Boundary local integrability of rational functions in two variables
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910072689393664 |
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| author | Knese, Greg |
| author_facet | Knese, Greg |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05042 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary local integrability of rational functions in two variables Knese, Greg Functional Analysis Algebraic Geometry Classical Analysis and ODEs Complex Variables 32A40, 13H10, 14H45 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. |
| title | Boundary local integrability of rational functions in two variables |
| topic | Functional Analysis Algebraic Geometry Classical Analysis and ODEs Complex Variables 32A40, 13H10, 14H45 |
| url | https://arxiv.org/abs/2404.05042 |