Monge-Ampere Equations and Bellman Functions: The Dyadic Maximal Operator
When proving a sharp inequality in a harmonic analysis setting, one can sometimes recast the problem as that of ﬁnding the corresponding Bellman function. These functions often arise as solutions of Monge-Ampere PDEs on problem-speciﬁc domains; in such a case, the optimizers in the inequality can be found using the straight-line characteristics of the equation. I will show how to ﬁnd the Bellman function for one important example – the dyadic maximal operator on Lp. This function has been previously found by A. Melas in a diﬀerent way. The approach presented can be generalized to other well-localized operators and function classes. Joint work with Leonid Slavin and Vasily Vasyunin.
International Conference on Harmonic Analysis and Partial Differential Equations
"Monge-Ampere Equations and Bellman Functions: The Dyadic Maximal Operator."
Mathematical Sciences Faculty Presentations.