Extremal Length and Precise Functions
Sobolev spaces of first order are spaces of functions that are integrable together with their distributional p-th derivatives. An element of a Sobolev space is itself an equivalence class of functions, which mutually agree almost everywhere. A single function in this equivalence class is called a representative. It is often desirable to distinguish between ‘good’ and ‘bad’ representatives as many statements admit finer formulations for the good representatives. Following Beppo Levi, classes of functions that are absolutely continuous along almost all lines parallel to coordinate axes (ACL functions) are often considered. A refinement of this idea leads to the definition of AC functions. These are intended to be functions that are absolutely continuous on almost all curves. Therefore we need to know which families of curves can be regarded to be exceptional. Let Γ be a family of curves. A function ρ is called admissible if the integral of ρ over any curve in Γ is at least one. Then the p-modulus of Γ is defined as the infimum of integrals of the p-th power of ρ where ρ runs over all Γ-admissible functions. The reciprocal of the modulus is called the extremal length. We can neglect a family of curves that has vanishing p-modulus and define p-precise functions as functions absolutely continuous along p-almost every curve (in the sense of modulus) with p-power integrable gradient.
The presentation begins with an introduction of the notion of extremal length and related topics. An interesting section is the treatment of tubes and the computation of their extremal length. Next, the theory of various types of absolutely continuous and precise functions is developed with their comparison and analysis of exceptional sets. Inequalities for Hardy-Littlewood maximal functions, Calderón-Zygmund operators and Riesz potentials, Sobolev and Poincaré inequalities, extension theorems and trace theorems are presented. Finally, it is shown how extremal distance leads to an approach to capacity and both qualitative and quantitative properties of capacities are studied. In particular, various symmetrization results for capacities are given. Most of the material is presented in the setting of weighted spaces. The book is a valuable source for a considerable amount of useful material on Sobolev functions and their representatives and on important inequalities in analysis. It is worth noticing that the extremal length approach to representatives is not often fully recognized and thus this book fills a gap in expository literature.