Embedding theorems for generalized Sobolev-Nikol'skij-Wiener spaces and applications to differential equations.
type: Item
instance of: scholarly article
title: Embedding theorems for generalized Sobolev-Nikol'skij-Wiener spaces and applications to differential equations.
zbMATH Open document ID: 1081.46506
author: S. V. Uspenskij
author: E. N. Vasil'eva
author: S. I. Yanov
published in: Doklady Mathematics
publication date: 2004-06-15T00:00:00Z
review text: Functions defined on the entire space \({\mathbb R}^n\) and belonging to spaces of the Sobolev--Nikol'skii and Besov--Triebel--Lizorkin type have been the subject of intensive investigation. One of their properties is that, under some conditions, functions belonging to these classes tend to polynomials as \(| x| \to \infty\). When the domain of definition of these functions is not the whole space, then their behaviour at infinity is more complex. In the paper under review, cylindrical domains \(G=\{(x,t)\, :\, x=(x_1,\dots,x_n)\in g\), \(0\leq t<\infty \}\) are considered, where \(g\) is a smooth bounded domain in \({\mathbb R}^n\). On~\(G\), a special class of functions is defined which generalizes the Paley--Wiener and Sobolev--Nikol'skii function spaces. For functions from these classes, the authors find oscillation conditions with respect to~\(t\) and a condition of stabilizations to zero as \(t\to\infty\). The results are then applied to the analysis of the solutions to a Sobolev problem at infinity.
reviewed by: Walter Farkas
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35G15
zbMATH DE Number: 2074641
zbMATH Keywords: embedding theorems
zbMATH Keywords: generalized Sobolev spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2074641