Discrete functional inequalities on lattice graphs

TL;DR

This work develops a rigorous theory of discrete functional inequalities on lattice graphs, focusing on Hardy-type and rearrangement inequalities on $\mathbb{Z}^d$. It introduces two complementary analytical frameworks: a discrete supersolution approach and a Fourier-analytic method, enabling sharp constants and optimality results for 1D and higher-dimensional settings, as well as higher-order operators. A key finding is that discrete Hardy constants grow as $C_H(d)\sim d$ in dimension, while discrete Rellich constants grow as $C_R(d)\sim d^2$, highlighting fundamental differences from their continuous counterparts. The thesis further extends rearrangement theory to discrete graphs, developing a 1D rearrangement with weighted Polya–Szegő inequalities and a Fourier rearrangement that preserves spectral advantages, and connects discrete isoperimetric properties to Polya–Szegő-type results. Collectively, these results deepen the understanding of discrete analysis on lattices and have implications for spectral theory and graph isoperimetry.

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 $n^α$. 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-Szegő inequality. These inequalities are also applied to prove some weighted Hardy inequalities on integers. Finally, we study Rearrangement inequalities (Polya-Szegő) 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-Szegő inequality and various isoperimetric inequalities on graphs.

Figures (12)

• Figure 1: The spiral-like labelling on the $3$-regular infinite tree (only the first three layers of the infinite tree are shown).
• Figure 2: Part of the two dimensional lattice graph $(\mathbb{Z}^2, \ell^1)$
• Figure 3: Left: Solution of edge isoperimetric problem. Right: Solution of vertex isoperimetric problem.
• Figure 4: Part of one-dimensional lattice graph
• Figure 5: $u^*(1) \,$ is the largest value of $\,u, \, u^*(2) \,$ is the second largest, and so on.

Theorems & Definitions (189)

• Theorem 1.1: E.B. Davies and A.M. Hinz davies
• Remark 2.1
• Remark 2.2
• Theorem 2.1
• Remark 2.3
• Corollary 2.1
• Remark 2.4
• Corollary 2.2
• Remark 2.5
• Definition 2.1