Quasiconformal Mappings and Sobolev Spaces (Hardcover, 1990)

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. Integral Representations for Elements of the Space W?1,locl.- 4.3. The Imbedding Theorem.- 4.4. Corollaries of Theorem 4.2. Normalization of the Spaces Wpl(U).- 4.5. Approximation of Functions from Wpl by Smooth Functions.- 4.6. Change of Variables for Functions with Generalized Derivatives.- 4.7. Compactness of the Imbedding Operators.- 4.8. Estimates with a Small Coefficient for the Norm in Lpl.- 4.9. Functions of One Variable.- 4.10. Differential Description of Convex Functions.- 4.11. Functions Satisfying the Lipschitz Condition.- 5. Theorem on the Differentiability Almost Everywhere.- 5.1. Definitions.- 5.2. Auxiliary Propositions.- 5.3 The Main Result.- 5.4. Corollaries of the General Theorem on the Differentiability Almost Everywhere.- 5.5. The Behaviour of Functions of the Class Wpl on Almost All Planes of Smaller Dimensionality.- 5.6. The ACL-Classes.- 3. Nonlinear Capacity.- 1. Capacity Induced by a Linear Positive Operator.- 1.1. Definition and the Simplest Properties.- 1.2. Capacity as the Outer Measure.- 1.3. Sets of Zero Capacity.- 1.4. Extension of the Set of Admissible Functions.- 1.5. Extremal Function for Capacity.- 1.6. Comparison of Various Capacities.- 2. The Classes W(T, p, V).- 2.1. Definition of Classes.- 2.2. Theorems of Egorov and Luzin for Capacity.- 2.3. Dual (T, p)-Capacity, p 1. Definition and Basic Properties.- 2.4. Calculation of Dual (T, p)-Capacity.- 3. Sets Measurable with Respect to Capacity.- 3.1. Definition and the Simplest Properties of Generalized Capacity.- 3.2. (T, p)-Capacity as Generalized Capacity.- 4. Variational Capacity.- 4.1. Definition of Variational Capacity.- 4.2. Comparison of Variational Capacity and (T, p)-Capacity.- 4.3. Sets of Zero Variational Capacity.- 4.4. Examples of Variational Capacity.- 4.5. Refined Functions.- 4.6. Theorems of Imbedding into the Space of Continuous Functions.- 5. Capacity in Sobolev Spaces.- 5.1. Three Types of Capacity.- 5.2. Extremal Functions for Capacity.- 5.3. Capacity and th