Coarea properties of Sobolev functions. (English) Zbl 1036.46025

Haroske, Dorothee (ed.) et al., Function spaces, differential operators and nonlinear analysis. The Hans Triebel anniversary volume. Based on the lectures given at the international conference on function spaces, differential operators and nonlinear analysis, FSDONA-01, Teistungen, Germany, June 28–July 4, 2001, in honor of the 65th birthday of H. J. Triebel. Basel: Birkhäuser (ISBN 3-7643-6935-3/hbk). 371-381 (2003).
Summary: The Lusin \(N\)-property is well-known as a criterion for validity of theorems on change of variables in integrals. Here we consider related properties motivated by the coarea formula. They also imply a generalization of Eilenberg’s inequality. We prove them for functions with gradient in the Lorentz space \(L_{m,1}\). This relies on estimates of Hausdorff content of level sets for Sobolev functions and analysis of their Lebesgue points. A significant part of the presented results has its origin in joint work with David Swanson and William P. Ziemer.
For the entire collection see [Zbl 1011.00045].


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems