Embedding theorems for generalized Sobolev-Nikol'skij-Wiener spaces and applications to differential equations. (Q1432337)

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.