The Bellman Function Technique in Harmonic Analysis

Description

The Bellman function, a powerful tool originating in control theory, can be used successfully in a large class of difficult harmonic analysis problems and has produced some notable results over the last thirty years. This book by two leading experts is the first devoted to the Bellman function method and its applications to various topics in probability and harmonic analysis. Beginning with basic concepts, the theory is introduced step-by-step starting with many examples of gradually increasing sophistication, culminating with Calderón–Zygmund operators and end-point estimates. All necessary techniques are explained in generality, making this book accessible to readers without specialized training in non-linear PDEs or stochastic optimal control. Graduate students and researchers in harmonic analysis, PDEs, functional analysis, and probability will find this to be an incisive reference, and can use it as the basis of a graduate course. Contents Introduction Preface Acknowledgments The Short History of the Bellman Function The Plan of the Book Notation 1 Examples of Exact Bellman Functions 1.1 A Toy Problem 1.2 Buckley Inequality 1.3 John–Nirenberg Inequality 1.4 Homogeneous Monge–Amp` ere Equation 1.5 Bellman Function for General Integral Functionals on BMO 1.6 Dyadic Maximal Operator 1.7 Weak Estimate of the Martingale Transform 1.8 Burkholder’s Bellman Function 1.9 On the Bellman Function Generated by a Weak Type Estimate of a Square Function 1.10 More about Buckley’s Inequality 1.11 Hints and Answers 2 What You Always Wanted to Know about Stochastic Optimal Control, but Were Afraid to Ask 2.1 Disclaimer 2.2 Stochastic Integrals Are Not That Simple 2.3 Itˆ o’s Definition of Stochastic Integral 2.4 Stochastic Differential and Itˆ o’s Formula 2.5 Bellman Functions of Stochastic Optimal Control Problems and Bellman PDEs 2.6 Almost Perfect Analogy between Stochastic Optimal Control and Harmonic Analysis: Functions of One Variable 2.7 Almost Perfect Analogy between Stochastic Optimal Control and Harmonic Analysis: Functions on the Complex Plane 2.8 A Problem of Gohberg–Krupnik from the Point of View of Stochastic Optimal Control 3 Conformal Martingale Models: Stochastic and Classical Ahlfors–Beurling Operators 3.1 Estimates of Subordinated Martingales 3.2 Conformal Martingales and the Ahlfors–Beurling Transform 3.3 Proof of Theorem 3.2.3: Right-Hand Side Conformality, 2 < p < ∞ 3.4 Proof of Theorem 3.2.3: Left-Hand Side Conformality, 1 < p < 2 3.5 Burkholder, Bellman, and Ahlfors–Beurling Operator in Lp for Large p 4 Dyadic Models: Application of Bellman Technique to Upper Estimates of Singular Integrals 4.1 Dyadic Shifts and Calder´ on–Zygmund Operators 4.2 Sharp Weighted Estimate of Dyadic Shifts 4.3 Universal Sufficient Condition: Boundedness of All Calder´ on– Zygmund Operators in Two Different Weighted Spaces 5 Application of Bellman Technique to the Endpoint Estimates of Singular Integrals 5.1 Endpoint Estimate 5.2 The Bellman Function of Weak Weighted Estimate of the Martingale Transform and Its Logarithmic Blow-Up 5.3 Sharp Weak Weighted Estimate for the Martingale Transform 5.4 Obstacle Problems for Unweighted Square Function Operator: Burkholder–Gundy–Davis Function 5.5 Bollob´ as Function 5.6 The Weak Norm of the Square Function 5.7 Saturation of Estimates by Extremal Sequences 5.8 An Obstacle Problem Associated with the Chang–Wilson– Wolff Theorem 5.9 Strong Weighted Estimate of the Square Function 5.10 Weak Weighted Estimates for the Square Function 5.11 Restricted Weak Weighted Estimate of the Square Function References Index