Fractional Hardy type inequalities on homogeneous Lie groups in the case $Q<sp$

TL;DR

This paper establishes new fractional Hardy and related inequalities on homogeneous Lie groups for the case where the homogeneous dimension is less than sp, extending the theoretical framework beyond the well-studied case Q>sp.

Contribution

The paper introduces fractional Hardy, Hardy-Sobolev, logarithmic Hardy-Sobolev, and Nash inequalities on homogeneous Lie groups specifically for Q<sp, a less explored parameter range.

Findings

Derived fractional Hardy inequality for Q<sp

Established uncertainty principle as an application

Proved fractional Hardy-Sobolev and Nash inequalities

Abstract

In this paper, we obtain a fractional Hardy inequality in the case $Q<sp$ on homogeneous Lie groups, and as an application we show the corresponding uncertainty principle. Also, we show a fractional Hardy-Sobolev type inequality on homogeneous Lie groups. In addition, we prove fractional logarithmic Hardy-Sobolev and fractional Nash type inequalities on homogeneous Lie groups. We note that the case $Q>sp$ was extensively studied in the literature, while here we are dealing with the complementary range $Q<sp$.