Discrete functional inequalities on lattice graphs

Abstract

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.