From a dichotomy for images to Haagerup's inequality

TL;DR

This paper introduces a dichotomy principle for continuous maps on topological spaces, with applications to holomorphic functions and a proof of Haagerup's inequality related to Rademacher variables.

Contribution

It establishes a new topological dichotomy principle and applies it to derive classical results in complex analysis and probability theory, including Haagerup's inequality.

Findings

Dichotomy principle for continuous maps on topological spaces

Applications to maximum and minimum modulus principles in holomorphic functions

Proof of Haagerup's inequality for Rademacher random variables

Abstract

Let X be a compact topological space, and let D be a subset of X. Let Y be a Hausdorff topological space. Let f be a continuous map of the closure of D to Y such that f(D) is open. Let E be any connected subset of the complement (to Y) of the boundary of D. Then f(D) either contains E or is contained in the complement of E. Applications of this dichotomy principle are given, in particular for holomorphic maps, including maximum and minimum modulus principles, an inverse boundary correspondence, and a proof of Haagerup's inequality for the absolute power moments of linear combinations of independent Rademacher random variables.