ON GEOMETRIC PROPERTIES OF FUNCTIONS WITH GENERALIZED FIRST DERIVATIVES

Abstract

CONTENTS Introduction Chapter 1. Integral representations and differential properties of functions § 1. Integral representations of Sobolev type (general scheme) § 2. Application of the general scheme. Three types of integral representation § 3. Differentiability of functions in Chapter 2. Change of variable in an integral § 1. The formula for a change of variable in an integral for mappings of class § 2. Sufficient conditions for monotonicity and continuity for functions of class § 3. Oriented volumes. An isoperimetric inequality Chapter 3. Removable singularities for -spaces § 1. Capacity of condensers. Extremal functions for capacity § 2. “Almost orthogonal” expansion of functions of -classes § 3. -equivalent domains and sets of removable capacity § 4. Removability of -sets for certain classes of mappings Chapter 4. Mappings that leave -classes invariant § 1. The -spaces as partially ordered spaces. Order-preserving operators § 2. Invariance of spaces under quasi-conformal (quasi-isometric) mappings § 3. A theorem on lattice isomorphism Chapter 5. Extension theorems for -classes § 1 . The capacity metric § 2. Sufficient conditions for extension of -classes § 3. Extension criteria for the space in plane domains References