Some extremal problems for martingale transforms. III

V. I. Vasyunin

Saint Petersburg State University

Abstract: This paper is a direct continuation of two papers with the same title published in the preceeding issues of this journal (vol. 527 (2023) and 537 (2024)). For this reason neither the introductory part of the paper nor the list of references is duplicated. However for the reader's convenience, the formulas from the preceding papers that are cited here are collected in a special addendum at the end of the paper with their original numbers. At this paper new local foliations are investigated: major and medium pockets and some bilinear domains. The emergence of such local foliations is illustrated by the further investigation of the examples where the boundary functions are polynomials of the third degree. At the end of the paper a survey is given, where all possible foliations are discribed, when the boundary functions are polynomials of the third degree.

Key words and phrases: Bellman function, martingale transform, diagonally concave function.

UDC: 517.51, 519.216.8

Received: 20.09.2025