In this thesis, we study problems at the interface of analysis and discrete mathematics. We discuss analogues of well known Hardy-type inequalities and Rearrangement inequalities on the lattice graphs $\mathbb{Z}^d$, with a particular focus on behaviour of sharp constants and optimizers.In the first half of the thesis, we analyse Hardy inequalities on $\mathbb{Z}^d$, first for $d=1$ and then for $d \geq 3$. We prove a sharp weighted Hardy inequality on integers with power weights of the form $n^\alpha$. This is done via two different methods, namely super-solution and Fourier method. We also use Fourier method to prove a weighted Hardy type inequality for higher order operators. After discussing the one dimensional case, we study the Hardy inequality in higher dimensions ($d \geq 3$). In particular, we compute the asymptotic behaviour of the sharp constant in the discrete Hardy inequality, as $d \rightarrow \infty$. This is done by converting the inequality into a continuous Hardy-type inequality on a torus for functions having zero average. These continuous inequalities are new and interesting in themselves. In the second half, we focus our attention on analogues of Rearrangement inequalities on lattice graphs. We begin by analysing the situation in dimension one. We define various notions of rearrangements and prove the corresponding Polya-Szeg\H{o} inequality. These inequalities are also applied to prove some weighted Hardy inequalities on integers. Finally, we study Rearrangement inequalities (Polya-Szeg\H{o}) on general graphs, with a particular focus on lattice graphs $\mathbb{Z}^d$, for $d \geq 2$. We develop a framework to study these inequalities, using which we derive concrete results in dimension two. In particular, these results develop connections between Polya-Szeg\H{o} inequality and various isoperimetric inequalities on graphs.

翻译：本论文研究分析与离散数学交叉领域中的问题。我们讨论了格图 $\mathbb{Z}^d$ 上经典Hardy型不等式和重排不等式的类似形式，特别关注尖锐常数和最优函数的性态。论文前半部分分析了 $\mathbb{Z}^d$ 上的Hardy不等式，首先考虑 $d=1$ 的情形，然后推广至 $d \geq 3$。我们证明了整数上形如 $n^\alpha$ 的幂权重的尖锐加权Hardy不等式，采用超解方法和傅里叶方法两种不同途径实现。同时利用傅里叶方法证明了高阶算子的加权Hardy型不等式。讨论一维情形后，我们研究了高维（$d \geq 3$）Hardy不等式，特别计算了当 $d \rightarrow \infty$ 时离散Hardy不等式尖锐常数的渐近行为。这一结果通过将原不等式转化为环面上关于零平均值函数的连续Hardy型不等式得到，这些连续不等式本身具有新颖性和重要价值。后半部分聚焦于格图上的重排不等式类似形式。首先分析一维情形，定义了多种重排概念并证明了相应的Polya-Szegő不等式，这些不等式进一步用于证明整数上的某些加权Hardy不等式。最后，研究了一般图（特别关注 $d \geq 2$ 的格图 $\mathbb{Z}^d$）上的重排不等式（Polya-Szegő）。我们建立了研究此类不等式的框架，并以此导出二维情形的具体结论，这些结果揭示了Polya-Szegő不等式与图上各类等周不等式之间的内在联系。