Theory of locally concave functions and its applications to sharp estimates of integral functionals

TL;DR

This paper introduces a duality framework for locally concave functions, leading to sharp estimates of integral functionals and norms of monotonic rearrangements, using a novel martingale approach.

Contribution

It develops a new duality theorem for locally concave functions and applies it to obtain sharp integral functional estimates via a specialized martingale class.

Findings

Established a duality theorem for locally concave functions

Derived sharp bounds for norms of monotonic rearrangements

Introduced a novel martingale-based extremal problem

Abstract

We prove a duality theorem the computation of certain Bellman functions is usually based on. As a byproduct, we obtain sharp results about the norms of monotonic rearrangements. The main novelty of our approach is a special class of martingales and an extremal problem on this class, which is dual to the minimization problem for locally concave functions.