Chapter
4.1 The Fourier Transform
4.1.1 The Fourier Transform in L[sup(1)] and L[sup(2)]
4.1.2 Fourier Transform on Measures
4.1.4 The Hecke Identities
4.2 The Complex Riesz Transforms R[sup(k)]
4.2.1 Potentials Associated with R[sup(k)]
4.3 Quantitative Analysis of Complex Potentials
4.3.1 The Logarithmic Potential
4.3.2 The Cauchy Transform
4.4 Maximal Functions and Interpolation
4.5 Weak-Type Estimates and L[sup(p)]-Bounds
4.5.1 Weak-Type Estimates for Complex Riesz Transforms
4.5.2 Estimates for the Beurling Transform S
4.5.3 Weighted L[sup(p)]-Theory for S
4.6 BMO and the Beurling Transform
4.6.1 Global John-Nirenberg Inequalities
4.6.3 Orthogonality Properties of S
4.6.4 Proof of the Pointwise Estimates
4.6.6 The Beurling Transform of Characteristic Functions
4.7.1 Hölder Bounds for the Beurling Transform
4.7.2 The Inhomogeneous Cauchy-Riemann Equation
4.8 Beurling Transforms for Boundary Value Problems
4.8.1 The Beurling Transform on Domains
4.8.3 Complex Potentials for the Dirichlet Problem
4.9 Complex Potentials in Multiply Connected Domains
5 The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mappings
5.1 The Basic Beltrami Equation
5.2 Quasiconformal Mappings with Smooth Beltrami Coefficient
5.3 The Measurable Riemann Mapping Theorem
5.4 L[sup(p)]-Estimates and the Critical Interval
5.4.1 The Caccioppoli Inequalities
5.4.2 Weakly Quasiregular Mappings
5.5 Stoilow Factorization
5.6 Factoring with Small Distortion
5.7 Analytic Dependence on Parameters
5.8 Extension of Quasisymmetric Mappings of the Real Line
5.8.1 The Douady-Earle Extension
5.8.2 The Beurling-Ahlfors Extension
6 Parameterizing General Linear Elliptic Systems
6.1 Stoilow Factorization for General Elliptic Systems
6.2 Linear Families of Quasiconformal Mappings
6.3 The Reduced Beltrami Equation
6.4 Homeomorphic Solutions to Reduced Equations
6.4.1 Fabes-Stroock Theorem
7 The Concept of Ellipticity
7.1 The Algebraic Concept of Ellipticity
7.2 Some Examples of First-Order Equations
7.3 General Elliptic First-Order Operators in Two Variables
7.3.2 Homotopy Classification
7.3.3 Classification; n = 1
7.4 Partial Differential Operators with Measurable Coefficients
7.5 Quasilinear Operators
7.7 Fully Nonlinear Equations
7.8 Second-Order Elliptic Systems
7.8.1 Measurable Coefficients
8 Solving General Nonlinear First-Order Elliptic Systems
8.1 Equations Without Principal Solutions
8.2 Existence of Solutions
8.3 Proof of Theorem 8.2.1
8.3.1 Step 1: H Continuous, Supported on an Annulus
8.3.2 Step 2: Good Smoothing of H
8.3.3 Step 3: Lusin-Egoroff Convergence
8.3.4 Step 4: Passing to the Limit
8.4 Equations with Infinitely Many Principal Solutions
8.6.1 Uniqueness for Normalized Solutions
9 Nonlinear Riemann Mapping Theorems
9.1 Ellipticity and Change of Variables
9.2 The Nonlinear Mapping Theorem: Simply Connected Domains
9.3 Mappings onto Multiply Connected Schottky Domains
9.3.2 Proof of the Mapping Theorem 9.3.4
10 Conformal Deformations and Beltrami Systems
10.1 Quasilinearity of the Beltrami System
10.1.1 The Complex Equation
10.2 Conformal Equivalence of Riemannian Structures
10.3 Group Properties of Solutions
10.3.2 Sullivan-Tukia Theorem
10.3.3 Ellipticity Constants
11 A Quasilinear Cauchy Problem
11.1 The Nonlinear [Omitted]-Equation
11.2 A Fixed-Point Theorem
11.3 Existence and Uniqueness
12.2 Two Compelling Examples
12.2.1 Limit Sets of Kleinian Groups
12.2.2 Julia Sets of Rational Maps
12.3 The Extended λ-Lemma
12.3.1 Holomorphic Motions and the Cauchy Problem
12.3.2 Holomorphic Axiom of Choice
12.4 Distortion of Dimension in Holomorphic Motions
12.5 Embedding Quasiconformal Mappings in Holomorphic Flows
12.7 Deformations of Quasiconformal Mappings
13.1.1 Initial Bounds for Distortion of Area
13.1.2 Weighted Area Distortion
13.1.4 General Area Estimates
13.2 Higher Integrability
13.2.1 Integrability at the Borderline
13.2.2 Distortion of Hausdorff Dimension
13.3 The Dimension of Quasicircles
13.3.1 Symmetrization of Beltrami Coefficients
13.3.2 Distortion of Dimension
13.4 Quasiconformal Mappings and BMO
13.4.1 Quasiconformal Jacobians and A[sub(p)]-Weights
13.5 Painlevé's Theorem: Removable Singularities
13.5.1 Distortion of Hausdorff Measure
13.6 Examples of Nonremovable Sets
14 L[sup(p)]-Theory of Beltrami Operators
14.1 Spectral Bounds and Linear Beltrami Operators
14.2 Invertibility of the Beltrami Operators
14.2.1 Proof of Invertibility; Theorem 14.0.4
14.3 Determining the Critical Interval
14.4 Injectivity in the Borderline Cases
