Monge--Amp\`ere equation and Bellman optimization of Carleson Embedding Theorems

TL;DR

This paper explores solving the Monge--Ampère equation using a method of characteristics to find Bellman functions, linking optimal control and harmonic analysis to determine sharp constants and extremal sequences.

Contribution

It introduces a novel approach to solving Monge--Ampère equations via characteristics for Bellman functions in harmonic analysis.

Findings

Derived explicit Bellman functions for classical problems

Identified extremal sequences and sharp constants

Connected Monge--Ampère equations with stochastic control methods

Abstract

Monge--Amp\`ere equation plays an important part in Analysis. For example, it is instrumental in mass transport problems. On the other hand, the Bellman function technique appeared recently as a way to consider certain Harmonic Analysis problems as the problems of Stochastic Optimal Control. This brings us to Bellman PDE, which in stochastic setting is often a Monge--Amp\`ere equation or its close relative. We explore the way of solving Monge--Amp\`ere equation by a sort of method of characteristics to find the Bellman function of certain classical Harmonic Analysis problems, and, therefore, of finding full structure of sharp constants and extremal sequences for those problems.