(Weak) Hardy and Poincar\'e inequalities and criticality theory

TL;DR

This paper investigates Hardy and Poincaré inequalities for quadratic forms satisfying the Beurling-Deny criterion, developing a criticality theory to understand their properties and implications.

Contribution

It introduces a criticality framework for quadratic forms based on Hardy and Poincaré inequalities under the Beurling-Deny criterion, extending existing theories.

Findings

Established weak and strong Hardy and Poincaré inequalities for quadratic forms.

Developed a new criticality theory connecting inequalities to form properties.

Provided applications to the analysis of quadratic forms satisfying the Beurling-Deny criterion.

Abstract

In this paper we study Hardy and Poincar\'e inequalities and their weak versions for quadratic forms satisfying the first Beurling-Deny criterion. We employ these inequalities to establish a criticality theory for such forms.