Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces

Abstract

A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric measure space without any geometric constraint on the potential function. The adopted approach highlights some criteria for a smooth metric measure space to admit Hardy inequalities related to Witten and Witten p-Laplace operators. The results in this paper complement in several aspect to those obtained recently in the non-compact setting.

Keywords

• Riemannian manifold
• Hardy inequality
• uncertainty principle
• elliptic operators
• Rellich inequality