格图中的离散Schwarz重排
Discrete Schwarz rearrangement in lattice graphs
摘要：
本文证明了$\mathbb{Z}^d$上广义Riesz不等式的一个离散形式.因此，我们将得到推广的Hardy-Littlewood和Pólya-Szegö不等式。我们还将在后者中建立平等的案例。我们的方法是完全新颖和独立的。特别是，我们发明了一个定义的离散重排在更高的维度。此外，我们表明，定义“建议”的Pruss不工作。我们解决了Alexander Pruss在[Pru 98，p494]，Duke Math Journal中提出的一个长期存在的开放问题，并在2009-2010年的几次通信中与他讨论，[Pru 10]。我们的方法也提供了一条攻击线，以证明其他离散重排不等式，并打开大门，建立优化器的许多重要的离散功能不等式$\mathbb{Z}^d,$ $d\geq2$。我们还将讨论我们的研究结果的一些应用。据我们所知，我们的结果是第一个在文献中处理离散重排$\mathbb{Z}^d,$ $d\geq2$。
英文摘要：
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 $\mathbb{Z}^d,$ $d\geq2$.
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