Brézis, Haïm How to recognize constant functions. Connections with Sobolev spaces. (English. Russian original) Zbl 1072.46020 Russ. Math. Surv. 57, No. 4, 693-708 (2002); translation from Usp. Mat. Nauk 57, No. 4, 59-74 (2002). A new characterization of Sobolev spaces is given by means of integral conditions. The question of when a measurable function is a constant under various integral conditions is discussed and a new criterion is derived for a function \(f \in L^p\) to belong to \(W^{1,p}\) or to \(BV\). Interesting connections are derived for the space of functions with vanishing mean oscillation and open problems are outlined. This is a nicely written paper which includes a wealth of interesting ideas. Reviewer: Thomas Sonar (Braunschweig) Cited in 5 ReviewsCited in 51 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26A45 Functions of bounded variation, generalizations Keywords:Sobolev spaces; functions of bounded variation PDF BibTeX XML Cite \textit{H. Brézis}, Russ. Math. Surv. 57, No. 4, 693--708 (2002; Zbl 1072.46020); translation from Usp. Mat. Nauk 57, No. 4, 59--74 (2002) Full Text: DOI