Best constants for Hardy inequalities in Triebel--Lizorkin spaces

TL;DR

This paper determines optimal constants for fractional Hardy inequalities within weighted Triebel--Lizorkin spaces, extending previous results to new settings and providing sharp bounds for these inequalities.

Contribution

It introduces the first sharp constants for fractional Hardy inequalities in weighted Triebel--Lizorkin spaces, generalizing prior work on Gagliardo seminorms and unweighted cases.

Findings

Sharp constants found for fractional Hardy inequalities

Results apply to weighted spaces on entire space and half-spaces

Generalizes previous inequalities for Gagliardo seminorms

Abstract

We find sharp constants in fractional Hardy inequalities for weighted Triebel--Lizorkin seminorms on the whole space and half-spaces. Our results generalize recently obtained weighted fractional Hardy inequalities for Gagliardo seminorms, but are new even for the unweighted case.