14.4.1 Failure of Factorization in W[sup(1,q)]
14.4.2 Injectivity and Liouville-Type Theorems
14.5 Beltrami Operators; Coefficients in VMO
14.6 Bounds for the Beurling Transform
15 Schauder Estimates for Beltrami Operators
15.2 The Beltrami Equation with Constant Coefficients
15.3 A Partition of Unity
15.5 Hölder Regularity for Variable Coefficients
15.6 Hölder-Caccioppoli Estimates
15.7 Quasilinear Equations
16 Applications to Partial Diffierential Equations
16.1.1 Equations of Divergence Type: The A-Harmonic Operator
16.1.2 The Natural Domain of Definition
16.1.3 The A-Harmonic Conjugate Function
16.1.4 Regularity of Solutions
16.1.5 General Linear Divergence Equations
16.2 Topological Properties of Solutions
16.3 The Hodographic Method
16.3.1 The Continuity Equation
16.3.2 The p-Harmonic Operator div|[Omitted]|[sup(p–2)][Omitted]
16.3.3 Second-Order Derivatives
16.3.4 The Complex Gradient
16.3.5 Hodograph Transform for the p-Laplacian
16.3.6 Sharp Hölder Regularity for p-Harmonic Functions
16.3.7 Removing the Rough Regularity in the Gradient
16.4 The Nonlinear A-Harmonic Equation
16.4.1 ʈ-Monotonicity of the Structural Field
16.4.2 The Dirichlet Problem
16.4.3 Quasiregular Gradient Fields and C[sup(1,α)]-Regularity
16.5 Boundary Value Problems
16.5.1 A Nonlinear Riemann-Hilbert Problem
16.6 G-Compactness of Beltrami Diffierential Operators
16.6.1 G-Convergence of the Operators [Omitted] — μ[sub(j)][Omitted][sub(z)]
16.6.2 G-Limits and the Weak*-Topology
16.6.3 The Jump from [Omitted][sub(2)] –V[Omitted][sub(z)] to [Omitted][sub(z)] – μ[Omitted][sub(z)]
16.6.4 The Adjacent Operator's Two Primary Solutions
16.6.5 The Independence of [Omitted][sub(z)](z) and [Omitted][sub(z)](z)
16.6.6 Linear Families of Quasiregular Mappings
16.6.7 G-Compactness for Beltrami Operators
17 PDEs Not of Divergence Type: Pucci's Conjecture
17.1 Reduction to a First-Order System
17.2 Second-Order Caccioppoli Estimates
17.3 The Maximum Principle and Pucci's Conjecture
17.5 Equations with Lower-Order Terms
17.5.1 The Dirichlet Problem
18 Quasiconformal Methods in Impedance Tomography: Calderón's Problem
18.1 Complex Geometric Optics Solutions
18.2 The Hilbert Transform Hσ
18.3 Dependence on Parameters
18.4 Nonlinear Fourier Transform
18.6 Subexponential Growth
18.7 The Solution to Calderón's Problem
19 Integral Estimates for the Jacobian
19.1 The Fundamental Inequality for the Jacobian
19.2 Rank-One Convexity and Quasiconvexity
19.2.1 Burkholder's Theorem
19.3 L[sup(1)]-Integrability of the Jacobian
20 Solving the Beltrami Equation: Degenerate Elliptic Case
20.1 Mappings of Finite Distortion; Continuity
20.1.1 Topological Monotonicity
20.1.2 Proof of Continuity in W[sup(1,2)]
20.2 Integrable Distortion; W[sup(1,2)]-Solutions and Their Properties
20.4 Distortion in the Exponential Class
20.4.1 Example: Regularity in Exponential Distortion
20.4.2 Beltrami Operators for Degenerate Equations
20.4.3 Decay of the Neumann Series
20.4.4 Existence Above the Critical Exponent
20.4.5 Exponential Distortion: Existence of Solutions
20.4.6 Optimal Regularity
20.4.7 Uniqueness of Principal Solutions
20.4.8 Stoilow Factorization
20.4.9 Failure of Factorization in W[sup(1,q)] When q < 2
20.5 Optimal Orlicz Conditions for the Distortion Function
20.8 Applications to Degenerate PDEs
21 Aspects of the Calculus of Variations
21.1 Minimizing Mean Distortion
21.1.1 Formulation of the General Problem
21.1.2 The L[sup(1)]-Grötzsch Problem
21.1.3 Sublinear Growth: Failure of Minimization
21.1.4 Inverses of Homeomorphisms of Integrable Distortion
21.1.5 The Traces of Mappings with Integrable Distortion
21.2 Variational Equations
21.2.1 The Lagrange-Euler Equations
21.2.2 Equations for the Inverse Map
21.3 Mean Distortion, Annuli and the Nitsche Conjecture
21.3.3 Lower Bounds by Free Lagrangians
21.3.4 Weighted Mean Distortion
21.3.5 Minimizers within the Nitsche Range
21.3.6 Beyond the Nitsche Bound
21.3.7 The Minimizing Sequence and Its BV-limit
Appendix: Elements of Sobolev Theory and Function Spaces
A.1 Schwartz Distributions
A.2 Definitions of Sobolev Spaces
A.4 Pointwise Coincidence of Sobolev Functions
A.5 Alternate Characterizations
A.7 Duals and Compact Embeddings
A.9 Reverse Hölder Inequalities
A.10 Variations of Sobolev Mappings
Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)
：Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)
( Princeton Mathematical Series )
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