A note on best constants for Weighted Integral Hardy inequalities on homogeneous groups

TL;DR

This paper establishes sharp weighted integral Hardy inequalities on homogeneous Lie groups with quasi-norms, providing exact constants and extending previous results to a broader setting.

Contribution

It proves sharp weighted integral Hardy inequalities on homogeneous groups with explicit constants, improving prior results for this class of groups.

Findings

Derived sharp weighted integral Hardy inequalities on homogeneous groups.

Calculated exact constants for these inequalities.

Extended known results to more general homogeneous group settings.

Abstract

The main aim of this note is to prove sharp weighted integral Hardy inequality and conjugate integral Hardy inequality on homogeneous Lie groups with any quasi-norm for the range $1<p\leq q<\infty.$ We also calculate the precise value of sharp constants in respective inequalities, improving the result of $[19]$ in the case of homogeneous groups.