arXiv subjects

Roman Panenko

Publications and source records attributed to Roman Panenko.

3 recordsLinked to original sources

arXiv2025

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

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.

math.DG↗

arXiv2022

A Lipschitz version of de Rham theorem for $L_p$-cohomology

We focus our attention on the de Rham operators' underlying properties which are specified by intrinsic effects of differential geometry structures. And then we apply the procedure of regularization in the context of Lipschitz version of de Rham calculus on metric simplicial complexes with bounded geometry.

math.DG↗

arXiv2013

$\Phi$-Harmonic Functions on Discrete Groups and First $\ell^\Phi$-Cohomology

We study the first cohomology groups of a countable discrete group $G$ with coefficients in a $G$-module $\ell^\Phi(G)$, where $\Phi$ is an $N$-function of class $\Delta_2(0)\cap \nabla_2(0)$. In development of ideas of Puls and Martin--Valette, for a finitely generated group $G$, we introduce the discrete $\Phi$-Laplacian and prove a theorem on the decomposition of the space of $\Phi$-Dirichlet finite functions into the direct sum of the spaces of $\Phi$-harmonic functions and $\ell^\Phi(G)$ (with an appropriate factorization). We also prove that if a finitely generated group $G$ has a finitely generated infinite amenable subgroup with infinite centralizer then $\overline{H}^{1}(G,\ell^{\Phi}(G)) = 0$. In conclusion, we show the triviality of the first cohomology group for a wreath product of two groups one of which is nonamenable.

math.GR↗