Hardy inequalities in Triebel-Lizorkin spaces

TL;DR

This paper establishes a Hardy inequality for functions in Triebel-Lizorkin spaces relative to Ahlfors regular sets, and explores applications to operator boundedness and extension problems.

Contribution

It introduces a Hardy inequality tailored for Triebel-Lizorkin spaces with respect to Ahlfors regular sets, advancing the understanding of function behavior near fractal-like sets.

Findings

Proved a Hardy inequality for Triebel-Lizorkin functions relative to Ahlfors sets

Analyzed boundedness of pointwise multiplication operators

Addressed extension problems in the context of the inequality

Abstract

We prove an inequality of Hardy type for functions in Triebel-Lizorkin spaces. The distance involved is being measured to a given Ahlfors d-regular set in R^n, with n-1<d<n. As an application of the Hardy inequality, we consider boundedness of pointwise multiplication operators, and extension problems.