How to recognize constant functions. Connections with Sobolev spaces. (English. Russian original) Zbl 1072.46020

Brézis, Haïm

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)

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