Vasyunin V., Volberg A. The Bellman Function Technique in Harmonic Analysis

Cambridge: Cambridge University Press, 2020. — 465 p.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.The Short History of the Bellman Function
The Plan of the Book
Notation
Examples of Exact Bellman Functions
A Toy Problem
Buckley Inequality
John–Nirenberg Inequality
Homogeneous Monge–Amp` ere Equation
Bellman Function for General Integral Functionals on BMO
Dyadic Maximal Operator
Weak Estimate of the Martingale Transform
Burkholder’s Bellman Function
On the Bellman Function Generated by a Weak Type Estimate of a Square Function
More about Buckley’s Inequality
Hints and Answers
What You Always Wanted to Know about Stochastic Optimal Control, but Were Afraid to Ask
Disclaimer
Stochastic Integrals Are Not That Simple
Itˆ o’s Definition of Stochastic Integral
Stochastic Differential and Itˆ o’s Formula
Bellman Functions of Stochastic Optimal Control Problems and Bellman PDEs
Almost Perfect Analogy between Stochastic Optimal Control and Harmonic Analysis: Functions of One Variable
Almost Perfect Analogy between Stochastic Optimal Control and Harmonic Analysis: Functions on the Complex Plane
A Problem of Gohberg–Krupnik from the Point of View of Stochastic Optimal Control
Conformal Martingale Models: Stochastic and Classical Ahlfors–Beurling Operators
Estimates of Subordinated Martingales
Conformal Martingales and the Ahlfors–Beurling Transform
Proof of Theorem : Right-Hand Side Conformality, < p < ∞
Proof of Theorem : Left-Hand Side Conformality, < p <
Burkholder, Bellman, and Ahlfors–Beurling Operator in Lp for Large p
Dyadic Models: Application of Bellman Technique to Upper Estimates of Singular Integrals
Dyadic Shifts and Calder´ on–Zygmund Operators
Sharp Weighted Estimate of Dyadic Shifts
Universal Sufficient Condition: Boundedness of All Calder´ on– Zygmund Operators in Two Different Weighted Spaces
Application of Bellman Technique to the Endpoint Estimates of Singular Integrals
Endpoint Estimate
The Bellman Function of Weak Weighted Estimate of the Martingale Transform and Its Logarithmic Blow-Up
Sharp Weak Weighted Estimate for the Martingale Transform
Obstacle Problems for Unweighted Square Function Operator: Burkholder–Gundy–Davis Function
Bollob´ as Function
The Weak Norm of the Square Function
Saturation of Estimates by Extremal Sequences
An Obstacle Problem Associated with the Chang–Wilson– Wolff Theorem
Strong Weighted Estimate of the Square Function
Weak Weighted Estimates for the Square Function
Restricted Weak Weighted Estimate of the Square Function