Fine properties of functions: an introduction
Résumé
These lecture notes, intended as support to an intensive course at Scoala Normala Superioara din Bucuresti, cover classical properties of function spaces such as: a.e. differentiability of Lipschitz functions (Rademacher's theorem), maximal functions (the Hardy-Littlewood-Wiener theorem), functions of bounded variation, area and coarea formulae, Hausdorff measure and capacity, isoperimetric inequalities, Hardy and bounded mean oscillations spaces (and their duality), trace theory, precise representatives. Several textbooks cover part of these topics: Herbert Federer, Geometric measure theory, Springer, 1969 Vladimir Maz'ja, Sobolev spaces, Springer, 1980 Leon Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathemat- ical Analysis, Australian National University, 1983 William P. Ziemer, Weakly differentiable functions, Springer, 1989 Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, 1992 Elias M. Stein, Harmonic Analysis: real variable methods, orthogonality, and oscillatory inte- grals, Princeton University press, 1993 However, there is no single source covering the basic facts the working analyst needs. This is the main purpose of the notes. These notes are also an invitation to reading the above wonderful books. The background required is a good knowledge of standard measure theory: Radon-Nikodym and Hahn decomposition theorems, Riesz representation theorem. Also, the standard theory of distributions and basics about Sobolev spaces are presumed known. All other standard tools (Vitali and Whitney covering theorems, for example) are proved in the text.