Bellman Functions on Simple Non-Convex Domains in the Plane

This memoir provides a generalization of the authors' previous work on Bellman functions for integral functionals on $\mathrm{BMO}$. Those Bellman functions are the minimal locally concave functions on parabolic strips in the plane. We describe an algorithm for constructing minimal locally concave functions on a planar domain that is a difference of two unbounded convex domains. This leads to many sharp estimates for functions in the classes like $\mathrm{BMO}$, $A_p$, or the Gehring classes.