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.
- Enables researchers in harmonic analysis to use Bellman function techniques in their own work
- Introduces theories step-by-step using classical examples, including the work of Burkholder, as well as recent solutions of several outstanding problems
- Demonstrates interdisciplinary interactions between stochastic control, non-linear PDE, and harmonic analysis
- Offers a geometric perspective, translating linear infinite dimensional problems in functional analysis to finite dimensional non-linear problems