Bellman functions and their method in harmonic analysis

This work uses the method of the Bellman function to show a new proof of Hardy's inequality for Carleson measures. Bellman functions come from the theory of stochastic optimal control and there is a method to prove theorems about inequalities over dyadic trees (which have applications in harmonic analysis) that takes inspiration from concepts from the theory of the Bellman functions. The work will display the important concepts of the theory of Bellman functions in stochastic analysis, will show how to use the method of the Bellman function to prove the estimate over dyadic trees for Carleson measures for Hardy spaces (while also showing the connections between the stochastic theory and the harmonic analysis) and will give a new proof of Hardy's inequality for dyadic trees (which is related to the characterization of Carleson measures in Besov spaces) using the Bellman function method.