论文标题
晶格图中的离散Schwarz重排
Discrete Schwarz rearrangement in lattice graphs
论文作者
论文摘要
在本文中，我们证明了$ \ mathbb {z}^d $上的广义riesz不等式的离散版本。结果，我们将得出扩展的Hardy-Littlewood和Pólya-Szegö的不平等现象。我们还将在后者中建立平等案例。我们的方法是完全新颖和独立的。特别是，我们在更高维度中发明了一个定义，以定义。此外，我们表明普鲁斯“建议”的定义不起作用。我们解决了亚历山大·普鲁斯（Alexander Pruss）在[PRU98，P494]，杜克数学杂志（Duke Math Journal）中提出的一个长期的公开问题，并在2009 - 2010年的多次通讯中与他讨论了[PRU10]。我们的方法还提供了一系列攻击线，以证明其他离散的重排不平等，并为建立许多重要离散功能不平等的优化者打开了$ \ Mathbb {z}^d，$ $ $ d \ geq2 $的优化者。我们还将讨论我们发现的一些应用。据我们所知，我们的结果是在$ \ Mathbb {z}^d，$ $ $ d \ geq2 $上处理离散重排的文献中的第一个结果。
In this paper, we prove a discrete version of the generalized Riesz inequality on $\mathbb{Z}^d$. As a consequence, we will derive the extended Hardy-Littlewood and Pólya-Szegö inequalities. We will also establish cases of equality in the latter. Our approach is totally novel and self-contained. In particular, we invented a definition for the discrete rearrangement in higher dimensions. Moreover, we show that the definition "suggested" by Pruss does not work. We solve a long-standing open question raised by Alexander Pruss in [Pru98, p494], Duke Math Journal, and discussed with him in several communications in 2009-2010, [Pru10]. Our method also provides a line of attack to prove other discrete rearrangement inequalities and opens the door to the establishment of optimizers of many important discrete functional inequalities in $\mathbb{Z}^d,$ $d\geq2$. We will also discuss some applications of our findings. To the best of our knowledge, our results are the first ones in the literature dealing with discrete rearrangement on $\mathbb{Z}^d,$ $d\geq2$.
