Fractional Hardy inequalities and visibility of the boundary

TL;DR

This paper establishes fractional Hardy inequalities on open sets using combined boundary conditions, highlighting the importance of visibility alongside fatness, and explores their implications for fractional Sobolev space extension operators.

Contribution

It introduces a new boundary condition combining fatness and visibility for fractional Hardy inequalities, supported by counterexamples and applications to Sobolev space extensions.

Findings

Fatness alone is insufficient for Hardy inequalities.

Visibility conditions are crucial for the inequalities.

Applications to boundedness of extension operators.

Abstract

We prove fractional order Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary. We demonstrate by counterexamples that fatness conditions alone are not sufficient for such Hardy inequalities to hold. In addition, we give a short exposition of various fatness conditions related to our main result, and apply fractional Hardy inequalities in connection to the boundedness of extension operators for fractional Sobolev spaces.