Some Poincar\'{e}--Sobolev inequalities for differential forms

TL;DR

This paper investigates Poincaré--Sobolev inequalities for differential forms, providing estimates for embedding operators and exploring their compactness in Euclidean balls and bi-Lipschitz images.

Contribution

It extends previous work by establishing new norm estimates for embeddings of Sobolev spaces of differential forms, especially for specific parameter ranges.

Findings

Derived estimates for embedding operator norms when p=q.

Analyzed compactness of embedding operators in Euclidean balls.

Extended inequalities to bi-Lipschitz images of Euclidean domains.

Abstract

We continue the~study of embeddings between different classes of Sobolev spaces of differential forms started in 2006 in a~paper by Gol$'$dshtein and Troyanov. As in this paper, our study is based on relations between $L_{q,p}$-cohomology and Sobolev type inequalities. The~main results are estimates for the norms of the embedding operators for $q=p$ and $p>\frac{n-1}{k-1}$ in the~Euclidean $r$-ball $B(r)$ and its bi-Lipschitz images. We also study the~compactness of such operators.