Hardy Spaces and Boundary Conditions from the Ising Model

TL;DR

This paper characterizes Hardy space functions on multiply-connected domains using boundary conditions inspired by the Ising model, leveraging operator positivity to connect complex analysis with statistical physics.

Contribution

It introduces a novel boundary condition for Hardy spaces based on the Ising model, providing an explicit characterization for multiply-connected domains.

Findings

Boundary conditions derived from the Ising model characterize Hardy spaces.

Positivity of an inductively constructed operator is central to the characterization.

The approach links complex analysis with statistical physics models.

Abstract

Functions in Hardy spaces on multiply-connected domains in the plane are given an explicit characterization in terms of a boundary condition inspired by the two-dimensional Ising model. The key underlying property is the positivity of a certain operator constructed inductively on the number of components of the boundary.