Addendum to "BMO: Oscillations, Self Improvement, Gagliardo Coordinate Spaces and Reverse Hardy Inequalities"

TL;DR

This paper offers a detailed proof of a Sobolev inequality, including higher order and fractional cases, clarifying and extending the results from the original work.

Contribution

It provides a precise, self-contained proof of a Sobolev inequality and extends the results to higher order and fractional inequalities.

Findings

Confirmed the validity of the Sobolev inequality with rigorous proof

Extended the inequality to higher order derivatives

Included fractional Sobolev inequalities

Abstract

We provide a precise statement and self contained proof of a Sobolev inequality (cf. [A, page 236 and page 237]) stated in the original paper. Higher order and fractional inequalities are treated as well.