete Hardy inequality. It has been widely studied and several applications and new versions have been shown. In this work, we use a characterization of a weighted version of this inequality to exhibit a sufficient condition for the existence of solutions of the differential equation ${\rm div}\,{\bf u}=f$ in weighted Sobolev spaces over a certain plane irregular domain. The solvability of this equation is fundamental for the analysis of the Stokes equations. The proof follows from a local-to-global argument based on a certain decomposition of functions which is also of interest for its applications to other inequalities or related results in Sobolev spaces, such as the Korn inequality.
