Equivariant Borel liftings in complex analysis and PDE

TL;DR

This paper develops Borel equivariant analogues of classical theorems in complex analysis and PDE, including an equivariant Weierstrass theorem and the existence of Borel right inverses for key operators, using approximation and ergodic theory techniques.

Contribution

It introduces the concept of equivariant Borel liftings in complex analysis and PDE, providing new existence results under freeness conditions and employing novel Borel toasts and approximation methods.

Findings

Existence of Borel equivariant entire functions with prescribed divisors.

Borel right inverses for Laplacian, heat, and ar operators exist on the free part of the range.

Freeness assumptions are essential; Borelness cannot be improved to continuity.

Abstract

We establish Borel equivariant analogues of several classical theorems from complex analysis and PDE. The starting point is an equivariant Weierstrass theorem for entire functions: there exists a Borel mapping which assigns to each non-periodic positive divisor $d$ an entire function $f_d$ with divisor of zeros $\mathrm{div}(f_d)=d$ and which commutes with translation, $f_{d-w}(z)=f_d(z+w)$. We also examine the existence of equivariant Borel right inverses for the distributional Laplacian, the heat operator, and the $\bar{\partial}$-operator on the space of smooth functions. We demonstrate that Borel equivariant inverses for these maps exist on the free part of the range. In general, the freeness assumptions cannot be omitted and Borelness cannot be strengthened to continuity. Our positive results follow from a theorem establishing sufficient conditions for the existence of equivariant Borel liftings. Two key ingredients are Runge-type approximation theorems and the existence of Borel toasts, which are Borel analogues of Rokhlin towers from ergodic theory.