Some extremal problems for martingale transforms. II

TL;DR

This paper extends previous work on extremal problems for martingale transforms by exploring new local foliations, specifically minor pockets and rectangles, and analyzing boundary functions with third-degree polynomials.

Contribution

It introduces and investigates two new types of local foliations for martingale transforms, expanding the understanding of extremal problems in this area.

Findings

Analysis of minor pockets and rectangles as local foliations.

Application to boundary functions with third-degree polynomials.

Further development of extremal problem techniques.

Abstract

This paper is a direct continuation of the paper arXiv:2401.00053. By this reason neither introductory part of the paper nor the list of references are not duplicated. However for the reader convenience, the formulas from the first paper that are cited here are collected in a special addendum at the end of the paper with their original numbers. At this paper two new local foliations are investigated: minor pockets and rectangles. The appearance of such local foliations is illustrated by the further investigation of the examples with the boundary functions being the polynomials of the third degree.