On a Bellman function associated with the Chang--Wilson--Wolff theorem: a case study

TL;DR

This paper studies a Bellman function related to the Chang--Wilson--Wolff theorem, providing improved tail estimates for functions with bounded dyadic square functions, revealing a complex structure with dense derivative jumps and smoothness properties.

Contribution

It introduces a detailed analysis of the Bellman function for tail estimates, uncovering its intricate structure and employing computer-assisted rigorous proofs.

Findings

Improved tail distribution estimates over previous bounds

Discovery of dense derivative jumps in the Bellman function

Demonstration of the Bellman function's smoothness properties beyond a certain point

Abstract

In this paper we estimate the tail of distribution (i.e., the measure of the set $\{f\ge x\}$) for those functions $f$ whose dyadic square function is bounded by a given constant. In particular we get a bit better estimate than the estimate following from the Chang--Wilson--Wolf theorem. In the paper we investigate the Bellman function corresponding to the problem. A curious structure of this function is found: it has jumps of the first derivative at a dense subset of interval $[0,1]$ (where it is calculated exactly), but it is of $C^\infty$-class for $x>\sqrt3$ (where it is calculated up to a multiplicative constant). An unusual feature of the paper consists in the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to computer.