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

Abstract

The tail of distribution (i.e., the measure of the set { f ≥ x } \{f\ge x\} ) is estimated for those functions f f whose dyadic square function is bounded by a given constant. In particular, an estimate following from the Chang–Wilson–Wolf theorem is slightly improved. The study of the Bellman function corresponding to the problem reveals a curious structure of this function: it has jumps of the first derivative at a dense subset of the interval [ 0 , 1 ] [0,1] (where it is calculated exactly), but it is of C ∞ C^\infty -class for x > 3 x>\sqrt 3 (where it is calculated up to a multiplicative constant). An unusual feature of the paper consists of the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to a computer.