Hardy inequalities

Abstract

The main theme of this thesis is further scrutinizing classic Hardy inequalities and expanding the study on "Optimal Hardy Inequality for General Elliptic Operators with Improvement". We rediscovered explicit integral form of Hardy inequality with the main focus on its functional aspects, including density of Sobolev space. We discuss our motivation for exploring the main Hardy inequality, with a focus on the quadratic case. The fundamental solution of Laplacian is applied for illustrating the proof of classical Hardy inequality. Furthermore, we conclude the classic Hardy inequalities for a bounded domain and Hardy's boundary inequality for smooth boundary of domain. Then, the inequalities are considered for operators more general than Laplacian and various Hardy inequalities are explored in terms of different boundary weight and interior weight of positive function E. Also The generalization of Cafferelli-Kohn-Nirenberg inequality are examined to find the optimal and not attained constant. Furthermore, the possibility of more general weighted inequalities is investigated. Finally, we consider one common type of improvement for the above-mentioned inequalities.