Quasiconformal Mappings and Sobolev Spaces [Paperback]

1. Preliminary Information about Integration Theory.- ?1. Notation and Terminology.- 1.1. Sets in Rn.- 1.2. Classes of Functions in Rn.- ?2. Some Auxiliary Information about Sets and Functions in Rn.- 2.1. Averaging of Functions.- 2.2. The Whitney Partition Theorem.- 2.3. Partition of Unitiy.- ?3. General Information about Measures and Integrals.- 3.1. Notion of a Measure.- 3.2. Decompositions in the Sense of Hahn and Jordan.- 3.3. The Radon-Nikodym Theorem and the Lebesgue Decomposition of Measure.- ?4. Differentiation Theorems for Measures in Rn.- 4.1. Definitions.- 4.2. The Vitali Covering Lemma.- 4.3. The Lp-Continuity Theorem for Functions of the Class Lp,loc.- 4.4. The Differentiability Theorem for the Measure in Rn.- ?5. Generalized Functions.- 5.1. Definition and Examples of Generalized Functions.- 5.2. Operations with Generalized Functions.- 5.3. Support of a Generalized Function. The Order of Singularity of a Generalized Function.- 5.4. The Generalized Function as a Derivative of the Usual Function. Averaging Operation.- 2. Functions with Generalized Derivatives.- ?1. Sobolev-Type Integral Representations.- 1.1. Preliminary Remarks.- 1.2. Integral Representations in a Curvilinear Cone.- 1.3. Domains of the Class J.- 1.4. Integral Representations of Smooth Functions in Domains of the Class J.- ?2. Other Integral Representations.- 2.1. Sobolev-Type Integral Representations for Simple Domains.- 2.2. Differential Operators with the Complete Integrability Condition.- 2.3. Integral Representations of a Function in Terms of a System of Differential Operators with the Complete Integrability Condition.- 2.4. Integral Representations for the Deformation Tensor and for the Tensor of Conformal Deformation.- ?3. Estimates for Potential-Type Integrals.- 3.1. Preliminary Information.- 3.2. Lemma on the Compactness of Integral Operators.- 3.3. Basic Inequalities.- ?4. Classes of Functions with Generalized Derivatives.- 4.1. Definition and the Simplest Properties.- 4.2. Inl3³