Preface
1 Potential Theory and Weighted Norm Inequalities for Singular Integrals
1.1 Singular Integrals and Weighted Inequalities
2 Degenerate Elliptic Equations, Subelliptic Operators, and Monge-Ampére Equations
2.1 Two Weight Norm Inequalities for Fractional Integrals
2.2 Fefferman-Phong and Hörmander Regularity
2.3 The Monge-Ampére Equation
3 Papers by Richard Wheeden Referred to in the Preface
References
4 Other Papers Referred to in the Preface
References
Contents
On Some Pointwise Inequalities Involving Nonlocal Operators
1 Introduction
2 Some New Inequalities
2.1 A Pointwise Inequality for Nonlocal Operators in Non-divergence Form
2.2 The Case of Translation Invariant Kernels
2.3 Some Integral Operators on Geometric Spaces
2.3.1 The Case of Lie Groups
2.3.2 The Case of Manifolds
3 A Review of the Extension Property
3.1 The Extension Property in Rn
3.2 The Extension Property in Bounded Domains
3.3 The Extension Property in General Frameworks
4 Proofs of Theorems 1.1 and 1.2
4.1 Proof of Theorem 1.1
4.2 Proof of Theorem 1.2
4.3 The Results in Bounded Domains
5 Geometric Ambient Spaces
5.1 The Case of Manifolds
5.2 The Case of Lie Groups
5.3 The Case of the Wiener Space
Appendix
References
The Incompressible Navier Stokes Flow in Two Dimensions with Prescribed Vorticity
References
Weighted Inequalities of Poincaré Type on Chain Domains
1 Introduction
2 Preliminaries
3 Proof of Main Theorems
References
Smoluchowski Equation with Variable Coefficients in Perforated Domains: Homogenization and Applications to Mathematical Models in Medicine
1 Introduction
2 The Problem at ε-Scale: Existence and Regularity
3 Homogenization
4 A Mathematical Model in Medicine
References
Form-Invariance of Maxwell Equations in Integral Form
1 Introduction
2 Maxwell Equations
2.1 Maxwell Equations in Integral Form
3 Changes of Coordinates
4 Invariance Properties of (5)–(8)
4.1 Invariance of (5) and (6) by Changes of Coordinates
4.2 Invariance of (7) and (8) by Changes of Coordinates
4.3 Conclusion
References
Chern-Moser-Weyl Tensor and Embeddings into Hyperquadrics
1 Introduction
2 Chern-Moser-Weyl Tensor for a Levi Non-degenerate Hypersurface
3 Transformation Law for the Chern-Moser-Weyl Tensor
4 A Monotonicity Theorem for the Chern-Moser-Weyl Tensor
5 Counter-Examples to the Embeddability Problem for Compact Algebraic Levi Non-degenerate Hypersurfaces with Positive Signature into Hyperquadrics
6 Non-embeddability of Compact Strongly Pseudo-Convex Real Algebraic Hypersurfaces into Spheres
References
The Focusing Energy-Critical Wave Equation
References
Densities with the Mean Value Property for Sub-Laplacians: An Inverse Problem
1 Introduction
2 Sub-Laplacians and Related Triples
3 Results on the Inverse Problem for L
4 Proofs of the Results on the Inverse Problem
Appendix: L-Superharmonic Functions
References
A Good-λ Lemma, Two Weight T1 Theorems Without Weak Boundedness, and a Two Weight Accretive Global Tb Theorem
1 Introduction
1.1 Quasicubes
1.2 Standard Fractional Singular Integrals and the Norm Inequality
1.2.1 Defining the Norm Inequality
1.3 Quasicube Testing Conditions
1.4 Quasiweak Boundedness and Indicator/Touching Property
1.5 Poisson Integrals and A2α
1.5.1 Punctured A2α Conditions
1.6 A Weighted QuasiHaar Basis
1.7 The Strong Quasienergy Conditions
2 The Good-λ Lemma
2.1 Corollaries
2.1.1 Boundedness of the Cauchy Integral on C1,δ Curves
3 Proof of the Good-λ Lemma
3.1 Good/Bad Technology
3.1.1 Q-Good Quasicubes and Q-Good Quasigrids
3.2 Control of the Indicator/Touching Property
3.2.1 Three Critical Reductions
3.2.2 Elimination of Small Pairs
3.2.3 NTV Surgery
3.2.4 Higher Dimensions
Appendix
References
Intrinsic Difference Quotients
1 Introduction
2 Notations and Definitions
2.1 Carnot Groups
2.2 Complementary Subgroups and Graphs
3 Intrinsic Lipschitz Functions
3.1 General Definitions
3.2 Intrinsic Difference Quotients
3.3 Examples of Difference Quotients and of Intrinsic Derivatives
3.4 Horizontal Difference Quotients and Lipschitz Functions
References
Multilinear Weighted Norm Inequalities Under Integral Type Regularity Conditions
1 Introduction
2 Preliminaries and Statement of the Main Result
3 The Classes A(P,q,r)
4 Proof of Theorem 2.3
5 Multipliers
References
Weighted Norm Inequalities of (1,q)-Type for Integral and Fractional Maximal Operators
1 Introduction
2 Integral Operators
2.1 Strong-Type (1,q)-Inequality for Integral Operators
2.2 The One-Dimensional Case
2.3 Weak-Type (1,q)-Inequality for Integral Operators
3 Fractional Maximal Operators
3.1 Strong-Type Inequality
3.2 Weak-Type Inequality
4 Carleson Measures for Poisson Integrals
References
New Bellman Functions and Subordination by Orthogonal Martingales in Lp, 1<p≤2
1 Introduction: Orthogonal Martingales and the Beurling-Ahlfors Transform
2 New Questions and Results
3 Orthogonality
3.1 Local Orthogonality
4 Subordination by Orthogonal Martingales in L3/2
5 Bellman Functions and Martingales, the Proof of (4)
6 Special Function B=29 ( y112 + y122) + 3 (y212 + y222)1/2)3/2 +29 (( y112 + y122))3/2
7 Hessian of a Vector-Valued Function
7.1 Positive Definite Quadratic Forms
7.2 Example
7.3 Verifying (27)
8 Explanation of How We Found This Special Function B: Pogorelov's Theorem
9 Explanation: Pogorelov's Theorem Again
10 The Case When p=3 and q=32
11 The Proof of Theorem 7 for General q(1,2]
References
Bounded Variation, Convexity, and Almost-Orthogonality
1 Introduction
2 The Proof of Theorem 1
3 The Proofs of Theorems 2 and 3
Appendix
References