Quasiconformal Mappings and Sobolev Spaces

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 the Hausdorff h-Measure.- 5.4. Sufficient Conditions for the Vanishing of (l, p)-Capacity.- §6. Estimates of [l, p]-Capacity for Some Pairs of Sets.- 6.1. Estimates of Capacity for Spherical Domains.- 6.2. Estimates of Capacity for Pairs of Continuums Connecting Concentric Spheres.- §7. Capacity in Besov-Nickolsky Spaces.- 7.1. Preliminary Information.- 7.2. Capacities in bl,p,?,G,hl. Simplest Properties.- 7.3. Comparison of Capacity of a Pair of Points to Capacity of a Point Relative to a Complement of a Ball.- 7.4. Capacity of the Spherical Layer.- 4. Density of Extremal Functions in Sobolev Spaces with First Generalized Derivatives.- §1. Extremal Functions for (l, p)-Capacity.- 1.1. Simplest Properties of Extremal Functions.- 1.2. The Dirichlet Problem and Extremal Functions.- 1.3. Extremal Functions for Pairs of Smooth Compacts.- §2. Theorem on the Approximation of Functions from Lpl by Extremal Functions.- 2.1. Auxiliary Statements.- 2.2. The Class Extp(G).- 2.3. Proof of the Theorem on Approximation.- 2.4. Representation in Form of a Series.- §3. Removable Singularities for the Spaces Lpl (G).- 3.1. Two Ways of Describing Removable Singularities.- 3.2. Properties of NCp-Sets. Localization Principle.- 5. Change of Variables.- §1. Multiplicity of Mapping, Degree of Mapping, and Their Analogies.- 1.1. The Multiplicity Function of Mapping.- 1.2. The Approximate Differential.- 1.3. The K-Differential.- 1.4. The Change of Variable Theorem for the Multiplicity Function.- 1.5. The Degree of Mapping.- §2. The Change of Variable in the Integral for Mappings of Sobolev Spaces.- 2.1. The Change of Variable Theorem for Continuous Mappings of the Class Lnl.- 2.2. The Linking Index.- 2.3. The Change of Variable Theorem for Discontinuous Mappings of the Class Lnl.- §3. Sufficient Conditions of Monotonicity and Continuity for the Approximation Functions of the Class Lnl.- §4. Invariance of the Spaces Lpl(G)(Lnl(G)) for Quasiisometric (Quasiconformal) Homeomorphisms.- 4.1. Preliminary Information on the Mappings.- 4.2. Differentiation of Composition.- 4.3. Representation of Operators Preserving the Order.- 6. Extension of Differentiate Functions.- §1. Arc Diameter Condition.- 1.1. Analysis of the Ahlfors Condition.- 1.2. The Arc Diameter Condition.- 1.3. Properties of Domains Satisfying the Arc Diameter Condition.- §2. Necessary Extension Conditions for Seminormed Spaces.- 2.1. The Extension Operator. Capacitary Extension Condition.- 2.2. Additional Properties of Capacity.- 2.3. The Invisibility Condition.- 2.4. The Extension Theorem.- 2.5. Verification of the Conditions of the Theorem for the Spaces Lpl (G), Wpl(G).- §3. Necessary Extension Conditions for Sobolev Spaces.- 3.1. Necessary Extension Conditions for Lpl, Wpl at lp=n.- 3.2. Necessary Conditions for Lpl, Wpl at 1 ? lp ? 2 in Plane Domains.- 3.3 Necessary Conditions Different from the Arc Diameter Condition.- 3.4. Refinement for the Space Wpl.- §4. Necessary Extension Conditions for Besov and Nickolsky Spaces.- 4.1. Extension Theorem for lp > n.- 4.2. Extension Conditions for lp = n.- §5. Sufficient Extension Conditions.- 5.1. Quasiconformal Extension.- 5.2. Extension Conditions for Sobolev Classes.- 5.3. Example of Estimating the Norm of an Extension Operator.- 5.4. The Extension Condition for Nickolsky-Besov Spaces.- Comments.- References.