Boundary local integrability of rational functions in two variables

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 Kollár, 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.

Revision based on referee report. To appear in TAMS. Corrected typos