Fractional Korn and Hardy-type inequalities for vector fields in half space

TL;DR

This paper establishes a new fractional Hardy-type inequality for vector fields in half space, improving classical inequalities and linking specialized function spaces with fractional Sobolev spaces.

Contribution

It introduces a modified fractional semi-norm that yields a stronger Hardy inequality and proves a fractional Korn's inequality in half space.

Findings

The modified semi-norm is not equivalent to the standard fractional semi-norm.

The new inequality improves classical fractional Hardy inequalities.

It demonstrates the equivalence of certain function spaces with fractional Sobolev spaces.

Abstract

We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to proving a fractional version of the classical Korn's inequality.