Boundary local integrability of rational functions in two variables

TL;DR

This paper characterizes when rational functions in two variables are locally integrable near a boundary singularity, providing necessary and sufficient conditions, and applies these results to problems involving bounded rational functions and stable polynomials.

Contribution

It offers a complete characterization of local $L^{p}$ integrability for rational functions near boundary singularities, extending previous work and providing new proofs of conjectures.

Findings

Necessary and sufficient test for local $L^{p}$ membership.

Complete description of numerators for local $L^{p}$ integrability.

Proof that bounded rational functions on the bidisk have $L^1$ derivatives.

Abstract

Motivated by studying boundary singularities of rational functions in two variables that are analytic on a domain, we investigate local integrability on $\mathbb{R}^2$ near $(0,0)$ of rational functions with denominator non-vanishing in the bi-upper half-plane but with an isolated zero (with respect to $\mathbb{R}^2$) at the origin. Building on work of Bickel-Pascoe-Sola, we give a necessary and sufficient test for membership in a local $L^{p}(\mathbb{R}^2)$ space and we give a complete description of all numerators $Q$ such that $Q/P$ is locally in a given $L^{p}$ space. As applications, we prove that every bounded rational function on the bidisk has partial derivatives belonging to $L^1$ on the two-torus. In addition, we give a new proof of a conjecture, started in Bickel-Knese-Pascoe-Sola and completed by Koll\'ar, characterizing the ideal of $Q$ such that $Q/P$ is locally bounded. A larger takeaway from this work is that a local model for stable polynomials we employ is a flexible tool and may be of use for other local questions about stable polynomials.