Title: Novel Bellman Estimates for Ap Weights

Abstract:

The Bellman function method is an assortment of tools for obtaining sharp inequalities in harmonic analysis. To handle an inequality, one fixes a set of parameters, called Bellman variables, and maximizes (or minimizes) the left-hand side subject to these constraints. The solution of the corresponding extremal problem is called the Bellman function. Often, a choice of variables can be made so as to obtain — and solve — a PDE for the function. In others, much less studied settings, there is no PDE and one has to design a different way of solving the extremal problem. Sometimes, in lieu of the actual function, one has to settle for a majorant, which may or may not give sharp results.

In this talk, I will present two different implementations of these ideas. The first part deals with a general family of dyadic sequences related to $A_2$ weights. The Bellman function here does not satisfy any PDE, but we’re able to obtain a special majorant that gives the sharp constant. The second part presents sharp lower bounds for logarithms of $A_\infty$ weights as a means of estimating the John–Nirenberg constant of the space ${\rm BMO}^p, 0<p<1$. The corresponding Bellman function solves the homogeneous Monge–Amp\’ere equation, but the geometry of the solution goes beyond established theory due to the lack of regularity in the boundary condition. This is joint work with Leonid Slavin.