## 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.

### 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
Full Text: