Some Poincaré--Sobolev inequalities for differential forms

TL;DR

The paper develops Poincaré–Sobolev type inequalities for differential forms by linking L_{q,p}-cohomology to Sobolev-type inequalities. It extends the Iwaniec–Lutoborski homotopy approach to balls and their bi-Lipschitz images, providing explicit norm bounds for the homotopy operators and establishing compactness in the relevant range. A key result is a concrete estimate for the embedding norm when p=q and for p > (n-1)/(k-1), plus a general Lipschitz-transfer construction; an explicit simplex example demonstrates the method on non-ball domains. These results have implications for PDEs and geometric analysis, including applications to Laplace-Beltrami, Maxwell, and p-Laplacian equations, as well as Stokes-type theorems on manifolds.

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.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Theorem 1.1
• proof
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Remark 1
• proof